# Biaxially curved glass with large radii—determination of strength using the coaxial double ring test

## Abstract

The coaxial double ring test is used for determining the surface strength of flat glass e.g. as it is described in existing standards. The main idea of this test setup is to generate a defined area of surface tensile stress, which is uniform in all directions within the load ring. Thus, the orientation of surface flaws does not influence the test results. Furthermore, the defined area with uniform stress allows a statistical evaluation with regard to the flaw distribution and therefore a prediction for deviating area sizes is possible. For biaxially curved glass, the generation of such an area of uniform tensile stress is not easily possible in a test setup. However, for large radii of curvature and low glass thicknesses it is possible to flatten the specimen during the test and to generate an area of uniform tensile stress with increasing load. This contribution shows under which geometrical conditions the coaxial double ring test can also be used for biaxially curved glass.

## Keywords

Curved glass Coaxial double ring test Bending strength of glass Annealed glass Spherically curved glass## 1 Introduction

Testing the strength of multiaxially curved glass is much more difficult. The coaxial double ring test (CDR) is a rotationally symmetrical experimental setup which is particularly suitable for testing brittle materials, in which the specimen should be ideally plane. Pećanac et al. (2011) examined spherically curved solid oxide fuel cells (SOFCs) by using the CDR. The study on this very thin material (the thickness of SOFCs is about 310 \(\mu \)m) were carried out on a very small experimental scale of the CDR with a diameter of the load ring of 9 mm, a diameter of the support ring of 19 mm and a sample diameter of 25, 1 mm. It was found that similar stress distributions as for flat specimens can be obtained from this experimental setup. However, restrictions on the smallest radius of curvature and on the load level are given. The advantage of the spherically curved glass is that it always has full geometrical contact with the support ring and the load ring of the CDR. This is not the case with multiaxially bent glass. Figure 2 shows a quarter model of the test setup, which was developed with a diameter of the load ring of 80 mm, a diameter of the support ring of 160 mm and a square sample with a side length of 250 mm and a thickness of 4 mm as this represents a common glass thickness. In addition, the shown specimen is biaxially curved with different radii \(R_{\mathrm{x}}\) and \(R_{\mathrm{y}}\). In Fig. 2a, the setup without load is shown. Only the specimen on the sectional plane, facing the viewer touches the support ring, while the load ring on this plane has no contact with the specimen. The loaded specimen is shown in Fig. 2b. In the loaded state, the specimen is fully in contact with the support ring and the load ring. With an increasing load, the stress profile of a flat test specimen superimposes the stress profile resulting from the aforementioned structurally non-linear behaviour, which has ceased until the full contact. If the stress profile in the moment of the first complete contact is of lower significance, the CDR can also be used for biaxially curved glass.

In this paper, it is systematically examined, in which range the considered experimental setup for curved glass fulfils the actual intention of the double-ring bending test for flat glass. The main focus lays on the influence of different radii of curvature on the generation of a uniform stress field within the load ring. By parameter studies, limitations are discussed in which the CDR can be appropriately used for curved glasses.

## 2 Methods

### 2.1 The coaxial double ring test

*L*denotes the edge length of the square sample.

Dimensions of the CDR for testing glass, which is going to be used in the building sector, are given in EN 12882. Here, the area of equibiaxial tensile stress is 240.000 mm\(^{2}\). This results in large radii of the load ring and the support ring. Because of these large dimensions, the resulting deformations in this test setup are very high, whereby the behaviour is strongly geometrically non-linear. According to the standard, this non-linear behaviour is to be counteracted by air pressure, applied onto the glass pane within the loading ring. This additional step results in a complex setup which may be vulnerable to application errors. On the basis of the aforementioned principle of effective stresses, Pisano et al. (2015) proposes a smaller setup of the CDR in which no air pressure is necessary. However, even here there are geometric non-linear effects that lead to a nonuniform stress profile. These non-linear effects are taken into account here with correction factors depending on the specimen thickness.

*L*= 250 mm and the thickness of the glass pane was set to 4 mm. The characteristic tensile strength of float glass as given in standards (EN 572-1 2012) is 45 MPa, so that with the selected dimensions and for the relevant stress levels of about 40–140 MPa, geometrically non-linear effects in the test procedure with planar glass only have a larger influence for higher loads. In addition, it is useful for practical reasons to allow deviations of the stresses within the load ring in a certain range. EN 1288-1 (2000) specifies a limit value of 2%. The transfer of the results to larger areas according to Eq. (4) is also possible here.

