### III. EXPERIMENTAL EVALUATION OF THE BENDING STRENGTH IN THE BOREHOLE AREA

#### A. Test set-up

To avoid all uncertainties of the theoretical determination of the glass strength caused by the surface condition after the drilling process and by the tempering process, the experimental set-up shown in figure 4 was used. The advantage of this set-up is a constant distribution of tensile stresses at the edge or chamfer of a hole so that the maximum surface flaw and/ or minimum temper stress will always be detected. The different types of holes in annealed and tempered soda-lime-glass and borosilicate glass from table 1 were tested at different stress rates. Figure 5 shows the boundary conditions of a finite-element-model of the test. Figure 6 shows the stress distribution of the principal tensile stress around the hole.

<figure><img src="image_96.png" /><figcaption>Figure 4: Section of experimental set-up: Modified coaxial double-ring bending test [mm]</figcaption></figure>

<figure><img src="image_97.png" /><figcaption>Figure 5: Finite-element model of the coaxial double ring bending test for a conical hole with static boundary conditions (1/4 of a sample, symmetry)</figcaption></figure>

<figure><img src="image_98.png" /><figcaption>Figure 6: Stress distribution at the chamfers of a cylindrical hole in a coaxial double-ring bending test</figcaption></figure>
## B. Test results

Failure occurred at the edges or chamfers for nearly all test samples. Table 5 gives the results of the bending strength. 5%-fractiles were calculated for a 95% confidence interval [PLATE 1993]. Note that a logarithmic normal distribution was used instead of a Weibull-distribution to calculate fractiles, because the evaluation of the experimental data for both float and tempered glass clearly showed that a Weibull-distribution – especially for low fractiles - does not fit the data as well as a logarithmic normal distribution (Figures 7, 8).

<figure><img src="image_99.png" /><figcaption>Figure 7: Fit of failure stress of series A.F4 to the Weibull distribution with linear regression and confidence intervals</figcaption></figure>

<figure><img src="image_100.png" /><figcaption>Figure 8: Fit of failure stress of series A.F4 to the logarithmic normal distribution with linear regression and confidence intervals</figcaption></figure>
<table><caption>TABLE 5:
BENDING STRENGTH AT THE EDGES/ CHAMFERS OF ANNEALED AND TEMPERED GLASS WITH HOLES</caption><thead><tr><th>Test Series</th><th></th><th>stress rate<br>[MPa·s<sup>-1</sup>]</th><th>mean<br>[MPa]</th><th>standard deviation<br>[MPa]</th><th>coefficient of variation<br>[-]</th><th>5% fractile<br>[MPa]</th></tr></thead><tbody><tr><td>A.F1</td><td rowspan="6">annealed</td><td>2,0</td><td>59,5</td><td>4,6</td><td>0,08</td><td>49,8</td></tr><tr><td>A.F2</td><td>2,0</td><td>56,7</td><td>6,1</td><td>0,11</td><td>44,3</td></tr><tr><td>A.F3</td><td>0,2</td><td>51,5</td><td>4,6</td><td>0,09</td><td>41,5</td></tr><tr><td>A.F4</td><td>2,0</td><td>57,1</td><td>6,3</td><td>0,11</td><td>43,4</td></tr><tr><td>A.F5</td><td>20,0</td><td>66,6</td><td>6,2</td><td>0,09</td><td>53,4</td></tr><tr><td>A.F6</td><td>2,0</td><td>45,8</td><td>9,4</td><td>0,21</td><td>25,3</td></tr><tr><td>A.E1</td><td rowspan="6">tempered</td><td>2,0</td><td>193,2</td><td>14,9</td><td>0,08</td><td>162,1</td></tr><tr><td>A.E2</td><td>2,0</td><td>167,3</td><td>11,2</td><td>0,07</td><td>143,9</td></tr><tr><td>A.E3</td><td>0,2</td><td>173,0</td><td>11,1</td><td>0,07</td><td>146,1</td></tr><tr><td>A.E4</td><td>2,0</td><td>185,5</td><td>8,8</td><td>0,05</td><td>166,3</td></tr><tr><td>A.E5</td><td>20,0</td><td>187,9</td><td>10,0</td><td>0,05</td><td>166,7</td></tr><tr><td>A.E6</td><td>2,0</td><td>195,2</td><td>14,6</td><td>0,07</td><td>163,8</td></tr><tr><td>B.F1</td><td rowspan="6">annealed</td><td>0,2</td><td>61,3</td><td>6,2</td><td>0,10</td><td>48,3</td></tr><tr><td>B.F2</td><td>2,0</td><td>67,3</td><td>6,5</td><td>0,10</td><td>52,9</td></tr><tr><td>B.F3</td><td>20,0</td><td>67,5</td><td>6,1</td><td>0,09</td><td>54,4</td></tr><tr><td>B.F4</td><td>0,2</td><td>79,0</td><td>12,6</td><td>0,16</td><td>52,2</td></tr><tr><td>B.F5</td><td>2,0</td><td>94,9</td><td>11,3</td><td>0,12</td><td>70,8</td></tr><tr><td>B.F6</td><td>20,0</td><td>105,6</td><td>10,7</td><td>0,10</td><td>82,9</td></tr><tr><td>B.E1</td><td rowspan="6">tempered</td><td>0,2</td><td>204,2</td><td>10,1</td><td>0,05</td><td>182,6</td></tr><tr><td>B.E2</td><td>2,0</td><td>215,8</td><td>8,1</td><td>0,04</td><td>198,5</td></tr><tr><td>B.E3</td><td>20,0</td><td>221,4</td><td>10,6</td><td>0,05</td><td>198,2</td></tr><tr><td>B.E4</td><td>0,2</td><td>234,6</td><td>14,0</td><td>0,06</td><td>204,2</td></tr><tr><td>B.E5</td><td>2,0</td><td>250,1</td><td>16,4</td><td>0,07</td><td>214,6</td></tr><tr><td>B.E6</td><td>20,0</td><td>265,2</td><td>13,9</td><td>0,05</td><td>236,4</td></tr><tr><td>C.F1</td><td rowspan="3">annealed</td><td>0,2</td><td>42,5</td><td>3,0</td><td>0,07</td><td>36,1</td></tr><tr><td>C.F2</td><td>2,0</td><td>45,3</td><td>5,0</td><td>0,11</td><td>34,5</td></tr><tr><td>C.F3</td><td>20,0</td><td>54,9</td><td>5,3</td><td>0,10</td><td>42,9</td></tr><tr><td>C.E1</td><td rowspan="3">tempered</td><td>0,2</td><td>163,0</td><td>10,6</td><td>0,06</td><td>145,6</td></tr><tr><td>C.E2</td><td>2,0</td><td>167,7</td><td>10,0</td><td>0,06</td><td>140,1</td></tr><tr><td>C.E3</td><td>20,0</td><td>177,2</td><td>11,6</td><td>0,07</td><td>151,9</td></tr></tbody></table>
### C. Discussion of the results

For float glass, the bending strength mainly depends on the drilling process. Holes manufactured by producer A show lower mean values than glasses without holes from literature (approx. 60–120 MPa, [FINK 2000, LAUFS 2000, SCHNEIDER 2001]) but 5 %-fractiles are in the range of the bending strength defined in the standards (45 MPa, [DIN 1249]) because the scattering is very low. So the drilling process reduces the mean value but at the same time reduces scattering. Only for holes drilled with water-jet without a chamfer, mean values and 5 %-fractiles are significantly lower which corresponds to bigger surface flaws at the edges resulting from this drilling process [WÖRNER 2001].

Glass of producer B gives higher values and even 5 %-fractiles are higher than 45 MPa. This phenomenon could be explained by the different drilling process or by a “better” basis glass; the origin of the soda-lime-glasses of both producers A and B were unknown. It is also interesting with the results from series B that significantly lower values result from holes with the cylindrical part under tension for both float and tempered glass. A reason for this could be that in this drilling process the driller goes through the whole plate from the conical side and the chamfer on the cylindrical side is done in a second process. This could cause bigger surface flaws. The holes from producer A are drilled from both sides simultaneously.

Float glass from borosilicate glass (producer C) shows lower values than the soda-lime float glasses which is in contrast to results from literature [EXNER 1982]. The reason for this could also be the drilling process.

For all tempered glasses, the bending strength in the hole area is higher than the 5 %-fractile defined in the standards (120 MPa, e.g. [DIN 1249]) - even for tempered borosilicate glass although the amount of temper stress was significantly lower than for soda-lime-glass due to a lower thermal expansion coefficient. It is not possible to clarify if the higher compression

stresses at the edges/ chamfers calculated in the numerical models (Figure 3) are the reasons for the relatively high bending strength. Also a – for glass - very low scattering of the test results causes higher 5 %-fractiles, so both effects cannot be separated.

From the difference in the results from different stress rates, the crack propagation factor *N* for sub-critical crack growth could be calculated for the annealed glasses and compared to data from literature. It resulted in approx. *N*=17 for series A and C and *N*=21 for series B. Usual values for annealed soda-lime-glass are *N*=18 and for annealed borosilicate-glass *N*=30. For tempered glasses, sub-critical crack growth only occurs at stresses above the temper stress. As design stresses of tempered glasses are well below the temper stress, the determination of the crack propagation factor for tempered glasses is not interesting from an engineering point of view.

It is interesting to note for holes drilled with water-jet (test series A.F6 and A.E6, table 5) that after tempering the bending strength is in the same range as for holes drilled with diamond drillers whereas before tempering this samples showed the lowest values. It is assumed that crack healing originating from the tempering process plays an important role for the glass strength of tempered glass [DANNHEIM 1991, LAWN 1993, STAVRINIDIS 1983, WIEDER-HORN 1970].

## IV. CONCLUSION AND FURTHER RESEARCH

Results show that finite-element-calculations in combination with photoelastic measurements give a good approximation about the qualitative distribution of temper stresses in the borehole area. Parameter studies and strength tests can be used where the heat transfer coefficient is unknown. For more detailed numerical simulations of thick glasses the effect of heat radiation, especially in the holes, should be considered.

Simulations show that temper surface stresses at the chamfers of holes are at least as high as in the “infinite” part of the plate whereas the
surface stresses are presumably lower in the
centre part of the holes. This was corroborated
by the results of the bending strength. Still, it
could be shown by the comparison of the
bending strength of annealed glasses and
tempered glasses (identical samples, annealed or
tempered) that the influence on the scattering of
the bending strength caused by a variation of the
temper stresses from the technical process and
caused by the surface condition of different
drilling processes cannot be identified
separately. Therefore, only with the large
number of test samples and bending tests, design
values for engineers can be calculated on a
statistical basis for different glasses, drilling and
tempering processes. In this study, the bending
strength in the borehole area of tempered glass
was higher than the 5%-fractile for tempered
glass according to German or European
standards (120 MPa) for all glasses and
tempering processes.

Samples of a soda-lime-silica glass with water-drilled holes gave the lowest strength values for float glass but reached the same strength as diamond drilled holes for tempered glass for an identical basis glass and tempering process. Therefore it is assumed that crack healing while tempering plays an important role for tempered glass strength.

Further research should concentrate on crack
healing from tempering, glass strength for in-
plane loading and the interaction of glass fittings
with glass for the determination of load-bearing
capacity of glazing systems.

REFERENCES

[ABEN 1993]

ABEN, H., GUILLEMET, C., *Photoelasticity of Glass*, Springer, Berlin 1993

[ANSYS 1999]

ANSYS 5.5.2, *Theory Reference 4.6: Viscoelasticity*, Eigth Edition, 1999.

[BATHE 1990]

BATHE, K.J., *Finite-Elemente-Methoden*, Berlin, Springer, 1990.

[CARRE 1997]

Carré H., *Le verre trempé, un nouveau matériau de structure*. Cahiers du CSTB, Livraison 385, Cahier 3003, 1997.

[CARRE 1999]

CARRÉ H., DAUDEVILLE, L., "Load-bearing capacity of tempered structural glass", Journal of Engineering Mechanics, Vol. 125 (1999) No. 8, pp. 914-921.

[DANNHEIM 1991]

DANNHEIM H., OEL H.J., PRECHTL W., "Einfluß der Vorspannung auf die Oberflächendefektentstehung bei thermisch vorgespannten Flachgläsern" Glastechnische Berichte, Vol. 54 (1981) No. 10, pp. 312-318.

