Elsevier

Engineering Structures

Volume 220, 1 October 2020, 110983
Engineering Structures

Cold bending of vertical glass plates: Wind loads and geometrical instabilities

https://doi.org/10.1016/j.engstruct.2020.110983Get rights and content

Highlights

  • Vertically oriented plates can be bent into symmetric hyperparaboloid shapes.

  • Geometrical instabilities are triggered for large deflections.

  • Either optical distortion or tensile strength limit the curvature of glass.

  • Rectangular plates accommodate large curvatures than square plates.

  • Snap-through buckling of curved façade glazing can be triggered from wind pressure.

Abstract

The study investigates the mechanical response of glass shells cold bent into hyperparaboloid shapes used in building technology. Focus is addressed on two subjects: the application of the cold bending procedure to vertically oriented rather than horizontal plates, which allows to minimize the disturbance produced by the initial deflection due to self-weight, and the effect of wind load on curved glass panels installed in glazing façades.

The particular case of a vertical monolithic glass plate (2000 × 2000 × 10 mm) loaded symmetrically on two diagonally opposite corners is studied experimentally and numerically, developing a finite element model, and the results compared each other. Both experiments and numerical analyses show that a mechanical disturbance, introduced e.g., on the centre of the plate, promotes a change in the deformation mode during the bending process. Geometrical instabilities which impair the optical quality of the glass, and trigger a serviceability limit state failure, are evidenced.

A parametric analysis is conducted to investigate the influence of the size and the aspect ratio of the glass plate on the cold forming procedure. The examined response parameters include the geometrical distortion of the glass plate, involving the possible occurrence of optical distortion and the maximum tensile stress in the plate.

In the second part of the study, the effect of wind load on curved glass shells is numerically evaluated. The results point out the possible trigger of a limit equilibrium stability, with a “snap-through” transition from an equilibrium state to another non-adjacent equilibrium configuration which can result in a sudden break of the glass plate. This phenomenon is affected from the size of the glass plate and the curvature imposed to the plate during cold bending. The proposed methodology could represent a convenient tool to assist practitioners in selecting the appropriate thickness of façade glass panels.

Introduction

Curved glass has been used for building purposes since the early nineteenth century, yet it has far from exhausted its potential, and its use is steadily increasing in contemporary architecture. In the context of the so called “fluid design”, fostered by the development of computer aided design technologies in the past twenty years, the use of curved glass has been more than ever favoured, since it facilitates the creation of unique free-form surfaces that are characterized by a combination of aesthetic appeal, transparency and use of natural light within buildings [1]. Typical uses include both interior and exterior applications, such as façades and display windows, skylights and cupolas, skywalks, entrances, revolving doors, canopies, winter gardens and conservatories, railings for staircases and elevated walkways, elevators, and partitions.

There are two main technologies for production of curved glass shells, depending on whether heat is involved in the process: hot bending and cold bending. Hot bent glass is obtained by heating glass sheets above glass transition temperature (600 °C), and curving them into the desired shape using moulds. Then the panel is cooled down in a controlled manner, resulting in a tension free end product. Both single-and double-curvature surfaces can be obtained, but a major limit is represented by the need of a negative form: when using this method to produce many different façade panels, as one has to do for a free form design, equally many different moulds have to be set up. In recent years, numerically-controlled rolling machines have been developed that allow to obtain hot-bent glass sheets with single or even with small bi-directional curvature, and radius that varies in a continuous manner, without a mould. However, hot bending technologies are neither energy nor cost effective. Curved panels are indeed prepared at the factory, thereby requiring long delivery time and high transport costs to carry panels with different curvatures. Additionally, the optical quality of hot formed shells may be sometimes unsatisfactory. The thickness of the glass sheet can be locally affected from imperfections in the mould or from straightness and distortion of the rollers, resulting in optical distortions and reflections. To conclude, hot bending techniques fit for small projects with strong curvature, but large-scale projects need other techniques [2], [3], [4], [5], [6].

