There is a risk that excessive vibration in long span cantilever grandstands can be triggered by the spectators synchronising their jumps to the music played. If the jumping frequency excites a resonance of the grandstand, large force could be generated.
This thesis studies human-structure interaction in cantilever grandstands, with emphasis on modelling the passive and jumping crowds, and analysing the response of a single degree-of-freedom (SDOF) structural system. Preliminary work on analysing a cantilever occupied by seated humans shows that it is acceptable to use a SDOF structural system for analysis which meant emphasis of later work could be placed on understanding the interaction between a passive crowd and the structure.
Human dynamic models from published biomechanics studies are used to develop a passive crowd model. A transfer function, fitted to the crowd apparent mass, is used to define the crowd model. It is found that the passive crowd can be approximated well by using a single 2DOF system.The combined passive crowd-structure system is modelled as a feedback system and a parametric study is conducted.
It is found that the passive crowd adds significant mass and damping to the structure and these effects vary with the natural frequency of the structure. Records of forces of people jumping to a beat are used to develop a probabilistic model of crowd jumping loads. Key parameters are introduced to characterised the timing and shape of the jumping impulses. An analytical function is used to approximate the impulse shape. All parameters are characterised with probability distribution functions.
Using the fitted probability distribution functions, the Monte Carlo method is used to simulate individual jumping load-time histories and to obtain the structural responses due to group jumping loads. The variations of the structural response with the natural frequency of the empty structure and the size of the active crowd are presented in charts. As expected, the worst response is found on structures with natural frequencies coinciding with the first three harmonics of the crowd jumping loads.
For structures occupied by passive crowds, a significant reduction in the structural response is found at resonance excited by the second and third harmonics, due to high levels of damping provided by the passive crowds. On variation of the structural response with the crowd size, it is found that the structural response becomes asymptotic for groups larger than 16 people.
Experimental individual jumping and bobbing tests are conducted at six distinct beat frequencies to look at the variations of the impulse shape and degree of synchronisation with the beat frequency. The bobbing action is found to have a higher inherent variability between individuals compared to jumping. Jumping tests involving two people facing each other are also conducted. The results show that there is a better synchronisation when two people are jumping together compared to when jumping alone.
Source: University of Oxford
Author: Jackie H.H. Sim