Laminated Glass Plates Research Articles

Post-Failure Behavior of Point-Fixing Laminated Glass Plates under Out-of-Plane Uniform Pressure

This paper presents the results of an experimental and numerical investigation of the mechanical response of undamaged and damaged 2-ply laminated glass plates. Two types of glass plies, thermally toughened and heat-strengthened, coupled with SentryGlas (SG) were considered. Laminated glass plates supported with articulated point fixing bolt under out-of-plane uniform pressure were investigated under four different damage configurations: (i) undamaged; (ii) partially damaged, with the bottom ply broken; (iii) partially damaged, over-flipping the specimen of mode II; (iv) both two glass plies broken, which was applicable for LG plates made by heat-strengthened glass. In the above four modes, the top ply is always subjected to compression while the bottom one is in tension. The different responses of each configuration (different glass types and damage modes) were discussed and compared in this study. In addition, numerical models were adopted to reproduce the experimental results. The influence of the hitting location and glass types on the mechanical behavior of LG plates was analyzed. The results showed that the contribution of the broken glass ply could not be disregarded in the evaluation of the global stiffness of partially damaged LG plates and their bearing capacity.

Sensitivity to intensity and distribution of the temperature field in the nonlinear thermo-mechanical analysis of laminated glass plates

Glass laminates consist of stiff glass plies permanently shear-coupled by polymeric interposed layers. When an external temperature rise occurs, the interlayers undergo a dramatic stiffness decay. As a consequence, not only the sectional warping typical of alternating stiff/soft composites is intensified, but also the overall behavior may evolve counter-intuitively. When slender elements prone to geometric nonlinearities are involved, even small thermal variations in intensity or distribution may act as uncertainty factors, strongly affecting the outcome. This paper proposes an efficient, robust, and accurate numerical framework to perform the sensitivity analysis to thermo-mechanical actions in glass plates. A large deformation isogeometric Kirchhoff-Love shell model enriched with through-the-thickness warping is employed, together with a generalized arc-length method involving a suitable temperature parameter as an additional unknown, namely the thermal amplifier or a spatial distribution coefficient. Numerical experiments are presented to highlight the effects that even small temperature variations produce on the equilibrium paths and the influence of the stiffness loss in the interlayer on the structural behavior and the accuracy of the models.

Mechanical Analysis of the Quasi-Static and Dynamic Composite Action in PV Modules with Viscoelastic Encapsulant.

The mechanical analysis of photovoltaics and building integrated photovoltaics is a key step for their optimal design and certification, and requires careful consideration, alongside solar power, durability and functionality issues. The solar cells are encapsulated in thin interlayers that are usually composed of a viscoelastic Ethylene-Vinyl Acetate compound, and protected by thin glass and/or plastic layers. This paper investigates the out-of-plane bending response of a full-scale commercial PV module and focuses attention on the shear bonding efficiency of the thin encapsulant for quasi-static and dynamic mechanical considerations. The parametric analytical analysis, carried out in this study for a laminated glass plate, highlights the possible consequences of the viscoelastic shear coupling on the cross-section load-bearing demand in the covers. As a direct effect of severe operational conditions (i.e., ageing, non-uniform/cyclic thermal gradients, humidity, extreme mechanical/thermal loads, etc.) the shear rigidity and adhesion of these films can suffer from repeated/progressive modification and even degradation, and thus induce major stress and deflection effects in the out-of-plane mechanical response of the PV module components. The minimum shear bond efficiency required to prevent mechanical issues is calculated for various configurations of technical interest. Accordingly, it is shown how the quasi-static and dynamic mechanical performance of the system modifies as a function of a more rigid or weak shear coupling.

