24 Predicting Footbridge Vibrations Using a Probability-Based Approach 199 Basically the load model is built such that it includes both main harmonics occurring at frequencyifs (see Eq. 24.3) and a set of subharmonics (see Eq. 24.4). Fi.t/ DW˛i iC0:25 Xf jDi 0:25 ˛i f j cos 2 f jfst C f j (24.3) FS i .t/ DW˛ S i i 0:25 X f S j Di 0:75 ˛ S j f S j cos 2 f S j fst C f S j (24.4) For explanation of the parameters of the load model (the parameters included in the two equations above) and for a description of how the model is operated, reference is made to [11]. Here it suffices to mention that load model II includes the main harmonics also considered in load model I. Furthermore load model II includes subharmonics occurring at frequencies in between the main harmonics and it also models a leakage of energy around these frequencies. 24.2.3 Parameters of the Models For the studies of this paper, both load model I and load model II are employed in a way accounting for the fact that different pedestrians will generate different dynamic forces. This is done by modelling at least some of the load model parameters as random variables. Mean values, , and standard deviations, , assumed for step frequency, fs, pedestrian weight, W, and step length, ls, are listed in Table 24.1. The stochastic nature for the load model parameters reported in Table 24.1 are those suggested in [11]. It is seen that the pedestrian weight, W, is handled as a deterministic property in that the standard deviation for this parameter, , is set to zero. For the random variables (step frequency and step length) normal distributions are assumed and furthermore step frequency and step length are considered to be independent parameters. Although the step length does not appear in the load models introduced above it defines the position of the pedestrian when crossing the bridge and hereby the modal load generated by the pedestrian. Apart from the parameters listed above, assumptions are also made about the stochastic nature of dynamic load factors. The mean value of the first dynamic load factor is modelled to depend on step frequency in accordance with Eq. (24.5). ˛1 D 0:2649f 3 s C1:3206f 2 s 1:7597fs C0:7613 (24.5) This equation models a relationship calibrated in [6], which also suggests a standard deviation, , of 0.16 for this parameter, which is the stochastic nature of the action assumed for the studies of this paper. Load model II models load harmonics beyond the first load harmonic, and Table 24.2 shows the numbers assumed for the mean values and standard deviations for these load harmonics. These load harmonics are not included when employing load model I. However, they form an integrated part of load model II. Phases between the load harmonics are modelled as random variables following a uniform distribution in the range [ , ]. For a description of how parameters of load model II not uniquely defined above are computed see [11]. Table 24.1 Mean values and standard deviations fs (Hz) 1.87 0.186 W(N) 750 0 ls (m) 0.71 0.071 Table 24.2 Mean values and standard deviations [11] ˛2 ˛3 ˛4 ˛5 0.07 0.05 0.05 0.03 0.030 0.020 0.020 0.015
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