Glass Belis, Bos & Louter (Eds.) Conference on Architectural and Structural Applications, Ghent University, September 2020. Copyright author. all rights reserved. ISBN 978-94-6366-296-3, https://doi.org/10.7480/cgc.7.4484
Laura Galuppi a Gianni Furio Mario Royer-Carfagni a,b Luca Barbieri b Massimo Maffeis ba Department of Engineering and Architecture, University of Parma, Italy b Maffeis Engineering SpA, Italy
The double-glazed unit (DGU) consists of two glass panels fixed together by structural edge seals. The calculation method of DGU believes that due to the coupling of entrained gas, the operation applied to one pane will have an effect in all panes. Several methods are proposed in the standard to evaluate this load distribution, depending on the stiffness of the glass sheet, the thickness of the spacer, and the size of the DGU.
Recently, a comprehensive analysis formula, the Betti Analysis Method (BAM), was proposed to calculate the load sharing in any shape of DGU composed of glass plates of any thickness, with various supporting conditions and various types of external Functions include concentrated load and line load. A simple expression can determine the gas pressure as a function of a general shape function that coincides with the deformed surface of a simply supported plate with the same DGU shape under a uniformly distributed load.
Here, a special program is executed in the software Straus7 developed by Maffeis Engineering to compare the numerical analysis, in which the deflection of the glass panel is calculated iteratively until the enclosed volume reaches a value compatible with the applied pressure. gas. Numerical procedures are part of the integrated parametric method of exterior wall design, allowing accurate calculations for any type of building, panel shape and load conditions.
The hollow glass unit (IGU) is composed of two or more glass plates, whether integral or laminated, connected to each other by spacers, which are bonded to the structural edge seals along their periphery. The resulting cavity is usually filled with argon, krypton, or xenon to reduce thermal conductivity and improve sound insulation. When composed of two glass plates, they are often referred to as double glazing units (DGU). Due to the interaction between the glass plate and the gas, all panes share part of the load applied to one pane, whether it is uniform, linearly distributed or concentrated.
Various methods are proposed in the standard to evaluate load distribution under uniformly distributed loads (wind, snow, dead weight, changes in internal pressure relative to external pressure). A simpler formula is the "thickness method" proposed by ASTM E-1300 in the United States. The draft proposal of the European norm prEN 16612 considers the gas compressibility through the insulation unit factor, which is only listed for rectangular IGUs. A more refined method is proposed by Feldmeier (1996), which can also consider the distribution of lines and concentrated actions. This method needs to calculate the volume enclosed by the deformed plate under the action of a single line or concentrated load.
Recently, Galuppi and Royer Carfagni (2019) proposed a comprehensive analysis formula to calculate the load distribution in the DGU of any shape and any thickness of glass plates. There are various support conditions and general external effects at the boundary, including concentrated loads and linear loads. Special cases. This formula for considering gas compressibility relies on Betti's customized use of the classic reciprocal work theorem in linear elasticity. Therefore, it is called DGU's Betti Analysis Method (BAM).
A simple expression can determine the gas pressure as a function of a general shape function that coincides with the deformed surface of a simply supported plate with the same DGU shape under a uniformly distributed load. For laminated glass, it has been verified (Galuppi 2020) that if the laminate is treated as a whole with a flexural effective thickness, the result is accurate, provided that the calculation is performed using the EET method (Galuppi and Royer Carfagni 2012).
Here, a special program is executed in the software Straus7 developed by Maffeis Engineering to compare the numerical analysis, in which the deflection of the glass panel is calculated iteratively until the enclosed volume reaches a value compatible with the applied pressure. gas. Numerical procedures are part of the integrated parametric method of exterior wall design, which can accurately calculate load distribution for any type of building, panel shape and load condition. The results obtained in the example confirm the accuracy of the proposed analysis method for rectangular and triangular DGU. The comparison also considers the case of a DGU whose panes are made of laminated glass.