A von Kármán-type model for two-layer laminated glass plates, with applications to buckling and free vibration under in-plane edge loads

This article presents a layerwise geometrically nonlinear two-dimensional model for the undamped dynamical response of thin two-layer plates with partial shear interaction. The model is tailored for the analysis of laminated glass plates of the type most commonly used in building structures (two glass layers bonded by a polymeric interlayer). The geometrical nonlinearities are introduced via the so-called von Kármán strains. The process of dimensional reduction is based on Podio-Guidugli’s method of internal constraints. An unconventional choice of generalised displacements highlights several direct similarities with the single-layer plate theories of von Kármán and Mindlin.The model is used to study the flexural buckling and free vibration of two-layer plates under in-plane edge loads. Analytical solutions are obtained for rectangular plates with (i) all edges simply supported without shear restraint and (ii) a two-parameter system of uniform in-plane normal edge loads. These solutions degenerate into those of single-layer von Kármán plates in the two limiting cases of zero and full shear interaction. A conforming rectangular finite element is formulated and shown to perform effectively. When using uniform meshes, the observed asymptotic rate of convergence for the finite element approximation of the lowest buckling load and vibration frequency is quadratic.

Gradual fracture of layers in laminated glass plates under low-velocity impact

Modeling of gradual fracture of individual layers of laminated glass produced by successive low-velocity impacts is presented. This subject is first examined experimentally focusing on fracture sequence of the laminate composed of three or four glass layers bonded by polyvinyl butyral (PVB) interlayers. While both laminate geometries, limited to four test samples, show a significant scatter in impact energies and breakage forces causing fracture as well as differences in fracture sequence with increasing impact height, they consistently support ability of partially damaged samples to withstand contact forces exceeding those leading to fracture of the previous glass layer. Also, the four-glass layer laminates seem to provide the post-fracture response consistent with the concept of a sacrificial-glass-ply design. However, many questions remain concerning stochastic behavior of glass laminates further promoted by the present experimental results. To open the door to probabilistic modeling, the present topic is addressed also computationally adopting the LS-DYNA software. It is illustrated that properly tuned computational model may provide results, which are comparable to experimental observations both in terms of contact forces and the resulting fracture patterns.

A constrained solid-shell model for the geometric nonlinear finite-element analysis of laminates with alternating stiff/soft layers. Applications to laminated glass

Solid-shell models are developed for the geometrically nonlinear analysis of multi-layered composite structures made of alternating layers with large difference in material properties. Exemplificative applications are presented for laminated glass, in which a number of stiff plies of glass are permanently shear-coupled by soft interlayers. The sectional warping due to significant transverse shear strains in the soft layers makes theories of laminated plates based on the plane-section hypothesis unreliable. The proposed approach is based on a geometrically exact solid-shell finite element model with one element per layer in the thickness direction, as alternative to solid discretization. The element approximation is based on the displacement nodal values at the top and bottom surfaces of the layers, with a natural C0 continuity. An alternative solid-shell model with fewer parameters is derived imposing the equal finite rotation of the stiff layers at each surface point by a local rotation-free re-parametrization of the nodal displacements and enforcing the plane stress condition. The approach permits an easy coupling with a fully solid discretization, e.g. to model connections, and is based on a simple strain measure quadratic in the displacement unknowns and suitable for finite strains. Extensive numerical examples for laminated glass plates and curved shells susceptible to large deflections and buckling are provided, comparing the results with those from a fully solid approach.

Effect of delamination size, location and boundary conditions on the behavior of a laminated glass plate

Laminated glass is made by combining glass layers with polyvinyl butyral (PVB) or a different adhesive with the help of heat and pressure. Its popularity is increasing in the building industry as a protective material. However, despite the popularity of laminated glass as a structural material, delamination is recognized as a highly undesirable problem inducing major damage to laminated glass. The effect of boundary conditions on the delamination strength of laminated glass is observed in this study. It is concluded that the location and size of delamination have a considerable effect in determining the delamination strength of laminated glass. The results of the previously conducted experiment and the developed finite element model are utilized to validate the model used to determine the delaminated strength of laminated glass. The simulation results and measurements are in good agreement with the results of the mathematical model. The results of the current research can be utilized by design engineers while designing laminated glass for structural applications.

