Influence of the random dynamic parameters of the human body on the dynamic characteristics of the coupled system of structure–crowd
Introduction
The presence of human occupants may change the dynamic behaviour of structures considerably. This can be seen in Fig. 1 which, based on the peak-picking method for the Fourier transforms of measured accelerations for time windows of 13.653 s, displays the change of the natural frequency of a simply supported reinforced slab during the period of filling a stand in a stadium. During this phase, the size of the crowd increases, leading to a more or less constant decrease of the effective natural frequency until the slab is completely occupied.
It is important to note that during the filling process, usually some people leave shortly after their arrival, e.g. to get food and drinks for themselves and others. Furthermore, although the stand has permanent seats, the audience is neither permanently seated during the filling process nor during the match. This unrest of the crowd, i.e. the change of postures and activities, leads to a large scatter of the effective natural frequency which is reflected in the change of the probability densities of the identified dominant frequencies sampled over sub-periods of 10 min.
All dominant frequencies in Fig. 1 which are below 3 Hz indicate rhythmic activities of the crowd. These frequencies have not been considered in the basic trend of the natural frequency and the corresponding probability densities. Altogether, the slab with a mass of 15.72 tons provides 63 seats. Considering only the additional mass of the audience with an average weight of 80 kg per person, the frequency is predicted to drop from 6.73 to 5.86 Hz. However, the observed drop of the natural frequency in Fig. 1 is much larger. The drop of the natural frequency below 5 Hz points out the need for an appropriate method to describe the dynamic behaviour of the coupled structure–crowd system for all possible stages during the filling process and during the match. Realistic values of the effective natural frequency are required for both the Serviceability and Ultimate Limit State, since the dynamic loads and the human sensation of accelerations strongly depend on this frequency.
The vibration amplitudes are influenced by the effective damping of the coupled structure–crowd system. It is important to note that the presence of human occupants also influences the damping capacity. This has been documented for civil engineering applications already in the 1960s, e.g. by Lenzen [1], who found that the presence of occupants leads to a decrease in the natural frequency and an increase of the damping. The influence of occupants can be modelled by mass–spring–damper representations of the human body. The first models have been developed since the 1950s [2]; however, the main application field was not civil engineering but mechanical engineering (interaction between driver seat and driver) and biomechanics (potentially damaging effects of vibrations).
Based on measurements of the impedance of the human body, Dieckmann published in 1957 a first approach [3] using a linear single-degree-of-freedom model (sdof) which was able to reproduce approximately the dynamic behaviour of the human body under sinusoidal vertical excitation up to frequencies of 10 Hz. The influence of the second natural frequency around 12 Hz was considered later in a two-degree-of-freedom model (2dof) [4]. The results of the following decades of intensive international research were summarised and condensed in ISO 5982 [5], presenting a deterministic two-degree-of-freedom model for standing and sitting posture. In 1987, ISO 7962 [6] published a four-degree-of-freedom model, specifying also minimum and maximum values of the impedance curves. However, application of dynamic models of the human body in civil engineering remained scarce with only a few exceptions.
For the dynamic analysis of floor responses, in Ref. [7] a sdof-system was used and in Ref. [8] a 2dof and additionally an 11dof system were used to model the increased damping due to human occupants. In regard to the natural frequency, it was generally believed that for the prediction of the natural frequency, the mass-only model was appropriate [9], [10], i.e. occupants were considered only as additional masses.
A first more consistent approach was developed by Ellis and Ji [11] who recommended to model passive persons as spring–mass–damper systems and to consider active persons (walking or jumping) as external loads. Strictly speaking, this concept requires at least one further degree of freedom for each passive individual in the crowd, assuming that the dynamic characteristics of the human body are, to some extent, random variables. If the random scatter of the dynamic characteristics of the human body can be neglected and the dynamic behaviour of the structure is modelled by an equivalent modal system a two-degree-of-freedom system can be achieved. Sachse et al. [12] tried to identify an equivalent sdof-system for the dynamic influence of up to five sitting persons based on measurements on a pre-stressed concrete slab with a weight of 15 tons and a lowest natural frequency of 4.51 Hz when empty. Sim adopted this idea [13] and obtained equivalent 2dof-systems for the coupled system for standing or sitting posture of the audience. The dynamic behaviour of the crowd was modelled as the average apparent mass, considering the raw data of the dynamic characteristics of the human body of 60 sitting [14] or 12 standing persons [15]. For seated posture, Sim specified separate models for men, women and children, while for standing posture, only a model for men was given. With the averaged models for only male persons, the change of the lowest natural frequency and the effective damping was determined for a range of crowd to structural mass ratios varying from 5% to 40%.
