Equivalent spectral model and maximum dynamic response for the serviceability analysis of footbridges

Modern footbridges can be very sensitive to walking-induced vibrations. The assessment of footbridge vibrations due to normal unrestricted pedestrian traffic is a topical problem in the serviceability analysis of these structures. In this paper real pedestrian traffic conditions are modeled probabilistically, considering several sources of randomness among which pedestrian arrivals, step frequencies and velocities, force amplitudes and pedestrian weights. Based on such a probabilistic model and on the analysis of peculiar non-dimensional parameters, a spectral model for the modal force induced by pedestrian groups, modeled as a stationary random process, is analytically deduced under suitable simplifying assumptions. Fully probabilistic numerical simulations with different values of the involved parameters are performed, in order to assess the validity and the limits of the proposed model. Starting from the proposed analytical spectral model of the loading, vibration serviceability analysis can be dealt with the classic methods of linear random dynamics. Simple closed-form expressions for the evaluation of the maximum dynamic response are provided. Comparisons with numerical and experimental results seem very promising, also when the step frequency is not in resonance with the dominant mode of the structure. A simple step-by-step flow-diagram is proposed in order to assist the designer in the calculation of the maximum expected acceleration.