An alternative, energy- and cost-efficient method to produce curved glass panels is cold bending, in which elastic deformation of glass is induced at ambient temperature by applying out-of-plane loads. The process starts by fixing the flat glass unit to a frame that mechanically bends it into the desired shape. The frame is then ready for installation into the building. The process can be performed after the glass has been tempered, therefore allowing for enhanced mechanical properties of the end product. The bending radius is normally quite high, and the thinner the glass, the lower it is the required force to bend. Moreover, cold bending can be executed on site with a relatively small amount of equipment, and without any requirements for moulds, which reduces the costs and makes this an attractive technique for creating curved glass surfaces. Eventually, the surface appearance is very smooth as it is not affected by distortions created by the mould or the rollers. For shaped glazing units having a shallow curvature, cold bent glass is therefore more and more used instead of hot formed glass panels. Another recent technique is cold lamination bending [7], in which unbonded glass sheets and adhesive interlayers are firstly bent in the desired shape, and then laminated in autoclave. The interlayer preserves the curved shape of the glass in place, though a certain springback is triggered when the restraints are removed after lamination [8], [9]. One of the advantages of using this process is the possibility of making very flexible shapes, though at a higher cost than simple cold bending. Remarkably, application of the cold bending technology is today not limited to glass, but it is used as well for composite plates used in aerospace, civil, marine and other fields [10], [11], [12], [13], [14].

However, cold bending has also some limitations. Since the flat sheet is deformed elastically, permanent stresses are induced inside unit. The maximum curvature that can be achieved is limited by the maximum elastic stresses that can be safely resisted by the glass panel, accounting for both the stress produced during the forming and the stress induced by service loads. Therefore, toughened glass in the form of heat treated (heat strengthened or fully tempered) or chemically toughened glass is typically used. Also, there can be some relaxation in the glass over its lifetime, which can cause it to change its shape over time.

Cold bending can be used to create either single or double curved glass surfaces. Single (cylindrical) curved surfaces are easier to form, but double curved surfaces are more popular in architectural design as they provide more freedom and the possibility to create free-form façades. Double curved surfaces are defined by two radii of curvature and can be divided in synclastic and anticlastic surfaces. In synclastic surfaces (e.g., paraboloid geometry), both centres of curvature lie on the same side of the surface, while in anticlastic surfaces (e.g., hyperparaboloid geometry) the centres of curvature lie on different sides. Depending on the pane dimension and shape, synclastic and anticlastic shapes can be produced having radii from 17 to 40 m, and spans up to 9–14 m [15]. Some of the most notable examples of application of cold bent double curved glass shells are e.g., the TGV railway station in Strasbourg [2], the glass roof for the medieval and renaissance gallery of the Victoria and Albert Museum, London [16], the Ocean Heights Tower in Dubai and the IAC Headquarters in New York [17].

The internal stress is not the only mechanical factor that limits the maximum curvature that can be achieved with cold bending. Geometrical instabilities can indeed occur during the warping of the glass sheet. Such phenomenon was firstly reported by Staaks and Eekhout [18], [19], [20], [21], who considered square plates fixed at three corners, while the fourth corner was forced out-of-plane. This load condition results in a hyperparaboloid form of the plate, where both diagonals are curved, and the edges remain straight. However, when the displacement at the forced corner reaches a certain value, namely 16.8 times the thickness of the sheet [18], a noticeable change in the deformation mode is triggered, as one of the diagonals tends to straighten, while the second diagonal becomes more curved.

This kind of buckling, in which the geometry of the plate changes from an anticlastic double curved to an approximatively single curved shape, has been observed in a number of experimental and numerical works published in the last years, accounting for multiple panel sizes and aspect ratios, and different boundary conditions [7], [8], [17], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30]. Notably, this change in the deformation mode is accompanied by two phenomena: (a) the edges of the plate, which remain straight in the early stage of bending, become curved; and (b) local instability appears on the stiffened diagonal in form of ripples [25]. Curved edges endanger the assembly of glass panes as they make it difficult to fix the plate to a frame and/or to align the edges of adjacent plates. The ripples observed in cold bending (generally referred to as cold bending distortion) can impair the optical quality of glass by producing optical distortion and therefore trigger a serviceability limit state failure. In accordance with the provision of the European standard EN 12150-1 [31] that limits the amplitude of roller wave distortion in fully toughened glass to 0.5 mm over a length of 300 mm, some authors [32], [33] recommend to apply the same limit to cold bending distortion in order to determine whether the optical quality of the curved plate is acceptable.

The instability may be triggered from several factors, including the plate orientation and the support conditions. A first distinction can be made indeed between glass plates linearly supported along the edges and glass plates restrained at some points [17]. The situation of horizontally oriented plates supported at two diagonally opposite corners was extensively investigated by Datsiou [25], [26], who concluded that in this condition it is the forced diagonal that, once buckling is triggered, acquires more curvature, while the curvature of the other diagonal diminishes. Another remarkable finding is that this change of deformation mode occurs gradually with increasing load, whereas a snap-through instability in the form of a sudden reversal of the direction of curvature of the diagonal is triggered when the loaded corners are forced to displace in the direction contrary to the initial deflection of the plate resulting from e.g., its self-weight or surface imperfections [25], [26].

