**1. Introduction**

The application of curved Insulating Glass Units (IGUs) in facades has emerged as a novel solution to meet aesthetic and energy performance objectives [1]. The increased stiffness resulting from curvature has been identified as a remarkable advantage over flat glass, decreasing support requirements and improving the aesthetics or increasing spans. However, there are many constraints regarding its design, production, and performance during operation [2]. In general, as the temperature of the gas entrapped in the IGU's gap changes its value, the gas volume inside varies relative to its original volume. The bending stiffness of the component panes limits their deformations (pillowing effect of the panes) and causes a change in the pressure of the gas inside the IGU gap. Changes in internal pressure result in unfavourable visual effects that can be observed as distorted images reflected in windows. Due to improved stiffness, curved IGUs cannot equalise internal and atmospheric pressure changes by pillowing, as flat IGUs do. Therefore, the climatic loads in curved IGUs can be several times higher compared to flat units, leading to glass fracture or failure of the silicone seal, particularly in combination with mechanical stresses [3].

Although research that focuses on the performance of flat IGUs under climatic loads has been presented in many studies, research on curved IGUs is still limited. Some results from numerical studies have been reported, but there are practically no experimental studies. Nizich et al. [4] proved numerically that the equalisation rate of the curved IGU is significantly reduced compared to a flat unit. Furthermore, it was found that the curvature and resulting increased internal pressure require increasing the horizontal bite of the seal several times. In the paper by Bao and Gregson [5], a thorough sensitivity study on the internal pressure of cylindrically curved IGUs was presented, considering a series of geometric variables. The main finding was that, given the same climatic action and for the same dimension, curved IGUs always generate significantly higher internal pressure than flat IGUs.

**Citation:** Kozłowski, M.; Zemła, K. Numerical Modelling of Structural Behaviour of Curved Insulating Glass Units. *Mater. Proc.* **2023**, *13*, 12. https://doi.org/10.3390/ materproc2023013012

Academic Editors: Katarzyna Mróz, Tomasz Tracz, Tomasz Zdeb and Izabela Hager

Published: 14 February 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

This paper deals with experiments and numerical simulations of cylindrically curved IGUs. It presents the results of an experimental campaign designed to collect data that validate a numerical model of curved IGU developed within this study. The validated model is sequentially used to analyse a case study comparing the resulting internal pressure and other static quantities in flat and curved IGUs subjected to characteristic climatic actions given by DIN18008-1 [6].

#### **2. Research Methodology**

#### *2.1. Experiments*

In the experimental part of the work, a curved IGU specimen was produced, with dimensions of 500 × 1000 mm2 (width × arch length) and a 1500 mm radius of curvature (Figure 1a). The specimen was composed of two 4 mm panes made of toughened glass, a 16 mm wide flexible silicone foam spacer, and a 6 mm wide hot-melt secondary sealant. Before testing, two holes were drilled in the spacer, and copper tubes, 6 mm in diameter, were glued into the holes (Figure 1b). Subsequently, silicone conduits were mounted at the ends of the tubes and connected to some laboratory equipment (a set of syringes with a maximum capacity of 300 mL and a pressure sensor). Insulating glass units are typically mounted in frames through flexible gaskets, which partly support the glass. The specimen was tested in a vertical position in a simply supported set-up that allowed for its unrestrained deformation during testing. This set-up was chosen to detect any deformation of the IGU specimen under loading.

**Figure 1.** Experimental study: (**a**) Curved IGU sample; (**b**) Copper tube glued into a hole drilled in the spacer.

A single series of tests involved injecting or withdrawing a defined air volume into/from the IGU cavity [7]. After the procedure, pressure measurements were performed for 5 min, until the stabilisation of the pressure. The test campaign consisted of eight tests, including air injection and withdrawal into/from the cavity ranging from ±75 mL to ±300 mL with a 75 mL margin. More details on the experimental campaign can be found in [8].

### *2.2. Development of the Numerical Model of the Curved IGU*

Numerical models were developed in the ABAQUS CAE finite element analysis programme to investigate the behaviour of IGUs in the experiments described in Section 2 [9]. A script in Python language was developed to improve the speed of the model generation process. The numerical models differ only by the volume of injected/withdrawn gas, whereas other parameters, such as the geometry of the sample, material properties, and boundary conditions, remain the same. Since creating eight numerical models in the ABAQUS GUI (Graphical User Interface) would consume significant time, a Python script was developed to accelerate the process, in which each syntax reflected the ABAQUS program commands. The variable parameter, in this case the volume of gas, was assigned

as input data. The script was developed in the loop, taking another set of input data each time while passing through the code. Applying the Python script to the modelling process significantly reduces the time of the process. The numerical analysis time was limited to the execution of the Python code and computing of the data. As a result, a simple 3D model was created, with dimensions of 500 × 1000 mm2 and a 1500 mm radius of curvature. The model consists of shell parts with square 4-node finite elements with full integration (S4 from the ABAQUS library [9]) and a fluid cavity. Two parallel shells represent the glass panes, while four perpendicular shells correspond to the foamed spacer bar and the secondary sealing (see Figure 2). The fluid cavity was implemented for gas simulation in the air cavity of the IGU. It was vital to create fluid exchange as an injected/withdrawn volume of gas similar to the test conditions.

**Figure 2.** Numerical model of the curved IGU: (**a**) Finite element mesh; (**b**) Detail of the glass–spacer interface. Note: glass panes are shown in green, while spacers are grey.

The following gas constants described the fluid cavity: the universal gas constantRu = 8.314 J/ (K·mol) [10] and the molecular weight of dry air: Mair = 28.97 × <sup>10</sup>−<sup>3</sup> kg/mol [11]. In addition, it was crucial to assign the appropriate physical parameters, such as the thermal expansion coefficient and density, to the air, which were necessary to perform the temperature load analysis. The thermal expansion coefficient was equal to 0.00343 1/K, and the density was 1.204 kg/m3. The model was simply supported and free to deform. The material properties for the glass parts were taken from EN 572-1 (E = 70 GPa, μ = 0.23) [12], while for the spacer, E = 4.6 GPa and μ = 0.49 were used according to [13]. Kozłowski et al. [8] reported a more detailed description of the material properties and the parameters of the fluid cavity. In the same article, a mesh convergence study for maximal stress was performed for similar model parameters, and the model converged to an acceptable degree (change in stress not greater than 2% compared to the previous iteration step). Consequently, in the following research, the same mesh size was assumed, which was equal to 34 points of division along the longer edge of a glass component.
