Development of an Analytical Method to determine Cavity Pressure in Cold-Bent Insulating Glass Units

Abstract

Cold-bent insulating glass units (IGU) have the potential to be one of the answers to current needs and wishes of modern and sustainable architecture. Curved glass is stiffer against out-of-plane loads and therefore the structural behaviour of cold-bent IGU’s is different compared to flat IGU’s. Structural behaviour of flat glass in buildings is assessed using standards such as EN 16612 and NEN 2608. Within these norms, isochoric and load sharing cavity pressures are assessed using simplified analytical equations. In case of cold-bent IGU’s, the additional assessment of increased stiffness, permanent cold-bending stresses and optical distortions are of importance. Currently, no standards are available for building envelopes containing curved glass elements. Methods from current standards have proved to be sufficient to quantify the complex role of cavity pressure in flat IGU’s. The goal of this research is to develop an analytical method to determine the effect of cavity pressure on the structural behaviour of cold-bent insulating glass units.

For the structural assessment of cold-bent IGU’s three stages are defined: ’Bending phase’, ‘Fixation moment’ and ‘Use phase’. Numerical models are developed to gain insights into the structural behaviour of cold-bent glass plates. Insights are then used to derive and validate simplified analytical methods.

The bending phase is numerically modelled with a geometrically nonlinear analysis in which curvatures are realized by prescribed displacements along two opposite edges of the plate. Results show that maximum cold-bending stresses are independent of plate size. By defining a tensile bending strength, insights are gained into the maximum radius of curvature of a plate with a specified thickness. A closed form equation is derived, which is used to calculate cold-bending stresses considering plate thickness and radius of curvature as input parameters. Resulting principal stresses have an average accuracy of 99.1% compared to numerical outcomes. Validity of the cold-bending stresses is limited to linearly fixed plates, up to a size of 6x3m, with equivalent plate thicknesses ranging from 4-20mm and design radii ranging from 3m to 25m.

Numerical results from the fixation moment show an anticlastic double or triple (w-shape) curvature in single curved plates, that cause optical distortions. The extent and shape of anticlastic bending depend on the Poisson’s ratio, plate thickness, radius of curvature and plate size. Based on regression between the parameters of influence, a result-based approach is developed to determine anticlastic bending displacements at the centre of curved plates. The approach is simplified to a curved plate with a sinusoidal anticlastic bending curvature. The displacements are expressed in a parameter for the radius at midspan and are calculated using tabulated coefficients with an average accuracy of 99.3%. Validity of the radius at midspan is limited to the previously mentioned boundary conditions.

For the use phase, the calculation of cavity pressures depends on the volume of deformation per pane of a linearly supported cold-bent IGU. Interaction between the panes is modelled using pressure loads derived from Boyle’s law. Using numerical models, volumes of deformation (v) from curved plates subjected to pressure loads (p) are presented in P-V diagrams. These diagrams are used calculate cavity pressures ranging from -1 to 1 kN/m2, considering plate sizes up to 6x3m, plate thickness of 8, 10 and 12mm and design radii up to 8m. Due to the increased stiffness of curved IGU’s, a higher load is shared by the pane onto which an external pressure is applied. Furthermore, as a result of anticlastic bending, a permanent isochoric pressure is present inside cavities of asymmetric IGU’s, which increases as curvatures increase.

A simplified analytical method to calculate effective cavity pressures is derived from shell theory. Load sharing pressures of symmetric IGU’s are calculated with an average accuracy of 95%, compared to results from the numerical P-V diagrams. Load sharing pressures of asymmetric IGU’s (91% accuracy) are summed with isochoric pressures due to cold-bending (92% accuracy) to calculate combined effective pressures with an average accuracy of 87%.

The equations are summarized in a guideline, which is complemented by an interactive calculation tool. The developed analytical method is effective in calculating effective cavity pressures and can be used to quickly explore various design options in a wide range of cold-bent IGU’s.