The performance of laminated glass, which consists of two or more glass plies bonded together by polymeric interlayers, depends upon shear coupling between the plies through the polymer. This is commonly considered by defining the effective thickness, i.e., the thickness of a monolithic beam with equivalent bending properties in terms of stress and deflection. General expressions have been proposed on the basis of simplified models by Newmark and Wölfel-Bennison, but they are either difficult to apply or inaccurate. Here, a variational approach to the problem is presented. By choosing appropriate shape functions for the laminated-beam deformation, minimization of the strain energy functional gives new expressions for the effective thickness under any constraint-and load-conditions, embracing the classical formulations as particular cases. Comparisons with numerical experiments confirm the better accuracy of the proposed approach with respect to the previous ones.Keywords: Structural glass, laminated glass, composite structures, bending strength, effective thickness, variational approach. IntroductionIn order to reduce the risk of catastrophic collapse of structures made of glass, the brittle material par excellence, an effective technique is to bond two or more glass plies with 1 2 L. Galuppi & G. Royer-Carfagni thermoplastic polymeric interlayers with a treatment in autoclave at high pressure and temperature. This bond is quite strong because it is chemical in type, being due to the union between hydroxyl groups along the polymer and silanol groups on the glass surface. The resulting laminated glass is a safety glass because, after breakage, the fragments remain attached to the interlayer: risk of injuries is reduced and the element maintains a certain consistency that prevents detachment from fixings. But the iterlayer affects also the pre glass-breakage response because it allows the transfer of shear stresses among glass plies, at the price of a relative sliding due to the deformation of the polymer. The assessment of the degree of connection offered by the polymer is crucial for the design of glass structures in the serviceability limit state and this is why a great number of studies, including this one, have considered the response of the composite laminated package before first cracking occurs.Indeed, the polymeric interlayers are too soft to present flexural stiffness per se, but they can provide shear stresses that play an important role for the glass-layer interaction [7]. In general, the degree of coupling of two glass layers depends upon the shear stiffness of the polymeric interlayer, as first mentioned by Hooper in 1973 [3] while studying the bending of simply supported laminated-glass beams. Since then, the problem has been considered by many authors [15], one of the most recent contribution being the careful finite element analysis of [14], which includes an updated list of the most relevant literature.In the pre glass-breakage modeling no distinction has to be made for what the type of ...
a b s t r a c tWe analytically solve the time-dependent problem of a simply-supported laminated beam, composed of two elastic layers connected by a viscoelastic interlayer, whose response is modeled by a Prony's series of Maxwell elements. This case applies in particular to laminated glass, a composite made of glass plies bonded together by polymeric films. A practical way to calculate the response of such a package is to consider also the interlayer to be linear elastic, assuming its equivalent elastic moduli to be the relaxed moduli under constant strain, after a time equal to the duration of the design action. The obtained results, that are confirmed by a full 3-D viscoelastic finite-element numerical analysis, emphasize that there is a noteworthy difference between the state of strain and stress calculated in the full-viscoelastic case or in the aforementioned ''equivalent'' elastic problem.
The flexural performance of laminated glass, a composite of two or more glass plies bonded together by polymeric interlayers, depends upon shear coupling between the glass components through the polymer. This effect is usually taken into account, in the design practice, through the definition of the effective thickness, i.e., the thickness of a monolith with equivalent bending properties in terms of stress and deflection. The traditional formulas à la Bennison-Wölfel are accurate only when the deformed bending shape of the plate is cylindrical and the plate response is similar to that of a beam under uniformly distributed load. Here, assuming approximating shape function for the deformation of laminated plates variously constrained at the edges, minimization of the corresponding strain energy furnishes new simple expressions for the effective thickness, which can be readily used in the design. Comparisons with accurate numerical simulations confirm the accuracy of the proposed simple method for laminated plates.
Due to deformability of the polymeric interlayer, stiffness and strength of laminated glass are usually less than those corresponding to a monolith with same total thickness. A practical design tool consists in the definition of the "effective thickness", i.e., the thickness of an equivalent monolithic glass that would correspond to the same deflection and peak stress of the laminated glass, under the same constraint and load conditions. Very recently, a new model has been proposed for the evaluation of the effective thickness. Here, a comparison is made with the classical approach by Wölfel-Bennison and the new method is specialized to the most common cases of the design practice, providing synthetic tables for ease of reference and immediate applicability.
The stiffness and strength of laminated glass, a composite of glass layers bonded together by polymeric interlayers, depend upon shear coupling between the glass plies through the polymer. In the design practice, this effect is commonly considered by defining the effective thickness, i.e., the thickness of a monolith with equivalent bending properties. Traditional formulations have been proposed for a package of two layers of glass and one polymeric interlayer, but their extrapolation to a higher number of layers gives in general inaccurate results. Here, the recently-proposed Enhanced Effective Thickness method is extended to the case of laminated glass beams composed i ) by three layers of glass of arbitrary thickness, or ii ) by an arbitrary number of equally-thick glass layers. Comparison with numerical experiments confirms the accuracy of the proposed approach.
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