# Vertical Vibrations of Footbridges Due to Group Loading: Effect of Pedestrian-Structure Interaction
## 1 Introduction  
Footbridges are designed for the conveyance of pedestrians. Vibrations induced by the latter during the normal use of the structure have long been recognized as a design problem in terms of a serviceability limit state. The features of walking are such that pedestrians can cause vibrations in the vertical or lateral direction. Several case reports of footbridges that presented excessive vibrations can be found in the literature [1; 2; 3; 4; 5; 6; 7; 8]. Two major groups of such structures can be distinguished: the group in which footbridges presented excessive vibration in the lateral direction [1; 2; 3; 4; 6], and the other group presenting excessive vibration in the vertical direction [1; 5; 6; 7; 8]. In the former, the lateral natural frequencies of such lively footbridges are within the usual range of half of pedestrian pacing rates. This is because a complete cycle of body movement in the lateral direction requires two steps to be taken. Such critical lateral natural frequencies are around 1.0 Hz and are usually associated with long-span footbridges, above 100 m long. In the latter, the structures presented naturalfrequencies in the vertical direction within the usual range of the pacing rates of pedestrians. These natural frequencies may be associated with footbridges having spans of around 60 m or associated with lightweight, easily excited structures [9; 10]. Detailed, advanced studies on human movement when walking or running on footbridges can be found, e.g., in [11; 12]. This vibration problem has been dealt with by codes of practice. One such code that first tackled vibrations in the vertical direction was the British code BS 5400 [13]. Two features of this code are worth calling attention to: a single pedestrian was considered as the design load, and the action of the pedestrian was represented as an oscillatory moving force. Later on, more realistic design conditions were conceived, in which the load case was now due to a group of pedestrians walking together or a stream (crowd) of pedestrians [14; 15; 16]. However, pedestrians continued to be considered as moving forces acting on the structure. A compromise of the effect regarding the presence of pedestrians not only as loads is considered by the Setra [15] and Hivoss [16] guidelines, in which a fraction of the mass of the pedestrians is added when calculating the natural frequency of the structure. A collection of results from recent research has emphasized the role of pedestrians as dynamic systems that interact with the vibrating structure [17]. From a historical perspective, one of the first works that proposed the inclusion of pedestrians as dynamic systems was that of [PERSON] [18]. This was more of a procedure than an analytical formulation, in which the moving force was applied together with a single-degree-of-freedom spring-mass-damper (SMD) model to consider the pedestrian action as a dynamic system. Elaborate analytical formulations for SMD models of single pedestrians have been further proposed [19; 20; 21; 22; 23; 24], in which the interaction between pedestrian and structure can be included in the formulation. These models have been discussed, and a basic SMD model has been appointed as potentially representative of the pedestrian's dynamic action [25]. The presence of pedestrians affects at least the mass and damping of the vibrating system and has resulted in reductions in the overall vibration levels, when compared to results obtained by considering pedestrians as loads only [25; 26]. However, the dynamic effect of several pedestrians crossing the structure simultaneously still remains a less investigated aspect of the design, particularly with the support of experimental data. The aim of this paper is to deepen the investigations of pedestrian-structure interaction, applicable to the vibration serviceability of footbridges in the vertical direction. The focus is on loads due to groups walking together, which is one of the load cases considered in the design [14].