[DIN 1249]

DIN 1249-10, Flachglas im Bauwesen, *Chemische und physikalische Eigenschaften*. Berlin, Beuth, 1990.

[DIN EN 1288]

DIN EN 1288-5, Glas im Bauwesen - Bestimmung der Biegefestigkeit von Glas - Teil 5, *Doppelring-Biegeversuch an plattenförmigen Proben mit kleinen Prüfflächen*. Berlin, Beuth, 2000.

[DUFFRENE 1997]

DUFFRÈNE L., GY R., "Viscoelastic constants of a soda-lime-silica glass" Journal of Non-Crystalline Solids, Vol. 211 (1997), pp. 30-38.

[EXNER 1982]

EXNER G., LINDIG O., "Bestimmung des Widerstandswertes der Spannungsrißkorrosion n an Borosilicatglas DURAN", Glastechnische Berichte, Vol. 55 (1982) No. 5, pp. 107-117.

[FINK 2000]

FINK A., *Ein Beitrag zum Einsatz von Floatglas als dauerhaft tragender Konstruktionswerkstoff. Dissertation (PhD-thesis)*, Institut für Statik, Technische Universität Darmstadt, 2000.

[GY 1994]

GY R., DUFFRÈNE L., LABROT M., "New insights into the viscoelasticity of glass" Journal of Non-Crystalline Solids, Vol. 175 (1994) pp. 103-117.

[KURKJIAN 1963]

KURKJIAN C.R., "Relaxation of torsional stress in the transformation range of a soda-lime-silica glass", Physics and Chemistry of Glasses, Vol. 4 (1963) No. 4, pp. 128-136.

[LAUFS 2000]

LAUFS W., *Ein Bemessungskonzept zur Festigkeit thermisch vorgespannter Gläser*. Dissertation (PhD-thesis), Lehrstuhl für Stahlbau der RWTH Aachen, Shaker-Verlag, 2000.

[LAWN 1993]

LAWN B., *Fracture of brittle solids*. 2nd edn., New York, Cambridge University Press, 1993.

[NARAYANASWAMY 1971]

NARAYANASWAMY O.S., "A Model for Structural Relaxation in Glass", J. Am. Ceram. Soc., Vol. 54 (1971) No. 10, pp. 491-498.

[NARAYANASWAMY 1978]

NARAYANASWAMY O.S.: "Stress and Structural Relaxation in Tempering Glass", J. Am. Ceram. Soc., Vol. 61 (1978) No. 3, pp. 146-152.

[OEL 1984]

OEL H., HELLAND G., DANNHEIM H., "Thermisches Vorspannen von Glasscheiben", Glastechnische Berichte, Vol. 57 (1984) No. 1 pp. 1-6.
[ÖKSOY 1994]

ÖKSOY D., PYE L.D., BOULOS E.N., "Statistical analysis of viscosity-composition data in glassmaking." Glass Science & Technology, Vol. 67 (1994) No. 7, pp. 189-195.

[PLATE 1993]

PLATE E.J., *Statistik und angewandte Wahrscheinlichkeitslehre für Bauingenieure*. Berlin, Ernst & Sohn, 1993.

[SCHERER 1986]

SCHERER G.W., *Relaxation in Glass and Composites*. New York, Wiley, 1986.

[SCHNEIDER 2001]

SCHNEIDER J., *Festigkeit und Bemessung punktgelagerter Gläser und stoßbeanspruchter Gläser*. Dissertation (PhD-thesis), Institut für Statik Technische Universität Darmstadt, 2001.

[SCHNEIDER 2004]

SCHNEIDER J., "Glass Strength in the Borehole Area of Annealed Float Glass and Tempered Float Glass", The International Journal of Forming Processes (IJFP), Special Issue on Glass (2004), in print

[SEDLACEK 1999]

SEDLACEK G. LAUFS W. "Stress distribution in thermally tempered glass near the edge, corner and hole, part 2: Distribution of thermal stresses." Glass Science & Technology, Vol. 2 (1999), pp. 42-48.

[SOULES 1987]

SOULES T.F., BUSBEY R.F., REKHSON S.M., MARKOVSKY A., BURKE M.A., "Finite Element Calculation of Stresses in Glass Parts Undergoing Viscous Relaxation", J. Am. Ceram. Soc., Vol. 70 (1987) No. 2, pp. 90-95.

[STAVRINIDIS 1983]

STAVRINIDIS B., HOLLOWAY D.G., "Crack Healing in Glass." Phys. and Chem. Glasses, Vol. 24 (1983) pp. 19-25.

[WIEDERHORN 1970]

WIEDERHORN S.M., TWONSED P.R., "Crack Healing in Glass." Proc. 72nd Ann. Meeting, The Am. Ceram. Soc., 1970, p. 486-489.

[WÖRNER 2001]

WÖRNER J.D., SCHNEIDER J., "Glass Strength of Annealed an Tempered Structural Glass in the Area of Drilled holes." Proc. 7th Glass Processing Days Tampere Finland, 2001, p. 193 – 198.
<figure><img src="image_101.png" /></figure>
# Buckling-related problems of glass beams

Jan Belis and Rudy Van Impe, Laboratory for Research on Structural Models, Ghent
University,
Ghent, Belgium

At different research institutes in and outside Europe, research is in progress to find pieces of the structural glass puzzle. At Ghent University, the focus is on glass beams.

In the present contribution, the authors want to highlight some experiences of the past and current activities concerning buckling problems of glass beams. It is the authors' opinion that at the current state of technological development, glass beams with a rectangular cross-section are by far the most realistic starting-point. Such geometry implies a slender cross-section, which is sensitive to lateral-torsional buckling.

Some concepts are compared in order to deal with lateral torsional buckling of laminated glass beams.

By means of experimental tests on monolithic glass beams and numerical simulations on corresponding models, it has become clear that stability (buckling) instead of strength can be the limiting factor of the load-carrying capacity of glass beams, especially if strengthened or tempered glass is used.

It is also shown that prevention of buckling can increase the load-bearing capacity considerably. Several configurations for buckling prevention are proposed and compared, varying from fixed local supports to continuous supports comparable to elastic foundations. In many cases, the elastic sealants used to connect the beam to the supported glass plates suffice to realize the desired buckling prevention. This support is not brought into account in today's practice.

**Keywords:** glass, beams, buckling, laminate

## I. INTRODUCTION

The material glass is developing towards a very popular all-round building material, able to fulfil an infill, cladding and structural role. The structural use of glass in the sense of primary load-carrying components however, lacks design recommendations, codes and standards.

Although some researchers are currently experimenting with alternative geometries, the basic (and actually by far the most-used) glass beams have an approximately rectangular cross-section. In many cases this cross-section is quite slender, which makes it rather sensitive to lateral torsional buckling.

## II. LATERAL TORSIONAL BUCKLING

### A. General

In the scarce literature on load-bearing glass beams, the failure mechanism that is usually examined is brittle fracture due to exaggerated tensile stresses at the edge. These stresses are induced by simple bending along the strong axis, so the beam is supposed to deform only in its own plane. In experiments described in literature [Hess 2000], [Veer et al. 2001], precautions are taken in order to prevent lateral torsional buckling: lateral supports are provided along the length of the beam, excluding any out-of-plane movement.

Due to the slenderness of the rectangular cross-section, however, the risk of instability—especially lateral torsional buckling—increases [Kasper et al. 2003], [Belis et al. 2003], [Luible 2004]. In favour of general comprehensibility, lateral torsional buckling is briefly illustrated in Figure 1, in which the combined action of out-of-plane displacement $u$, in-plane displacement $v$ and torsion angle $\phi$ due
to a post-critical in-plane load P is indicated.

<figure><img src="image_102.png" /><figcaption>Fig. 1. Principle of lateral torsional buckling of a beam</figcaption></figure>

Lateral torsional buckling can be the factor that limits the load-bearing capacity instead of fracture due to in-plane bending.

## B. Structural Analysis

Numerical analyses are performed with the finite elements software Abaqus [HKS 2002]. The mesh and element type have been chosen after an optimisation study.

The numerical analysis of lateral torsional buckling can be performed on different levels of complexity, which are briefly explained below.

### 1) Elastic buckling analysis

The most simple approach is the elastic analysis, which assumes the glass to be perfectly straight, free of any residual stresses, and without set-up imperfections (e.g. loading eccentricity, initial inclination,...).

Analytical expressions for the elastic buckling approach can be found in literature, e.g. in [Timoshenko et al. 1961].

The elastic buckling load (theoretical critical load) and the corresponding buckling modes can be determined using elastic buckling analyses, which are basically eigenvalue calculations.

### 2) Non-linear buckling analysis

The buckling analysis becomes more realistic – and more complex– when realistic imperfections are taken into account. Non-linearity covers geometrical aspects like initial shape imperfections and loading eccentricity, as well as material-related aspects like residual stresses and visco-elastic

interlayer behaviour. Only a few authors have published data on some of these parameters, so values should be adopted with care and only for a corresponding area of applicability [Laufs 2000], [Luible 2004], [Belis 2004.1]. Wherever possible, more data should be collected in a systematic way.

### 3) "Buckling strength" analysis

Some glass beams can show considerable lateral displacements without glass fracture when the critical buckling load has been reached. In that case, the glass beam failed due to lateral torsional buckling, but is still able to carry a load which is slightly higher than the critical load: the limited post-critical stability can offer some residual load-bearing capacity.

To indicate this phenomenon using the expression “buckling strength” is rather confusing, since it refers at the same time to an instability phenomenon (buckling) as well as to a material characteristic (strength). The authors would prefer to replace the expression “buckling strength” with “post-buckling strength”.

### 4) Material properties

#### a) Glass

The glass is modelled as a linear elastic material in both the elastic and the non-linear analysis. At normal serviceability conditions regarding to temperature and time, this is in good correspondence with experimental results. The assumed values of 70 GPa for Young’s module and 0.23 for Poisson’s ratio are also generally accepted.

#### b) Interlayer

The material chosen for the adhesive layers in the numerical work is polyvinyl butyral (PVB), because it is probably the most common interlayer material at present. This polymer shows a more complex, visco-elastic behaviour with sensitivity to creep under long-term loadings and temperature-dependent reologic properties. The PVB properties used in the simulations can e.g. be adopted from [Van Duser et al. 1999].

For eigenvalue calculations the time effect plays no role, so “instantaneous values” are used for the PVB material properties. They are displayed in Table 1:
<table><caption>TABLE 1. PROPERTIES OF BUTACITE ACCORDING TO [VAN DUSER ET AL. 1999]</caption><tr><td>G<sub>0</sub></td><td>471 MPa</td></tr><tr><td>E<sub>0</sub></td><td>1310 MPa</td></tr><tr><td>ν<sub>0</sub></td><td>0.391 [-]</td></tr></table>

Hence that these values correspond only to the Butacite type, which can vary from other types of PVB.

For non-linear analyses, a real relaxation curve can be followed [Van Duser et al. 1999], [Luible 2004], [Belis 2004.1].

### c) Boundary conditions

The beam is supported by fork bearings, which exclude any lateral displacement or sideways rotation of the cross-sections above the supports. Loading takes place as a concentrated load at mid span, because this loading case can be simulated as a reference in destructive experiments without too many practical problems. This concentrated load is divided equally amongst the different glass leafs in order to avoid initial eccentricity.

## III. LAMINATED GLASS BEAMS

### A. General

From a safety point of view, the practical use of monolithic glass beams is limited to some cases of glass fins used as stiffeners for glass facades. In general, laminated beams are obligatory for reasons of residual strength and avoidance of falling glass pieces, especially when they are used to support floors or roofs.

The actual structural component is a composite of glass alternated with transparent interlayers, usually PVB or resin. Although the total thickness of the composite can be higher compared to monolithic sections and the slenderness is possibly reduced, a study of the buckling behaviour of laminated glass beams seems very useful here too.

### B. Parameters

The following geometric parameters are taken into account:

#### 1) Span

The span of the beams is varied in a range from 2m to 4m. The authors are aware of the fact that standard production limits allow lengths up to 6m, but taking into account the relatively high cost of such specimens, lengths are limited to more “affordable” beams in order to allow comparative experimental tests in a later phase. The span-parameter is varied with incremental steps of 0.10 m.

