4.1. One-Dimensional (1D) Numerical Modelling
presents a comparison of numerical and experimental results for the MG specimen T4. According to Table 2
, the figure illustrates the T4 thermal response, as obtained from variations in some key input features of glass, such as (i) emissivity, (ii) film surface coefficient (constant or temperature dependent values, respectively) and (iii) glass thickness.
In general, the FE numerical results were observed to slightly overestimate the experimental data, see Figure 7
a–d. For the reference 1D model of Figure 7
a, as well as for the FE parametric models in general, a close correlation was found for the Exp node especially at the beginning of the collected temperature histories, rather than at the later stage of the analyses (where the FE temperature values presented, in any case, less than 12% scatter, with respect to the test measurements). For the UnExp node of the same FE models, through the overall simulation time, the numerical results were indeed found to overestimate the experimentally measured temperatures, by approximately 10% the test values.
For all the comparative plots of Figure 7
, at ≈500 s of thermal loading, a drop of temperature can be also observed, which was caused by the sudden shutting off of the radiant panel (see Section 2
and the T4 heat flux history of Figure 4
). This phenomenon, as expected, proved to be more evident for the Exp node of the parametric FE models (≈25–30°). The UnExp node, even in presence of a relatively limited thickness for the T4 glass panel, was less affected by such a drop in the assigned loading history, due to the thermal inertia of the T4 glass volume, hence resulting in a less pronounced variation of the calculated temperature-time history. In terms of sensitivity of FE estimations to input parameters, see Table 2
and Figure 7
b,c, minor effects due to variation of glass emissivity or film surface coefficient were found, and such an outcome applies especially to the early stage of the analyses. A change in the emissivity value resulted, up to ≈800 s of thermal loading, in less than 2% the calculated scatter between the parametric FE plot and the reference 1D model, see Figure 7
c. For the final loading phase (>800 s), an average increase up to 4% was numerically observed, with respect to the reference assembly, with higher temperature measurements for lower glass emissivity values. In terms of film coefficient, see Figure 7
b, a mostly negligible variation of temperatures (less than 1%) was indeed predicted, when replacing the US input with a constant value.
In Figure 7
d, finally, the effects of glass thickness variations are shown. The temperature variations, compared to the reference 1D model, were found to be directly proportional to the glass panel thickness, both at the Exp and UnExp nodes, with an average scatter of ±0.6% and ±1.5% on the front and backside of the T4 sample.
a, in this context, presents the temperature gradient ΔT
calculated from the past experimental data, as well as obtained from the parametric FE simulations on the T4 sample, being representative of the temperature scatter, in time, between the Exp and UnExp nodes of the specimen. As shown, compared to test predictions, the numerical results were observed to underestimate the experimental values until ≈1200 s of exposure, while the opposite effect was found for the following time instants, up to the conclusion of the simulations. Since the temperature gradient is strictly related to potential failure of glass due to thermal shock phenomena (see Table 1
), careful attention should be spent on this aspect. Worth of interest, in this regard, is that the experimentally measured and numerically simulated ΔT
values were found to be much larger than the allowable temperature gradient given by the prEN thstr:2004 document (40 °C for 15 mm thick AN glass panels with polished edges).
At the time of the past experiment, the T4 sample gave evidence of severe cracks in the glass surface, see Figure 8
b. The time instant of first glass fracture in the test, however, is not available for comparative purposes. From a qualitative point of view, given the FE estimations of Figure 8
a and the sensitivity of glass mechanical properties to temperature variations [7
], it is also expected that the overall uncoupled thermomechanical analysis of the same sample could also result in crack propagation. At the current stage of the FE study, see Figure 8
a, emissivity and film coefficient modifications proved to have negligible effects on the ΔT
values, as also in accordance with Figure 7
(with up to 1.3% the measured scatter). A higher sensitivity of the collected FE results was observed when varying the glass thickness (with ±7.2% the effect of product tolerance values on the nominal thickness of the sample).
As far as LG specimens are taken into account in the FE investigations, further interesting conclusions can be derived on the reliability and accuracy of 1D models. Figure 9
, for example, presents a comparison of numerical and experimental results for the T5 sample. The FE analyses, in this case, were specifically focused on the influence of varying the thickness of the PVB interlayer foils (with 0.76 mm the nominal value for the T5 specimen and 1.52 mm a numerical value of technical interest in the design of glazing systems). As in the case of the T4 specimen briefly discussed in Figure 7
and Figure 8
, much better agreement with the past experimental values was observed for the initial stage of the FE analysis, for the model with nominal cross-sectional features (0.76 mm thick foils). This includes the temperature evolution at both the Exp and UnExp model nodes, with respect to the test. During the past experiment, similarly to the T4 sample, a sudden shutting off of the radiant panel took place. This accidental drop can be clearly perceived in the recorded time-temperature histories of Figure 9
, at ≈950 s of loading. Major effects were experimentally and numerically perceived at the Exp node of the sample, as also in accordance with the T4 observations. In this latter case, the presence of multiple layers for the T5 sandwich cross-section typically resulted in a progressively increasing protection level for the middle and unexposed glass layers, respectively. Such a finding can be also perceived from comparative plots of Figure 10
, where thermal gradients are proposed for the same sample.