### 2.2 Finite-element model

The material parameters of the glass were set to E = 70000 MPa and \(\nu \) = 0, 23. The load ring and the support ring were modelled with the material properties of structural steel. The bottom of the support ring was fixed. The load was applied at the entire top surface of the load ring. Furthermore, the load ring was modelled very stiff. This was done because in the case of biaxially curved glass, the load ring touches the glass surface only at two points during the beginning of the simulation. Thus, the addition of material prevents deformation in the load ring due to loading. The radii of the semicircles of the support ring and the load ring were 5 mm.

## 3 Results and discussion

### 3.1 Spherically curved glass

In a first step, spherically curved glass is examined. The radial stresses \(\sigma _r \) and the tangential stresses \(\sigma _t \) are shown in Fig. 7 for three different radii at the same load level. All results were generated by FE analyses. The used load level of 1616 N was chosen by the condition that an analytically determined tensile stress of \(f_k =45 \hbox {N}/\hbox {mm}^{2}\) occurs. Setting a very large radius in the numerical simulation, the profile of the tensile stress becomes similar to the analytical solution for flat glass. With increasing curvature, two effects occur: On the one hand, the values of the tensile stress within the load ring are reduced and, on the other hand, the tensile stress differences within the load ring increase.

It can be seen in Fig. 8a, that the relative stress difference \({\Delta }\sigma \) within the load ring depends both on the radius and on the load level: The flatter the curvature (bigger radius), the lower the load dependency. Thus, it is possible to specify stress ranges for a specific radius in which the differential tensile stresses do not exceed a certain limit. The shown average stress values (Fig. 8b) were calculated with the maximum and minimum values of \(\sigma _r \) and \(\sigma _t \). Considering for example a stress range of approx. 40–125 MPa and a radius of 3000 mm the relative stress differences do not exceed a value of 2% as it is requested in EN 1288-1 (2000). For smaller radii, a higher load is needed, since the specimen has to be flattened more to generate a homogeneous stress distribution. The load at failure measured in the test must be converted into the breaking stress. As already shown, the correlation between the applied load and the resulting stress is non-linear. The deflection in the centre of the specimen shows the same behaviour. In Fig. 9 it can be seen, that this non-linearity decreases with larger radii. Hence, the analytical solution for the stress and the deflection [Eqs.(1) and (2)] delivers sufficiently accurate results. For smaller radii, the geometrical non-linear behaviour should be respected. Considering for example a spherically curved glass with a thickness of *t* = 4 mm, a radius of 3000 mm and breakage load of 2838 N, the strength can be determined to 70,9 MPa (Fig. 9a). Furthermore, the reference Area \(A_{0 }\) is 5027 mm\(^{2}\) (area within the load ring) as the relative stress differences are lower than 1% (Fig. 8) and thus the orientation of the flaws is considered to have no influence.

### 3.2 Biaxially curved glass

As with spherically curved glass, a parameter study was carried out using Eq. (5) for a more accurate examination of the stress differences \(\Delta \sigma \) within the load ring. Thus, all stress differences between the two principal stresses relative to each other as well as the differences in the stress distribution within the load ring are firstly considered. The results of this parameter study are shown in Fig. 12 for different combinations of the dimensions of the two radii in dependence of the average stress which was calculated with the minimum and maximum principal tensile stress values. It turns out that even for very large radii and slight differences between the two radii \(R_{\mathrm{x}}\) and \(R_{\mathrm{y}}\), the stress differences within the load ring are very high. A statistically meaningful evaluation with a defined reference area is not possible here.

For the investigation of the geometrically non-linear behaviour, the load was increased for different combinations of the radii \(R_{\mathrm{x}}\) and \(R_{\mathrm{y}}\), and the corresponding stress \(\sigma _1 \) and the deflection *w* was evaluated in the center of the specimen. Fig. 14 shows the results for the ratio 10/9 of the two radii for three different values of \(R_{\mathrm{y}}\). It can be observed that the non-linear behaviour occurs only at loads which result in stress levels well above the typical strength of annealed float glass. Therefore, the analytical solution for the tensile stress is very suitable for large radii. For the deflection, non-linear behaviour can be noticed for smaller loads in comparison to the stress reaction. Furthermore, the non-linear behaviour of the deflection increases for higher radii. This is because the influence of membrane effects due to deflection occurs earlier. For smaller radii, non-linear effects have to be taken into consideration, comparable to spherically curved glass (see Fig. 9).