A linear two-dimensional mathematical model for thin two-layer plates with partial shear interaction, with a view towards application to laminated glass

This work presents a consistent derivation, from three-dimensional linear elasticity, of a two-dimensional mathematical model describing the bending and in-plane stretching behaviours, under a general system of quasi-static distributed loads and prescribed support displacements, of thin two-layer plates with partial shear interaction. This layerwise model is specifically tailored for the requirements posed by the analysis of laminated glass plates commonly used in building structures (consisting of two thin glass layers bonded together by an adhesive interlayer). Our approach, based on Podio-Guidugli’s method of internal constraints, avoids mutually contradictory assumptions (not uncommon in the literature on structural mechanics) and yields a complete two-dimensional characterisation of displacement, strain and stress fields that exactly satisfy the field equations of three-dimensional linear elasticity and the boundary conditions at the end faces. The choice of generalised variables is designed to bring to light the following fundamental conclusion and physical insight: the resulting two-dimensional boundary value problem is a combination of the equations of Kirchhoff and Mindlin plates (with specified rigidities).Two examples illustrate the application of the proposed model: (i) the cylindrical bending of plate strips and (ii) a family of problems with Navier-type analytical solution. The solutions exhibit continuity across the whole range of zero, partial and full interaction between the layers. Moreover, the Navier-type solutions are consistently in close agreement with the results of three-dimensional finite element analyses. On the contrary, analogous results previously reported in the literature exhibit considerable deviations. An explanation for these discrepancies is discussed in detail.

Three-stage breakage model for laminated glass plate under low-velocity impact

Two failure modes of laminated glass under impact loading are summarized, after which a series of experiments are carried out and a three-stage breakage model for laminated glass plate is proposed. The three-stage breakage model consists of an elastic stage (stage Ⅰ), a stage of total breakage of the back glass layer (stage Ⅱ) and a stage of total breakage of both layers of glass (stage Ⅲ). Furthermore, a combined TCK-JH2 constitutive model is proposed for the numerical analysis conducted with Finite Element Method (FEM). In this constitutive model, the TCK model is employed when the material is under tensile condition whereas the JH2 model is employed when the material is under compressive condition. Moreover, a stiffness recovery behavior is defined. The criterion of tension and compression is studied as well.

Post-failure behavior of 2-ply laminated glass plates with different interlayers

Modern design codes, such as the new Eurocode on the design of structural glass components, consider the prediction of the post-failure behavior of these elements of paramount importance for a proper and safe design. In this view, an experimental investigation is presented in the paper, on the mechanical response of undamaged and damaged 2-ply Laminated Glass (LG) under quasi-static loads, with different interlayers, namely polyvinyl butyral (PVB), SentryGlas (SG) and a plasticized version of PVB, Saflex DG41 (DG41). Firstly, undamaged specimens (UDLG) were tested in simply supported configuration (configuration 0) with a vertical load in the middle increased up to the failure of the bottom glass ply. damaged LG specimens (PDLG) were then tested into two different configurations: configuration I, with broken ply below (bottom ply) and configuration II, with broken ply above (top ply). All the presented tests were performed under displacement control. For both configurations, the interlayers influence on the post-breakage behavior was discussed. It is underlined that the global response of the two configurations was completely different: configuration II, which was characterized by a deformed shape opposite to the load direction, showed a good residual stiffness and reached high failure loads. On the contrary, in configuration I, the response was characterized by a low stiffness and load carrying capacity. In addition, also the influence of different glass types was discussed, focusing attention on tempered and toughened glass. The results showed that the contribution of the fractured layer cannot be disregarded in the evaluation of the global stiffness of a PDLG and its contribution is strictly related to the interlayer and glass typologies.

