In recent years, several equations have been proposed to calculate deflections and stresses in laminated-glass beams and plates under static loading using the concept of effective thickness, which consists of calculating the thickness of a monolithic element with equivalent bending properties to a laminated element. Recently, an effective thickness for the dynamic behaviour of laminated-glass beams has been proposed to enable the modal parameters (natural frequencies, loss factors and mode shapes) to be determined using an equivalent monolithic model. In the present paper, the technique has been extended to the two-dimensional case of rectangular laminated-glass plates and the steps needed to estimate the modal parameters of laminated-glass elements using this methodology are presented. The dynamic effective thickness concept has been validated by experimental tests made on a laminated-glass beam and a laminated-glass plate. The results show that good accuracy is achieved in the natural frequencies and mode shapes but high scatter is encountered in the loss factors.

The mechanical behavior of laminated glass elements is governed by material properties of the interlayer, the Polyvinyl Butiral (PVB) being the most used interlayer material in these elements. PVB is a viscoelastic material whose mechanical properties (Young's modulus, shear modulus, etc.) depend mainly on the load application time and the temperature. Thus an adequate mechanical characterization of the PVB must be performed in order to predict the response of laminated glass elements with a good accuracy In this work, PVB specimens were subjected to static relaxation tests and to dynamic experimental tests (frequency domain) at different temperatures from −15 to 50 using a DMTA equipment. Then the curves at different temperatures were related using the William-Landel-Ferry (WLF) Time-Temperature Superposition (TTS) model to obtain the mastercurve of both the time and frequency domain Young's moduli of the PVB. Finally, a viscoelastic Prony based model was fitted to the experimental data and used, afterwards, to simulate numerically the static and dynamic behaviour of different laminated glass elements at different temperatures. The numerical simulations were compared with the static and dynamic experimental results achieving a good accuracy in both the static deflections and the natural frequencies. With respect to the damping, the discrepancies are less than 22%.

The mass change method is used to estimate the scaling factors, the uncertainty is reduced when, for each mode, the frequency shift is maximized and the changes in the mode shapes are minimized, which in turn, depends on the mass change strategy chosen to modify the dynamic behavior of the structure. On the other hand, the aforementioned objectives are difficult to achieve for all modes simultaneously. Thus, a study of the number, magnitude and location of the masses must be performed previously to the modal tests. In this paper, the mass change method was applied to estimate the scaling factors of a steel cantilever beam. The effect of the mass change strategy was experimentally studied by performing several modal tests in which the magnitude, the location and the number of the attached masses were changed.

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When operational modal analysis (OMA) is used to estimate modal parameters, mode shapes cannot be mass normalized. In the past few years, some equations have been proposed to scale mode shapes using the mass-change method, which consists of repeating modal testing after changing the mass at different points of the structure where the mode shapes are known. In this paper, the structural dynamic modification theory is used to derive a set of equations, from which all the existing formulations can be derived. It is shown that the known equations can be divided into two classes, the exact and the approximated equations, where the former class does in fact fulfill the equations derived from the theory of structural modification, whereas the remaining equations do not, mainly because the change of the mode shapes of the modified structure is not taken properly into account. The paper illustrates by simulations the large difference in accuracy that exists between the approximate and the exact formulations. The paper provides two new exact formulations for the scaling factors, one for the non-modified structure and -as the first time in the literature -one for the modified structure. The paper illustrates by simulation the influence of errors on the measured natural frequencies and mode shapes on the estimation of the scaling factors using the two exact formulations from the literature and the new exact formulation proposed in this paper. Further, the paper illustrates statistics of the errors on mode shape scaling. All simulations were carried out using a plate with closely spaced modes.