24 Predicting Footbridge Vibrations Using a Probability-Based Approach 201 24.5 Results Sections 24.5.1 and 24.5.2 presents results of calculations. 24.5.1 Influence of Choice of Load Model Table 24.4 presents acceleration quantiles derived having applied load models I and II for computing responses of the two bridges (bridges A and B). In both load models, the step length, ls, is handled as a random variable.as defined in Table 24.1. Focusing on the acceleration quantiles obtained for the 1.9 Hz-bridge (bridge A) it can be seen that there is not much difference between the results obtained employing load model I (the simple load model) and load model II (the advanced load model). The opposite conclusion is drawn for the 2.85 Hz-bridge (bridge B). For this bridge, the simple load model (model I) underestimates the acceleration response. This is explained by the circumstance that the simple load model does not model subharmonics of load action. Hereby load model I can only excite the 2.85 Hz-bridge as a result of the modelled randomness of step frequency. That the advanced load model can excite the 2.85 Hz-bridge to resonance is illustrated in Fig. 24.1 which shows a power spectrum of the load generated by a pedestrian. The frequency representation of the load shown in Fig. 24.1 is constructed by analysing a time history generated by pedestrian forces. The time history was produced assuming that all random variables of the load model assumed their mean value (only the phases between load harmonics have been modelled as a random parameter when producing the time history). It is apparent that this load model contains load energy close to the frequency of bridge B (2.85 Hz) which can put the bridge into resonance vibrations. It is also apparent that the energy that can excite bridge A (1.9 Hz) is higher than that at 2.85 Hz, which explains why the acceleration response of bridge A is higher than that found for bridge B. Returning to the results for bridge A, it can be observed that for the quantile a75, there is a noticeable difference between results obtained assuming load model I and load model II. It is believed that most often it would be the higher quantiles that would be of interest (a95 or a90). If the task is to produce estimates of these quantiles, the result appears to be insensitive to the choice of load model (i.e. whether the simple or the advanced load model is chosen is not important). This is not the case if the task is to produce estimates of response of a bridge with a natural frequency at about 2.85 Hz. For that bridge, the application of the advanced load model is recommended. Table 24.4 Acceleration quantiles Bridge A B Load model I (m/s2) II (m/s2) I (m/s2) II (m/s2) a95 0.3524 0.3425 0.0257 0.0754 a90 0.2764 0.2672 0.0213 0.0672 a75 0.1099 0.1297 0.0152 0.0560 Fig. 24.1 Normalized power spectrum for load model II. ( : frequency of bridge A, *: frequency of bridge B)
RkJQdWJsaXNoZXIy MTMzNzEzMQ==