A spectral density approach for modelling continuous vertical forces on pedestrian structures due to walking

Abstract

A limitation of existing walking models used for vibration serviceability assessment of structures carrying pedestrians is that they are typically based on direct measurements of single footfalls replicated at precise intervals. This assumption of 'perfect periodicity' allows walking forces to be modelled as a Fourier series of sine waves having frequencies of the walking pace and its integer multiples. The true imperfection and randomness in walking is currently not taken into account even in more advanced dynamic loading codes of practice and leads to an unknown degree of conservatism.Having this in mind, this paper examines real continuous walking forces obtained from an instrumented treadmill and the effect of their random imperfection through time and frequency domain simulations of structural response.The main conclusions are that there are significant differences between responses due to the imperfect real walking forces and the 'equivalent' perfectly periodic simulation.These differences are most significant for higher harmonics where the simulated vibration response overestimates the real-life behaviour, sometimes significantly. This is mainly due to random imperfections in real walking. Given that a more realistic representation of imperfect walking is an auto-spectral density function, the random character naturally leads to a stochastic approach to treatment of pedestrian loading applied in the frequency domain. The approach can be used for single pedestrians but the 3 benefits are greater for crowd loading where correlation between pedestrians as well as statistics of their pacing rates can be applied directly.

Equivalent Spectral Model for Pedestrian-Induced Forces on Footbridges: A Generalized Formulation

“…A critical review of the different approaches for the estimate of the dynamic response in such a loading condition can be found in [3]. They may be classified into three categories: Monte Carlo simulations, simplified procedures ( [4]- [7]) and spectral approach ( [8], [9]). Monte Carlo simulations may be very time consuming and not practical from the usual design of footbridges.…”

Dynamic Analyses of a Curved Cable-Stayed Footbridge Under Human Induced Vibrations: Numerical Models and Experimental Tests

“…As such, the human-induced force is weighted by the mode shape  n in order to take into account that the force moves across the footbridge and it has a limited duration. The walking force F(t) is represented through the pedestrian load model proposed by [25] and based on the results obtained from [27]. By analyzing walking force time-histories measured on a treadmill, [27] found that human-induced walking force is not periodic, but rather it is a narrow band random process, meaning that there is a leaking of energy around the main harmonics in the force spectrum.…”

“…The walking force F(t) is represented through the pedestrian load model proposed by [25] and based on the results obtained from [27]. By analyzing walking force time-histories measured on a treadmill, [27] found that human-induced walking force is not periodic, but rather it is a narrow band random process, meaning that there is a leaking of energy around the main harmonics in the force spectrum. Based on the results provided from Brownjohn [27], Živa-nović [25] developed a multi-harmonic force model for the calculation of the multi-mode structural response to a single person walking across a footbridge.…”

“…By analyzing walking force time-histories measured on a treadmill, [27] found that human-induced walking force is not periodic, but rather it is a narrow band random process, meaning that there is a leaking of energy around the main harmonics in the force spectrum. Based on the results provided from Brownjohn [27], Živa-nović [25] developed a multi-harmonic force model for the calculation of the multi-mode structural response to a single person walking across a footbridge. The proposed model takes into account the inter-and intra-subject variability in the induced walking force.…”

“…As regards the intra-subject variability, imperfections in the human walking can be described through slight changes in the walking frequency and, consequently, in the spectrum of the walking force, resulting in a narrow band spectrum. Using walking force time-histories measured on the treadmill, [27] investigated the frequency content of the force and observed that main harmonics appear at frequencies equal to the pacing frequency and its integer multiples but some sub-harmonics appear at frequencies between the main harmonics. Starting from the frequency content of the walking force, a model representation of the spectrum amplitudes and phases was achieved and the multi-harmonic force model was formulated.…”