In the scope of the research task carried out in this work, a basic analytical model for calculating deflection and stress in rectangular IGUs with an unlimited number of gaps was adopted, taking into account the possibility of a non-vertical positioning of the unit. Next, an analytical model for determining static quantities in circular IGUs was developed. In parallel, a numerical model for determining static quantities in square and circular IGUs was developed and compatibility with the analytical model was verified.
For example, using IGUs with different dimensions, the static quantities (resultant load, deflection, stress) were compared in square and circular IGUs loaded with changes in atmospheric pressure and wind pressure.
Finally, a numerical model was used to identify and compare static quantities in rectangular and elliptical IGUs.
2.1. Analytical Model of a Rectangular Multi-Glazed IGU
The base model for determining static quantities in a rectangular IGU is described in [
7,
20]. The unit consists of
n slots and (
n + 1) glass plates; the subscript symbols for the gaps and plates are described in
Figure 1.
It was assumed that the initial parameters of gas
p0,
T0 are known and are the same for all gaps; with these parameters and the vertical positioning of the IGU, the component plates are undeformed and parallel to each other. Under operating conditions, the glass panes may be subject to temporary changes in atmospheric pressure, changes in gas temperature in the gaps, as well as surface loads per area (mainly wind) marked as
qz,ex,
qz,in [kN/m
2] in
Figure 1. It was also assumed that the deflections of the panes in the unit are small, i.e., the relationship between the deflection and the resultant load of individual component panes is linear; according to [
21] the linear approximation is sufficient if the deflection does not exceed the plate thickness.
The convention is that loads and deflections are positive when facing the interior (left to right as in
Figure 1, or top down for horizontally placed panes).
The model was experimentally validated for double-glazed IGUs [
22]. In the tests, the asymmetric surface load was simulated with a water column 0–35 cm high (the IGU formed the bottom of the water tank). Based on the measured deflections, the percentage value of the load transferred to the gas-coupled plate (not directly loaded) was estimated. The experimental values differed from the theoretical maximum by 7.44%, and on average by 3.15%.
In operating conditions, each of the panes in the unit bends under the action of the resultant load
q [kN/m
2] acting on the surface. This causes a change in the volume of the gaps adjacent to the given pane. Assuming the coordinate system as shown in
Figure 2, the change in the gap volume Δ
v [m
3] caused by the deflection of one of the limiting glass plates is expressed by:
where:
a, b—width and length of glass panes [m],
w(x,y)—function of deflection [m],
αv—proportionality factor [m5/kN],
In the analysis, it was assumed that the change in volume Δv is positive if it causes an increase in the volume of the adjacent gap.
The methodology of calculating Δ
v is described in more detail in [
20]. After applying the function of deflection from Timoshenko and Woinowsky-Krieger [
5] and the appropriate transformations, the following was obtained:
where:
α′
v—dimensionless coefficient [-], see
Table 1,
D—flexural rigidity of glass pane [kNm],
d—glass pane thickness [m],
E—Young’s modulus of glass [kPa],
μ—Poisson’s ratio [-].
Formula (1) gives the equation of state which for each gap in the set can be formulated as:
where:
∑Δv—change in gap volume caused by deflection of both panes limiting the gap [m3].
The equations of state for all the gaps in the unit form a system of n quadratic equations, the solution of which allows the operating pressure in each IGU gap to be determined. The system of equations for IGUs located vertically is presented in [
7].
Horizontal and diagonal glazing is more and more often used in practice—for example, winter garden coverings, glazed patios, or glass roofs over utility rooms. In the case of double-glazed IGUs, taking into account the self-weight of glass plates is also not a problem—the perpendicular component
qγ [kN/m
2] can be included in the loads
qz,ex and
qz,in. For each of the plates, the value of
qγ is:
where:
γ—the specific gravity of the glass [kN/m3],
φ—the angle of slope of the IGU to the horizontal [deg].
It is different in the case of multi-glazed IGUs—the self-weight should be included in the system of equations. Then it can be written as follows:
where:
p1, …, pn—unknown operating pressure of gas in the gaps [kPa],
pa—current atmospheric pressure [kPa],
T1, …, Tn—operating gas temperature in the gaps [K],
v01, …, v0n—the gap volume [m3],
αv,ex, αv,1−2, …, αv,(n−1)−n, αv,in—proportionality factors for individual components of the unit, [m5/kN].
In the above formulas there are auxiliary parameters:
The system of equations for
n > 1 has no analytical solutions—the system should be solved numerically by iteration. For
n = 1 (double-glazed IGU) there is an analytical solution:
The knowledge of the operating pressure in the gaps allows the resultant load
q for each of the component panes to be determined:
Then, the static quantities in the individual panes of the unit based on generally known equations of plate theory can be estimated. Deflection
w [m] at the center of the glass pane is:
where:
α′
w—dimensionless coefficient [-], see
Table 1.
