Effective thickness of laminated glass beams: New expression via a variational approach

Abstract

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 ...

Enhanced Effective Thickness of multi-layered laminated glass

“…The formulation based upon the original approach by Wölfel [12], later developed by Bennison et al [10,18] and mentioned in the American standard ASTM-E1300 [9], relies upon several simplifying assumptions that, as discussed at length in [13], render it accurate for the case of simply supported beams under uniformly distributed loading.…”

“…An approximation can be found by choosing an appropriate class of shape functions for the unknown fields v(x) and u i (x), defined up to a few parameters that will be determined from energy minimization. This procedure has already been used in [13] for the case of three-layered beams.…”

“…Remarkably, these data are identical to those recorded in [17] for a three-layered beam. Indeed, (3.17) represents the natural extension of the corresponding expression obtained in [13].…”

“…The first formulation condenses all the effects of the interlayer in one coefficient ω provinding a linear interpolation between the layered and the monolithic limit; however, as it will be demonstrated later on in Section 4.2, this method is quite crude and leads to contradictory results. On the other hand the method of ASTM E1300, as discussed in [13,14], gives excellent results for laminated beams under uniformly distributed loading, but is inaccurate in other cases. Very recently, an alternative method [13], called Enhanced Effective Thickness, has been proposed by the authors.…”

“…On the other hand the method of ASTM E1300, as discussed in [13,14], gives excellent results for laminated beams under uniformly distributed loading, but is inaccurate in other cases. Very recently, an alternative method [13], called Enhanced Effective Thickness, has been proposed by the authors. This is based upon a variational approach where, through minimization of the strain energy functional, the best approximation for the response of laminated glass is selected among a restricted class of shape functions for the deflection surface.…”