PSI - Issue 14

Angitha Vijayan et al. / Procedia Structural Integrity 14 (2019) 696–704 Angitha Vijayan et al. / Structural Integrity Procedia 00 (2018) 000 – 000

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properties of the empty structure, determination of crowd properties, development of model for CS system and analysis to determine the properties and response of CS system.

2.1. Determination of modal properties of the empty structure

The structure considered for this study is a simply supported concrete slab strip shown in Fig. 1(a) and (b). The structure geometry is adopted from literature (Shahabpoor et al., 2017). In order to find the modal properties of its dominant mode, the empty structure is analyzed as a lumped mass 2DoF system as represented in Fig. 1(c).

Fig. 1. Structure considered for analysis (a) Plan; (b) Elevation; (c) 2DoF lumped mass model.

The flexibility matrix is formulated using geometric and material properties of the structure, from which stiffness matrix is obtained. Modal analysis is carried out to obtain the natural frequencies and corresponding mode shapes of the empty structure. The two vertical modes are obtained from which the dominant (fundamental) mode is selected for further analysis. Once the dominant mode is obtained, the empty structure is considered as a SDoF system with the modal properties of the dominant mode for further analysis. From the natural frequency ( ) and damping ratio ( ζ ) of the structure, modal mass ( ), modal stiffness ( ) and modal damping coefficient ( ) are obtained. 2.2. Determination of crowd properties Each person may be conceptualized as an independent single degree of freedom (SDoF) system. The combined effect of all the people may be again represented as an equivalent SDoF system for the crowd. The properties of crowd is obtained from the characteristics of the individuals in the crowd. 2.2.1. Modeling single person The entire mass of the body is concentrated at the center of mass (CoM). The human body can be compressed from standing upright position and released back to the same position like a spring, where movement is allowed at hip, knees and ankles. Hence, it possesses stiffness. Any such movement of human body once initiated, stops after some time. This is because of the presence of viscous damping in the body that helps in absorbing shocks and maintaining stability. Therefore, an individual can be modeled as a SDoF system having mass ( ), stiffness ( ) and damping ( ) as depicted in Fig. 2 (Zhang et al., 2000). 2.2.2. Modeling crowd as SDOF system The properties of an equivalent SDOF model of crowd such as mass ( ), stiffness ( ) and damping ( ) are obtained from each individual’s properties and location on the structure using the following equations (Shahabpoor et al, 2017): = ∑ 2 (1) = ∑ 2 (2) = ∑ 2 (3)