#### 2) Height

The height parameter is expressed in relation to the span. The aspect ratio is based on the acceptability of the dimensions and on the perception of built examples. The three aspects ratios that are withheld are 0.10, 0.15 and 0.20 times the span.

#### 3) Glass thickness

The basic assumption is that only glass leafs with a load-bearing function are taken into account, which means that eventual thin glass panes with a solely protective function are abandoned. The following series of standard production thicknesses is implemented in the parametric study: 8, 10, 12, 15 and 19 mm.

In this study, a laminated beam consists always of two identical glass layers interconnected by PVB. In the following, the value of the thickness of a beam consisting of e.g. two panes of 10mm will be referred to with “10mm”, even if the total glass thickness comes to 20mm.

#### 4) Interlayer thickness

The standard production thickness of PVB interlayers is 0.38 mm. The combination of several stacked layers allows multiplication of this value. This is in particular useful for glass panes that have been subjected to thermal treatments, since residual stress-caused shape imperfections need to be intercepted by the thickness of the PVB when laminated beams are composed.

### C. “Ideal” laminated beams

#### 1) Parameter combination

The results of the analyses can be put in a series of three-dimensional graphs, representing the values of the critical load $P_{cr}$ corresponding to different combinations of span and thickness for a given aspect-ratio of the height [Belis 2004.2].
A quick growth of the critical load is noticed when the thickness increases. Moreover, a steady decrease of $P_{cr}$ becomes visible for higher values of the span. A less important influence of the height parameter can be mentioned. More details on the study of “ideal” laminated glass beams can be found in [Belis 2004.2].

## 2) Buckling or fracture?

For practical purposes, it is very useful for designers to know which failure mechanism is more critical: instability or strength. The results for the buckling loads are compared to the loads which cause tensile bending stresses that correspond to the strength of different types of glass. The strength of glass is a complex characteristic, which should be applied with great care and with respect to the type of glass, the way the glass element is used (as a beam or as a plate), the finishing of the edges, etc. A good overview is given in [Haas et al. 2004]. As an example, some design values are given of the own strength (i.e. the failure stress minus the pre-stress) of toughened glass according to [Laufs 2000], [Haas et al. 2004] in Table 2:

<table><caption>TABLE 2.</caption><caption>DESIGN VALUES FOR THE OWN STRENGTH OF GLASS, USED AS DIFFERENT STRUCTURAL COMPONENTS, ACCORDING TO (LAUFS 2000)</caption><thead><tr><th>Glass used as</th><th>Strength [MPa]</th></tr></thead><tbody><tr><td>Plate</td><td>17.1</td></tr><tr><td>Beam</td><td>18.3</td></tr></tbody></table>

In the example of a beam with 2m span and 0.2m height, values of the buckling loads are clearly higher than loads corresponding to fracture due to in-plane bending. For this geometry, it seems that buckling will not even happen for float glass.

However, the authors want to draw the reader's attention to the fact that until now a perfect geometry and instantaneous material properties are assumed. Results will change considerably for beams with realistic geometric imperfections and long-term effects. The picture is different for beams with higher values for span and height.

For thermally treated glass with the given ideal geometry, failure will always be due to buckling. The bending strength is only of importance in case

of float glass.

## 3) Variation of PVB thickness

Curves for different PVB thicknesses showed analogous results. More of interest is the comparison of the effect of the interlayer thickness on the buckling loads. A higher thickness seems to cause a slight augmentation of the critical load. This effect is more accentuated for higher beams. However, again the authors remind the reader of the initial assumptions of perfect geometry and instantaneous material properties made here.

## D. Imperfect laminated beams

Two major imperfections have been taken into account: shape imperfections and temperature/time effects.

### 1) Shape imperfections

Shape imperfections have been implemented in the FE model, based on the first buckling eigenmode. A range of amplitudes has been tested, in order to characterise the influence on the buckling load.

The results are of major importance. For reasonable shape imperfections –i.e. with amplitudes smaller than or equal to 1/400 of the span– the critical load is reduced to 75% of the value corresponding to “perfectly” shaped beams.

### 2) Temperature effects

The visco-elastic behaviour of PVB has been mentioned before. The main consequences in a structural context are creep and temperature dependency.

Practical values of Young's module and Poisson's ratio are derived from a Williams-Landell-Ferry equation. These values allow to estimate the structural behaviour under a load of long duration.

The effect is important: Buckling loads drop to only one quarter of their instantaneous value. It should be clear however, that this may vary when other types of PVB than Buctacite are used.

### 3) Combined effect of shape imperfections and long term loadings

Taking the two previous parameters together results in a quite realistic modelisation of glass beams. On the long term, the critical load drops to only 16% of its initial value. For more details the reader is referred to [Belis 2004.1].
## IV. MONOLITHIC EQUIVALENTS FOR LAMINATED BEAMS?

A finite elements model of a laminated glass beam would initially be three dimensional, as was the case in the previous chapters. In this research in which a considerable amount of calculations is made, parametric studies last for the order of magnitude of one day. For this reason, alternative numerical models have been sought in order to reduce the required computing time.

Two different approaches are mentioned briefly below.

### A. First approach - Virtual stiffness

#### 1) Principle

The laminated beam is considered to be a fictitious beam: an imaginary monolithic section made of a homogeneous and isotrope material. The bending stiffness of this fictitious beam represents the combined effect of the bending stiffness of the glass, the deformation of the PVB, and the duration of loading. It is called the “virtual stiffness” E’I’, in which E’ represents the virtual Young’s modulus and I’ the virtual second moment of area.

The virtual bending stiffness is found from the comparison of the theoretical deflection of the fictitious monolithic material and the deflection of the laminate measured in the experiments

$$E' I' = \frac{11}{768} \frac{FL^3}{u_{\text{midspan, experimental}}} \quad (1)$$

With Equation 1, the virtual bending stiffness can be determined on the basis of a four-point bending test, since both the force F and the deflection can be measured accurately.

By means of bending tests on laminated glass specimens, measured deformations and loadings are used to calculate the overall bending stiffness of the laminated glass beam.

In certain cases, the necessity of experimental tests can be replaced by a theoretical expression for bending of laminated glass sections, as proposed by Hooper [Hooper 1973]. The bending at mid span, valuable for four-point bending, is given by:

$$u_{\text{midspan, theoretical}} = \frac{11}{384} \frac{PL^3 K_3}{E_g I} = \frac{11}{768} \frac{FL^3 K_3}{E_g I} \quad (2)$$

(For the calculation of the factor K3 the reader is referred to [Hooper 1973].)

The good correspondence between theoretically and experimentally obtained Young’s moduli validates this operation.

The virtual bending stiffness can then be calculated as follows:

$$E' I' = \frac{E_g I}{K_3} \quad (3)$$

Since I' follows directly from the geometry of the glass beam, E' can be obtained easily from equation (3). The virtual Young’s modulus found can directly be imported as a material property of a single layer shell finite elements model.

The stiffness of the interlayer and the corresponding structural behaviour of laminated glass beams is influenced by temperature conditions and duration of loading. The tests are performed at a constant room temperature and at different speeds of loading. The loading duration has been taken into account by performing continued bending tests during more than a daytime.

Next, the virtual stiffness is implemented in the finite elements computer model, built with ordinary single-layer Lagrange shell elements.

Due to the simplicity of the elements chosen, exaggerated computing times are avoided, even if very fine meshes are used. Since the basic numerical model properties are obtained experimentally, the accordance between numerical and experimental results for bending out-of-plane are very good.

#### 2) Limitations

The virtual stiffness model works well for out-of-plane bending of the geometries tested. However, when the focus is on buckling of glass beams, in-plane and out-of-plane bending combined with torsion have to be dealt with simultaneously. In most proposed expressions for lateral torsional buckling, torsional stiffness and bending stiffness
are data that need to be inserted separately.

In an analogous way, virtual torsional stiffness
properties can be determined experimentally in the
laboratory.

In-plane bending experiences no significant
influence of the presence of interlayers, since their
bending stiffness is negligible. This means that the
stiffness of the virtual monolith equals the actual
in-plane bending stiffness as discussed above.
Finding a suitable law to determine a correct
balance between the two to unify them into one
value that is needed in buckling expressions, is a
difficult problem that has not been solved at this
moment.

In principle the virtual stiffness model can be
used to study second order effects as well. It does
not allow, however, to include material non-
linearity: residual stresses in a shell elements
model cannot correctly simulate the residual
stresses which are present in the different glass
panels of a laminate. The simulation of visco-
elasticity of the interlayer is limited to a single
elastic value, corresponding to a fixed temperature
or load duration.

B. Second approach - Laminated shell
elements

1) *Principle*

Special shell elements with laminated section
properties can be used in Abaqus [HKS 2002].
Properties and dimensions of the different glass
and interlayers can be inserted in the program and
connected to the laminated shell elements. The
model is built with one set of elements, that is
virtually planar, but which has the internal
mechanical properties of the laminate.

2) Limitations

Like most simplified methods, this one too has
its limitations regarding boundary conditions and
applicability.

A direct consequence of working with laminated
shell elements, is that shear deformations of the
soft interlayers cannot be taken into account. The
model will only give realistic results for loading
cases where straight cross sections of the laminate
will stay straight during and after deformation.

In terms of practical applications, the model

could be useful for elastic buckling simulations of
beams under normal or low temperature
conditions, but above all for short-term loadings.
As confirmed in many codes on laminated glass,
long-term loadings coincide with considerable
shear deformations in the laminated glass as a
whole and in the interlayer in particular.

On the level of complexity of the numerical
simulations, the same remarks are valid as for the
virtual stiffness model (cfr. Supra).

V. MONOLITHIC GLASS BEAMS

A. General

The authors are aware of the fact that for
practical applications, laminated glass beams will
often be required. Since laminates consist of
several individual glass panes held together with
transparent interlayers, it is useful to examine
single panes first. In the following chapter, the
focus will be on the effect and relevance of
common structural devices in the context of
avoidance of lateral torsional buckling. Even if the
obtained buckling loads for single and laminated
beams may differ considerably, the general
principle of buckling prevention stays valid for
laminated glass beams as well.

Glass fins, which are meant to take horizontal
loads acting on a glass façade, are often designed
as a monolithic piece. In relation to wind pressure,
which can be considered perpendicular to the
surface it is acting on, the horizontal or vertical
glass stiffeners will act like beams. For reasons of
completeness, the authors mention that lateral
torsional buckling could also occur to that kind of
structural glass components.

B. Parameters

Basically, the same geometric parameters of span,
height and glass thickness are used in the analysis
as those already mentioned in §III.B (cfr. Supra).
However, the range of these properties was rather
limited, as will become clear below. Especially for
the study of buckling prevention, extra parameters
were introduced, like the type, number and
position of lateral supports (cfr. Infra).
### C. "Ideal" monolithic glass beams: possible actions in order to avoid buckling?

Most of the time a glass roof, ceiling or floor is connected to glass beams by means of point-wise fixations like bolted connections, or continuous elastic joints like sealants. Generally, such joints are applied to avoid a direct glass-on-glass contact and to introduce forces in a smooth way into the beam. However, both point-wise and continuous fixations fulfil a very important secondary structural role, which is often not counted on in the structural designing process. The horizontal support against out-of-plane movements of the beam, as realised at the same time by such joints, is of underestimated structural importance. Without any horizontal support, the compressed rim of the beam can buckle out-of-plane at a lower critical value of the applied load.

Numerical and experimental analyses show clearly that the overall load-bearing capacity of glass beams can be increased significantly when elastic joints are applied, as demonstrated in [Belis 2003].

However, caution is needed since most, but not all combinations of lateral restraints of the compressed rim result in a synergetic improvement of the total load-bearing capacity of the beam. Moreover, the numerical part published there covers only the elastic buckling analysis, which does not take the initial shape imperfections etc. into consideration (Cfr. Supra). In spite of this, a good correspondence was found with the analytical (elastic) expressions. A reason could be a very limited amplitude of the initial shape imperfections ("global bow") of the tested glass specimens.

In the numerical simulations, small models (span $x$ height $x$ thickness = 1000 mm $x$ 100 mm $x$ 50 mm) are used as well as larger models (2100 mm $x$ 400 mm $x$ 10 mm). Loads are applied point-wise at mid span as well as uniformly distributed along the upper rim.

In order to simulate the effect of a continuous elastic joint with a varying stiffness, a continuous spring support along the upper rim is modelled. The spring stiffness $k$ could be understood as the ability of the elastic joint to resist against lateral

displacements of the glass beam.