A fractional viscoelastic model for laminated glass sandwich plates under blast actions

A model is proposed to evaluate the effects of explosions on laminated glass plates, made of glass plies bonded by thin polymeric interlayers, in the pre-glass-breakage phase. The novelty consists in the description of the viscoelastic properties of the interlayer, based on fractional calculus. This has noteworthy advantages because the relaxation function of most commercial polymers can be well approximated by power laws in the time interval of interest: in the fractional approach, this trend can be directly described by only two material parameters, whereas the traditional Prony series requires many more coefficients to be calibrated. Modelling of the plate is layerwise and the coupling with an elastic supporting back structure is considered. The glass plies are Kirchhoff–Love plates, whereas the impinging pressure from the explosion follows Friedlander equation. The governing equations, found via Hamilton’s principle, are solved à la Galerkin through a spatial expansion in Fourier series and a step-by-step integration in time using the Grünwald–Letnikov operators. Numerical experiments aim at describing how the fractional parameters of the interlayer, as well the compliance/mass of the back structure, affect the stress and deflection of the laminated plate. Comparisons are made with the commonly used quasi-elastic approximation, which considers the interlayer as linear elastic. The results show that the viscosity of the interlayer cannot in general be neglected and how a compliant back structure can help to safeguard the glass, although its displacement may exceed the serviceability limits. The optimal design shall thus represent a compromise between different effects: despite its limitation, the proposed model can represent a tool to make a correct choice.

Overall response of 2-ply laminated glass plates under out-of-plane loading

The results of an experimental and numerical investigation on the mechanical response of undamaged and damaged 2-ply Laminated Glass (LG) plates with different interlayers are presented in the paper. Three different interlayers, polyvinyl butyral (PVB), SentryGlas (SG) and Saflex DG41 (DG41) characterized by different rheological properties and fully tempered glass plates were considered. Simply supported plates under out of plane loads were tested in three different configurations: configuration 0, undamaged; configuration I, partially damaged, with a broken ply below (bottom ply) and configuration II, partially damaged, with a broken ply above (top ply). In the three configurations the top ply is always subjected to compression while the bottom one is in tension. The main aspects and the different response of each configuration were discussed and compared. Experimental results highlighted the influence of the interlayer properties on both the pre- and post-breakage behavior. Moreover, numerical models were developed to reproduce the experimental results. For the post-breakage response two different strategies were developed: a) the equivalent temperature variation and b) elastic-brittle constitutive law. Differences and advantages associated with both solutions are discussed. Finally, extensive parametric numerical analyses are proposed to underline the influence on the mechanical response of different parameters: i) plate geometry (size effect); ii) interlayer thickness; iii) glass thickness; iv) additional glass ply (3-ply LG plate). It is underlined that the size effect plays an important role on the mechanical response of LG plates and it should be always considered in the design. The numerical findings were used to define a critical length from which the plates behave as monolithic, independently of the stiffness of the interlayer.

Long-term loading and recovery of a laminated glass slab with three different interlayers

The mechanical behaviour of laminated glass plates and beams is affected in different ways by the shear stiffness of interlayer materials. Polymeric interlayers are viscoelastic materials, and therefore experience creep when subjected to long-term loading. In this paper, three laminated glass panels, each with a different interlayer material (PVB Clear, PVB ES, and SentryGlas), were placed between two laminated glass beams. A uniformly distributed load was applied during four months to study the effect of creep, and then was removed to see the deflection recovery during one month. The midspan vertical displacement of the two laminated glass beams remained almost constant over time, with a small variation that was attributed to the rubber sheets placed in the supports. The plate with SentryGlas had the lowest elastic deformation, creep, and residual deformation. The plate with PVB Clear had the highest elastic and total deformation. The plate with PVB ES had a similar initial deflection as SentryGlas, but was the one that experienced the highest creep. At the end of the test the plates still had a small residual deflection, which could be due to an incomplete deflection recovery. The flexural behaviour of the three laminated glass plates was simulated using a Finite Element Model, representing the loading, the creep, the unloading, and the deflection recovery stages.

