Enhanced Effective Thickness of Multi-Layered Laminated Glass

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

Energetic Balance in the Debonding of a Reinforcing Stringer: Effect of the Substrate Elasticity

An effective way to strengthen deteriorated concrete or masonry structures is to glue to them, at critical regions, strips or plates made of Fiber Reinforced Polymers (FRP). The reliability of this technique depends upon interfacial adhesion, whose performance is usually evaluated through an energetic balance, assuming that the support is rigid. The present study analyzes the contact problem between reinforcement and substrate, both assumed to be linear elastic. The solution of the resulting integro-differential equations is expressed in terms of Chebyshev polynomials. A generalization to this problem of the Crack Closure Integral Method developed by Irwin allows to calculate the energy release rate associated with the debonding of the stiffener. Energetic balance `a la Griffith emphasizes the role played by the length of the stiffener and the deformation of the substrate, predicting load vs. displacement curves that, in agreement with experimental measurements, exhibit a snap-back phase

Composite beams with viscoelastic interaction. An application to laminated glass.

A practical way to calculate the response of laminated glass is to consider both glass and polymeric interlayer as linear elastic materials; the viscoelastic behavior of the polymer is evaluated assuming equivalent elastic moduli, that is, the relaxed moduli under constant strain after a time equal to the duration of the design action. Here, we analytically solve the time-dependent problem of simply-supported laminated-glass beams, modeling the response of the polymer by a Prony’s series of Maxwell elements. The obtained results, in agreement with 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 second approach gives in general results that are on the side of safeness, but the design may be too conservative for short-time actions, whose duration depends upon the polymer type

Cohesive debonding of a stiffener from an elastic substrate

To strengthen concrete or masonry, a modern technique uses adherent strips made of Fiber Reinforced Polymer (FRP). A model problem for this is here considered, represented by an elastic stiffener pulled at one end, in adhesive contact with an elastic half space in generalized plane stress. An analytical solution is developed under the hypothesis `a la Baranblatt that cohesive adhesion forces remain active between the two materials when relative slip occurs (provided this is less than a critical value), so that the stress singularity predicted by the theory of elasticity in the case of perfect bonding is removed. We find that the bond length beyond which no further increase of strength could be achieved, referred to as the effective bond length, coincides in practice with the ultimate length of the cohesive zone, i.e., its maximal extension prior that the critical slip limit is attained. The debonding process in a pull-out experiment is analyzed in detail. Results are in better agreement with experimental data than those obtainable with traditional models, which neglect as a rule the deformation of the substrat

Effective Thickness of Laminated Glass Beams.New expression via a variational approach.

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 diffcult 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

Plastic hinges as phase transitions in strain softening beams

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Post-breakage Tensile and Bending Response of Laminated Glass

Laminated glass, composed by glass plies sandwiching polymeric interlayers, can provide a safe post-glass breakage response, in compliance with the fail-safe approach used in the structural design. In fact, when glass breaks, shards remain attached to the polymer, preventing danger from falling materials and imparting a "tension stiffening" effect to the interlayer, so that the broken panel maintains a certain residual load-bearing capacity. Here, a homogenized approach is presented to describe the mechanical properties of broken heat-treated laminated glass under tensile stresses. The model accounts for the stress diffusion from the delaminated zones, where shards are bridged by the interlayer-ligament only, to the regions where glass is bonded to the interlayer. The model provides a simple but accurate estimate of the effective tensile properties of the cracked laminate. Here, the influence of the interlayer thickness, the size of the glass shards and the glass-polymer delamination on the post-critical response is accurately investigated, and analytical results are compared with numerical ones. The obtained expression for the tensile modulus is used to predict, in more general terms, the response of cracked laminated glass under in-plane and out-of-plane bending. In both cases, a key point is the correct evaluation of the tension stiffening in the polymeric interlayer due to the adhesion with the glass shards

Practical expressions for the design of laminated glass

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

Sharing of general loading in double glazed units. The BAM analytical approach

Double Glazed Units (DGUs) consist of two glass panes held together by structural edge seals. Calculation methods for DGUs consider that actions applied on one pane develop effects in all the panes, due to the coupling from the entrapped gas. Various methods have been proposed in standards to evaluate this load sharing, which depends upon the stiffness of the glass panes, the thicknesses of spacer and the size of the DGU. A comprehensive analytical formulation, the Betti’s Analytical Method (BAM), has been recently proposed to calculate the load sharing in DGUs of any shape, composed by glass panes of arbitrary thickness, with various support conditions at the borders and various types of external actions, including concentrated and line loads. Simple expressions can determine the gas pressure as a function of a universal shape function, which coincides with the deformed surface of a simply supported plate, of the same shape of the DGU, under uniformly distributed load. Here, comparisons are made with numerical analyses, performed by implementing an ad hoc routine in the software Straus7, developed by Maffeis Engineering, where the deflection of the glass panels is iteratively calculated, until the volume enclosed reaches a value compatible with the pressure exerted by the gas. The numerical routine, that is part of an integrated parametric approach to the façades design, allows precise calculations for any kind of build-up, panel shapes and load conditions

Boundary Layer Effects in a Finite Linearly Elastic Peridynamic Bar

Abstract The peridynamic theory is an extension of the classical continuum mechanics theory. The peridynamic governing equations involve integrals of interaction forces between near particles separated by finite distances. These forces depend upon the relative displacements between material points within a body. On the other hand, the classical governing equations involve the divergence of a tensor field, which depends upon the spatial derivatives of displacements. Thus, the peridynamic governing equations are valid not only in the interior of a body, but also on its boundary, which may include a Griffith crack, and on interfaces between two bodies with different mechanical properties. Near the boundary, the solution of a peridynamic problem may be very different from the classical solution. In this work, we investigate the behavior of the displacement field of a unidimensional linearly elastic bar of length L near its ends in the context of the peridynamic theory. The bar is in equilibrium without body force, is fixed at one end, and is subjected to an imposed displacement at the other end. The bar has micromodulus C, which is related to the Young's modulus E in the classical theory and is given by different expressions found in the literature. We find that, depending on the expression of C, the displacement field may be singular near the ends, which is in contrast to the linear behavior of the displacement field observed in the classical linear elasticity. In spite of the above, we show that the peridynamic displacement field converges to its classical counterpart as a length scale, called peridynamic horizon, tends to zero