Simply Supported Rectangular Glass Units

It is well known in civil engineering that for simply supported rectangular plates, concentrated corner loads will be generated in the case of constant surface loads. In addition, these concentrated corner loads are oriented in the direction oppo­site that of the surface loads, i. e., pressure loads evoke tensile corner forces, whereas suction loads lead to compressive corner forces. In the case of SSG applications, the significant impact of the corner forces on bond loading was highlighted in the 1990s, for instance, by Vallabhan [2] and Krueger [3]. Please note that in order to ensure the overall equilibrium condition in the plate for deformations in the out-of-plane direction (i. e., the sum of the surface loads must equal the support loads in magnitude), the concentrated corner loads lead to increased distributed loads along the plate edges. The following partial differ­ential equation links the vertical plate displacements w of the rectangular plate in the xy-plane to the surface loads p, assigning D as the plate bending stiffness, as discussed, for instance, in detail by Blaauwendraad [4].

d4 d4

Diax4 + 2+ dy4 )w "p

For a complete description, boundary conditions have to be added in order to obtain a solution for the displacement w in the z-direction. In order to get a rough impression for the case of structural glass facades featuring rectangular units, the vertical displacement w is set to zero along the plate edges in the x – and y-directions. Furthermore, it is assumed that the bending moments along the edges can be neglected, leading to the following boundary conditions for the second derivatives of w along the edges in the x-direction (Eq 2),

and in the y-direction (Eq 3),

Please note that the mixed partial derivative d2wl(dxdy) along the plate edges is still existent as the slopes dwldy along the edges in the x-direction and dwldx in the y-direction vary, approaching zero in the corners. The mixed partial derivative is linked to twisting moments mxy according to Eq 4, with t as Poisson’s ratio.

This twisting moment is the key element for understanding the existence of the concentrated corner loads. Due to the mechanical equilibrium conditions for in­finitesimal corner segments—as shown in Fig. 4—the twisting moments are

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FIG. 4—Concentrated comer force for simple support.

related to vertical forces in the corners. Figure 5 underlines this behavior by illustrating a torsion panel relating dedicated corner loads to a constant twisting moment within the panel. In the case of isotropic plates (mxy = myx), the related concentrated corner reaction force F = mxy + myx = 2mxy.

For the quantification of the corner loads, tables are usually applied for civil engineering problems (see, for instance, those published by Czerny [5] or Stiglat and Wippel [6]). Two major cases are typically treated: one-directional support and bi-directional support. In the case of one-directional support, e. g., when only compressive loads are transferred, the plate edges lift off, leading to a re-distribution of the support loads as shown for a plate aspect ratio of 1.2 in Fig. 6. It is obvious that for this case, no corner forces will be generated.

Closely representing the situation of SSG is the case of bi-directional sup­port established by the bonding adhesive. As already noted, the corner forces quantified in Fig. 7 are acting in opposite directions. In addition to the pressure load magnitude and the size of the plate unit, the magnitude of the corner forces

FIG. 5—Torsion panel featuring constant twisting moment.

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is also affected by the aspect ratio of the plate unit, as shown in Fig. 8. Conse­quently, the aspect ratio of glass units is one of the key parameters investigated in this paper.