Stress
σ [m] in the center of the pane in the direction of the x and y axes:
where:
kx,
ky—dimensionless coefficients [-], for
μ = 0.2 see
Table 1.
The presented model allows the estimation of the deflection and stress in IGUs of various structures subjected to loads related to the temporary changes of atmospheric pressure and temperature, as well as the effects of wind.
2.2. Analytical Model of a Circular Multi-Glazed IGU
To create an analogous calculation model for IGUs with a shape other than rectangular, it is necessary to formulate the function of deflection in an analytical form and integrate it over the surface of the pane. This is possible with the circular shape of the IGU. In this case, it is convenient to consider a single glass pane in the radial system with the coordinate r (
Figure 3). The pane has the radius
R [m] and the diameter
f = 2
R.
Timoshenko and Woinowsky-Krieger [
5] give the deflection function in the form:
Then, the maximum deflection
wr [m] and the stress
σr [kPa] in the center of the pane can be formulated:
where:
αw,r, kr—dimensionless coefficients [-].
For
μ = 0.2, the values of the coefficients are as follows:
α′
w,r = 0.004231;
kr = 0.3. A comparison with the data from
Table 1 shows that for the same surface load
q, the maximum deflection in a circular pane with a diameter
f is 4.2% greater, and the maximum stress is 13% greater than in a square plate with a side dimension
a =
f.
Of course, this fact does not translate directly into static quantities in IGUs. The greater deflection vulnerability results in increased gas interaction in the gaps, whereby the resultant loads for the panes are different for the same external load.
It should be noted that less rigid and larger panes are more susceptible to deflection—the problem is, therefore, complex and requires analysis each time the size, shape and structure of the IGU is changed.
In order to determine the proportionality coefficient
αv,r for the change in the gap volume caused by the deflection of the loaded circular glass, the function of deflection should be integrated.
After the appropriate transformations:
where:
α′v,r—dimensionless coefficient [-]; for μ = 0.2 → α′v,r = 0.001534.
The further procedure is the same as in the case of a rectangular glass pane. In order to determine the operating gas pressure in each gap and the resultant load for each component pane, formulas 6 ÷ 10 can be used, however, for each pane, instead of the
αv factor,
αv,r should be calculated from the formula:
Maximum deflection and stress should be calculated from Formulas (15) and (16).
2.3. Numerical Model for Determining Static Quantities in IGUs
A reference (rectangular) finite model (FE) was developed and is shown in
Figure 4. It is composed of two shells representing the glass panes and a set of shells (representing a gasket) that closes the unit around its periphery. The glass panes and the spacer were modeled using shell elements with reduced integration (S4R element type from Abaqus library [
23]). This was performed for a practical reason—to include the possible transmission of loads from one glass pane to the other only via the gap. In the reference model, all edges of the glass panes were simply-supported. The same assumptions were used for the circular and elliptical models. In the analyses, large deflection theory was adapted.
Glass was represented using linear elastic material properties with the density ρ = 2500 kg/m
3, the nominal Young’s modulus
E = 70 GPa and Poisson’s ratio
μ = 0.20 [
24]. For the spacer, equivalent linear-elastic materials were considered with
E = 3.0 MPa and
μ = 0.30 [
25].
Special attention was paid to the description of load share effects as a result of a sealed, air-filled gap between the panes. A fluid cavity interaction (pneumatic gas law) was applied in the models [
23], the approach provides the coupling between the deformation of the fluid-filled assembly and the pressure exerted by the contained fluid on the gap boundary of the structure [
26]. The boundary of the fluid cavity was defined by element-based surfaces (the glass panes and spacer) with normal directions pointing to the inside of the gap. This approach requires a definition of a single node (cavity reference node), which is associated with the fluid cavity. The reference node has a single degree of freedom representing the pressure inside the fluid cavity. In the FE simulations for a fluid cavity, calculations are performed using volume elements (internally created by Abaqus) using the surface facet geometry and the cavity reference node. In the simulations, the following physical constants were taken into account: the universal gas constant Ru = 8.314 J/(K·mol) [
27] and the molecular weight of dry air M
air = 28.97 g/mol [
28]. It should be noted that the type of gas does not affect the calculation results, as the pressure is the parameter that matters. However, declaring this data is necessary for the correct operation of the software.
Following a mesh convergence study, which was aimed at the verification of the mesh quality, a regular mesh pattern was applied to the glass panes and the spacer.
Figure 5 shows the relationship of the relative change of the normalized maximum principal stress in glass to the number of divisions of the edge along the periphery of the glass panes. From the mesh convergence study, it was found that the model with 100 finite elements along the edge of the pane converges to a sufficient degree. The same mesh was used for the circular and elliptical models.