A basic structural sealant is used in the experiments to realise the elastic joint. Different stiffnesses of the joint are simulated by testing connections at different stages of the curing process. A relationship is found between spring stiffness and curing time, so experimental results could be linked to numerical values.

Experimental results show that even a weak continuous connection results in a remarkable gain of buckling load, which is in complete agreement with numerical results.

Elastic joints, which connect glass beams with its superstructure, have an important positive effect on the buckling load. Especially for beams with a slender cross-section, the overall load-bearing capacity can be improved considerably.

Even joints with a relatively low stiffness, like e.g. sealants at a rather early stage in the curing process, can influence the results in a positive way. More details are given in [Belis 2003].

Results for "imperfect" glass beams covering second-order effects are not available yet, but they will be published by the authors soon.

## VI. CONCLUSIONS

Geometric parameter analyses have been performed on "ideal" laminated beams, consisting of two glass layers and one PVB interlayer. The influence of the basic dimensions of the whole composite beam have been examined in a range that is relevant to practice.

The parameter that influences the buckling load most is the glass thickness. To a lesser extent, the height of the beam plays a similar role. Longer spans increase the buckling risk.

Elastic buckling of ideal glass beams seems to be the failure mechanism for beams with a slender cross-sections and a long span in case they are composed of thermally treated glass (heat-strengthened or fully tempered).

The buckling risk is considerably higher when imperfections are taken into account. Especially the "weakening effect" of PVB under long term loadings or under higher temperatures are important. Moreover, the influence of initial shape imperfections (which are present for fully
tempered glass in particular) is not to be neglected. An important reduction factor has to be taken into account on computing the lateral torsional buckling load of realistic laminated glass beams.

Elastic joints, which connect glass beams with its superstructure, have an important positive effect on the buckling load. Especially for beams with a slender cross-section, the overall load-bearing capacity can be improved considerably.

## VII. ACKNOWLEDGEMENT

The authors wish to acknowledge their colleagues at the Laboratory for Research on Structural Models, as well as Solutia and Glaverbel for their interest and support. In addition, Dr. Andreas Luible is acknowledged for his critical review and valuable comments.

## REFERENCES

**[Belis et al. 2003]**

BELIS, J., VAN IMPE, R., LAGAE, G. AND VANLAERE, W., *Enhancement of buckling-load bearing capacity of glass beams by means of lateral restraints*, Struct. Eng. Mech., Vol. 15(5), pp. 495-511, 2003.

**[Belis et al. 2004.1]**

BELIS, J., VAN IMPE, R., DE MEESTER, B., LAGAE, G., KATNAM, K., “Stability Approach of the Dimensioning of Glass Beams”, ISAAG International Symposium on the Application of Architectural Glass, Munich, 15-16 november 2004.

**[Belis et al. 2004.2]**

BELIS, J., VAN IMPE, R., LAGAE, G., “Parametric approach of buckling of load-bearing glass beams”, Proceedings of ASEM’04 Advances in Structural Engineering and Mechanics, paper published on CD-ROM, pp. 1599-1608, Seoul, 2004.

**[Haas et al. 2004]**

HAAS, C., HALDIMANN M., *Entwurf und Bemessung von Tragelementen aus Glas – Wissensstandsbericht*, ICOM rapport 493, Lausanne, 2004.

**[Hess 2000]**

HESS, R., *Glastraeger*, vdf Hochschulverlag AG an der ETH Zürich, Zürich, 2000.

**[HKS 2002]**

HKS, *Abaqus User's Manuals version 6.3*, Hibbitt, Karlsson & Sorensen, USA, 2002.

**[Hooper 1973]**

HOOPER, *On the bending of architectural laminated glass*, International Journal of Mechanical Science, Vol. 15, pp. 309-329, 1973.

**[Kasper et al. 2003]**

KASPER, R., SEDLACEK, G., Structural use of glass

beams, Glass Processing Days, pp. 312-315, Tampere, 2003.

**[Laufs 2000]**

LAUFS, W., *Ein Bemessungskonzept zur Festigkeit thermisch vorgespannter Gläser*, Schriftenreihe Stahlbau – RWTH Aachen Heft 45, Shaker Verlag, Aachen, 2000

**[Luible 2004]**

LUIBLE, A., *Stabilität von Tragelementen aus Glas*, dissertation, Thèse N° 3014(2004), EPFL, Lausanne, 2004

**[Timoshenko et al. 1961]**

TIMOSHENKO, S.P., GERE, J.M., *Theory of elastic stability*, McGraw-Hill Book Company, Inc., New York, Toronto, London, 1961.

**[Van Duser et al. 1999]**

VAN DUSER, A., JAGOTA, A., BENNISON, S.J., “Analysis of Glass Polyvinyl Butyral Laminates Subjected to Uniform Pressure”, *J. Eng. Mech.*, april issue, pp. 435-442, 1999.

**[Veer et al. 2001]**

VEER, F., RIEMSLAG, TING, “Structurally efficient glass laminated composite beam”, Glass Processing Days, pp. 363-367, Tampere, 2001.
# Stability of laminated glass beams

Ruth Kasper and Gerhard Sedlacek

RWTH Aachen – Institute of Steel Construction, Germany

The following paper summarizes the actions and the results of a research project concerning the behaviour of laminated glass beams.

**Keywords:** glass, laminated glass, glass beams, material data of PVB-foil, analytic calculation method

## I. INTRODUCTION

Beams consisting of single glass panels either laminated glass panels are popular elements in modern architecture (figure 1). The mechanical viewpoint is focused on the stability of laminated glass beams that is influenced by the behaviour of the interlayer. The interlayer is mostly consisting of PVB-foils that have a time and temperature dependent material characteristic.

<figure><img src="image_103.png" /><figcaption>Fig 1: Example</figcaption></figure>

The stability behaviour of steel beams can be described by buckling curves. Buckling curves include geometrical and material imperfection and are based on tests or calculations. Buckling curves are evaluated by using the ultimate load carrying

capacity.

The use of buckling curves is not appropriate for glass beams, in particular for beams consisting of laminated glass, as the ultimate load carrying capacity of a laminated glass beam depends on the loading rate and loading duration caused by the visco-elastic material properties of the PVB-foils. Additionally the influence of loading with different loading durations (e.g. self weight and wind load) can not be taken into consideration.

Caused by these reasons a design concept and a calculation method based on the theory second order have been developed to take the mentioned aspects into account [Kasper 2005].

## II. PROBLEM AND PROCEEDING OF THE RESEARCH PROJECT

The idea of the calculation model was to describe the sandwich with equivalent stiffness $I_z^{eq}$ (second moment of area) and $I_T^{eq}$ (torsional stiffness). $I_z^{eq}$ and $I_T^{eq}$ depend on the actual PVB-foil-stiffness $G_F(t,T)$ that is influenced by the temperature and loading duration. The global behaviour of the beam can be described with the elastic bending and torsional theory. $I_z$ and $I_T$ of a beam consisting of a single glass panel can be replaced by the equivalent values $I_z^{eq}$ and $I_T^{eq}$.

The proceeding of the research project was divided into the following steps:

1. Small scale tests to determine the time and temperature depending behaviour of the PVB-foil.

2. Original size lateral torsional buckling tests with beams (single glass panels and laminated glass panels).

3. Solving the equations of the elastic bending and torsional problem to describe the
buckling behaviour, the deformation of the
beam in the space and to determine the
additional stress determinant M<sub>z</sub><sup>II</sup> of second
order theory.

4. Application of the extended bending and torsional theory to determine the equivalent stiffness I<sub>z</sub><sup>eq</sup> and I<sub>T</sub><sup>eq</sup> of the glass sandwich and to calculate the stress distribution over the sections caused by M<sub>z</sub><sup>II</sup>.

5. Numerical analysis to verify the theoretical model.

6. Development of a design concept which take into account different loading durations.

The investigations were focused on the load
carrying behaviour of laminated glass beams with
a visco-elastic interlayer. That is why no
evaluation concerning the resistance of the glass
has been carried out.

III. PROPERTIES OF THE PVB-FOIL

A. *Introduction*

The interlayer of laminated glass beams is mostly consisting of PVB-foils. PVB-foils have a significant visco-elastic material behaviour that is characterised by a time and temperature dependent material stiffness. Because of too little test results [Sobek 1998, Duser 1999, Schuler 2003] further tests were necessary to receive available data for the evaluation of the lateral torsional buckling tests with laminated glass beams.

B. *Testing and results*

A new test method has been used for the
determination of material data. The test specimens
were laminated glass panels (1100 mm x 360 mm).
Comparable specimens are normally used for four-
point-bending tests to determine the bending
strength of glass [DIN EN 1288-3]. Compared with
other test methods the advantage of the used test
method is that the evaluated material data are
based on a larger surface.

The test specimens were distorted to a constant
angle θ. The torsional moment M<sub>T</sub> was measured
over the time. By using the equations derived by

the extended bending and torsional theory the
effective PVB-foil shear stiffness G<sub>F</sub>(t,T) could be
determined depending on the angle θ and the
torsional moment M<sub>T</sub>. The tests have been carried
out for different loading durations (up to 60 h) and
different temperatures (-10°C up to 40°C) (figure
2).

<figure><img src="image_104.png" /><figcaption>Fig 2: Test set-up for torsional tests</figcaption></figure>

The evaluation of the test results shows the
following:

1. The shear stiffness G<sub>F</sub>(t,T) depends highly on the temperature during the first hour of loading (figure 3 and 4). After a loading duration longer than e.g. 20 hours the influence of temperature is anymore existing (figure 5).

<figure><img src="image_105.png" /><figcaption>Fig 3: G<sub>F</sub>(t,T) (Variation of the angle and the temperature) – loading duration 1 h (T = -10°C–15°C)</figcaption></figure>
<figure><img src="image_106.png" /><figcaption>Fig 4: G<sub>F</sub>(t,T) (Variation of the angle and the temperature) – loading duration 1 h (T = 15°C–40°C)</figcaption></figure>

<figure><img src="image_107.png" /><figcaption>Fig 5: G<sub>F</sub>(t,T) (Variation of the temperature) – loading duration 60 h</figcaption></figure>

2. The scatter of the PVB-foil shear stiffness G<sub>F</sub>(t,T) for a constant temperature was significant (figure 6). The test results of repeated tests with one test specimen (equal test conditions) showed also a scatter (figure 7). The shear stiffness G<sub>F</sub>(t,T) was not influenced by the order of the tests.

<figure><img src="image_108.png" /><figcaption>Fig 6: G<sub>F</sub>(t,T) – Statistical evaluation of all test results (T = 23°C)</figcaption></figure>

<figure><img src="image_109.png" /><figcaption>Fig 7: G<sub>F</sub>(t,T) – Repeated loading of one test specimen (T = 35°C)</figcaption></figure>

3. The effective shear stiffness G<sub>F</sub>(t,T) was influenced by the temperature at the beginning of the tests. A following change of the temperature had only an influence on the relaxation time and not on the effective value of G<sub>F</sub>(t,T) (figure 8)

<figure><img src="image_110.png" /><figcaption>Fig 8: G<sub>F</sub>(t,T) – loading duration 9 h</figcaption></figure>

### C. Numerical description of the visco-elastic behaviour

For numerical simulations the visco-elastic behaviour can be described by using a prony series consisting of exponential elements. Additionally the property of the thermorheological simplicity can be used. Thermorheological simplicity means that the time-dependent material properties must be determined for one reference temperature T<sub>Ref</sub> (“master curve”), e.g. a prony series, and can be converted to all further temperatures by using a shift function (e.g. William-Landels-Ferry) (figure 9).

$$ \text{Shift function} \qquad a_s = \frac{\tau(T)}{\tau_{\text{Ref}}(T_{\text{Ref}})} $$

$$ \text{WLF-equation} \qquad \log a_s(T) = - \frac{C_1(T - T_{\text{Ref}})}{C_2 + (T - T_{\text{Ref}})} $$
with

$C_1, C_2$: calibration constants

$\tau$: relaxation time at temperature T

$\tau_{ref}$: relaxation time at the reference temperature $T_{Ref}$

[Duser 1999] gives the elements of a prony series for a reference temperature $T_{Ref}$ evaluated by small scale tests and the coefficients for the shift function of William-Landels-Ferry for PVB-foils.

