1. University of Oxford Modelling of joint crowd-structure system using equivalent reduced- DOF system Jackie Sim, Dr. Anthony Blakeborough, Dr. Martin Williams Department of Engineering Science Oxford University

2. University of Oxford Cantilever grandstands

3. University of Oxford Dynamic analysis of cantilever grandstand Human-structure interaction Passive crowd Crowd model Active crowd Load model

4. University of Oxford Full model m s F x Total mass of crowd =  m s m s F x Crowd as 2DOF system Structure as SDOF system

5. University of Oxford Equivalent reduced DOF systems Equivalent SDOF system Equivalent 2DOF system m s F x Full model

6. University of Oxford Contents Crowd model Response of full model Equivalent SDOF model Equivalent 2DOF model

7. University of Oxford DOF 2 m2m2 k 2 c 1 k 1 m1m1 c 2 m 0 F y2y2 y 1 DOF 1 m2m2 k 2 c 1 k1 k1 m1m1 c 2 F y 2 y 1 DOF 2 Seated model Standing model Individual models – Griffin et al.

8. University of Oxford Crowd response

9. University of Oxford Crowd model Transfer functions: Seated: Standing: Fourth order polynomial i.e. 2DOF system

10. University of Oxford Dynamic analysis (1) 2% structural damping, Natural frequency of 1 to 10 Hz. 50% seated and 50% standing crowds  = 0%, 5%, 10%, 20%, 30% and 40% m s F x Crowd mass =  m s

11. University of Oxford Dynamic analysis (2) DMF = Peak displacement / Static displacement SDOF structure Seated / standing crowd Displacement Excitation force Interaction force Acceleration +

12. University of Oxford Results – DMF vs Frequency 2 Hz structure4 Hz structure

13. University of Oxford Summary of results (1): Resonant frequency reduction factor F.R.F. = Change in frequency / Frequency of bare structure

14. University of Oxford Summary of results (2): DMF reduction factor DMF R.F. = Change in DMF max / DMF max of bare structure

15. University of Oxford Why reduced-DOF system? Full crowd-model: 2DOF crowd + SDOF structure A simplified model for  Easier analysis  Insight into the dynamics

16. University of Oxford Equivalent SDOF system SDOF system transfer function: Curve-fit DMF frequency response curve over bandwidth

17. University of Oxford Dynamic properties

18. University of Oxford Error analysis (1) Peak DMF relative error Resonant frequency relative error

19. University of Oxford Error analysis (2)

20. University of Oxford Equivalent 2DOF system ms ms F x Crowd modelled as SDOF system Structure remains the same SDOF system

21. University of Oxford SDOF crowd model

22. University of Oxford Dynamic analysis SDOF structure SDOF Seated / standing crowd Displacement Excitation force Interaction force Acceleration +

23. University of Oxford Error analysis

24. University of Oxford Bode diagrams

25. University of Oxford Conclusions Passive crowd adds significant damping 1 to 4 Hz – behaviour of a SDOF system > 4 Hz – behaviour of a 2DOF system