We extend, to the case of equibiaxial states of stress, the variational approach originally proposed by Hashin for the analysis of cracked three-layer laminates under uniaxial tensile loading. The ...type of damage considered here is complementary to that analyzed by Hashin, in the fact that the external plies of the laminate, rather than the inner layer, are those that are supposed to break, now according to a regular 2D crack pattern. This renders the model particularly suitable for the post-breakage response of laminated glass, composed by a polymeric interlayer sandwiched by two brittle glass plies: the resulting shards partially delaminate but remain attached to the polymer, so to impart to the system a residual load bearing capacity. Within a class of admissible stress systems that verify the equilibrium equations, variational minimization of the complementary energy functional is used to find the optimal approximation, which represents a lower bound for the stiffness. The major complication here consists in the definition of such class, which must take into account the biaxial nature of the problem. Comparisons with numerical experiments confirm the very good accuracy of the proposed approach, at least when applied to laminated glass.
The stability of monolithic glass beams is reasonably well defined; as an elastic material it behaves in a similar manner to other elastic materials such as steel, for which there are many equations ...of different forms which give similar results. Special care is required for continuous restraint to the tension flange. Equations presented in Australian Standard AS1288 Glass in Buildings – Selection and Installation have been used successfully for many years for monolithic fins when used with the strength model of AS1288 but require a more comprehensive approach when using laminated fins and/or strength models that allow higher levels of stress. A review of equations for cantilevers results in a wider range of approaches with significant variance between the outcomes of various published steel and glass standards. AS1288 has been used as the default standard for stability of glass fins, however for cantilevers it appears to have a misprint which has existed for decades. This paper presents strategies for determining the moment capacity of beams and cantilevers made of laminated glass with continuous flexible buckling restraints, such as structural silicone, which have initial imperfections and a known design strength capacity. Where multiple wave lengths form, the warping stiffness may contribute and formulations for rectangles are presented. The accuracy and validity of the approach is also assessed by means of comparisons with the outcomes of Finite Element numerical analyses.
•The structural response of DGUs is governed by the load sharing between glass panes.•The BAM allows to evaluate the load sharing under the most various loading conditions.•Compact formulas and ...tables are provided, to design DGUs of any size and shape.•The BAM approach is validated by means of comparison with numerical results.•A practical method for the design of DGUs made of laminated glass is proposed.
The structural response of Double Glazing Units (DGUs), composed by two glass panes held together by structural edge seals, entrapping a gas for thermal and acoustic insulation, is governed by the load sharing between the glass elements, due to a complex interaction with the gas in the interpane space. Various methods are proposed by standards to evaluate it when DGUs are subjected to climatic loads and/or uniform pressure, but the case of line distributed and punctual load usually requires the use of FEM softwares.
Very recently, a new approach -the Betti’s Analytical Method (BAM)- has been proposed for the evaluation of the load sharing in DGUs under the most various boundary and loading conditions. Here, the method is revised, providing compact formulas and synthetic tables for ease of reference and immediate applicability, for both rectangular and triangular DGUs. Furthermore, the BAM approach is validated by means of comparison with numerical results obtained with MEPLA ISO software. For DGUs made of laminated glass panels, a practical approach, based on the use of the Enhanced Effective Thickness (EET) model, is suggested, and its accuracy is validated by means of numerical comparisons.
A critical issue in the design of structural glass elements in buildings is represented by the evaluation of thermally induced stresses and strains. For both climatic actions and fire, thermal ...stresses represent one of the main causes of premature failure, due to the high sensitivity of glass to temperature gradients. Thermal loads pose a severe safety risk for glass, due to their uneven distribution but also the lack of knowledge on the modification of mechanical properties with temperature. In design practice, approximate tools are used to describe temperature fields in glazing, which do not adequately estimate the thermally induced stresses. Additionally, the existing standards prescribe different methods for the calculation of both the temperature field and the consequent stress, usually based on strong simplifying assumptions, and there is a lack of uniformly defined procedures. Here, an accurate review of the state of the art on glass elements exposed to thermal actions, from both the scientific and the regulatory perspectives, is presented. Reference is made first to the evaluation of the thermal actions, and further to the proper assessment of both the temperature distribution and the consequent thermal stress. The paper also emphasizes open problems and future perspectives related to these topics, to evidence areas of research that should be strengthened and possible future enhancements to the current design and assessment methodologies, which should also be introduced in a regulatory framework.