As shown in Figure 9
and Figure 10
, in addition, the parametric FE simulations emphasized that even small variations in the thickness of the bonding PVB foils can typically result in increased temperatures at the Exp node, and reduced temperatures at the UnExp node, compared to the reference geometrical configuration, as a direct effect of the added thermal inertial for the composite section. In terms of recorded temperature gradients, finally, the TCs for the past T5 sample were unfortunately mounted at the external surfaces (front and backside) of the panel only, thus no direct comparison of test data with the numerically predicted gradients in each glass ply can be performed. In any case, the FE models can offer some useful background, for a further extension and interpretation of the experimental results.
a presents in fact the numerical temperature gradients for each ply, as obtained for the T5 assembly. The exposed surface, as shown, obviously heats up fastest, and a high temperature gradient (up to ≈35 °C) is achieved at the early stage of the simulation (≈130 s). The middle glass ply, being protected from direct radiant heating, shows ΔT
values of slightly lower magnitude (≈30 °C the maximum value) and a certain delay in the temperature increase (≈250 s of loading for the gradient peak), compared to the exposed glass ply. At the same time, the middle glass layer further insulates the unexposed ply, which shows ΔT
values that are in the order of 50% of the previous glass layers, and present a rather smoothed, linear trend. In Figure 10
a, according to Figure 9
, a sudden drop of ΔT
values (up to 15°) for the exposed ply can be then observed, at approximately 950 s of the simulation. This finding, being related to the previously discussed experimental accident, is reduced to a minimum for the middle glass layer (≈8 °C), and fully vanishes in the temperature history for the unexposed ply.
According to standards, see Table 2
, the thermal fracture is conventionally ensured in a given LG assembly as far as the allowable nominal gradient for the weakest ply is not exceeded. In the case of the T5 sample, the experimental crack was observed to initiate after ≈200 s of exposure. In this regard, interesting feedback for the specimen object of investigation can be derived from its absolute/total gradient evolution. In Figure 10
b, such an absolute temperature variation is shown, as a function of time, as obtained by comparing the Exp and UnExp nodal temperatures (i.e., assuming an equivalent, fully monolithic performance for the LG cross-section). Compared to the reference limit value of 45 °C (see Table 2
), the experimental cracks were typically observed to propagate for absolute thermal gradients in the order of 60–70 °C. At this stage, and up to ≈350 s of exposure, the corresponding FE model overestimates the test data in the order of 15–20%. The actual role and effect of the intermediate PVB foils for the thermal performance of the T5 nominal layered section, however, still requires further extended investigations. Figure 11
, finally, presents a comparison of 1D numerical and experimental results for the 10 mm thick, MG specimen T2. For the FE assembly calibration, the same input features of the reference 1D model in Table 2
were taken into account. In the past experiment, compared to the T4 and T5 samples, the T2 specimen was exposed to a rather constant heat flux, slightly decreasing in time (see Figure 4
). Such a constant loading configuration resulted in a more stable, progressive increase of temperatures in time, see Figure 11
a. However, the FE results were found to overestimate the experimental predictions, for the full loading stage, on both the Exp and UnExp sides. A better numerical-to-experimental correlation was observed especially at the beginning of the thermal loading phase, see Figure 11
a. Later on, however, the temperature scatter was found to lie in the range of ≈12%, which may be related to the sensitivity of FE results to the input parameters assumed in the numerical study (and in particular, the thermo-physical properties of glass), as well as to the intrinsic simplified assumptions of the 1D modeling approach.
b, in conclusion, presents the gradient calculated for the MG specimen T2. In general, as shown, the numerical results were observed to strongly overestimate the test values, by approximately 50%. Such a constant overestimation was found for most of the ΔT
-time history, and may be partly caused by the thermal parameters assumed for glass in the FE study, but most probably by defects of the past test measurement methods, as also described in [16
]. Both the experimental and numerical temperature gradients of Figure 11
b, in addition, lie below the allowable gradient of 45 °C given by standards for AN glass panels with polished edges and a nominal thickness of 10 mm (see Table 1
). Such a reference limit value was found to be mostly twice the thermal gradient for the experimentally observed cracks in glass, after ≈1150 s of exposure.
