## 2 Glass Structures and Vibrations
### Vibration Analysis and Human-Structure Interaction (HSI) Phenomena  
The vibration response of glass structures is a relatively recent research topic. Most of the literature studies include experimental, analytical and numerical analyses voted to explore the dynamic behavior  
Figure 3: The role of transparency on magnified human perception of vibrations and discomfort feelings for the glass walkway object of investigation (photo by [PERSON], courtesy of So.Co.Ba). of single monolithic or LG members, with limited efforts for the performance assessment of complex assemblies, see for example [25; 26; 27; 28; 29; 30]. As a general rule, glass slabs should be verified against vibrations under operational conditions in the same way of pedestrian structures composed of other constructional materials. Possible critical aspects of glass pedestrian structures, in this regard, can be represented by typically small thickness-to-size ratios, high flexibility and slenderness, unconventional or limited number/size of restraints (i.e., point supports, etc.). Additional uncertainties can then derive from the possible/progressive degradation of materials and restraints (i.e., Figure 1), hence resulting in even marked variations of the fundamental dynamic properties. Finally, HSI effects may be crucial [18; 31; 32; 33; 34]. From a mathematical point of view, a single stationary human body represents in fact a dynamic system able to interact with a given structure (see for example Figure 4a). Especially in presence of moving occupants or crowds (Figure 4b), HSI phenomena can hence involve complex effects, due to the occurrence of reciprocal motion forces and to the capacity of the occupants to change the dynamic features of a structure, over which they are [18]. Key parameters are represented by the motion features (i.e., pacing frequency and phase, walking speed, stride length, etc. ), and dedicated tools are necessarily required for reliable analyses. A given pedestrian structure must in fact satisfy the general equation of motion:  
\[M\ddot{x}(t)+C(t)+Kx(t)=P(t) \tag{1}\]  
with \(M\), \(C\) and \(K\) denoting the modal mass, damping and stiffness matrices, \(P(t)\) the imposed external (periodic) force reproducing the motion of occupants and \(x(t)\) the vertical displacement vector. As far as the pedestrian systems of Figure 3 are occupied (\"os\"), it is expected that the fundamental frequency and the corresponding modal damping modify, with respect to the empty structure (\"es\"), that is:  
\[f_{\text{os}}=\frac{1}{2\pi}\cdot\sqrt{\frac{k_{\text{os}}}{m_{\text{os}}}}\
eq f _{\text{es}} \tag{2}\]  
and  
\[\xi_{\text{os}}=\frac{c_{\text{os}}}{2\sqrt{k_{\text{os}}m_{\text{os}}}}\
eq \xi_{\text{es}} \tag{3}\]  
Several literature contributions demonstrated that [18]:  
* standing people commonly reduces the frequency of an occupied system, that is \(f_{\text{os}}<f_{\text{es}}\), while;
* walking occupants generally manifest in a frequency increase, \(f_{\text{os}}>f_{\text{es}}\). Figure 4: Mathematical models for human-structure interaction (HSI) calculations of pedestrian structures: (**a**) Inverted pendulum approach and (**b**) crowd effects on stadium structures. Figures reproduced from [18] under CC BY 4.0 license. Moreover:  
* when the natural frequency of walking bodies (\(f_{p}\)) is less than the frequency of the empty structure (\(f_{\rm es}\)), it is expected that \(f_{\rm os}>f_{\rm es}\);
* when \(f_{p}>f_{\rm es}\), otherwise, it is typically observed that \(f_{\rm os}<f_{\rm es}\);
* the effects of walking human bodies are more pronounced as the number of pedestrian increases;
* and finally, damping is always expected to increase for an occupied system, i.e., \(\xi_{\rm os}>\xi_{\rm es}\). The common assumption of the general considerations listed above is that they are strictly related to a series of multiple aspects, including the mechanical features of the structure under analysis and the role that human bodies can induce in the vibration response of the empty system.