<figure><img src="image_111.png" /><figcaption>Fig 9: Principle of the time shift</figcaption></figure>

For one specimen tested between -10°C and 40°C the theoretical approach has been applied. Figure 10 shows the test results for all measured temperatures represented in a logarithmic scaling.

<figure><img src="image_112.png" /><figcaption>Fig 10: $G_F(t,T)$ (Variation of the angle and the temperature) – loading duration 1 h – logarithmic scaling</figcaption></figure>

By using the shift function of Williams-Landel-Ferry (WLF) calibration constants have determined for the temperatures between 10°C and 40°C. Figure 11 shows the results of the calibration. The reference temperature was chosen to $T_{Ref} = 22°C$.

In the temperature field near to the glass temperature ($T_g = 10°C-15°C$) the calibration constants had to be modified and the thermorheological simplicity was not valid.

<figure><img src="image_113.png" /><figcaption>Fig 11: Application of the thermorheological simplicity on the test results</figcaption></figure>

## D. Conclusions

The test results can be used for further calculation of laminated glass panels with the interlayer PVB. The tests showed that the material properties of PVB scatter in a large range and that a temperature dependency after a loading duration longer than 10 h is not useful.

The tests and the evaluation of the results showed that numerical material descriptions can be used only with defined restrictions (e.g. temperature field and loading duration).

A comparison with existing test results showed that the formula of [Sobek 1998] give a good approach to describe the time and temperature behaviour for $T > T_g$. The formula is written as:

$$G_F(t,T) = 0,008 \cdot (100 - T) - 0,0011 \cdot (50 + T) \cdot \log(t)$$

with

$T$: temperature [°C]

$t$: time [sec]
# IV. LATERAL TORSIONAL BUCKLING
BEHAVIOUR OF GLASS BEAMS

## A. Introduction

For the investigation of the lateral torsional buckling behaviour of laminated glass beams tests with single glass panels as well as laminated glass panels have been carried out.

The tests with single glass panels were useful to assess the functionality of the test set-up and to verify the used finite element model. The advantage of this approach was that the test results could be evaluated without the influence of the PVB-foil. The results of this investigations are not demonstrated in this paper.

In the second step lateral torsional buckling tests with laminated glass have been carried out. The loading rate $v_B$ [mm/sec] and the loading duration have been varied.

## B. Testing and evaluation

For lateral torsional buckling tests the applied load F has to follow vertically and horizontally the load application point during the test. Caused by the restraints of the testing machine in Aachen the problem was solved by a test set-up with supports which can move horizontally whereas the load application point was horizontally fixed. At both ends of the beams the rotation about the vertical axis (z-axis), the rotation about the beam-axis (x-axis) and the horizontal displacement were free (figure 12). During the tests the vertical deformation of the load application point, the horizontal deformations of the supports, the inclination at midspan and the strains at midspan were measured (figure 13).

A disadvantage of the test set-up was that the results could not directly compared with the analytic solution based on second order theory caused by the changed boundary conditions.

<figure><img src="image_114.png" /><figcaption>Fig 12: Boundary conditions of the test set-up</figcaption></figure>

<figure><img src="image_115.png" /><figcaption>Fig 13: Test set-up of the lateral torsional buckling tests</figcaption></figure>

The beam was loaded by controlling the displacement of the force application point. The lowest possible loading rate was $v_B$ = 0,04 mm/sec. The fastest loading rate was chosen to $v_B$ = 1,0 mm/sec. By reaching the highest load level, perceptible from the moment when the load augmentation was stopped, the displacement of the force application point has been stopped and the system was de-loaded with the same loading rate. Caused by the high slenderness of the beams the load capacities were mostly determined by the deformation of the system and not limited by the resistance of the material. This course of action was called “short time loading”. For the “long time loading” the augmentation of the displacement at midspan was stopped before the highest load level was reached. The displacement was fixed and the augmentation of the inclination, the horizontal deformations of the support and the strains at midspan were measured. The measured temperature during the tests was 23°C.

The evaluation has been made in two different ways (figure 14). First, the tests were analyzed with a finite element simulation with an estimated elastic shear modulus $G_F(t,T)$=cte. This has been done for two different times: beginning and any time during the test. The load-deformation-curves and the load-stress-curves of the tests and the simulation have been compared (figure 16) and the estimated shear modulus $G_F(t,T)$ have been
compared with the results of the small scale tests. Second, the evaluation of the tests has been made by using the results of the small scale tests. The G<sub>F</sub>(t,T)-formula of Sobek was modified in the way that a formula for G<sub>F</sub>(t,23°C)<sub>upper</sub> and G<sub>F</sub>(t,23°C)<sub>lower</sub> was reached (figure 15). With these formulas a M<sub>ki</sub>(t,23°C)-curve or rather F<sub>ki</sub>(t,23°C)-curve could be calculated numerically and compared with the highest load levels and the load levels depending on the time.

<figure><img src="image_116.png" /><figcaption>Fig 14: Evaluation of the test results</figcaption></figure>

<figure><img src="image_117.png" /><figcaption>Fig 15: Modification of the G<sub>F</sub>(t,T)-formula of [Sobek 1998]</figcaption></figure>

Figure 16 shows an example for the evaluation method with the estimated shear modulus G<sub>F</sub>(t,T). Figures 17 and 18 show examples for the evaluation method with the M<sub>ki</sub>(t,23°C)-curves based on G<sub>F</sub>(t,T)<sub>upper</sub> and G<sub>F</sub>(t,T)<sub>lower</sub> for short and long time loading.

<figure><img src="image_118.png" /><figcaption>Fig 16: Test results (short time loading) – evaluation with an estimated G<sub>F</sub>(t,T)-value</figcaption></figure>

<figure><img src="image_119.png" /><figcaption>Fig 17: Test results (short time loading)</figcaption></figure>
<figure><img src="image_120.png" /><figcaption>Fig 18: Test results (long time loading)</figcaption></figure>

### C. Conclusions

The investigations showed that the shear modulus $G_F(t,T)$ evaluated on the basis of the small scale tests was related to the results of the lateral torsional buckling tests. The loading capacities of the laminated glass beams were influenced by the loading rate: the higher the loading rate the higher was the loading capacity. This phenomenon can be explained by the relaxation time of the PVB-foil that is lower than the loading rate.

The test specimens have been used for several tests. By measuring the imperfections of the beams before testing it could be identified that the imperfection of the beam became larger after repeated testing.

The evaluation with the finite element simulation showed that the buckling behaviour could be described by using an elastic shear modulus $G_F(t,T)$.

## V. ELASTIC BENDING AND TORSIONAL PROBLEM OF A BEAM SUBJECTED TO BENDING

### A. Introduction

The problems of stability and second order theory can be solved on the basis of the potential [Roik 1972]. The stability solution gives a value for the elastic critical moment $M_{ki}$ of a perfect beam subjected to bending without any imperfections (figure 19). In reality this value can not be reached caused by geometric imperfections of the beam. But $M_{ki}$ must be known to limit the loading. To determine the realistic deformations and the stresses in the beam additionally a solution for the deformations must be evaluated. This can be done on the basis of second order theory applied on an imperfect beam with the initial imperfections $v_0$ and $\vartheta_0$ (figure 19).

For the calculation of the stresses in the section based on second order theory the stress resultants $M_y^{\text{Th.I.}} = M_y^{\text{Th.II.}}$ and $M_z^{\text{Th.II.}} = M_z^{\text{Th.I.}} \cdot \vartheta$ must be taken into account.

<figure><img src="image_121.png" /><figcaption>Fig 19: Stability problem and stress problem</figcaption></figure>

Exemplary the solutions have been found for the load cases shown in figure 20.

<figure><img src="image_122.png" /><figcaption>Fig 20: Load cases</figcaption></figure>

The potential can be written as:
$$
\begin{align*}
\Pi &= \frac{1}{2} \int_{0}^{\ell} \left\{ E \cdot I_z \cdot v_{,xx}^2 + E \cdot I_y \cdot w_{,xx}^2 + G \cdot I_t \cdot g_{,x}^2 \right\} dx \\
&\quad - \frac{1}{2} \int_{0}^{\ell} \left\{ -2 \cdot M_y(x) \cdot v_{,xx} \cdot g - q_z(x) \cdot z_p \cdot g^2 + 2 \cdot q_z(x) \cdot w \right\} dx \\
&\quad - \frac{1}{2} \sum_j \left\{ P_{z_j} \cdot \left( 2 \cdot w - z_{p_j} \cdot g(x_j)^2 \right) \right\}
\end{align*}
$$

For the solution the deformations have been
attempt sinusoidal.

$$
v(x)^{\text{Th.II}} = a_1 \cdot \sin\left(\frac{\pi}{\ell} x\right)
$$

$$
w(x)^{\text{TH.II}} = a_2 \cdot \sin\left(\frac{\pi}{\ell} x\right)
$$

$$
\mathcal{G}(x)^{\text{Th.II}} = a_3 \cdot \sin\left(\frac{\pi}{\ell} x\right)
$$

B. *Elastic critical moment*

The following formulas give the solution for the
elastic critical moments for the considered load
cases.

$M_y = \text{cte}:$

$$
M_{ki} = \frac{\pi}{\ell} \sqrt{E \cdot I_z \cdot G \cdot I_t}
$$

$$
q_z = \text{cte and "P}_z \text{ at midspan":}
$$

$$
M_{ki} = M_y = \frac{1}{c_1} \cdot N_{ki} \left( \frac{1}{2} \cdot \frac{c_2}{c_1} \cdot z_p + \sqrt{\left( \frac{1}{2} \cdot \frac{c_2}{c_1} \cdot z_p \right)^2 + \frac{G \cdot I_t}{E \cdot I_z} \cdot \left( \frac{\ell}{\pi} \right)^2} \right)
$$

Table 1 gives the accompanying coefficients.

<table><caption>Tab 1: Coefficients $c_1$ and $c_2$</caption>
  <tr>
    <td>Load case</td>
    <td>c<sub>1</sub></td>
    <td>c<sub>2</sub></td>
  </tr>
  <tr>
    <td>q<sub>z</sub></td>
    <td>(2/3) + (2/π<sup>2</sup>)</td>
    <td>(8/π<sup>2</sup>)</td>
  </tr>
  <tr>
    <td>P<sub>z</sub></td>
    <td>(2/π<sup>2</sup>) + (1/2)</td>
    <td>(8/π<sup>2</sup>)</td>
  </tr>
</table>

C. Deformations based on second order
theory

The deformations based on second order theory

were evaluated by assuming sinusoidal initial
imperfections $v_0$ and $g_0$:

$$
v_0(x) = v_0 \cdot \sin\left(\frac{\pi}{\ell} x\right)
$$

$$
g_0(x) = g_0 \cdot \sin\left(\frac{\pi}{\ell} x\right)
$$

The solutions for the deformations v and g of
second order theory are:

$M_y = \text{cte}:$

$$
\mathcal{G}(x)^{\text{Th.II}} = \frac{\frac{M_y^2}{E \cdot I_z} \cdot g_0 + \left(\frac{\pi}{\ell}\right)^2 \cdot M_y \cdot v_0}{G \cdot I_t \cdot \left(\frac{\pi}{\ell}\right)^2 - \frac{M_y^2}{E \cdot I_z}} \quad (6.2)
$$

$$
v(x)^{\text{Th.II}} = \frac{\frac{G \cdot I_t}{E \cdot I_z} \cdot M_y \cdot g_0 + \frac{M_y^2}{E \cdot I_z} \cdot v_0}{G \cdot I_t \cdot \left(\frac{\pi}{\ell}\right)^2 - \frac{M_y^2}{E \cdot I_z}} \cdot \sin\left(\frac{\pi}{\ell} x\right)
$$

$$
q_z = \text{cte and "P}_z \text{ at midspan":}
$$

$$
\mathcal{G}(x)^{\text{Th.II}} = \frac{g_0 \left( \frac{c_1^2 \cdot M_y^2}{E \cdot I_z} - c_2 \cdot \left(\frac{\pi}{\ell}\right)^2 \cdot M_y \cdot z_p \right) + c_1 \cdot \left(\frac{\pi}{\ell}\right)^2 \cdot M_y \cdot v_0}{G \cdot I_t \cdot \left(\frac{\pi}{\ell}\right)^2 - \frac{c_1^2 \cdot M_y^2}{E \cdot I_z} + c_2 \cdot M_y \cdot z_p \cdot \left(\frac{\pi}{\ell}\right)^2} \cdot \sin\left(\frac{\pi}{\ell} x\right)
$$

$$
v(x)^{\text{Th.II}} = \frac{c_1 \cdot \frac{G \cdot I_t}{E \cdot I_z} \cdot M_y \cdot g_0 + c_1^2 \frac{M_y^2}{E \cdot I_z} \cdot v_0}{G \cdot I_t \cdot \left(\frac{\pi}{\ell}\right)^2 - \frac{c_1^2 \cdot M_y^2}{E \cdot I_z} + c_2 \cdot M_y \cdot z_p \cdot \left(\frac{\pi}{\ell}\right)^2} \cdot \sin\left(\frac{\pi}{\ell} x\right)
$$

The coefficients $c_1$ and $c_2$ are given in table 1.