The Betti analysis method (BAM) (Galuppi and Royer-Carfagni 2019) used to evaluate DGU load sharing relies on the assumption of the linear elastic response of the glass sheet from the material and geometrical perspective. There are no special assumptions regarding the type of boundary conditions for the out-of-plane displacement at the DGU boundary. It can be simply supported, continuous or point-by-point, elastically supported or free. In order to illustrate the basic equation of the BAM method, let us consider the arbitrary shape DGU shown in Figure 1, formed by glass plates 1 and 2 with thicknesses h1 and h2, respectively defining a sealed cavity of thickness s, filled with gas.
The internal glass pressure p0 in the reference undeformed state should correspond to the external atmospheric pressure during sealing. When an external load is applied, the reference volume V0 of the cavity may change, resulting in a change in gas pressure Δp, which is usually much lower than the atmospheric pressure, which adds to the applied load and causes the glass to strain.
Following the BAM method, Δp can be evaluated by considering the auxiliary problem shown in Figure 2. This corresponds to a single glass panel of thickness h, with the same shape as the considered DGU, simply supported at its edges, even though the DGU may have different boundary conditions and be subjected to an imaginary uniform pressure q.
Many textbooks have recorded the out-of-plane deflection w(x, y) of simply supported plates of various shapes under uniformly distributed loads. Use A to indicate the board area, and use
Its bending stiffness, out of plane
Among them, φ(x,y) is a dimensionless shape function, which only depends on the shape of the plate. The only quantity required to evaluate the load distribution in the DGU is the average value of the shape function on the board area, called φA, whose average value φL is on the line L where the linearly distributed load is applied, and its value φP is when the concentrated load is applied. At the coordinate point (xP, yP), that is,
The graphs and tables used to evaluate these coefficients are recorded in (Galuppi 2020) for rectangles (with different aspect ratios), equilateral triangles, and isosceles right-angled triangle plates.
The load distribution is greatly affected by the bending stiffness of the panel and the gas compressibility, which is provided by the pressure change Δp in the cavity between the panels. By considering the external load imposed on the glass plate 1, the DGU in Figure 1 can be used to estimate Δp using simple formulas for different types of external loads. In particular, some people think:
Is the bending stiffness of the i-th glass panel. As described in (Galuppi and Royer-Carfagni 2019), when the gas in the cavity is considered incompressible (isovolume gas transformation), the pressure change can be evaluated by setting V0/p0=0 with formula (5-7).
As described in (Galuppi and Royer-Carfagni 2019) and (Galuppi 2020), when the pane is made of laminated glass, D1 and D2 can be calculated by considering the effective thickness of the pane. Among the various methods of calculating effective thickness recorded in the literature, the enhanced effective thickness (EET) proposed in (Galuppi and Royer-Carfagni 2012) seems to be the method that provides the best approximation for various geometries.
The numerical model consists of displacement connections between multiple elements and nodes. In order to ensure that the aspect ratios of all elements remain below acceptable values, the main dimensions of the elements vary according to geometry, boundaries and loading conditions. The grid size changes follow user-defined rules, avoiding high size ratios between tight cells. A typical grid used for load sharing analysis is shown in Figure 3.
The most general case of a DGU composed of two laminated glass elements is shown in Figure 4.
When the glass elements that make up the DGU are laminated, they are modeled by using the multi-element cross section shown in Figure 5.
In order to simulate the effective behavior of the cavity, two (laminated) glass plates are connected by using kinematic elements, allowing relative rotation and linking the relative displacement between the two nodes. A sketch of this construction is shown in Figure 6.
The numerical iteration procedure is based on the assumption that there is no gas exchange between the cavity and the external environment. "Maffeis Glass Checker" considers the equal volume process in the load distribution process, that is, ignores the compressibility of the gas. Under this assumption, pressure changes will occur under glass deflection; this pressure can be evaluated by numerical iterative analysis, which follows the workflow shown in Figure 7. Here K1 represents the stiffness of the directly loaded panel, and K2 is the stiffness of the panel that is only affected by pressure changes.