## 4 Conclusion

The tensile strength of spherically or biaxially curved glass can be examined using the coaxial double ring test for specific thickness-to-radius relations and CDR geometries.

For the investigations on biaxially curved glass, an experimental setup of the double-ring bending test with a diameter of the load ring of 80 mm, a diameter of the support ring of 160 mm and a square sample with the edge length of 250 mm was chosen. The thickness of the glass plate was set to 4 mm as this represents a common glass thickness. With these dimensions of the test setup, geometrically non-linear effects only have to be taken into consideration when applying high loads, which lead to higher stresses than 140 MPa. In a first step, spherically curved glass was investigated. The advantage of this geometry is its rotational symmetry leading to a symmetrical tensile stress distribution. It was shown that spherically curved glass with radii larger than 3000 mm can be examined with this test setup. The generated stress field within the load ring shows differences lower than 2% for a load range of 40 to 125 MPa. In the case of biaxially curved glass, it is shown that, for small differences between the two radii, the differences of the maximum principal tensile stress are sufficiently small, thus the results can be related to a reference area.

Further investigations could be carried out in order to use an effective area which is representative for the applied stress distribution, thus the difference of the two principal stresses can be taken into consideration for the statistical evaluation. Furthermore, the use of smaller radii of the support and load rings could be investigated. The disadvantage here is the higher scattering of the test results since the probability that very large flaws occur on the surface within the load ring decreases. This leads to smaller 5% fractile values in the statistical evaluation.

A transfer to glass thicknesses which differ from the 4 mm examined here is possible with further studies as the basic mechanical behaviour remains the same. In the case of thinner glass plates, the geometrically non-linear effects increase, and in the case of thicker glass plates, these effects decrease. However, thinner glass plates adapt better to the test device and vice versa.

## Notes

### Compliance with ethical standards

### Conflicts of interest

The authors declare that they have no conflict of interest

## References

- Beason, W., Morgan, J.: Glass failure prediction model. J. Struct. Eng. ASCE
**110**(2), 197–212 (1984)CrossRefGoogle Scholar - Bundesverband Flachglas: Leitfaden für thermisch gebogenes Glas im Bauwesen, BF-Merkblatt 009/2011 (2011)Google Scholar
- EN 1288-1:2000: Glass in building. Determination of the bending strength of glass. Fundamentals of testing glass (2000)Google Scholar
- EN 572-1:2012: Glass in building—Basic soda lime silicate glass products—Part 1: Definitions and general physical and mechanical properties (2012)Google Scholar
- Ensslen, F., Schneider, J., Schula, S.: Produktion, Eigenschaften und Tragverhalten von thermisch gebogenen Floatgläsern für das Bauwesen - Erstprüfung und werkseigene Produktionskontrolle im Rahmen des Zulassungsverfahrens, Stahlbau Spezial 2010 - Konstruktiver Glasbau (2010), doi: 10.1002/stab.201001304
- Gulati, S., Helfinstine, J., Roe, T., Khaleel, M. et al.: Measurement of Biaxial Strength of New vs. Used Windshields, SAE Technical Paper 2000-01-2721 (2000), doi: 10.4271/2000-01-2721
- Overend, M.: Recent developments in design methods for glass structures. Struct. Eng.
**88**(14), 18–26 (2010)Google Scholar - Pećanac, G., Bause, T., Malzbender, J.: Ring-on-ring testing of thin, curved bi-layered materials. J. Eur. Ceram. Soc.
**31**, 2037–2042 (2011)CrossRefGoogle Scholar - Pisano, G., Royer-Carfagni, G.: Towards a new standardized configuration for the coaxial double test for float glass. Eng. Struct.
**119**, 149–163 (2015)CrossRefGoogle Scholar - Schneider, J., Kuntsche, J.K., Schula, S., Schneider, F., Wörner, J.-D.: Glasbau Grundlagen, Berechnung, Konstruktion. Springer, Berlin (2016)Google Scholar