VI. SANDWICH SOLUTION BASED ON THE
EXTENDED BENDING AND TORSIONAL
THEORY

For the use of the equations derived by elastic
bending and torsional theory for laminated glass
beams, formulas for the second moment of area $I_z^{eq}$
and the torsional stiffness $I_T^{eq}$ are needed. For a
monolithic section the formulas are known:

$$
I_z = \frac{b \cdot t^3}{12}
$$
$$I_t = \frac{b \cdot t^3}{3}$$

By using the “Extended bending and torsion theory” [Roik 1970, Scarpino 2002, Völling 2000] an equivalent stiffness $I_z^{eq}$ and $I_T^{eq}$ of the sandwich can be determined depending on the shear modulus $G_f(t,T)$ of the interlayer. To this end the basic degree of freedoms for monolithic sections were augmented by additional degrees of freedom due to the sandwich characteristics. Figure 21 shows the degree of freedoms of a monolithic section and the additional degree of freedoms of the sandwich for bending and torsion which have been used to solve the equations. The equations have been solved separately for bending and torsion for a laminated glass with two and three glass panels [Kasper 2004]. An extension of the solution is also possible for a laminated glass with more than three glass panels.

<figure><img src="image_123.png" /><figcaption>Fig 21: Basic and additional degree of freedom for bending and torsion</figcaption></figure>

The equivalent torsional stiffness $I_t^{eq}$ depends only on the geometry and the material characteristics of the sandwich. The second moment of area $I_z^{eq}$ depends additionally on the type of loading and the length of the beam.

Beneath the equivalent stiffness $I_z^{eq}$ and $I_T^{eq}$ the solution gives also the needed equations for the evaluation of the torsional tests and calculation of the stress distribution caused by $M_z^{Th.II}$.

The proceeding and the solutions are explained in detail in [Kasper 2005].

## VII. NUMERICAL INVESTIGATION

The numerical simulation for proving the theoretical model has been done with the FEM-program ABQAQUS. The glass and the PVB-foil have been modelled with 8-node-volume-elements. The material properties are for both materials (glass and PVB) elastic. The model was apart from the boundary conditions identically with the model used for the evaluation of the tests. Here, the force application point could move horizontally and the supports were fixed horizontally. The simulation has been carried out in two steps:

Step 1: Buckling of the system to determine the lowest eigenvalue and the associated eigenform.

Step 2: The displacements of the determined eigenform were scaled (e.g. L/1000) to the perfect beam. A geometric non-linear calculation has been carried out to determine the load-deformation-curves.

The theoretical model has been checked for single and laminated glass panels. The geometry of the beam (section values and length) as well as the stiffness of the interlayer have been varied. Figure 22 and 23 show exemplary a comparison between the finite element simulation and the theoretical model for single and laminated glass panels.

The theoretical model reach a good agreement on the numerical results for loading smaller than 80% of $M_{ki}$.

<figure><img src="image_124.png" /><figcaption>Fig 22: Comparison of the numerical results with the theoretical results (single glass panel, t = 10 mm, b = 500 mm)</figcaption></figure>
<figure><img src="image_125.png" /><figcaption>Fig 23: Comparison of the numerical results with the theoretical results (laminated glass panel, t = 2 x 8 mm, h = 1,52 mm, b = 300 mm, l = 3000 mm)</figcaption></figure>

VIII. DESIGN CONCEPT

The application of glass beams gives the problem that the loads are not active for the same loading durations. E.g. laminated glass beams in roof structures are loaded by the self loading of the glass construction and additionally by wind loading. For self loading the PVB-foil shear stiffness G<sub>F</sub>(t,T) is at the time t = ∞ equal to zero independent of the actual temperature. For wind loads with short loading duration the PVB-foil shear stiffness G<sub>F</sub>(t,T) can be chosen to e.g. 0,4 N/mm². Figure 24 shows the design model to take into account different loading duration with different effective shear modulus G<sub>F</sub>(t,T).

1a: The beam is loaded with the self weight of the structure M<sub>1</sub>. At the beginning the effective shear modulus is G<sub>F</sub> > 0.

1b: The deformations and stresses caused by self loading can be determined directly for the time t = ∞ with G<sub>F</sub> = 0.

2: Before loading the system with e.g. wind loading, the system is de-loaded. For the de-loading curve the effective shear modulus is G<sub>F</sub> > 0.

3: For the common effect of self loading + wind loading (M<sub>2</sub>) the system with the new initial imperfections v<sub>1</sub> and θ<sub>1</sub> must be taken into account.

The imperfections v<sub>1</sub> and θ<sub>1</sub> can be determined with the following conditions:

$$v(M_1, v_0, \theta_0, G_F = 0) + v_0 = v(M_1, v_1, \theta_1, G_F > 0) + v_1$$

$$\vartheta(M_1, v_0, \theta_0, G_F = 0) + \vartheta_0 = \vartheta(M_1, v_1, \theta_1, G_F > 0) + \vartheta_1$$

<figure><img src="image_126.png" /><figcaption>Fig 24: Consideration of loads with different loading duration</figcaption></figure>

The theoretical analyse of different load cases (different moment distribution over the beam) can also be done by using the theoretical formulations. The critical moment must be limited to:

$$\frac{M_1}{M_{ki,1}} + \frac{M_2}{M_{ki,2}} < 0,8$$

The calculation procedure is shown in figure 25.

IX. SUMMARY AND PROSPECTS

The paper summarizes the results of the DFG-project "Laminated glass beams" and the thesis "Tragverhalten von Verbundglasträgern". The work shows possibilities for the design of laminated glass beams which can be applied also on columns or shear fields consisting of laminated glass.

The theoretical model can be extended for further load cases, e.g. restraint beams.

The advantage of the theoretical model is that an application for several interlayer is possible and an optimizing parameter studies can easily done. In opposite to buckling curves the serviceability of the structure can be analyzed directly. This is useful caused by the high slenderness of glass beams where the application is (e.g.) limited by large deformations.
The used testing method (torsional tests) is a
very simple possibility to determine the material
data of different interlayer by using original size
test specimens. The testing data can also be used
for the design of laminated glass plates.

<figure><img src="image_127.png" /></figure>

<figure><img src="image_128.png" /></figure>

<figure><img src="image_129.png" /><figcaption>Fig 25: Consideration of loads effecting different stress
resultants along the x-axis of the beam</figcaption></figure>

X. ACKNOWLEDGMENT

The project has been carried out by the financial
support of the “Deutsche Forschungsgesellschaft”
(DFG). Saint Gobain Deutschland GmbH gave the
test specimens.

REFERENCES

[DIN EN 1288-3]
Glas im Bauwesen: Bestimmung der Biegefestigkeit von Glas. Teil 3 Prüfung von Proben bei zweiseitiger Auflagerung (Vierschneiden-Verfahren). September 2000

[Duser 1999]
Duser Van Alex, Jagota Anand, Bennison Stephen J.: Analysis of Glas/Polyvinyl Butyral Laminates subjected to uniform pressure, Journal of Engineering Mechanics, April 1999

[Kasper 2005]
Kasper, Ruth: Tragverhalten von Glasträgern. Dissertation 2005

[Roik 1970]
Roik, K.; Sedlacek, G.: Erweiterung der technischen Biege- und Verdrehtheorie unter Berücksichtigung von Schubverformungen, 1970, Die Bautechnik 47 Heft 1, S, 20-32.

[Roik 1972]
Roik K, Carl J, Lindner J: Biegtorsionsprobleme dünnwandiger Stäbe. Ernst & Sohn 1972

[Scarpino 2002]
Scarpino, P.: Berechnungsverfahren zur Bestimmung einer äquivalenten Torsionssteifigkeit von Trägern in Sandwichbauweise. Diplomarbeit RWTH Aachen, 2002, unveröffentlicht.

[Schuler 2003]
Schuler C: Einfluss des Materialverhaltens von Polyvinylbutyral auf das Tragverhalten von Verbundsicherheitsglas in Abhängigkeit von Temperatur und Belastung. Dissertation 2003 TU München Lehrstuhl für Stahlbau

[Sobek 1998]
Sobek W, Kutterer M, Messmer R: Bauen mit Glas - Rheologisches Verhalten von PVB im Schubverbund, Forschungsbericht 4/98, Universität Stuttgart, Institut für leichte Flächenträgwerke, April 1998

[Völling 2000]
Völling, B.: Berechnungsverfahren für Balken und Platten in Sandwichbauweise nach der erweiterten Biegetheorie. Diplomarbeit RWTH Aachen, 2000, unveröffentlicht.
<figure><img src="image_130.png" /></figure>
# Contribution to the use of reinforced glass loaded on its strong axis

Klaus Kreher, EPFL – IBOIS, Lausanne, Switzerland

Due to loading on the strong axis of glass panes, different fracture mechanical behaviour has to be admitted than due to the loading on the weak neutral axis. Existing models to describe the structural safety of glass can not easily be adopted.

The load-bearing of reinforced glass panes varies with the degree of prestressing due to increasing residual stresses of heat strengthened glass. Ductile load carrying-behaviour is achieved by using reinforced heat strengthened glass.

This article offers a model to calculate the stresses of I-Beams with and without the appearance of cracks (Mode I and II) and the load-bearing behaviour of the adhesive joint of reinforced glass, such as distances between cracks, number of cracks and load introducing length.

Safety-considerations taking into account the load carrying-behaviour of reinforced glasses with different residual stresses due to heat strengthening are shown.

**Keywords:** reinforced glass, strong neutral axis

## I. INTRODUCTION

Glass loaded on its strong axis offers excellent stiffness and load-bearing.

In reinforcing the stressed edge, the load-bearing varies to ordinary glass-panes. The panes do not collapse in a fragile way anymore; fractures appearing during loading preview the collapsing of a pane and an important post cracked load capacity can be achieved.

This offers the opportunity to modify the safety- considerations used for glass design and to increase the allowed stressing of glass.

The post-cracked load capacity is largely dependent on the quality of the used glass. Depending on the quality of the glass (residual stresses due to heat-strengthening, e.g. annealed glass, heat-strengthened glass, fully tempered glass), the ductility and the mode of failure of the panes change. As a result of its failure mode, annealed glass without internal prestressing offers the highest remaining load-carrying potential after the first crack appeared. This ductility, and thus the structural safety, diminishes with an increasing degree of internal prestressing due to thermal treatment. Heat-strengthened glass (with various degrees of prestressing, various residual stresses) shows a decrease in remaining load-carrying capacity with an increasing degree of prestressing until it fails in a brittle mode as fully toughened glass does.

In order to define the stressing of the glass, a calculation-method had to be developed taking into account the appearance of cracks and the deviation of strains and stresses.

This paper shows equations by solving the differential equations of a pulled slab to calculate the stressing of the glass, the adhesive and the reinforcing-material, which is suggested to be glued on the glass. The material stressing due to bending is transformed into a pulled slab.

Based on a slab model, the forces applied on the structure can be calculated as it is shown in [1] taking into account the non-cracked and the cracked structure.

## II. DEFINITION OF THE MATERIAL STRESSING

To obtain the structural safety the most stressed parts of the structure have to be focused on. On both sides of a crack the adhesive and the reinforcing-material show a maximum stressing. A model based on the differential equations of the non-rigid bond is defined in order to calculate the
loading-peaks of the adhesive joint and the material loading itself. The characteristic values of the non-rigid bond, such as the number and distances between cracks and the load introducing length, can be defined.

