## 1 Introduction  
Long-span structures, e.g., footbridges, have become prevalent all over the world, and the excessive dynamic vibrations induced by locomotive crowds are a crucial maintenance concern for their life span. Full-scale dynamic measurements on service structures has shown that humans can change structural vertical dynamic properties [1; 2], which has also been confirmed by a measurement experiment on a beam bearing different numbers of persons in [PERSON] et al. [3]. To explain the vibration mechanism induced by humans, some human-structure models have been proposed, including a one-person model [4; 5] and a crowd model [6; 7; 8; 9]. [PERSON] et al. [6; 7] studied vertical dynamic characteristics of structure under a modeled human oscillator. [PERSON] and [PERSON] [8] proposed a moving oscillator to simulate walking pedestrians and further studied the influence of structure frequency and damping variations. It is noted that crowd size may significantly affect variation of the dynamic properties of footbridges. Some crowd-structure interaction models have also been proposed to explore this mechanism. [PERSON] et al. [9] proposed two reduced-order models to approximate the dynamic behavior of a full-order coupled crowd-structure interaction system. [PERSON] et al. [10] also recommended a vertical crowd-structure interaction model to analyze the effect of modal properties variations of a footbridge. To consider gait details, a human-structure interaction with bipedal pedestrian model was proposed by [PERSON] et al. [4; 5]. [PERSON] and [PERSON] [11; 12; 13] extended this model to a crowd-structure interaction, based on a stability improvement technique. Although the bipedal pedestrian model is successfully applied for crowd-structure interaction in [PERSON] [PERSON] [11; 12; 13], the theoretical mechanisms between crowd size and structural modal properties by using this model have not yet been reported. [PERSON] et al. [3] studied the effect of crowd size on vertical modal parameters of an occupied structure by extensive Frequency Response Function (FRF)-based modal identification tests. Although they quantitatively researched the impact of the number of occupants on structural dynamic properties, one applicable formula which includes human dynamic properties for vibrational control between crowd scale and structural modal characteristics, is absent. Considering that the dynamic properties of humans are crucial to elucidating the variation of structural modal properties, it is therefore imperative to provide a quantitative estimation between crowd size and modal properties of structure.  
In this paper, the theoretical formulae between crowd size and vertical modal properties are proposed as an extension of the study in [PERSON] and [PERSON] [11; 12; 13]. Firstly, the dynamic equilibrium equation of a structure is established by considering the vertical interaction between bipedal crowd and structure. Secondly, an assumed uniform distribution of a walking crowd is used to calculate the crowd size by Taylor Expansion. Thirdly, the theoretical modal property formulas are derived, based on the assumption of a synchronous movement behavior. Finally, a numerical example is used to analyze the effect of crowd size on structural modal properties.