Additional deformation and increased stress of curved glass panels occur because of the external loads the units in service are subjected to. In roof constructions, nearly all permanent (self-weight) and accidental (snow) loads act in the vertical direction, more or less perpendicular to the shell surface, thus generating bending stresses that build up to the stress produced during the cold shaping. In façade panels the gravity load is mostly supported by diaphragm action according to vertical load paths to the ground, but, on the other side, the panels may be exposed to wind blowing normal to the surface. Being wind a short term load, long term effects are avoided; however high wind pressure can induce large stresses in glass panes, as well as modify their curved shape resulting in optical distortion of images and impaired aesthetic function [9], [34], [35]. Extensive research on the effect of out-of-plane loading on innovative cold bent glass-steel hybrid cells for building skins was conducted in the framework of the European Research Fund for Coal and Steel project called “S + G” [27], and results of full-scale experimental tests, performed on horizontally oriented elements, were presented by Hoffmeister [36].

The study aims at providing a deeper insight into the subject by investigating the forms of instability triggered in point-supported vertical glass panels that are cold formed into an anticlastic shape. The first part of the paper involves a combined experimental and numerical study, considering a 2000 × 2000 mm monolithic glass sheet, and explores the effect of an out-of-plane perturbation on the deformation mode of the vertical plate. Methods and results are presented in 2 Methods, 3 Experimental and numerical results, respectively. In section 4, the validated numerical model is used to perform a parametric study and assess the influence of the dimensions and the aspect ratio of glass units on the trigger of global and local instabilities during the cold bending of originally flat plates. The effect of wind load on curved façade glazing is hence investigated in Section 5 through the execution of numerical analyses on curved glass panels subjected to the application of an out-of-plane load. Eventually, in Section 6 the main results and the findings of the study are summarized and discussed.

Section snippets

Experimental investigation

The aim of the test was to assess physically the trigger of instability during the cold bending of a tempered glass panel arranged in vertical position.

A monolithic glass plate was examined. Monolithic cold bent glass is expected to also provide the base for future research on the more complicated subject of cold bent laminated glass [25]. Indeed, according to the Italian guideline CNR DT 210 [37] approximate methods, like e.g. the Wölfel-Bennison [38], or the Enhanced Effective Thickness (EET)

Experimental and numerical results

Fig. 4a illustrates the deflection of the centre of the plate (point E in Fig. 3) versus the load P applied on each of the two corners A and C in either experiment (Ver: vertical plate without disturbance; Ver-disturbed: vertical plate with out-of-plane force opposing to the centre point displacement).

In the first experiment the P–δE,z relationship between force and displacement is monotonic and, when the early stage of deformation is disregarded, almost linear (Fig. 4a). The change in

Parametric analysis

Numerical analyses were performed using the finite element model presented in Section 3 with the aim of studying the influence of the size and shape of the glass plate on the cold bending procedure.

Nine sizes of panels were studied (size in [mm]): five square plates, 1000 × 1000, 1500 × 1500, 2000 × 2000, 2500 × 2500, 3000 × x 3000; and four rectangular plates, 2000 × 1000, 3000 × 1500, 3000 × 1000, 4500 × 1500. The aspect ratio μ, defined as the ratio of long to short edge lengths, varies from

Effect of wind load on curved glass plates

Additional analyses were performed to study the effect of an out-of-plane uniformly distributed load on double curved glass panes, simulating the effect of wind pressure on façade glazing units. Two studies were performed, considering either a symmetric or an asymmetric geometry of the bent plate, depending on the prescribed corner displacement during cold bending.

Discussion and conclusions

A fabrication method to obtain double curved glazing for free-form building façades is cold bending. With this technique, hyperbolic paraboloid surfaces can be obtained from flat glass panes by forcing their corners in out-of-plane directions. However, increasing the distortion may trigger an instability phenomenon: the plate develops an asymmetric deformed shape, in which the bending occurs mainly through one of the two diagonal lines. Deep research has been conducted in this field in the

CRediT authorship contribution statement

Virginio Quaglini: Conceptualization, Methodology, Data curation, Visualization, Validation, Writing - original draft, Writing - review & editing. Sara Cattaneo: Conceptualization, Investigation. Carlo Pettorruso: Software. Luigi Biolzi: Conceptualization, Supervision.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The Authors wish to thank Mr. Roberto Minerva and the Materials Testing Laboratory of Politecnico di Milano for making available the experimental facilities and for supporting the execution of the tests, and in particular Mr. Antonio Cocco and Mr. Marco Cucchi for their valuable help in the preparation of the set-up.

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

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