## 4 Analysis of Relevant Influencing Parameters
### Derivation of Practical Fitting Curves for LG Members with Flexible Restraints  
According to literature, the definition of closed-form formulations accounting for the actual mechanical restraint in viscoelastic LG beams in free vibrations can involve complex mathematical problems (i.e., [43]). Otherwise, simple analytical fitting curves can offer robust support for reliable frequency estimations. Assuming that a given glass member is restrained via mechanical supports having partial rotational stiffness \(K_{r}\
eq 0\) (in kNm/rad) and \(K_{s}=\infty\), the restraints act as flexible clamps that must be properly taken into account for design. Moreover, following Section 4.1, it is convenient to express these rotational and axial stiffness contributions as:  
\[R_{r}=\frac{K_{r}L_{0}}{EI} \tag{9}\]  
and  
\[R_{s}=\frac{K_{s}L_{0}^{3}}{EI} \tag{10}\]  
where \(R_{r}=0\) corresponds to the limit condition of a S-S beam and \(R_{r}=\infty\) denotes the C-C configuration. As far as a monolithic or LG beam is taken into account in Equations (9) and (10), its bending stiffness \(EI\) can be expressed as a function of the total \(b\times h\) or equivalent \(b\times h_{\theta}\) section (Equation (3)). Thus, from a practical point of view, the vibration frequency of a given glass beam with non-ideal restraints can be conveniently calculated as (with \(R_{r}\geq 0\) and \(R_{s}=\infty\)):  
\[\overline{f}_{n}=k_{f}f_{n} \tag{11}\]  
where \(f_{n}\) is given by Equation (2) for a S-S beam and \(k_{f}\) represents a magnification factor, depending on \(R_{r}\). In this paper, a series of parametric FE numerical simulations was carried out in ABAQUS for glass beams with different geometrical features (\(b=0.1\)-\(0.5\) m, \(h=0.005\)-\(0.04\) m, \(L=L_{0}=1\)-\(4\) m) and  
Figure 13: Numerical (with PVB\({}_{A}\) interlayer properties) and experimental frequencies for the tested LG beams: (**a**) comparative frequency values and (**b**) corresponding percentage scatter \(\Delta\), as a function of \(L_{0}\). variable rotational stiffness \(K_{r}\) (with \(K_{s}=\infty\)). In doing so, beam (B31) type elements were used, with equivalent springs being able to reproduce the desired rotational stiffness \(K_{r}\) (Figure 14). Analytical fitting curves are thus proposed in this paper, in support of practical calculations (i.e., based on Equation (11)) on general LG beams with flexible restraints, see Figure 15 and Table 2.  
As shown in Figure 15, in particular, it was observed that the use of flexible restraints and their combination with different geometrical/mechanical LG properties can result in even marked variations of the expected vibration frequencies. This is especially the case of the fundamental mode (\(n=1\)), where relatively stiff restraints can amplify up to \(\approx\)2.25 times the S-S estimations. Certainly, the vibration shape is also expected to modify with the stiffness variation of restraints (i.e., Figure 14b). However, the current investigation was specifically focused on frequency estimates, and the accurate analysis of shape sensitivity would require more detailed experimental methods, as compared to the available test predictions. Figure 14: Parametric numerical analysis of monolithic glass beams (ABAQUS): (**a**) axonometric view of the expected vibration shape (\(n=1\)) and (**b**) qualitative shape variation, as a function of \(R_{r}\). Figure 15: Analytical fitting curves (see also Table 2) for the frequency estimation of glass beams with flexible restraints (\(K_{r}>0\) and \(K_{s}=\infty\)). Given that \(h_{\mathit{ef}}\) in Equation (3) modifies with \(G_{2}(\omega)\), another relevant outcome of Figure 15 is that the role of real restraints can be properly taken into account in the iterative analytical procedure recalled in Section 2. Marked variations of \(h_{\mathit{ef}}\) (and thus frequency estimates), as in Figure 16, can in fact be obtained within the \"_layered_\" and \"_monolithic_\" limit conditions, as far as \(R_{t}\) modifies.