Pedestrian load models of footbridges

The increase of vibration problems in modern footbridges shows that footbridges should no longer be designed for static loads only. Not only natural frequencies but also damping properties and pedestrian loading determine the dynamic response of footbridges and design tools should consider all of these factors. In this paper the pedestrian load models for serviceability verification of footbridges, which are missing in the current European codes, are presented. For simplicity reasons the proposed pedestrian load models are based on stationary pulsating loads instead of moving pulsating loads. It is shown that simplified procedure can be used in verification of the serviceability limit state related to vibration due to pedestrians. Footbridge vibrations don’t cause usually structural problems, but if the vibration behaviour does not satisfy the comfort criteria, changes in the design or damping devices could be considered. The most popular external damping devices are viscous dampers and tuned mass dampers (TMD). The efficiency of TMD is demonstrated on the example of a footbridge prone to vibrations induced by pedestrians. It is shown that if the TMD is tuned quite precisely the reduction of accelerations can be very significant.


Introduction
Modern footbridges are very often lightweight and flexible structures, where the first natural frequencies of vibration may fall close to dominant frequencies of the dynamic excitation due to walking or running.Such bridges are susceptible to vertical as well as to horizontal vibrations leading to a resonant response characterized by high levels of vibration and a dynamic design is necessary.
In this paper, the different models of dynamic loads caused by pedestrian crossing the bridge, which can be used in serviceability verification, are presented.Although footbridge vibrations do not cause usually structural problems, they can induce some uncomfortable sensation, and so many codes establish maximum acceptable values of acceleration.Provided that the vibration behaviour due to expected pedestrian traffic is checked with dynamic calculations and satisfies the required comfort, any type of footbridge can be designed and constructed.If the vibration behaviour does not satisfy some comfort criteria, changes in the design or damping devices could be considered.

Pedestrian loads
In order to verify serviceability limit state related to vibration due to pedestrians it is necessary to define dynamic pedestrian load.Numerous studies deal with determination of human walking, running or jumping force over the years, cf.e.g.[1].
The possible loading scenario can be divided into five categories: -Single person loading; -Normal traffic -spatially unrestricted traffic where each individual can move freely without having to change walking pattern to avoid contact with others; -Crowd loading -spatially restricted traffic where the walking of each individual is restricted due to limited space; -Group loading -a number of persons is walking closely together; -Vandal loading -a person, or a group of people, tries to excite the structure by moving in a correlated harmonic way in response-sensitive areas.
In addition to these five groups three different types of human motion are commonly considered to model the dynamic loads applied by pedestrians, namely walking, running and rhythmic jumping.All these load models can often be categorized into deterministic and probabilistic models.In this paper only the deterministic models of a single pedestrian, group and crowd loading will be considered.
Two types of analytical force models can be found in the literature: time-domain models (deterministic and probabilistic force models) and frequency-domain models -for a detailed review cf.[1] and [2].The suitable model of mutual interaction between human gait and elastic bridge has been developed in [3].
As an example the deterministic force model for walking is given.The vertical force component is greater than the horizontal one, but the lateral and longitudinal horizontal components can also cause vibration related problems of slender bridges.Frequency of lateral movement, which occurs as a result of moving the centre of mass from one foot to the other, is equal to half of the step frequency of vertical or longitudinal movement.
General shapes of the temporal evolution of the pedestrian loads -assuming a perfect periodicity of the force -can be performed using appropriate load-time functions, for a vertical periodic force Fp,ver (t), lateral periodic force Fp,lat (t) and longitudinal periodic force Fp,long (t): , ( ) 0.05 sin 2 2 , ( ) 0.20 sin 2 where G is the weight of the person (usually G = 700 N), fp is the pacing frequency, a1 = 0.4, a2 = a3 = 0.1 are the Fourier coefficients of the i-th harmonic for vertical, lateral and longitudinal forces, φ1 = 0 and φ2 = φ3 = π/2 are the phase shifts of the ith harmonic contributions.
The pacing frequency fp and the pedestrian forward speed vp are two parameters that play a fundamental role in terms of the characterisation of the excitation.The corresponding average values are presented in Table 1 for walking and running.A general proposal as to the typical frequency ranges for different human activities is given for walking 1.6-2.4Hz and 3.5-4.5Hz (first and second walking harmonics), for running 2.0-3.5 Hz, for jumping 1.8-3.4Hz and for bouncing 1.5-3.0Hz.Commonly adopted mean value frequency for running and jumping is 2.5 Hz [10].