Under normal load conditions acting on a pane, the cavity is affected by the volume change ΔV. The displacement field that produces the cavity volume change is obtained through geometric nonlinear finite element analysis and executed using the commercial software Straus7. Maffeis Glass Checker follows a step-by-step cyclic procedure, in which, according to the law of constant volume transformation, the volume change obtained at a given step will cause the pressure of the cavity to change Δp(ΔV), which will be added to the initial load condition Next step. The cycle continues until the cavity. At each step of the cycle, it starts from the undeformed state and reaches a sufficiently small volume change (ΔV<ε). Here, the convergence parameter ε has been set equal to 0.1%.
Here, compare the results obtained using the BAM method with the results obtained using the "Glass Checker" developed by Maffeis Engineering. Refer to the rectangular and triangular DGU, under distributed and concentrated load on the line. The simplest case of DGU under uniform pressure is not considered here. In this regard, the results given by the BAM method are completely consistent with the formula proposed by the current standard, as described in (Galuppi and Royer Carfagni 2019). The comparison is based on the pressure change Δp in the cavity between the plates and the out-of-plane displacement of the glass element. In particular, it is assumed here that an external load is applied to the glass plate 1, and the maximum deflection of the glass plate 2 is measured, which is only affected by Δp.
Consider the rectangular DGU shown in Figure 8, with width a and height b, cavity thickness s = 16mm, and panel thickness h1 = h2 equal to 8 mm, 10 mm, and 12 mm. The gas pressure during sealing is considered to be equal to the standard atmospheric pressure, and the cavity volume is assumed to be V0=sA.
4.1.1. Rectangular DGU bearing linear distributed load
First, consider the case of a 2m x 2m square DGU. In this case, the linear distributed load H‾=1 kN/m is applied to the horizontal line of height h, as shown in Figure 8a.
Figure 9a shows the pressure Δp generated in the cavity obtained by the BAM method for compressible (continuous line) and incompressible (dashed line) gases as a function of load position. Please note that, as described in (Galuppi and Royer Carfagni 2019), the pressure in the latter case is the highest and is independent of the thickness of the glass panel. However, because the glass plate is large and flexible, the influence of gas compressibility is very limited in these cases. Note that for compressible gas, the higher the deformability of the composite plate (that is, the smaller the thickness), the smaller the influence of gas deformation, and the pressure change is close to the value corresponding to the incompressible condition.
As mentioned in Section. 3. The Glass Checker of Maffeis Engineering considers the constant volume transformation of the gas in the cavity. The nonlinear modeling of the response of the glass plate leads to a slight dependence of the numerically calculated Δp on the plate thickness.
Figure 9b shows the comparison between the BAM method and the Glass Checker in terms of the out-of-plane displacement of the glass panel 2. It can be noticed that for panels with thicknesses of 10 mm and 12 mm, there is a very good agreement between the BAM and Glass Checker results, with an average difference of about 2%. On the other hand, for the case of h1=h2=8 mm, the geometric nonlinear effect is very relevant, and the BAM method tends to overestimate the maximum deflection, as shown in Figure 9b.
This problem can be overcome by using the BAM method to evaluate Δp and continuously performing FEM analysis, taking into account geometric nonlinearity to evaluate board deflection. The maximum out-of-plane displacement thus obtained was evaluated by using the ABAQUS program and is drawn with a dash-dotted line in Figure 9b. This allows to reduce the maximum difference between the BAM method and the numerical results from 14.5% to 2.5%.
In order to evaluate the influence of the aspect ratio of the board, in the case of rectangular DGUs, a=2m, b=3m (line load applied on the shorter side) and a=2 m, b=3 m (applied on the longer side) Line load) side) consideration. Since it has been verified that the influence of gas compression rate is limited in these cases, the analysis results are obtained by using the BAM method that considers the gas compression rate. For different plate thickness values, the results of the maximum deflection of plate 2 are plotted in Figure 10 as a function of the height of the applied line load.
Similarly, for the case of h1 = h2 = 8 mm, geometric nonlinear effects are relevant, and the BAM method tends to overestimate the plate deflection. As in the previous case, in order to consider the geometric nonlinearity of the response of the glass plate, the maximum deflection of the plate 2 has been evaluated, posteriorly, through nonlinear FEM analysis. The difference between the results obtained in this way and the results obtained using Glass Checker is because the latter does not consider the effect of gas compressibility.
