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 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.
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.
The stress state resulting from an non-uniform temperature distribution in architectural glass panes, consequent to environmental actions and, in particular, to the shadows cast, is one of the main ...causes of failures experienced in building façades. We propose a dedicated thermo-elastic model, implemented in a finite element framework, to overcome the weakness of standardized prescriptions, the limits of simplified practical rules, and the difficulty in using general-purpose thermal softwares, not always interfaced with a structural analysis, in the common design practice.
The formulation of the thermal problem is based on Biot’s variational principle for heat transfer, which facilitates the numerical implementation, via a 2D mesh in case of sharply varying temperature fields, expected for the case at hand. The elastic problem considers a kinematics à la Kirchhoff–Love for the glass plate, enriched by a term accounting for a non-linear thermal strain distribution in the thickness. The same shape functions can be used for the two problems, establishing a convenient formal analogy that facilitates the implementation. The proposed approach is here used to evaluate the stress distribution on paradigmatic case studies. The obtained results allow to recognize the main aspects that influence the state of thermal stress, a result that could be used to conceive reliable simplified methods, which may be incorporated in standards.
•Uneven heating from the environment is the root cause of glass failures in architecture.•Our thermoelastic model determines the thermal stress from environmental boundary conditions.•Biot’s variational principle for heat transfer provides a formal symmetry with the elastic formulation.•The FEM implementation uses the same shape functions for the heat and kinematic displacements.•Worked examples provide general indications for the parameters affecting the thermal stress state.
Laminated beams with viscoelastic interlayer Galuppi, Laura; Royer-Carfagni, Gianni
International journal of solids and structures,
09/2012, Letnik:
49, Številka:
18
Journal Article
Recenzirano
Odprti dostop
We 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.
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