The slab-model is based on the following infinitesimal part with the shown forces:

<figure><img src="image_131.png" /><figcaption>Fig 1: Infinitesimal part of the slab model with its forces</figcaption></figure>

The indices g is for material glass, h for the reinforcing material and kl for the adhesive (non-rigid bond).

### III. SUGGESTIONS

The differential equations are developed under suggestion of the following hypothesis:

* All materials have constant surfaces,

* All materials offer linear elastic material bearing,

* The stress-distribution is uniform in all cuts,

* The deformations are little and suggested for theory of 1st order,

The following strains can be defined:

$$ \delta'_{h} = \frac{d\delta_{h}}{dx} = \varepsilon_{h}, \text{ strain of reinforcing material} \quad (1) $$

$$ \delta'_{g} = \frac{d\delta_{g}}{dx} = \varepsilon_{g}, \text{ strain of the glass} \quad (2) $$

Horizontal equilibrium is:

$$ 2dF_{h} + dF_{g} = 0 \quad (3) $$

$$ 2dF_{h} - 2\tau_{kl} \cdot b_{kl} \cdot dx = 0 \rightarrow 2\tau_{kl} \cdot b_{kl} = 2 \frac{dF_{h}}{dx} \quad (4) $$

The stresses after Hook's law are:

$$ \sigma_{h} = E_{h} \cdot \varepsilon_{h} = E_{h} \cdot \delta'_{h} \quad (5) $$

$$ \sigma_{g} = E_{g} \cdot \varepsilon_{g} = E_{g} \cdot \delta'_{g} \quad (6) $$

Integration offers the internal forces as:

$$ F_{h} = \int_{A_{h}} \sigma_{h} \cdot dA_{h} = E_{h} \cdot A_{h} \cdot \delta'_{h} \quad (7) $$

$$ F_{g} = \int_{A_{g}} \sigma_{g} \cdot dA_{g} = E_{g} \cdot A_{g} \cdot \delta'_{g} \quad (8) $$

The difference between the materials displacements $\delta_h$ and $\delta_g$ are:

$$ \delta = \delta'_{h} - \delta'_{g} \quad (9) $$

The derivative shows the strain difference:

$$ \delta' = \delta'_{h} - \delta'_{g} \quad (10) $$

(7) and (8) calculate $\delta'_h$ and $\delta'_g$:

$$ \delta' = \frac{F_{h}}{E_{h} \cdot A_{h}} - \frac{F_{g}}{E_{g} \cdot A_{g}} \quad (11) $$

The second derivative is the equivalence to the stressing of the non-rigid bond:

$$ \delta'' = \frac{dF_{h}}{dx \cdot E_{h} \cdot A_{h}} - \frac{dF_{g}}{dx \cdot E_{g} \cdot A_{g}} = b_{kl} \tau_{kl} \left( \frac{1}{E_{h} \cdot A_{h}} + \frac{2}{E_{g} \cdot A_{g}} \right) \quad (12) $$

Taking into account a law for the bond $\tau_{kl}(\delta)$, which was evaluated by testing, the differential equations of the non-rigid bond in relation to the material stressing can be defined as the following:

$$ \delta'' = a \cdot \delta + b \quad (13) $$

The non-rigid bond can be described with the following equation:

$$ \tau(\delta) = \tau_{a} + k \cdot \delta \quad (14) $$

The factor k is to define the elasticity of the adhesive.

The constant value $\tau_a$ needs to be $\tau_a > 0$.
IV. THE NON-RIGID BOND FOR THE SLAB-
MODEL

<figure><img src="image_132.png" /><figcaption>Fig 2: Suggestions and distributions of stresses, strains and differences of the material displacements</figcaption></figure>

According to the infinitesimal part, the
distributions of the displacements for the whole
slab are:

$$
\delta_h(x) = \int_0^x \varepsilon_h(x) dx \qquad (15)
$$

$$
\delta_g(x) = \int_0^x \varepsilon_g(x) dx \qquad (16)
$$

The difference between the displacement is:

$$
\delta(x) = \delta_h(x) - \delta_g(x) \tag{17}
$$

At x=0 the force in the reinforcing material is:

$$
F_{h0} = \sigma_{h0} \cdot A_h = E_h \cdot \varepsilon_{h0} \cdot A_h \quad (18)
$$

For the whole length of the slab it can be written:

$$
\Delta F_h(x) = b_{kl} \cdot \int_0^x \tau_{kl}(x) \cdot dx \quad (19)
$$

With

$$
\Delta F_h(x) = \Delta \varepsilon_h(x) \cdot E_h \cdot A_h \quad (20)
$$

and

$$
\Delta \varepsilon_h (x) = \frac{b_{kl}}{E_h \cdot A_h} \cdot \int_0^x \tau_{kl} (x) \cdot dx \quad (21)
$$

the distribution of the strains can be defined as the
following:

$$
\epsilon_h(x) = \epsilon_{h0} + \frac{b_{kl}}{E_h \cdot A_h} \cdot \int_0^x \tau_{kl}(x) \cdot dx \quad (22)
$$

Equally, the glass parameters can be written:

$$
F_{g0} = \sigma_{g0} \cdot A_g = E_g \cdot \varepsilon_{g0} \cdot A_g \quad (23)
$$

$$
\Delta F_g(x) = \Delta \varepsilon_g(x) \cdot E_g \cdot A_g \quad (24)
$$

$$
\Delta F_h(x) + \Delta F_g(x) = 0 \qquad (25)
$$

$$
\Delta \varepsilon_g(x) = - \frac{2 \cdot b_{kl}}{E_g \cdot A_g} \cdot \int_0^x \tau_{kl}(x) \cdot dx \quad (26)
$$

$$
\epsilon_g(x) = \epsilon_{g0} - \frac{2 \cdot b_{kl}}{E_g \cdot A_g} \cdot \int_0^x \tau_{kl}(x) \cdot dx \quad (27)
$$

The differences of the material strains

$$
\begin{equation}
\begin{split}
\Delta \varepsilon(x) = \varepsilon_h(x) - \varepsilon_g(x) = \varepsilon_{h0} - \varepsilon_{g0} + & \\
& b_{kl} \left( \frac{1}{E_h \cdot A_h} + \frac{2}{E_g \cdot A_g} \right) \cdot \int_0^x \tau_{kl}(x) \cdot dx
\end{split}
\tag{28}
\end{equation}
$$

can be shown in relation to the stresses due to the
non-rigid bond:

$$
\begin{align}
\delta''(x) &= b_{kl} \left( \frac{1}{E_h \cdot A_h} + \frac{2}{E_g \cdot A_g} \right) \cdot \tau_a + \tag{29} \\
&\quad b_{kl} \left( \frac{1}{E_h \cdot A_h} + \frac{2}{E_g \cdot A_g} \right) \cdot k \cdot \delta \nonumber
\end{align}
$$

$$
\delta(x) = \int_{0}^{x} \Delta \varepsilon(x) \cdot dx \quad (30)
$$

Besides this equation, the difference of the
displacements is related due to the non-rigid law of
the bond. This allows developing the differential
equation for the non-rigid bond for the slab model:

$$
\begin{equation}
\begin{aligned}
\delta''(x) = b_{kl} & \left( \frac{1}{E_h \cdot A_h} + \frac{2}{E_g \cdot A_g} \right) \cdot \tau_a + \\
& b_{kl} \left( \frac{1}{E_h \cdot A_h} + \frac{2}{E_g \cdot A_g} \right) \cdot k \cdot \delta(x)
\end{aligned}
\tag{31}
\end{equation}
$$

A. Solution of the differential equation

The solution for this linear homogenous
differential equation of second order with constant
coefficient can now

$$
y''(x) - c_0 \cdot y(x) = r \quad (32)
$$

can be found with the following basic functions:

$$
y_{ho} = e^{\lambda x} \tag{33}
$$

the characteristic solution

$$
\lambda^2 - c_0 = 0 \tag{34}
$$

and the real solution:

$$
\lambda_{1,2} = \pm \sqrt{c_0} \qquad (35)
$$

With linear combination the basic solution is:

$$
y = c_1 \cdot e^{\lambda_1 x} + c_2 \cdot e^{\lambda_2 x} \quad (36)
$$

The particular solution can be found by approach
of the right part of the equation:
$$y_{pa} = c_3 \tag{37}$$

and defining the constant to:

$$c_3 = -\frac{r}{c_0} \tag{38}$$

The solution for the difference of the displacements is:

$$\delta(x) = c_1 \cdot e^{\lambda_1 x} + c_2 \cdot e^{\lambda_2 x} - \frac{\tau_a}{k} \tag{39}$$

with:

$$\lambda_{1,2} = \pm \sqrt{b_{kl} \left( \frac{1}{E_h \cdot A_h} + \frac{2}{E_g \cdot A_g} \right) \cdot k} \tag{40}$$

## B. Definition of constants with boundary conditions

To define the constants the known values can be evaluated and the equations solved. The two different states “not cracked” and “cracked” have to be taken into account.

At $x=0$ is

$$\delta(0) = c_1 + c_2 - \frac{\tau_a}{k} = 0 \tag{41}$$

for a non cracked slab can also be defined:

$$\delta'(0) = \pm \lambda \cdot (c_1 - c_2) = 0 \tag{42}$$

Introducing the length $s$ which is the equivalent to the needed length to introduce the force into the glass (defined by tests) it can be written:

$$\delta'(s) = \lambda \left\{ c_1 \cdot e^{[\lambda \cdot s]} - c_2 \cdot e^{-[\lambda \cdot s]} \right\} = \varepsilon_{hll} \tag{43}$$

For a cracked cut, a further compatibility condition can be chosen. The whole force has to be taken by the reinforcing material:

$$\delta'(s) = \varepsilon_{hll} = \frac{F_{hll}}{E_h \cdot A_h} \tag{44}$$

Solving the system of equations the constants are defined to:

$$c_1 = c_2 = \frac{\tau_a}{2 \cdot k} \tag{45}$$

## C. Defining the load introducing length

The load introducing length $s$ can be calculated in transforming:

$$e^x - e^{-x} = 2 \cdot \sinh(x) \tag{46}$$

and

$$ar \sinh(x) = \ln(x + \sqrt{x^2 + 1}) \tag{47}$$

to:

$$s = \frac{1}{\lambda} \cdot \ln \left[ \frac{\frac{F_{hrl}}{E_h \cdot A_h} \cdot k}{\tau_a \cdot \lambda} + \sqrt{\left( \frac{F_{hrl}}{E_h \cdot A_h} \cdot k \right)^2 + 1} \right] \tag{48}$$

## D. Solution

The stressing of the bond along the slab model can be calculated to:

$$\tau(x) = \frac{\lambda^2}{b_{kl} \cdot \left( \frac{1}{E_h \cdot A_h} + \frac{2}{E_g \cdot A_g} \right)} \cdot (c_1 \cdot e^{\lambda \cdot x} + c_2 \cdot e^{-\lambda \cdot x}) \tag{49}$$

with defined constants:

$$\tau(x) = \frac{\tau_a}{2} \cdot (e^{\lambda \cdot x} + e^{-\lambda \cdot x}) \tag{50}$$

The strains of the materials are:

$$\varepsilon_h(x) = \varepsilon_{hll} - \frac{\frac{1}{E_h \cdot A_h}}{\left( \frac{1}{E_h \cdot A_h} + \frac{2}{E_g \cdot A_g} \right)} \cdot \lambda \cdot \tag{51}$$

$$\left[ c_1 \cdot (e^{\lambda \cdot s} - e^{\lambda \cdot x}) - (c_2 \cdot e^{-\lambda \cdot s} - e^{-\lambda \cdot x}) \right]$$

and

$$\varepsilon_g(x) = \frac{\frac{2}{E_g \cdot A_g}}{\left( \frac{1}{E_h \cdot A_h} + \frac{2}{E_g \cdot A_g} \right)} \cdot \lambda \cdot \tag{52}$$

$$\left[ c_1 \cdot (e^{\lambda \cdot s} - e^{\lambda \cdot x}) - (c_2 \cdot e^{-\lambda \cdot s} - e^{-\lambda \cdot x}) \right]$$

