# Vertical Vibrations of Footbridges Due to Group Loading: Effect of Pedestrian-Structure Interaction
## 2 Materials and Methods  
### Test Structure and Pedestrian Tests  
The structure selected was a 68.0 m long two-span cable-stayed footbridge named Zlotnicka after its location in Zlotnicki Park in Wroclaw, Poland. It has central pylons to support the deck and to which the stays are attached (Figures 1 and 2). Selection of the structure adhered to both technical and logistical requirements. From the technical point of view, the structure had a natural frequency in the vertical direction within the range of the pacing rates of pedestrians and presented perceptible vibrations when crossed by them. In terms of logistics, it was in quiet environment with sparse use, enabling a controlled test program to be carried out. Figure 1: Technical drawing of the structure (side view, cross-section and top view), with relevant dimensions. Selected measurement point on the left span, 16.0 m from the support. Figure 2: Panoramic view of the footbridge. The following design situations were applied to this footbridge: from the Hivoss guideline [16], a traffic class TC1 (very weak traffic) was considered. On the other hand, an appointed comfort class CL3 (minimum degree of comfort) was targeted, which would be related to a limit of vertical peak accelerations between 1.0 and 2.50 m/s2. This corresponds to Traffic Class III (footbridge for standard use) and minimum comfort, according to the Setra guideline [15]. In the UK NA to BS EN 1991-2 [14], the accepted peak acceleration limit would be 1.3 m/s2, obtained by considering the structure as a suburban crossing and its respective height above ground. This vibration limit is compatible with the aforementioned range proposed in the Hivoss guideline [16]. The structure was the subject of previous investigations, and detailed information regarding its geometry and finite element (FE) modeling can be found elsewhere [27]. Herein, the focus is on the properties of interest for the purposes of this paper. A measured natural frequency of 2.07 Hz was identified, corresponding to the first vertical antisymmetric mode of vibration, with a node at the central support. Relevant mode shapes are shown in Figure 3 from the calibrated FE model. The modal mass related to this mode was also calculated from the calibrated FE model and had a value of 22,205.8 kg. By employing the natural frequency and the modal mass, the modal stiffness was calculated using the expression of the natural frequency of a single-degree-of-freedom system. This modal mass was scaled by considering a unity value for the maximum ordinate of the modal shape of the first mode. Regarding damping, the value was obtained from the tail end of the pedestrian tests and is discussed in the Section 3. Ten test subjects volunteered to take part in the experimental campaign (Figure 4). A metronome was employed during the crossings, and the reason for this was twofold: excitation in the resonance condition is the target of the codes of practice for designing footbridges against vibration serviceability, and the level of excitation should be as high as possible for the interest of this investigation. Therefore, the beating of the metronome was set to help the test subjects to walk with a constant pacing rate in order to excite the first mode of vibration in the resonance condition. The test subjects, eight males and two females, were arranged in military formation to help keep the distances between them constant and also characterize a group loading. The leading pedestrian was in front carrying the metronome and was instrumented with an HBM GmbH (type B12/200) accelerometer attached to his waist (Figure 4a), whereas the other accelerometers remained near the antinode of the first mode shape (see Figures 1 and 3). The other test subjects were arranged according to Figure 4d. The mass of each test subject is shown in Table 1 and was employed in the numerical simulations. In addition, each test subject crossed the footbridge alone using the metronome to control the step frequency.