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

700

5

2.4. Analysis to determine the properties and response of CS system

Frequency response function (FRF) analysis is performed to study the behaviour of the occupied structure. The following equations are used to calculate the FRF functions (Jones et al, 2011): ( ) = ( − 2 + ) −1 (6) ̈ ( ) = ( ) × −(2 ) 2 × [ 1,1 ( ) − 1,2 ( )] (7) ( ) = ( ) + ( ) (8) = ( ) (9) ( ) = −2( −1.8) 2 (10) = ∑ (11) ( ) in Eqn (6) is the displacement FRF of occupied structure subjected to a unit force at a frequency of rad/s or Hz. As shown in Eqn (7), the acceleration response of the occupied structure subjected to human dynamic loading is obtained by multiplying the difference between the elements in first row of the matrix ( ) with - 2 and the internal force. ( ) is the internal force represented in complex form as given by Eqn (8). is the phase lag and in this work, it is taken as zero for maximum force. is the overall amplitude of the force which is given in Eqn (9). ( ) is the effectiveness factor for a crowd of activity frequency . Here is the forcing mass of the structure which is obtained using Eqn (11) and is the generated load factor (GLF) of value 1.188, considering all the individuals active. is the acceleration due to gravity. Finally, to determine the physical accelerations experienced by the crowd at specific locations the acceleration response ̈ can be multiplied with the mode shape ordinate at that location. Table 1. Comparison of analytically obtained properties of CS system with experimentally obtained values.

Analytically obtained properties of CS system

Experimentally obtained properties of CS system (Shahabpoor et.al, 2017)

Input crowd properties (Shahabpoor et.al, 2017)

No. of people

f c (Hz)

ζ c (%)

c c (Ns/m)

k c (N/m)

f cs (Hz)

ζ cs (%)

f cs (Hz)

ζ cs (%)

Walking along the structure 3 2.504 32

1057 2053 3061 2301 4048 4586

25991 63980

4.445 4.465 4.475 4.450 4.480 4.500

1.10 1.65 2.30 2.11 2.90 3.40

4.448 4.460 4.480 4.454 4.485 4.520

1.00 1.55 2.20 1.80 2.92 3.00

6

2.778 2.900

28 24

10

116205

Walking in a circle at midspan 3 2.906 30

70012

6

2.950 2.962

26 22

144296 193963

10

3. Validation The modal properties of the crowd structure (CS) system considering the interaction between the two are analytically found. The input modal properties of the empty structure and crowd characteristics are taken from the experimental studies reported by Shahabpoor et al. (2017) for validation of results. The results obtained are compared with the experimental results and the same is given in Table 1. The obtained results are found to be in agreement with the experimental results.In order to understand the effect of the difference in physical characteristics of individuals in a group on the modal properties of the occupied structure, the same is obtained by inputting the crowd characteristics given by Shahabpoor et.al (2017) as well by inputting the properties of an average person for all the individuals (Table 2). The results are tabulated in Table 3 and are found to be almost same for both the cases.