4.1.2. Rectangular DGU with concentrated load
Now consider the rectangular DGU shown in Figure 8b, which bears a concentrated load F=1 kN and is applied to the coordinate points (d, h). As a representative example, the case where the load is applied to the diagonal of the plate is considered here, that is, the case of ℎ/b=d/a. Figure 11a and Figure 11b show a comparison of the maximum deflection of plate 2 between a square DGU with a side length of 2 m and a rectangular DGU 2 mx 3 m (equivalent to 3 mx 2 m).
Similarly, for high glass thickness values (ie, for h1 = h2 = 10 mm and h1 = h2 = 12 mm), there is very good agreement between the analysis (BAM) results and the results obtained using Glass Checker, that is, the average difference About 1.8%. On the other hand, in order to correctly evaluate the plate response when h1 = h2 = 8 mm, it is necessary to analyze and evaluate the gas pressure, and then perform numerical analysis to capture the effect of geometric nonlinearity.
Now consider the isosceles right triangle DGU shown in Figure 12, with a = 3 m. Please note that even though there are many architectural applications that use triangular DGUs, current standards (ASTM E-1300, prEN 16612) do not cover this situation. In this case, the graphs and tables used to evaluate the correlation coefficients φA, φL, and φP defined by equation (2-4) are recorded in (Galuppi 2020).
As in the previous example, assume that the cavity thickness is s = 16 mm, and different panel thickness values are considered. Similarly, the gas pressure at the time of sealing is considered to be equal to the standard atmospheric pressure, and the cavity volume is assumed to be V0 = sA.
First consider the distributed load per unit length of the triangular DGU H = 1 KN/m, and the general height h applied to the pane 1 along the horizontal line, as shown in Figure 12a. Note the length L of the line to which the load is applied, which is necessary to evaluate the pressure change Δp according to the equation. (6), corresponding to (ah). Figure 13a shows a comparison of the maximum deflection of pane 2 evaluated by the BAM method and the Glass Checker tool, plotted as a function of h/a, varying between 0.1 and 0.9.
For the second consideration, as shown in Figure 12b, the concentrated load F=1kN is applied on the symmetry axis of pane 1, and the distance from the triangular duct is h. The corresponding maximum out-of-plane displacement of pane 2 is plotted in Figure 13b as a function of h/a, varying between 0.1 and 0.4.
It can be confirmed that for the two cases considered, the effect of gas compressibility is very low. Therefore, the results provided by these two models are very consistent, at least for thick glass layers (the average difference is about 3%). On the other hand, for h1 = h2 = 8 mm, geometric nonlinear effects become relevant, and they can be explained by considering the BAM method to calculate Δp and then performing nonlinear FEM analysis.
4.3. Rectangular DGU made of laminated glass
When the panes that make up the DGU are made of laminated glass, the BAM formula can be used to consider the equivalent overall plate effective thickness, and the enhanced effective thickness (EET) method can be used to plate (Galuppi and Royer-Carfagni 2012, Galuppi et al. 2013) , 2014). In order to evaluate the bending stiffness of laminated glass panels, the effective flexural stiffness depends on the mechanical properties and thickness of the glass and the laminated glass, as well as the panel geometry, boundary and load conditions, which should be considered.
Since the effective thickness method is only equivalent to the monolith in terms of maximum deflection, there may be differences in other points between the deflection of the laminate and the deflection of the equivalent monolith. However, it has been proved in (Galuppi, 2020) that the resulting error of calculating the pressure change in the cavity is very low, so the error of the load distribution is very low, the error of the standard DGU geometry is about 1%.
In the numerical analysis using the Glass Checker tool, the different shear modulus values of the interlayer G are considered, corresponding to the most common situations in design practice, namely 0.1 MPa, 2.1 MPa, 4.2 MPa and 60 MPa. In addition, two different geometries of DGU are considered:
As in the previous sections, both square DGUs with a side length of 2 m and rectangles 2 m + 3 DGU m are considered. For the sake of brevity, only the DGU case where the linear distributed load acts on the centerline 1 of the plate and the concentrated load acts on the center of the plate are considered here.