Insulating Glass Units (IGUs), widely used in windows, building skins and facades, are composed of two or more glass panes, sealed by perimeter spacers entrapping a gas. The interaction between panes ...and gas is structurally beneficial, because it permits the sharing on the panes of the applied loads. However, the actual role of the spacers in the load sharing, in particular when the IGU is not supported at all sides, is not yet fully explained. Here, this problem is analytically solved by starting from the study of plates with two opposite edges simply supported, and the other two supported by elastic beams. Analytical solutions, obtained by using infinite series approaches, are provided for rectangular plates under uniform/linear/punctual loads, and parametric analyses have been carried out to evaluate the influence of the bending and torsional stiffness of the beams on the plate response. The obtained solutions are then used to evaluate the response of a full IGU, proposing an engineering approach providing practical values for the design. Comparisons with numerical analyses by finite element method, with reference to different IGU geometries and loading conditions, confirm the accuracy of the proposed approach: the mean gap is of the order of 1–2%.
•The paper provides analytical solutions and practical tables for IGUs design.•Square/rectangular glasses under uniform/linear/punctual loadings are considered.•Real influences of the spacers (bending/torsional stiffness) for IGUs are estimated.•Advanced analytical, parametric and numerical analyses are carried out.•Comparisons with numerical findings confirm the accuracy of the proposed model.
The structural performance of laminated glass is strongly dependent on the shear coupling offered by the interlayer between the bounding layered and monolithic limits of the glass plies. The most ...common simplified design approach consists of defining the effective thickness, i.e., the thickness of a monolithic section with equivalent flexural section properties. The Enhanced Effective Thickness (EET) method has been verified to estimate deflection in laminated glass for a range of load and boundary conditions for two-, three-, and multi-ply beams; however, for some static schemes, the EET method is less accurate for predicting stress. The recently proposed Conjugate Beam Effective Thickness (CBET) method, initially developed for cantilevered laminated glass balustrade applications, accounts for the relative displacement of glass plies across the interlayer for a range of loads and statically determinate boundary conditions. In this paper, the CBET method is extended to the evaluation of two-ply simply supported beams under concentrated, uniformly, and tapered distributed out-of-plane loads. Predicted deflection- and stress-effective thickness obtained from effective thickness methods is compared with finite element model results in illustrative examples, demonstrating improved accuracy. Closed-form formulas are summarized in tables to facilitate the practical application of the CBET method in the design practice.
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.
Any reliable use of glass for structural purposes cannot neglect that its breakage may be provoked by imponderable events, like impacts at critical spots or thermal shocks. Laminated glass, composed ...by glass plies sandwiching polymeric interlayer sheets, is used in architectural application thanks to its safe post-glass breakage response. When glass breaks, the interlayer retains the glass shards, and the cracked element maintain a certain residual load-bearing capacity, strongly influenced by the tension stiffening of the polymer due to the adhesion with the glass shards, which depends upon the size of the shards and of the debonded zones. Here, we review the most recent experimental results on the post-glass breakage response of laminated heat-treated glass elements, providing charts for the evaluation of such a stiffening effect. Based on this, simple formulas to analyze and interpret the experimental findings under both in-plane and out-of plane bending are proposed, providing analogies with the bending of bimodulus materials and the load-bearing mechanism of reinforced concrete, respectively.
A critical issue in the structural design of glazed surfaces is the evaluation of the strain consequent to temperature variations due to environmental actions such as solar radiation, which ...represents one of the main causes of breakage. In the practice, approximate solutions are used, where the temperature profile across the glass thickness is constant or linear, but the consequent thermal stress cannot be adequately estimated from these. On the other hand, sophisticated thermal software is available only for important tasks.
Here, we propose a semi-analytical approach, easily implementable in a simple FEM code, to evaluate the time-dependent temperature profile through the thickness of layered glazing, which is based on the variational method proposed by Biot in the Fifties. A prompt evaluation not only of the temperature field, but also of the heat flux, can be obtained. Compared to other numerical approaches, this method rigorously accounts for energy conservation and, since it does not involve temperature gradients in the formulation, it is particularly efficient for problems with steep temperature variations. Temperature profiles that are not necessarily linear can be approximated by Hermite–splines, for a precise evaluation of the thermally-induced stress. Comparisons with a direct numerical solution of the heat-conduction differential equations confirm the accuracy and the effectiveness of the proposed approach.
•Thermal strains induced by solar radiation can break the glass of building envelopes.•A semi-analytical FE approach is set from Biot’s variational method in heat transfer.•No temperature gradients appear; energy balance is rigorously set as a constraint.•Hermite–spline shape–functions accurately describe non-linear temperature profiles.•The time-dependent temperature profile in the laminate thickness is calculated.
