Validation of Models
To correctly create a structural model we need to understand the behavior of glass bending, its expected displacements and other factors that have an effect on the glass deformation. To understand the cold glass bending, it is useful to study simplified models of deformation. The shape of the cold-bent glass is more complex than the following two theoretical models, but they are the major contributors to the overall state of deformation.
For a cold-bent plate, the first idealized deformation shape is one where straight lines parallel to the edges remain straight after the plane is deformed. The deformed shape that follows the straight lines rule is presented in Fig. 16. Such deformation will create a state of stress in the glass such that:
Oxx (z) = Oyy(z) = Sxz(z) = Syz(z)=0 (1)
Sxy(t/2) = – Xxy(-tjl) = 0 (2)
This state of stress represents two-directional bending along x’-y0 directions, where x’-y0 are axes rotated 45° away from x-y (Fig. 16).
The second idealized state of deformation is unidirectional bending. There are two statically equivalent states of deformation with unidirectional bending
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FIG. 16—Idealization of purely cold-bent rectangular plate.
where the bending could occur about either of the two diagonals. Both of these states are presented in Fig. 17. Depending on the initial deformation, a structure can arrive at either state of equilibrium. The diagonal about which the glass bends can be selected by forcing the bend during the initial deformation. This effect is very difficult to obtain numerically. Depending on the initial deformation state, nonlinear FE analysis will return various outcomes. However, simple tests, such as bending a credit card by hand will reveal that unidirectional bending requires the least amount of energy to force four corners of a rectangular plate out of plane. Applying external pressure to the surface of such bent glass can cause an effect known in the literature as “snap through buckling" .
The major differences between the two cold-bent shapes described above
• In the bidirectional bending example, the edges of the rectangular glass remained straight (Fig. 16).
• In the pure unidirectional bending example, the edges of the glass deform freely (Fig. 17). One of the diagonal lines remains undeformed. The direction of bending is perpendicular to the undeformed diagonal.
FIG. 17—Two states of unidirectional bending.
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• In both pure unidirectional and bidirectional conditions, if the glass edge is framed the framing members are subject to torsion.
• The energy required to obtain unidirectional deformation is much smaller from the energy required in the bidirectional bending condition. Therefore the deformation state of bidirectional bending is possible but unstable.
The above states of deformation were observed with experimental testing
 , where the two-directional bending is observed with small deformations and the unidirectional bending is a post-buckling form.
The duality of the large deformation state was observed during the full – scale testing presented earlier in the paper. The application of pressure to the IGU in the cold-bent condition forced the deformed glass from one state of minimum energy to another. This is referred to as “snap through buckling" in the literature. This was quite a visual surprise during the full-scale testing applying the wind load to deformed unit.
An intuitive understanding of the principles laid out above would lead us to the following conclusions: a cold-bent plane with infinitely stiff edges would deform purely in a bidirectional manner, and a cold-bent plane with no frame at all would result in a simple unidirectional bend (about one of the diagonals). In our test, there are frame members that stiffen the sides of the rectangular IGU and the outcome was somewhere in between these two idealized cases.
Considering the above concepts, the deformation of the glass during the cold-bending process depends on the proportions between the flexural stiffness of the stiffening frame members and the glass panel itself. It should be noted here that the torsional deformation of the frame is a result in both of the idealized cases. Therefore, to allow for this deformation without high torsional forces in the frame, some members should be torsionally weak (i. e., thin-walled, open-section frames). The two vertical frame members in the full-size specimen are significantly weaker in torsion than the two horizontal members. A combination of framing members with different bending and torsional stiffness creates a complex system where the state of bending deformation may not be intuitive.
The physical tests of cold-bending of glass were intended to proceed to failure and large deformations were a part of the testing protocol. Whereas the behavior of many of the materials (such as glass or aluminum) had a linear physical behavior, the silicone connecting these parts had nonlinear physical behavior. Therefore, a model considering material nonlinearity and large deformation needed to be built.