# Authors

- Graziella Aghilone graziella.aghilone@uniroma1.it
- Massimo Cavacece cavacece@unicas.it

**Abstract**

Vibration serviceability of footbridges under human induced dynamic loading attracted a lot of attention of the research community. It is proposed an analysis of dynamic load factors proposed in the scientific literature. [DOI:10.12866/J.PIVAA.2016.20]

*Keywords*: Pedestrian action, Force Model, Serviceability, Cycle loading

**Introduction**

The pedestrian-induced load and the response of pedestrians to vibration is governed by changeability and depends on biological, mechanical and psychological parameters. The main effects of this changeability are that (i) each pedestrian within a group will induce different load and each pedestrian reacts differently to a vibratory environment (inter-subject variability); (ii) small variations in the walking pattern of each pedestrian (intra-subject variability) causes narrow-band random process rather than a perfectly periodic load and (iii) the pedestrian walking force is connected to accelerating and decelerating of the mass of human body (intra-subject variability). In addition, the intra-subject variability can cause the same pedestrian to operate differently in two nominally equal circumstances. In this research it is proposed an analysis of dynamic load factors proposed in the scientific literature.

**Review of the published data on modelling of human walking**

Bachmann and Ammann reported the first five harmonics for vertical walking force and also harmonics for the lateral and longitudinal direction. Bachmann and Ammann declared that the first and third harmonics of the lateral and the first and second harmonics of the longitudinal force are dominant. It is interesting that in the latter case some sub-harmonics appeared. Bachmann and Ammann explained it as a consequence of more pronounced footfall on one side [Bachmann et al., 1977]. The response can be, for vertical direction, divided into five harmonics: 2, 4, 6, 8 and 10 Hz. The first harmonic corresponds to the primary walking frequency. The five harmonics of the walking load offer the higher percentage contribution of static weight (Fig.1). Bachmann and Ammann considered the walking load along lateral (Fig.2) and along longitudinal direction (Fig.3). The response can be divided into two harmonics (2, 4 Hz) and three sub-harmonics (1, 3 and 5 Hz).

The relationship between α_{n} and step frequency was studied for walking rates from 1.0 to 3.0 Hz [Rainer et al., 1988]. Figure 4 shows that for walking, the dynamic load factor of the first harmonic α_{1} is the largest, at 2.4 Hz, and reaches an averaged maximum of 0.52. Magnitude of the dynamic force, induced by different people, is also an inter-subject variable and depends on walking frequency.

The results of first harmonic data [Kerr, 1998] had a trend that tended to follow a third order polynomial function:

where f_{p} is pacing frequency in Hz. The maximum value of the third order polynomial function assumed 0.48 and it was reached at about 2.4 Hz. Second harmonic values were considerably lower than the first harmonic. The average amplitude of second harmonic data was approximately 0.07. The remaining third and fourth harmonic values were even smaller (< 0.06).

Pernica evaluated the variation of the dynamic load factors respect to footstep rate and group size [Pernica, 1990]. Pernica proposed load factors suitable for floors subjected to pedestrian movements. Pernica evaluated the relationship between dynamic load factors and group size (Fig.6-7). The addition of people to the group reduced overall group coordination in performing the activity on the platform. The partecipants found more difficult to maintain the walking rate and the distance between themselves and other group members. If partecipants wish to walk in unison with and be part of a large group of walkers, members are forced to adjust their walking characteristics. In the range 0-10 Hz, maximum value of α_{1} decreased with group size, falling from 0.56 for one person to 0.46 for two people and to 0.36 for four people. Load factors α_{2} increased rapidly with footstep rate for one person. Load factors α_{2} increased slightly with footstep rate for four people. The footstep rate of the maximum value of α_{2} changed location with group size, moving from location above 3.0 Hz for one and two people to below 3.0 Hz for four people. Dynamic load factors for the third and fourth harmonics were relatively constant over the measured frequency range irrespective of group size. Amplitudes of the two harmonics dropped slightly as group size increased. The third harmonic fell from about 0.07 for one and two people to 0.04 for four people. The fourth harmonic from 0.05 for one and two people to 0.02 for four people.

**Periodic Load Models**

Periodic load models are based on an assumption that all pedestrians produce exactly the same force and that the pedestrian load is periodic force [Kala et al., 2012]. It is also assumed that the force produced by a single pedestrian is constant in time. Dynamic loading F(t), caused by a moving pedestrian can be represented as a Fourier series in which the fundamental harmonic f presents a frequency equal to the pacing rate:

where G is the pedestrian’s weight, α_{n} is the load factor of the nth harmonic, f is the frequency of the force, j_{n } is the phase shift of the *n*th harmonic, n is the number of the harmonic and N is the total number of contributing harmonics. The dynamic component of the activity force in Eq.\eqref{fouriermodel1} is represented by the summation term, which is a Fourier series with Fourier coefficients α_{n} , at the discrete frequencies (n f). The Fourier coefficients α_{n} and the footstep frequency f represent the key parameters in Eq.\eqref{fouriermodel1}, that describe the dynamic forces. The Fourier coefficients α_{n} , called *dynamic load factors* (DLFs), are defined as the ratio of the force amplitude of each harmonic to the weight of the person.