1 For rectangular panels, consider the case of a = 2m, b = 3 m, which corresponds to the linear load applied to the shorter edge.
For the first case, Figure 14 shows a comparison between the maximum deflection of plate 2 obtained using the BAM method (continuous line) and the Glass Checker (triangular). Figure 14a refers to a symmetrical DGU consisting of two laminates (case A), while Figure 14b refers to case B. It can be recognized that in these two cases, the response of the DGU lies between two limits: when the modulus of the shear middle layer is low, the laminated element reaches the so-called delamination limit, which corresponds to a free sliding glass layer. , Provide moderate bending stiffness, therefore, the maximum deflection value is very high.
Conversely, for high G values, the interlayer provides perfect coupling of the glass layers (monolithic limitation), and their stiffness increases. For the geometry under consideration, this limit is reached when the G value is higher than 10 MPa. As has been observed in (Galuppi 2020), the sandwich shear modulus of the rectangular DGU has a greater influence.
Figure 15 is an analogue of Figure 14 for the case of DGU under concentrated load.
You may notice a good agreement between the analytical results and the numerical results. In fact, the influence of geometric nonlinearity is very limited for the considered geometry and load. However, when the overall limit is reached, Glass Checker tends to overestimate the deflection relative to the BAM method. This may be because, in this case (harder glass plate), gas compressibility plays a relevant role, and the Glass Checker, which takes incompressible gas into account, tends to slightly overestimate the pressure change generated in the cavity.
The recently proposed BAM (Betti Analysis Method) method for calculating DGU is derived from the classic theorem in linear elasticity and has been revised to provide a compact formula for correlation coefficients. A typical case was analyzed, and the results obtained by the BAM method were compared with the numerical simulation performed by Maffeis Engineering's Glass Checker, which is part of the integrated parameterization method for exterior wall design. Based on iterative numerical procedures, Glass Checker can provide accurate calculations for any type of combination, panel shape and load conditions, and automatically update the results when the geometry and/or boundary conditions change.
When the influence of geometric nonlinearity plays an important role (large plates and thin plates), although the BAM method is based on linear elastic theory, it can still capture the pressure changes in the cavity very accurately, and therefore, the load distribution in the DGU. In order to evaluate the stress and deflection states of the panes, we recommend using the posterior standard geometric nonlinear FEM analysis, using the pressure changes obtained by BAM as input data. Our research confirms that for DGUs of different sizes and aspect ratios, there is good agreement between the results obtained using this program and the numerical results of other iterative methods under uniform, linear or concentrated external loads.
When the DGU is made of laminated glass panels, the enhanced effective thickness method can be used to easily estimate its bending stiffness, so that the response of the panel geometry can be correctly captured. The comparison with the numerical experiments of square and rectangular plates under online distribution and concentrated load shows that when the DGU is composed of two laminates or a laminate and a whole, it has good accuracy.
ABAQUS Analysis User Manual, version 6.12. Simulia (2012) ASTM E-1300-16: Standard Practice for Determining the Load Resistance of Building Glass. Standard, American Society for Testing and Materials (2016) prEN 16612: Building Glass-Determine the side load resistance of the glass panel by calculation. Standard, CEN/TC 129 (2016) Feldmeier, F.: Zur Berücksichtigung der Klimabelastung bei der Bemessung von Isolierglas bei Überkopfverglasungen. Der Stahlbau8, 285-290 (1996) Galuppi, L. and Royer-Carfagni, G. Effective thickness of laminated glass panels, J. Mech.Mater.Struct.7, 375-400 (2012) Galuppi, L., Manara, G. and Royer-Carfagni, G., Practical expression of laminated glass design, Part B45 of composite materials, 1677-1688 (2013) Galuppi, L., Manara, G. and Royer-Carfagni, G., errata and appendix to Practical expression of laminated glass design. Composite materials, part B56, 599–601 (2014), Galuppi, L. and Royer-Carfagni, G. Betti’s double-glazed unit load sharing analysis method, composite structure, 235, 111765 (2019) Galuppi, L., practical The expression is used in the design of the DGU. BAM method, Eng.Struct., submitted (2020)
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