Dynamics of contrasted stratified elastic and viscoelastic plates - application to laminated glass

This article deals with the dynamic behavior of highly contrasted stratified (HCS) plate made of elastic or visco-elastic, symmetric or non-symmetric, layers. Such composite structures may correspond to laminates plates or sandwich panels of often use in engineering. The study is based on the synthetic analytical HCS plate model derived by asymptotic approach (Boutin and Viverge 2016). The physical mechanisms within the plate are taken into account by the additional kinematic descriptor constituted by the internal sliding and a bi-torsor formulation.The first part focuses on the elastic HCS plates. After describing the three types of bending waves, and providing a simple assessment of their parameters through Padé approximant, the analytical expression of the harmonic Green’s function of the HCS plates is established. The classical tools of modal analysis are extended to such plates paying attention to the appropriate boundary conditions.The second part shows that these findings apply to HCS plates with visco-elastic constituents provided that they are expressed in the Fourier domain. This enables to determine the energy loss due to the viscous dissipation. The case of thermo-sensible interlayer is also investigated and the influence of the temperature on the plate behavior is accurately figure out. As a specific example, the laminated glass with PVB interlayer is considered. The damping of the harmonic bending waves is assessed, the procedure to perform a “quasi"-exact modal analysis is detailed and the effect of the temperature on the eigen modes, in terms of frequency and damping, is quantified.Finally, vibrational experiments performed on a laminated glass plate at different temperatures are shown to be in good agreement with the theoretical modeling.

Nonlinear Blast Responses of Thin Shell Roof Over Long Span Structures

This paper adopts both explicit and implicit finite element methods in a specialist package LS-DYNA to investigate the nonlinear, dynamic response of a long span shell roof structure when subjected to blast loading. Parametric studies have been carried out on blast loaded laminated glass plates with reference to experimental results obtained by European researchers. A case study that has been chosen is a light rail station in The Netherlands called The Erasmusline. Following the detonation of 15[Formula: see text]kg TNT charge, explicit analysis showed breakage surrounding the rigid supports along the edge beam where modal vibrations are restrained. An implicit analysis has confirmed the resonances in global eigen-frequencies where most blast damage is localized around the roof canopy hence producing cracking and potential glass detachment from the panels without full structural demolition. This insight from this study will inform structural engineers about the potential modes of failure and preventative strategies to minimize further secondary losses of life or assets from a terrorist attack.

Influences of temperature and impacting velocity on dynamic failure of laminated glass: Insights from peridynamic simulations

In this work, a non-local ordinary state-based peridynamic (PD) model for rate-dependent thermo-visco-elasticity was formulated to simulate the dynamic failure process of laminated glasses subjected to impact loads. The formulations of force states between two interacting material points within one ‘horizon’ were derived by taking the viscous and temperature-dependent mechanical responses of polyvinyl butyral (PVB) interlayers into consideration. A dynamic failure criterion used to determine the breakage of bonds was developed to model the failure process of glass sheets under impacting loads as well. The adhesion between glass and PVB layer was described by using a penalty-based method in numerical simulations. Two benchmark numerical examples, including (i) creep-recovery test of a PVB bar and (ii) dynamic fracture of a glass specimen subjected to different high strain rates, were performed to illustrate the accuracy and effectiveness of the proposed numerical model. After verification, the proposed model was employed to study the dynamic fracture of typical laminated glass plates under drop-weight loading conditions, in which numerical results show good agreement with experimental observations, and then the effects of temperature and impacting velocities on the impact resistance and failure mechanisms of laminated glass were investigated further.

Phase-Field Fracture Modelling of Thin Monolithic and Laminated Glass Plates under Quasi-Static Bending

A phase-field description of brittle fracture is employed in the reported four-point bending analyses of monolithic and laminated glass plates. Our aims are: (i) to compare different phase-field fracture formulations applied to thin glass plates, (ii) to assess the consequences of the dimensional reduction of the problem and mesh density and refinement, and (iii) to validate for quasi-static loading the time-/temperature-dependent material properties we derived recently for two commonly used polymer foils made of polyvinyl butyral or ethylene-vinyl acetate. As the nonlinear response prior to fracture, typical of the widely used Bourdin–Francfort–Marigo model, can lead to a significant overestimation of the response of thin plates under bending, the numerical study investigates two additional phase-field fracture models providing the linear elastic phase of the stress-strain diagram. The typical values of the critical fracture energy and tensile strength of glass lead to a phase-field length-scale parameter that is challenging to resolve in the numerical simulations. Therefore, we show how to determine the fracture energy concerning the applied dimensional reduction and the value of the length-scale parameter relative to the thickness of the plate. The comparison shows that the phase-field models provide very good agreement with the measured stresses and resistance of laminated glass, despite the fact that only one/two cracks are localised using the quasi-static analysis, whereas multiple cracks evolve during the experiment. It was also observed that the stiffness and resistance of the partially fractured laminated glass can be well approximated using a 2D plane-stress model with initially predefined cracks, which provides a better estimation than the one-glass-layer limit.

