In addition to the complex nature of the material strength discussed in Chapter 3, the engineer is also faced with the task of stress and deflection analysis. This adds another layer of complexity to the design, particularly when ‘unconventional’ support conditions and large deflections are involved. The ensuing sections provide some general guidelines and key references in this regard.
2.3.1 Geometric non-linearity
In contrast to most other building components, glass elements commonly experience large deflections (i. e. in excess of their thickness) prior to failure. In situations where the glass plate is loaded laterally and has translational restrains along its edges, the large displacements will cause the mid-plane to stretch thus developing in-plane or membrane stresses that increase the plate stiffness. An increase in plate stiffness may also observed when the ends of the glass element are not restrained, e. g. when circumferential membrane stresses are set up as the plate is constrained to deform into a non-developable surface.
In these large deflection situations, the assumptions of Kirchhoff’s plate theory are violated. Therefore a geometrically non-linear approach which is able to take membrane stresses into account must be used. A mathematical description for the non-linear be- haviour is provided by von Karman’s partial differential equations (which in the interest of brevity are not reproduced here), however the analytical solution of these equations is complex and unsuitable for manual calculation. Further information the equations and decoupling solutions are available in specialized literature such as[318]. In practice, it is common to use approximate computational methods, such as the finite difference method or the finite element method, to solve geometrically non-linear problems.
Failure to perform a geometrically non-linear analysis for large deflection situations will result in an overestimation of the lateral deformations. Therefore the actual tensile stresses for a given load are less and the actual tensile stresses for a given deflection are generally greater than those indicated by a linear analysis. An illustration of this non-linear behaviour is provided in Figure 2.10, which shows the uniform lateral load vs. maximum deformation of a 1676.4× 1676.4 × 5.66 mm thick fully tempered glass plate. Figure 2.10 shows that the non-linear finite element analysis provides a reasonably good prediction of this behaviour, however a linear analysis results in gross errors particulary at higher loads. 0 10 20 30 40 50 60 0 10 20 30 40 50 60 Experimental data (Norville et al., 1991)
Non-linear model predictions Linear model predictions
Lo ad (k N) Lateral displacement (mm) Figure 2.10:
Load vs. displacement relationship for a 1676.4× 1676.4 × 5.66 mm thick fully tempered glass plate.
2.3.2 Finite element analysis
For common geometries and loading conditions (e. g. a glass panel with simple supports along its edges and subjected to a uniform lateral load), hand calculations based on the tables and graphs in common design standards are usually sufficient for determining maximum stresses and maximum deflections. Unusual glass geometries and support conditions (e. g. curved glass, glass with re-entrant corners, point fixings, non-unform loading etc.) normally require a more detailed computational analysis.
Various software applications with finite element capabilities are now available and may run on single processor personal computers at relatively little cost. This accessibility and versatility of the finite element method means that virtually every engineering design office has the means to carry out some form of finite element analysis.
Incorrect modelling or misinterpretation of computer-generated results may result in an unsafe estimation of stresses and must therefore be avoided. Detailed advice on the correct use of the finite element method is beyond the scope of this publication and the
reader should refer to specialized publications on the subject (e. g.[72]). However, some general rules for the modelling of glass elements are given in the following:
u The mesh density should endeavour to match the expected stress concentrations,
i. e. a finer mesh should be adopted around bolt holes and other geometric disconti- nuities in the glass.
u The results for any given mesh density should be verified by carrying out conver-
gence tests to ensure that any further mesh refinement does not affect the magnitude of the stresses obtained from the analysis.
u Contact between glass and hard materials, such as steel, is normally prevented by
using a liner, gasket or bushing that has a lower modulus of elasticity than that of glass (e. g. Nylon, POM, aluminium, EPDM). One important consideration when modelling a fixing region is, therefore, to ensure that the contact surfaces and releases are modelled such that forces are transmitted in compression only and that no tension is transmitted through the gap. This can normally be achieved by using contact elements or by prescribing contact and non-contact surfaces. This approach requires a non-linear analysis.
u Details must be modelled with care. In a point fixing, for instance, the rotational
stiffness assigned to the model should match that of the specified bolt, i. e. whether the bolt is free to rotate as in fully articulated bolts or allows only partial rotation as in spring-plate type fixings.
Further advice on the buckling behaviour of glass structures is available in[34, 36, 241] and for the finite element modelling of point fixings in[310].
2.3.3 Simplified approaches and aids
Approximate solutions from tables or graphs provide a quick way to perform maximum stress and deflection calculations for glass panels in flexure[345] and stress concentrations around point supports[270]. These approximate analytical solutions also provide the means for verifying more complex finite element analyses.
Glass selection charts provided in ASTM E 1300-04[21] and prEN 13474-2:2000 [276] cater for a range of rectangular, circular and triangular flat plates and a variety of support conditions. In the case of a non-rectangular polygonal shape, an approximation of the stresses may be obtained by exercising some engineering judgement as suggested by Vallabhan et al.[328]. This method involves representing the polygonal glass element with an equivalent circular pane that circumscribes the polygonal plate. The maximum stresses and deflections for the equivalent circular plate may be obtained from[345].