Proposed load models
The current European standard for determination of traffic loads on bridges [4] does not recommend the load models for serviceability limit verification due to pedestrians.The Guidelines for the design of footbridges [5] gives the certain pedestrian load models.The load models are divided into three categories: Single pedestrian load model (DLM1), Group of pedestrians load model (DLM2) and Continuous pedestrian stream load model (DLM3).Instead of pulsating forces in vertical and lateral direction which move with the speed of 0.9 fp, the stationary pulsating forces applied at the most adverse position on the bridge are defined.
DLM1 defines vertical Fp,v (t) and horizontal (lateral) components Fp,h (t) as: DLM2 defines the effect of a group of 8 ÷ 15 persons walking across the bridge by vertical Fg,v (t) and horizontal components Fg,h (t) as: The effect of synchronisation of step frequencies and the phase shift between pedestrians is taken into account by coefficients kv and kh (Fig. 1).
Fig. 1.Coefficients kv and kh [5].The load should be applied in the way to produce the most unfavourable loading case (depending on the mode shape) and a uniformly distributed mass of 400 kN/m 2 (if unfavourable) should be applied at the same location.

Example -Footbridge in Čelákovice
The footbridge is a cable-stayed structure with 3 spans 43.0 + 156.0 + 43.0 meters made of Ultra-High Performance Concrete.The height of the steel pylons is 36 meters (Fig. 2 and Fig. 3).Natural frequencies are summarized in Table 2 and important modes of vibration are shown in Fig. 4 to 6.

Results
Pedestrian loading was modelled using Eq. ( 6) and Eq. ( 7) for three principal load casesa) horizontal excitation with pacing frequency corresponding to the fundamental lateral frequency; b) vertical excitation with pacing frequency corresponding to the fundamental vertical frequency; c) vertical excitation with pacing frequency corresponding to the commonly adopted mean value frequency for walking 2.0 Hz.The results of the analysis are given in Table 3.It can be seen that for the load case c) the accelerations are higher than the limit values taken from Eurocode EN 1990.In such that case the changing of vibration characteristics of the footbridge (natural frequencies) or damping devices should be considered.

Footbridge with TMD
To avoid undesirable vibrations of the structure it is a good idea to install tuned mass dampers (TMDs) on the footbridge to dissipate the energy from one or more modes.A TMD is often a much more lucrative solution when compared to changing the natural frequencies of the structure.
The theory of how a TMD works, and how to determine the optimal characteristics are summarized in [7].With respect to the antisymmetric shape with natural frequency of vertical bending 2.04 Hz (cf.As a result of the increased mass of the footbridge with two TMDs) the corresponding natural frequency changed to the value of 1.83 Hz.The response was calculated for the pacing frequency 1.83 Hz and results are given in Table 4.

Conclusions
In this paper the pedestrian load models for serviceability verification of footbridges, which are missing in the current European codes, are presented.For simplicity reasons the proposed pedestrian load models are based on stationary pulsating loads instead of moving pulsating loads.It is shown that simplified procedure can be used in verification of the serviceability limit state related to vibration due to pedestrians.Not only natural frequencies but also damping properties and pedestrian loading determine the dynamic response of footbridges and design tools should consider all of these factors.Footbridge vibrations don't cause usually structural problems, but if the vibration behaviour does not satisfy the comfort criteria, changes in the design or damping devices could be considered.The most popular of these are viscous dampers and TMDs.The efficiency of TMD is demonstrated on the example of a footbridge prone to vibrations induced by pedestrians.Is has been shown that if the TMD is tuned quite precisely (especially its frequency) the reduction of accelerations can be very significant.

Table 2 .
Footbridge Čelákovice -natural frequencies and modes of vibration.

Table 3 .
Footbridge Čelákovice -response due to pedestrian loading

Table 4 .
Footbridge Čelákovice -response due to pedestrian loading