Finite Element Models and Numerical Analysis of a Structural Glass Beam Reinforced with Embedded Carbon Fibre Rod

Due to the its brittleness and unpredictable failure behaviour, the glass is considered a structurally unsafe material. The poor tensile strength and the occurrence of brittle failures prevent the possible use of structural glass elements (i.e. floors, beams and columns). The present study focuses on the numerical analysis of glass beams reinforced with composite material (Fiber Reinforced Polymer) and the numerical analysis of the behaviour of glass plates after being punched. In particular, the method of reinforcement analysed consists in embedding the Carbon FRP-rod in the interlayer of laminated structural glass beams with the aim to increase their failure strength, but also the post-failure strength and ductility. A numerical analysis on different specimens has been carried out to evaluate the effect of the reinforcement. The response of the reinforced and un-reinforced beams has been compared and the results have been discussed. The results provide an estimation of the expected performance in order to propose a reinforcement method for glass beams using FRP composites. This study includes the numerical analysis on laminated glass plates with wide holes at both ends, the latter made for technological reasons to connect the glass beam to the structure. The obtained results are compared with integer specimens.

Blast loads and nonlinear vibrations of laminated glass plates in an enhanced shear deformation theory

The nonlinear dynamic response of a laminated glass plate under the pressure generated by a blast wave, described via the Friedlander’s model, is investigated. In order to account for the heterogeneity of the laminated package, composed by two glass plies sandwiching a thin and soft polymeric interlayer, a geometrically nonlinear, third-order shear deformation theory with rotary inertia is used. Within the quasi-elastic approximation, the constitutive response of the polymer is described by its secant elastic moduli for short-term loading at room temperature, while the dissipation associated with its viscosity is taken into account through an equivalent effective damping. The problem is discretized by a reduced-order model retaining 40 degrees of freedom and the first four vibration modes, which are the ones receiving most of the energy from the blast. A geometrically nonlinear damping model is introduced to simulate the increase in damping associated with large-amplitude oscillations of the plate, which results from the interaction between the large deflections of the glass plies and the viscosity of the interlayer. Forced nonlinear vibrations around the frequency of the fundamental mode are also investigated, to complete the study of the nonlinear dynamics. The results highlight the importance of the nonlinear effects of damping, which can almost halve the deformations and stresses in laminated glass with respect the linear case at high levels of excitation, while still remaining within the limits of material strengths.

Buckling of laminated glass plates using the effective thickness concept

The critical buckling loads of laminated glass panels are time and temperature dependent because the mechanical behavior of these elements is governed by the material properties of the interlayers, which exhibit a viscoelastic behavior. Although structural stability is one of the design requirements in laminated glass panels, the literature about buckling of these elements is sparse. The finite element method can be used to calculate the response of laminated glass plates, but the classical eigenvalue buckling analysis implemented in these programs does not consider the time and temperature dependency of the interlayers. In this paper, a simplified analytical method to calculate the buckling critical load of rectangular laminated glass plates is presented, where the equations corresponding to linear-elastic monolithic thin plates are modified with an effective stiffness [Formula: see text] dependent on the geometry, material properties, and boundary conditions of the plate. The analytical equations are validated by numerical simulations on simply-supported laminated glass plates subject to uniaxial, biaxial, and in-plane shear, the maximum discrepancies being less than 10% for all the cases studied in the paper.