28 A Critical Analysis of Simplified Procedures for Footbridges’ Serviceability Assessment 295 EMRL is provided by ISO [9] for a group of Np uncoordinated pedestrians. An approximate procedure for the estimate of the maximum acceleration based on an equivalent resonant moving load is also proposed by FIB [10]. Adopting EMRL and based on the results in [16], the normalized maximum acceleration becomes: RQpj norm, max D p j 2 2 Q 2 cmC 2 j Q cmexp jQtmax C Q cmcos Q cmQtmax j sin Q cmQtmax Qtmax D 1 Q cm h atan j Q cm C i Q cm D cm !jL (28.7) Based on ESM [13], the standard deviation and the maximum value of the normalized acceleration are given by: QRpj norm D p 4 p pQ .1/ RQpj norm,max D g QRpj p 4 p pQ .1/ (28.8) where g QRpj is the so-called peak factor of RQpj. It can be expressed as: g QRpj Ds2ln 2 e QRpj Q T C 0.5772 s2ln 2 e QRpj Q T (28.9) In Eq. (28.9), the following non-dimensional parameters appear [15]: QT D!j NL cm D N Q cm e RQp j D8< : 1.63q0.45 QRpj 0.38 QRpj Vanmarcke formulation .a/ QRpj ' Q nj D 1 2 Davenport formulation .b/ q QRpj '2r j (28.10) In Eq. (28.10), the first expression is based on the theory developed by Vanmarcke and Der Kiureghian for the analysis of the extreme distribution of narrow-band random processes (threshold up-crossings in clumps). The second expression corresponds to the classic formulation proposed by Davenport, and it is commonly applied for broad band random processes (independent threshold up-crossings). 28.3 Numerical Validation of the Approximate Procedures Monte Carlo simulations are carried out for different values of the non-dimensional mean step frequency Q m and coefficient of variation of the pedestrian weight VG, of the walking velocityVc and of the step frequencyV˝ . The simulations have two different aims: to check the sensitivity of the maximum dynamic response to stochastic walking parameters, and to assess the reliability of the simplified loading models in the estimate of the normalized maximum acceleration. For every case analyzed, the mean value among 104 simulations is estimated. Monte Carlo simulations carried out varying the coefficient of variation of the walking speed and of the pedestrian weight have confirmed the negligible influence of the statistical distribution of these random variables on the maximum dynamic response. For this reason, in the following, only the effects of variation of the non-dimensional mean step frequency and its coefficient of variation are shown. According to the literature (see, e.g., [14] for a brief summary), the coefficient of variation of the step frequency is assumed in the interval 0.06–0.1. Figure 28.1 plots the mean value of the normalized maximum acceleration derived from Monte Carlo simulations, as a function of the coefficient of variation of the step frequency V˝; the different gray scales correspond to different values of the non-dimensional mean step frequency Q m. The results of numerical simulations (symbols) are compared with the ESM closed-form expression (thin lines) based on the Vanmarcke (Fig. 28.1a) and Davenport (Fig. 28.1b) formulations. Furthermore, Fig. 28.1c compares the results of numerical simulations with the predictions by the EURL provided by Setra (‰D1) and BS (kD1), and by the EMRL provided by ISO, for different values of the non-dimensional mean step frequency Q m. Figure 28.1a, b point out the delicacy of the choice of the peak factor. In particular, the Davenport expression of the peak factor provides a large overestimation of the numerical results when the non-dimensional mean step frequency is
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