200 L. Pedersen and C. Frier Table 24.3 Modal properties of bridges and bridge length Bridge f1 (Hz) 1 (%) m1 (10 3 kg) L(m) A 1.9 0.05 43.8 45.3 B 2.85 0.05 19.5 30.2 24.2.4 A Variation of Study Assumptions A central objective of the studies of this paper is to compare the results of footbridge response calculations obtained using load models I and II. Both models handle a number of parameters as random variables. It is considered of interest to study whether one of the walking parameters might as well be handled as a deterministic property without loss in statistical information about relevant footbridge response data. In load model I as well as in load model II, the step length, ls, is modelled as a random variable. For calculations a scenario is constructed in which the step length is modelled as a deterministic property assuming the mean value (for any outcome of step length), thus not accounting for randomness in outcomes of step length. This in order to compare with results obtained accounting for randomness in step length. 24.3 The Footbridges Considered for the Study It is chosen to consider two-point pinned bridges with the modal properties (natural frequency, damping ratio, and modal mass) of the first vertical bending mode given in Table 24.3. Although the two bridges will have vertical modes of vibration beyond the first vertical bending mode, these are not considered for the present study, as the natural frequencies of those modes are so high that they are unlikely to be noticeably excited by pedestrian loading. The natural frequency of bridge A is chosen such that the first harmonic of pedestrian loading is likely to bring bridge A into resonant vibration, as the mean step frequency is fairly close to 1.9 Hz (according to mean values of step frequencies presented in [4, 11]). Bridge B, however, is not very likely to come into vibration due to the first harmonic load component but bridge B might resonate as a result of the first subharmonic above the step frequency. It can be seen that the bridges are very lightly damped which would suggest that the bridges respond quite lively if subjected to resonant excitation from a pedestrian. The bridge length, L, is not identical for the two bridges. Bridge B is shorter than bridge A, which is meaningful as bridge B is stiffer than bridge A. 24.4 Extracting Response Data Prior to computing bridge response, the load of the pedestrian was conveyed to a modal load acting on the first vertical bending mode of the bridge. This involves accounting for the mode shape of the bridge and the walking velocity of the pedestrian. The walking velocity is calculated based on outcomes of step frequency and step length. The mode shape, ·, of the first vertical bending mode of the bridges is assumed to have the shape of a half-sine normalized to 1 at the top point. The footbridge response parameter in focus in this paper is the vertical footbridge acceleration computed at footbridge midspan. During each pedestrian crossing of the bridge, the peak value of accelerations is extracted from a time domain representation of bridge accelerations at this position. The time history of the response is computed using a Newmark time integration approach. A large number of single person bridge crossings were simulated using the Monte Carlo approach respecting the modelled stochastic nature of parameters in the load models. This approach allows for sorting calculated acceleration response data and finding the probability distribution and then extracting quantiles of the response, hereby respecting that the response is in fact stochastic. The acceleration quantiles focused on in this paper are a95, a90, and a75. In order to assure confidence in estimates of acceleration response quantiles, 100,000 simulation runs were made for each studied scenario.
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