**Discussion**

A person will never produce exactly the same force-time history during repeated experiments. In the case of two persons it is even more so. In the case of a single person, the force is assumed to be periodic, but the distribution of person’s weight, pacing rate and different postures can provoke random effects [Setra, 2006]. In the case of several people, the probability distribution of time delay between people who perform a particular activity can be added. Therefore, inter-subject variability and intra-subject variability influence the trend of DLFs [Zivanovic, 2006].

As before mentioned, a group of design procedures is based on an assumption that human-induced forces are perfectly periodic loads. Therefore human-induced forces can be decomposed into harmonics by means of Fourier decomposition as given in Eq.\eqref{fouriermodel1}. Under this assumptions, only a single force harmonic can excite a resonance frequency of a footbridge. Usually, the first three or four excitation harmonics can provoke resonant. So serviceability should be checked in footbridges with fundamental natural frequencies f from 0 up to 5 Hz.

There are some sub-harmonics appearing at frequencies between the main harmonics. It has been widely accepted in the literature that the fundamental period of the walking load is equal to the time required to make two successive steps, rather than one. In this way, the fundamental period is actually approximately two times higher than when analyzing one step only. Consequently the fundamental frequency of the walking force is approximately two times lower than that for a single step. The reason for this aspect is that walking process for two legs can be described by slightly different parameters (walking frequency/period and step length). It is deduced that one leg is stronger than another.

Respect to vertical component, the horizontal component of the load presents less intensity. However, horizontal component can be a source of vibrations and it cannot be neglected. People are very sensitive to being moved sideways. The transverse component corresponds to changing from one foot to the other. The longitudinal component is mainly linked to the frequency of walking. When walking, transverse component occurs at a frequency of half that of the frequency of walking (1 [Hz] for f_{walking} = 2 [Hz]).

A comparison among Millenium bridge, steel bridges, concrete deck bridges, composite bridges, wooden bridges shows the lateral natural frequencies as a function of span (Fig.8) in frequency field 0-4.5 Hz. A comparison between concrete deck bridges and steel bridges illustrates vertical natural frequencies of bridges (Fig.9) in frequency field 0-8.0 Hz [Dallard et al., 2001].

In both vertical and horizontal directions, there are four frequency ranges, corresponding to a decreasing risk of resonance. Table 1 defines the frequency ranges for vertical vibrations and for longitudinal horizontal vibrations. Table 2 concerns transverse horizontal vibrations.

Critical range is related to first harmonic. The dominant contribution of the first harmonic leads to the following critical range for natural frequencies f_{i} : for vertical and longitudinal vibrations 1.7 [Hz] < f_{i} < 2.1 [Hz]; for lateral vibrations 0.5[Hz] < f_{i} < 1.1 [Hz]. There are situations in which natural frequencies lie in an interval susceptible of excitation by the second harmonic of pedestrian excitation. Under these circumstances, if it is considered relevant to investigate the effects associated with the second harmonic of pedestrian loads, the critical range becomes for vertical and longitudinal vibrations: 1.25 [Hz] < f_{i} < 4.6 [Hz]. Footbridges, which have natural frequencies f_{i} in the critical range, should be object of a dynamic assessment to pedestrian excitation. Lateral vibrations are not effected by the 2nd harmonic of pedestrian loads. Table 3 summarizes DLFs for single person force models after different authors in vertical walking.

**Conclusions**

Several modelling process proposed a walking force models moving across the bridge. A considerable improvement of design procedures was obtained with model of walking force that considered the effect of DLFs, dependent on step frequency, harmonic and duration time, limited by the length of the bridge.

**References**

H. Bachmann and W. Ammann. Vibration Problems in Structures. 1977. ISBN 3-7643-5148-9.

P. Charles and H. Wasoodev. Footbridges Assessment of vibrational behaviour of footbridges under pedestrian loading. Technical report, Setra, October 2006.

P. Dallard, T. Fitzpatrick, A. Flint, A. Low, R. Smith, M. Willford, and M. Roche. London Millenium Bridge: Pedestrian-Induced Lateral Vibration. 6:412–417, 2001.

J. Kala, V. Salajka, and P. Hradil. Dynamic Action Induced By Walking Pedestrian. World Academy of Science, Engineering and Technology International Journal of Civil, Environmental, Structural, Construction and Architectural Engineering V, 6(10):827–830, 2012.

S. Kerr. Human induced loading on staircases. PhD thesis, University of London, 1998.

G. Pernica. Dynamic load factors for pedestrian movements and rhythmic exercises. Canadian Acoustics / Acoustique Canadienne, 18(2):3–18, 1990.

J. Rainer, G. Pernica, and D. Allen. Dynamic loading and response of footbridges. Canadian Journal of Civil Engineering, 15(1):66–71, 1988.

S. Zivanovic. Probability-Based Estimation of Vibration for Pedestrian Structures due to Walking. Master’s thesis, February 2006.