## 4 Analysis of Relevant Influencing Parameters
### Effect of Delaminations
#### 4.3.1 Analytical Description of the Problem  
Given a composite beam with delaminations, the mathematical problem can be solved in accordance with [51], i.e., in the form of an effective longitudinal modulus (\(E_{\textit{eff}}\)) representative of the actual mechanical interaction between the constituent layers. For a laminated section. according to Figure 17a, such a modulus is given by:  
\[E_{\textit{eff}}=\frac{8}{h^{3}}\sum_{j=1}^{m/2}(E_{x})_{j}\left(z_{j}^{3}-z_{ j-1}^{3}\right) \tag{12}\]  
with \(E_{x}\) the MoE of the \(j\)-th layer, \(m\) is the total number of layers composing the beam section (with \(b\times h\) total dimensions); \(z_{j}\) is the distance between the outer face of the \(j\)-th layer and the neutral plane of the section. When \(m\) is an odd number--as in the case of the examined double LG sections, see Figures 2 and 17b--Equation (12) is still valid, given that the external \(j\)-th layer is subdivided into two symmetric parts.  
The effect of delaminations is, in fact, accounted for by reducing \(E_{\textit{eff}}\), as a function of the number and size of delaminated regions. Based on [51], the resulting longitudinal MoE of a partially delaminated composite section is, in fact, given by:  
\[E_{d}=\left(E_{\textit{zd}}-E_{\textit{eff}}\right)\frac{A_{d}}{A_{t}}+E_{ \textit{eff}} \tag{13}\]  
with:  
\[E_{\textit{zd}}=\frac{\sum_{j=1}^{s}E_{\textit{eff}}z_{j}}{z} \tag{14}\]  
Moreover:  
\(E_{\textit{zd}}\)--is the longitudinal MoE of a totally delaminated section (along one or more interfaces), representative of the so-called imperfect effective MoE;  
Figure 17: LG beams with delaminations: (**a**) reference analytical model for simply supported, delaminated composite beams (reproduced from [51]) and (**b**) detail example for a double LG section.  
\(S\)--is the number of sub-layers detected by the delamination; \(z_{j}\)--represents the thickness of the \(j\)-th sub-layers; and, \(A_{d},A_{t}\)--are respectively the delaminated and total interfacial area between the bonded layers.  
Key assumptions of Equations (12)-(14) are that:  
1. Plane sections are initially normal to the longitudinal axis of the glass beam, and remain plane and normal also during flexure;
2. The beam has symmetrical properties about the neutral axis (both geometrical and mechanical);
3. The sandwich beam section is composed of layers with a linear elastic behaviour; and,
4. Shear coupling between each ply can be disregarded.  
Based on point (d), it is expected that the analytical method herein recalled could overestimate the bending stiffness decrease of the delaminated LG beams, thus resulting in conservative frequency predictions. Another issue can be related to the location of delaminated regions (disregarded by the surface parameter \(A_{d}\)).  
For general LG members, which were included the tested specimens, delamination phenomena typically occur along the edges and close to restraints (Figure 9), where the LG layers are not protected and/or properly sealed. In some other cases, the spotted delaminated regions can be recognized over the surface of a given LG member. However, in both the cases, natural delaminated regions (and their quantitative effects for structural calculations) can be hard to visually detect and properly quantify.