Dynamic effective thickness in laminated-glass beams and plates
Introduction
Laminated glass is a sandwich or layered material consisting of two or more plies of monolithic glass with one or more interlayers of a polymeric material. Thus, a laminated-glass element (beam, plate, etc.) is a composite material which combines the properties of the glass with the benefits of a highly elastic polymeric material, i.e. the structural behavior of laminated glass is that of a composite structure. The glass layers may be either normal annealed, heat-strengthened, chemically strengthened or tempered glass. When the cross-section is made of different glasses (e.g. one layer of annealed glass and the other of tempered glass) it is called a hybrid laminated-glass element. However, the treatments affect the ultimate strength but not the Young modulus, and therefore no distinction is made concerning the type of glass if the calculations are made prior to glass breakage. All polymeric interlayers are viscoelastic in nature [1], i.e. their mechanical properties are frequency (or time) and temperature dependent. Polyvinyl butyral (PVB) is the most commonly used interlayer material and is marketed in thicknesses of 0.38 mm or a multiple of this value (0.76 mm, 1.12 mm, 1.52 mm). However, the new ionoplastic interlayers improve the mechanical properties of laminated glass and maintains a significant advantage (higher stiffness and strength) over the PVB for a large range of temperatures [1]. This interlayer material is now in flat sheet form, in thicknesses of 0.89, 1.52, 2.27, and 3.05 mm, and as rolled sheeting, at 0.89 mm thickness. The simplest laminated-glass configuration consists of three layers: two monolithic glass plies and a polymeric core (see Fig. 1).
The response of laminated-glass elements varies between two borderlines [2]: (1) The layered limit corresponding to the case when the beam consists of free-sliding glass plies and (2) the monolithic limit, when the Euler–Bernoulli assumptions hold (plane sections remain plane) for the entire section of the laminated-glass element (the response of the composite beam approaches that of a homogeneous glass beam with an equal cross-section) [3], [4]. As the tensile modulus of the PVB is far less in comparison with that corresponding to glass, significant transverse shear appears in the viscoelastic layer [1], [8], [10].
In the analytical and numerical models, glass mechanical behavior is usually modeled as linear-elastic prior to glass breakage, whereas the polymeric interlayer is characterized as linear-viscoelastic. Laminated glass is easy to assemble in a finite-element model but many small 3D elements are needed to mesh accurately, which, on the other hand, are very high time consuming. In the last few years, some papers have been published on the calculation of laminated-glass elements, examining the concept of effective thickness [1], [3], [4], [8]. The method consists of calculating the thickness of a monolithic element with bending properties equivalent to those of the laminated one. The effective thickness can then be used in analytical equations and simplified finite-element models instead of the laminated-glass element [9].
The aim of the present paper is to propose a simplified method to estimate the modal parameters of rectangular laminated-glass plates while avoiding the use of finite-element models or complicated analytical models. The method is based on the dynamic effective thickness proposed in a previous paper [9] for laminated-glass beams, which is here extended to the two-dimensional case of rectangular laminated-glass plates. An alternative to the effective thickness is the concept of effective Young modulus, which can be used interchangeably for laminated-glass elements with the same accuracy. This technique can be applied to three-layered laminated-glass plates with glass showing a linear elastic behavior and the polymeric core showing viscoelastic behavior. Thus the glass layers can be made of different types of glass (annealed, tempered, heat-strengthened, etc.), and the traditional cores (PVB, ionoplastic, etc.) can be considered in this model. In this paper, the modal parameters (natural frequencies, loss factors, and mode shapes) of a 1400 × 1000 × 16 mm laminated-glass plate pin-supported at the four corners, and of a beam 1 m long and 12 mm thick, both the beam and the plate with annealed glass plies and PVB core, were estimated using the effective thickness concept. For the validation of the model, operational modal tests were performed on the beam and the plate, and the modal parameters identified from the experimental responses were compared with those predicted using the effective thickness concept.
Section snippets
Viscoelastic behavior
The mechanical properties of a linear-viscoelastic material are frequency (or time) and temperature dependent [11]. In the frequency domain, the complex tensile modulus, , at temperature T is given by:where superscript ‘∗’ indicates complex, ω represents the frequency, i is the imaginary unit, and are the storage and the loss tensile moduli, respectively, andis the loss factor that relates the two moduli.
A model for the dynamic behavior of laminated-glass plates
In this section, we extend the model of Ross, Kerwin and Ungar to rectangular laminated-glass plates. Furthermore, a dynamic effective thickness and a dynamic effective Young modulus are derived to estimate the modal parameters in laminated-glass plates. The method is based on the relationship that exists between the static EI(t)S and the dynamic EI∗(t) stiffness in beams.
Laminated-glass plate
A rectangular laminated-glass plate 1400 × 1000 mm and thickness H1 = 7.82 mm, H2 = 0.76 mm, H3 = 7.828 mm, pinned supported at the four corners, were tested using operational modal analysis (OMA), with the temperature being T = 20.5 °C (Fig. 5). The interlayer was made of PVB whereas the faces were made of annealed glass. To study the effect of the supports on the damping, we made the experimental tests using two different supports. The first test was performed using four wooden balls 50 mm in diameter
Laminated-glass plate
The modal parameters of the plate described in the former section were also predicted using the dynamic effective concept. A Young modulus E1 = 72,000 MPa, Poisson ratio ν1 = 0.22 and density ρ1 = 2500 kg/m3 were considered for the glass [9]. With regard to the core of the beam, made of polyvinyl butyral (PVB), a density ρ2 = 1030 kg/m3 and the complex shear modulus indicated in Fig. 2 were considered [9].
In Section 3 two different methodologies were proposed to estimate the modal parameters of
Conclusions
In the practical calculations of laminated-glass elements, as well as in preliminary designs, it is very useful to consider simplified methods. In recent years, several equations have been proposed to calculate displacements, internal forces, stresses, etc., in laminated-glass beams and plates under static loading using the effective-thickness concept [1], [3], [4], [8].With this method, engineers can avoid the use of complex finite-element models with small 3D elements which are exceedingly
Acknowledgments
The economic support given by the Spanish Ministry of Education through the Project BIA2011-28380-C02-01 is gratefully appreciated.
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