A further refinement of the basic approach by Ellis and Ji has been published recently in Ref. [16]. Active persons are modelled as a self-generated dynamic action within a dynamic system representing the active human body. This leads to additional damping effects also for the active persons. The model has been calibrated to the activity bobbing, which is characterised by permanent contact of the active person to the ground. For jumping, however, there is a distinct flight phase, i.e. the human body (or the corresponding equivalent dynamic model) is not in contact with the structure. So far, there is no indication that the body of a jumping person contributes to the effective damping. Hence, for a conservative approach, the model proposed by Ellis and Ji is recommended for jumping crowds. Dougill's model has been adopted in the recommendations of the Joint IStructE working group [17]. The document also recommends deterministic values for a single-degree-of-freedom model for passive persons.
The approach by Sim and the refined approach by Dougill both suffer from the basic shortcoming that only averaged input values are used, i.e. the basic dynamic characteristics of the human body are the same for all individuals and, in the case of Dougill’s approach, the loads generated by each individual are the same. Especially for smaller groups of passive persons, the large scatter in the individual dynamic characteristics of the human body may lead to scatter in the position of the resulting lowest frequency (effective natural frequency) and the amplitude of the effective damping.
The refined analysis requires a probabilistic model of the basic dynamic characteristics of the human body for the two postures seating and standing. In Section 2, the basic features of the probabilistic model are explained. As input data, the results by Griffin [14], [15] are used. In the further analysis, each individual is considered with two degrees of freedom. The random characteristics of each individual are obtained by Monte-Carlo simulation. Finally, the dynamic behaviour of a specific deterministic structure coupled with the random crowd is analysed in terms of the dynamic amplification function. The randomness in the dynamic behaviour of the coupled system is evaluated based on 10,000 simulations for each combination of the basic variables which are the natural frequency of the empty structure, the crowd size, the mass ratio (defined as the ratio of the crowd’s total mass to the mass of the empty structure) and the posture. While in Sim’s study only a variation of the mass ratio from 0 to 0.4 has been considered, the actual study analyses the range from 0 to 1, thus covering also pre-stressed structures.
Section snippets
Dynamic characteristics of the human body
An efficient and simple description of the dynamic characteristics of the human body has been obtained by Griffin, modelling the human body as a two-degree-of-freedom system for sitting [14] and standing [15] persons. In Ref. [14] the test subjects were exposed to random vertical vibrations of 1.0 m/s² rms in the frequency range between 0.25 and 20 Hz, whereas in Ref. [15] the frequency range was from 0.5 to 30 Hz and the subjects were exposed to five random vertical vibration amplitudes from
Influence of the random body characteristics on the natural frequency and the effective logarithmic damping decrement
It is reasonable to assume that for very large groups of passive persons the average characteristics of the human body may be used to describe the influence of occupants on the dynamic behaviour of the coupled system. However, there are a lot of practical design cases where this scenario does not apply. For assembly halls, only wide-span floors meet the demand of a large influencing area which suppresses the individual influences. If, on the other hand, floor systems have smaller influence
Summary and conclusions
The paper studies the effect of the human body on the dynamic characteristics of the coupled structure–crowd system with special emphasis on the randomness in the effective dynamic characteristics. A probabilistic model of the random dynamic characteristics of passive persons is developed. Based on this probabilistic model extensive Monte-Carlo simulations are performed for different group sizes, postures (standing/sitting), frequencies of the empty structure, and mass ratios (mass of the empty
Acknowledgements
Part of this study was sponsored by the German Science Foundation (DFG) in the scope of the research project “Modellierung der dynamischen menscheninduzierten Einwirkungen und Einwirkungseffekte auf der Grundlage eines stochastischen Modells” (reference numbers KA 675/10-1 and KA 675/10-2). This support is gratefully acknowledged.
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