4.3. Two-Dimensional (2D) Numerical Modeling—Sensitivity Study
Given the FE observations briefly summarized in Figure 12
, the reference 2D parametric model was hence further explored, so to assess the sensitivity of the predicted thermal responses to selected influencing parameters, according to Table 3
and Figure 6
. Figure 13
, for example, presents a comparison of 2D results for different loading cases, and particularly emphasizes the influence of an additional heat flux at the top/bottom edges of the FE samples (i.e., Figure 6
c), with respect to the reference FE model (Figure 6
b). The temperature history at the Exp and UnExp nodes in the mid-height section, in particular, shows no variations in terms of thermal response, see Figure 14
a. However, marked effects due to the additional heat flux at the edges can be clearly observed in Figure 14
b, as far as the mid-height control point is replaced by control points at different distances from the center of the panel. The application at the edges of a 25% the nominal heat flux, as shown, resulted in approx. a 10% increase of temperature for both the Exp and UnExp nodes. The same magnitude of temperature increase was generally calculated from the FE models under different thermal exposures on the top/bottom edges (i.e., 5% and 15% in Figure 14
b), being the measured temperatures close to the top region of the FE assemblies directly proportional to the imposed heat flux amplitudes. In this regard, given the limited variations in the thermal loading scenarios of the FE models presented in Figure 14
b, a high sensitivity of thermal performance estimations can be perceived, together with the intrinsic limitations of 1D models. At the same time, the 2D simulations suggest the need of a larger number of TCs for experimental performance assessments.
As far as the distribution of the imposed heat flux modifies on the overall exposed surface of glass, the FE comparisons collected in Figure 14
are achieved. There, the 2D numerical estimations derived from the FE models according to the schematic drawings of Figure 6
c,d are shown. In both the FE models, an edge heat flux equal to 25% the nominal history was taken into account. As shown, the nonuniform heat flux distribution proved to have minor effects on the collected temperature histories for the FE nodes located at the middle height of the sample, see Figure 14
a. Until approx. 500 s of the FE analyses, no scatter can in fact be observed between the collected curves, at the Exp and UnExp surfaces.
However, some sensitivity of FE estimations (up to ≈4% of temperature variation) can be noticed by comparing the selected numerical curves, after 600 s of thermal loading and up to the end of the analyses. Such a numerical outcome is strictly related to the fact that—at the later stages of the analyses—the heat flux that grows up at the mid-section of glass progressively moves towards less heated regions of the panel (i.e., the top and bottom edges subjected to limited thermal exposure only).
In support of this statement, Figure 14
b collects in fact the temperature history at the top edge nodes, for the same FE models. As shown, a marked reduction of temperature (up to ≈43% the reference model) can be observed for the Exp node estimations, whereas a scatter in the order of ≈37% was numerically predicted from comparisons at the UnExp edge nodes.
and Figure 16
, finally, collect some further 2D results for the T2 specimen with the clamped upper edge, in accordance with Figure 6
e. As shown, the presence of a partially shaded upper edge for the glass sample proved to have (apparently) minor influence on the thermal performance of the specimen, and in particular for the temperature history recorded at the middle height control point (and bottom region, in general) of the T2 panel, see Figure 15
A more detailed analysis of the same FE results, however, gave evidence of a marked sensitivity of thermal predictions for the glass sample, especially when moving towards its top edge. In this latter case, variations up to 40% were numerically predicted, by comparing the selected FE models.
This finding is further confirmed by Figure 16
, where FE results are presented for the clamped glass panel, in the form of temperature contour plots at selected time instants of thermal exposure. There, it is possible to notice how the presence of the upper clamp—even limited in dimensions—can progressively affect the temperature evolution in the glass sample, as a function of time, hence requiring careful consideration for FE modeling purposes. Special care is then expected for the overall thermomechanical analysis of the same FE assembly, due to a combination of thermal and mechanical effects, especially in the vicinity of the mechanical restraint.
At a final stage of the FE study, numerical efforts were then spent for the T5 laminated sample discussed in Figure 9
and Figure 10
. Given the limited basic assumptions of the corresponding 1D model, a 2D assembly was described in accordance with Section 3
, so as to assess the possible sensitivity of 1D estimations to the model accuracy. The LG shell model for the T5 specimen was implemented by accounting for the reference input features of Table 3
. For the PVB layers, the thermophysical properties depicted in Figure 2
a,b were considered, in accordance with [14
]. In doing so, the mesh size of DC2D8 shell elements was still kept equal to 2 mm, with the exception of the PVB layers (0.76 mm in thickness), where two shell elements in the thickness were adopted, to ensure the consistency of thermal estimations. The final FE assembly hence resulted in ≈4400 elements and ≈13,800 DOFs.
Compared to the 1D predictions, see Figure 17
, the 2D model generally resulted in minor variations for the estimated temperatures, with less than 3% the average scatter. The added value of the 2D assembly, see Figure 18
, is indeed represented by the temperature estimation in the full cross-section of the T5 sample, hence including possible sensitivity to cross-sectional size effects and/or edge effects, as well as (if present) mechanical restraints. From the selected contour plots of Figure 18
, in particular, the progressive thermal protective contribution of the PVB foils can be perceived, as far as the temperature increases thorough the thickness of the glass panels. On the other hand—given the typical features of PVB foils—mostly negligible bonding contributions are expected from the same PVB layers, when assessing the stiffness and strength capacity of the same T5 panel under thermomechanical loads. In this sense, further extended investigations are required to explore the phenomena herein discussed.