With defined constants:

$$\varepsilon_h(x) = \frac{F_{hll}}{E_h \cdot A_h} - \frac{\frac{1}{E_h \cdot A_h}}{\left( \frac{1}{E_h \cdot A_h} + \frac{2}{E_g \cdot A_g} \right)} \cdot \lambda \cdot \tag{53}$$

and
$$ \epsilon_g(x) = \frac{\frac{2}{E_g \cdot A_g}}{\left(\frac{1}{E_h \cdot A_h} + \frac{2}{E_g \cdot A_g}\right)} \cdot \left[ \frac{\tau_a}{2 \cdot k} \cdot (e^{\lambda_s x} - e^{\lambda_x x}) - \frac{\tau_a}{2 \cdot k} \cdot (e^{-\lambda_s x} - e^{-\lambda_x x}) \right] \cdot \left[ \frac{\tau_a}{2 \cdot k} \cdot \left( e^{\left[ \frac{\tau_a}{2k}(e^{\lambda_s x} - e^{\lambda_x x}) - \frac{\tau_a}{2k}(e^{-\lambda_s x} - e^{-\lambda_x x}) \right]_x} - e^{\left[ \frac{\tau_a}{2k}(e^{\lambda_s x} - e^{\lambda_x x}) - \frac{\tau_a}{2k}(e^{-\lambda_s x} - e^{-\lambda_x x}) \right]_x} \right) - \frac{\tau_a}{2 \cdot k} \cdot \left( e^{\left[ \frac{\tau_a}{2k}(e^{\lambda_s x} - e^{\lambda_x x}) - \frac{\tau_a}{2k}(e^{-\lambda_s x} - e^{-\lambda_x x}) \right]_x} - e^{\left[ \frac{\tau_a}{2k}(e^{\lambda_s x} - e^{\lambda_x x}) - \frac{\tau_a}{2k}(e^{-\lambda_s x} - e^{-\lambda_x x}) \right]_x} \right) \right] \quad (54) $$

## E. Width of cracks and completed crack distribution

A completed crack distribution can be admitted when the following relation is achieved:

$$ s \le a_{r,\min} \le 2s \quad (55) $$

This shows that a maximum number of cracks can be calculated in relation of the load introducing length.

The theoretical width of a crack is defined to:

$$ b_r = 2 \cdot \delta(a) \quad (56) $$

## V. SAFETY CONSIDERATIONS

To design reinforced glass panes loaded on their strong axis, the design at the limit state of cracking governs the dimension (thickness of the panes) in relation to the post cracked load capacity. The author suggests a minimum of two cracks appearing before collapsing of the structure and a minimum ratio for post cracked load capacity of 1.5 (ultimate load / load of first crack).

The influence of the thermal treatment of the glass is also shown in [1].

Figure 3 shows the relation of post cracked load capacity in relation to glass with different residual stresses due to thermal treatment.

<figure><img src="image_133.png" /><figcaption>Fig 3: Post cracked load capacity in relation to glass with different residual stresses due to thermal treatment</figcaption></figure>

## VI. CONCLUSION/EXAMPLE

The load-bearing of glass panes loaded on their strong axis can be improved by reinforcing the stressed edge. In relation to the residual internal stresses due to thermal treatment of the glass, the post cracked load capacity varies.

To be able to calculate the loading peaks at the limit state of cracking, the stresses of glass, reinforcing material and adhesive need to be known.

The calculation method shown in this paper offers the opportunity to define these values to insure the design and the structural safety of reinforced glass panes loaded on their strong axis.

An estimation of the structural security taking into account the glass quality and failure of the reinforcing material shows good correlation with testing-results. Figure 4 shows the estimated structural safety (ultimate load/load at first crack; with: rigid and the real non-rigid bond; testing-results: points and line of linear correlation) and the corresponding stresses due to bending, calculated with the shown model.

<figure><img src="image_134.png" /><figcaption>Fig 4: Correlation between testing-results and calculated values</figcaption></figure>

Figure 5 shows a tested girder of heat strengthened glass reinforced by timber.
<figure><img src="image_135.png" /><figcaption>Fig 5: Reinforced girder after collapsing (left); failure of the reinforcing timber-slab (right)</figcaption></figure>

REFERENCE

[Kreher 2004]

KREHER, K., Tragverhalten und Bemessung von Holz-Glas-Verbundträgern unter Berücksichtigung der Eigenspannungen im Glas, These Nr. 2999, IBOIS, EPFL
# STABILITY OF LOAD CARRYING ELEMENTS OF GLASS

Andreas Luible, SCHMIDLIN AG, Aesch, Switzerland

Michel Crisinel, EPFL – ICOM, Lausanne, Switzerland

Glass is a material, that is able to resist very high compression stresses and which has special architectural appeal because of its transparency. For this reason, there is a growing trend to extend the use of glass sheets to load carrying elements such as beams, columns and shear panels. Due to their high slenderness and high compression strength, such load carrying elements tend to fail because of instability. The main objective of the research work is the experimental and theoretical study of the fundamental stability problems (column buckling, lateral buckling, plate buckling) for single layer and laminated glass.

Based on stability tests, the load carrying behaviour of simple and laminated glass in the foreseeable dimensions of application was examined and analytic and numeric models were developed. To simulate the buckling behaviour of laminated glass elements, the time and temperature-dependent behaviour of the PVB interlayer was modelled with viscoelastic finite elements.

The main objective of the work is to discuss possible design methods for single layered and laminated glass elements by means of the test results, the developed models and the parametric study.

**Keywords:** glass, stability, load carrying glass elements, buckling, second order

## I. INTRODUCTION

Modern architecture demands for more slender and lighter structures. Glass is a material that has been used for a long time in windows as a filling material and has much to offer in this regard due to

its very high compressive strength and transparency. For this reason, there is a growing trend to extend the use of glass to load carrying elements such as columns, beams and panels.

<figure><img src="image_136.png" /><figcaption>Fig 1: Stability problems.</figcaption></figure>

Due to their high slenderness, such elements tend to fail because of instability (e.g. column buckling, lateral torsional buckling or plate buckling) (Fig 1).

At present little knowledge exists about the load carrying behaviour of glass structural elements, and existing design methods for other materials (i.e. steel) cannot be directly transferred to glass panels, since the influence of the following aspects on the behaviour of glass must be investigated in a different manner:

* production tolerances (i.e. glass thickness)

* initial deformations,

* the visco-elastic Poly-Vinyl-Butyral interlayer (PVB) used for laminated safety glass,
* the ideal elastic material behaviour without plastic deformability, and

* the ultimate breaking stress in glass, which depends on the embedded compressive surface stress due to the tempering process, the degree of damage of the glass surface and the load duration.

With this in mind, the main objectives of the research project were:

* The theoretical and experimental study of the load carrying behaviour of glass elements which may fail due to lack of stability.

* Discussion of possible design methods for the three main stability problems (column buckling, lateral torsional buckling and plate buckling).

## II. STUDY OF THE PARAMETERS WITH THE MOST IMPORTANT INFLUENCE ON THE BUCKLING BEHAVIOUR

The dispersion of the **glass thickness** and the **initial deformation** of glass panels were measured for more than 200 test specimen from two different glass manufacturers.

The thickness *t* of annealed flat glass panels differs from the nominal value because glass manufacturers try to save material in making the most use of the thickness tolerances specified by the codes. The real glass thickness is often less than the nominal value, therefore reducing the moment of inertia of the cross section and, thus the buckling strength. The aforementioned measurements confirmed that the values follow a normal distribution. The 5% percentile value is 97.61% of the nominal glass thickness.

The initial geometric deformation $w_0$ (Fig 2) of flat glass, which is mainly caused by the tempering process was measured with a taut steel wire. For the measurements the glasses were supported in such a way that the deformation due to the dead load has no influence on their initial deformation. The results confirmed that non-tempered annealed flat glass has a very low initial deformation ($<$ 1/2500) while heat-strengthened and fully toughened glass can have a sinusoidal initial deformation up to 1/300 of the length *L*. The nominal thickness of the glasses have an influence

on the statistical distribution but for design in practice the simplified assumption of one single distribution might be sufficient. Laminated safety glass showed the same results. The measured values followed a normal distribution with a 95% percentile value of 1/386. However maximum initial deformations depend strongly on the quality of the furnace and can therefore vary between different glass manufacturers.

## III. COLUMN BUCKLING

### A. Introduction

Stability problems can be divided into two categories (Fig 2). The first includes perfect members that are subjected to an increasing load, where instability occurs suddenly when a critical load $N_{cr,K}$ is reached. The second more realistic category covers members with imperfections in the linearity of the bar (i.e. initial deformations $w_0$) and/or an eccentricity *e* of the applied load where the critical load can never be obtained. An increasing load *N* leads to disproportional increasing deformations already under very small loads until the strength of the material or a maximum deformation is exceeded (imperfect column buckling). The maximum load $N_K$ is the point where the maximum stresses in the material due to the lateral deformations are reached. For the design of structural elements under compression, the fundamental study of the difference between the critical load $N_{cr,K}$ and the maximum load $N_K$ is necessary.

<figure><img src="image_137.png" /><figcaption>Fig 2: Eccentrically loaded bar with initial deformation $w_0$.</figcaption></figure>
## B. Buckling Models for Glass

### 1) Single Layer Glass

The load carrying behaviour of single layered glass can be described using a second order differential equation [Hirt et al. 1998] for a bar with a length, $L_K$, under axial compression, $N$, the bar is pinned at the ends, has an initial sinusoidal deformation $w_0$ and the load is applied with an eccentricity, $e$. The solution for the elastic critical buckling load is given by:

$$N_{cr,K} = \frac{\pi^2 EI}{L_K^2} \quad \text{with:} \quad I = \frac{bt^3}{12} \qquad (1)$$

and the geometrical slenderness is defined as:

$$\lambda_K = \pi \sqrt{\frac{EA}{N_{cr,K}}} = \pi \sqrt{\frac{E}{\sigma_{cr,K}}} \qquad (2)$$

The maximum deformation is given by:

$$w = \frac{e}{\cos(L_K / 2\sqrt{N/N_{cr,K}})} + \frac{w_0}{1 - N/N_{cr,K}} \qquad (3)$$

and the maximum surface stress can be determined as:

$$\sigma = \frac{N}{A} \pm \frac{N}{W} \left[ \frac{e}{\cos L_K / 2\sqrt{N/EI}} + \frac{w_0}{1 - N/N_{cr,K}} \right] \qquad (4)$$

### 2) Laminated Safety Glass

The PVB interlayer in laminated glass behaves like a shear connection between the glass layers. The load carrying behaviour can be described using elastic “sandwich” theory [Zenkert 1997] or by means of a finite element model [ANSYS 2002] where the PVB interlayer can either be represented by elastic or visco-elastic elements [Van Duser et al. 1999].

The critical buckling load of a two layer elastic “sandwich” (Fig 3) with a width $b$ is given by [Stamm et al. 1974]:

$$N_{cr,K} = \frac{\pi^2 (1 + \alpha + \pi^2 \alpha \beta) EI_S}{1 + \pi^2 \beta L_K^2} \qquad (5)$$

For a laminated safety glass with two glass layers:

$$\alpha = \frac{I_1 + I_2}{I_S}; \beta = \frac{t_{PVB}}{G_{PVB} b(z_1 + z_2)^2} \frac{EI_S}{L_K^2} \qquad (6)$$

$$I_i = \frac{bt_i^3}{12}; I_S = b(t_1 z_1^2 + t_2 z_2^2) \qquad (7)$$

<figure><img src="image_138.png" /><figcaption>Fig 3: Laminated safety glass with two glass layers.</figcaption></figure>

For a laminated safety glass with three glass layers and symmetrical cross section (Fig 4)

$$\alpha = \frac{2I_1 + I_2}{I_S} \qquad (8)$$

$$\beta = \frac{t_{PVB}}{2G_{PVB} b z_1^2} \frac{EI_S}{L_K^2} \qquad (9)$$

$$EI_S = 2Ebt_1 z_1^2 \qquad (10)$$

<figure><img src="image_139.png" /><figcaption>Fig 4: Laminated safety glass with three glass layers and symmetric cross section.</figcaption></figure>

The geometrical slenderness is defined as

$$\lambda_{K,Sandwich} = \frac{L}{\sqrt{\frac{I_S (1 + \alpha + \pi^2 \alpha \beta)}{A (1 + \pi^2 \beta)}}} \qquad (11)$$

## C. Experimental Investigation

A total of 80 displacement- and load-controlled column buckling tests were carried out (Fig 5).

### 1) Single Layered Glass

#### a) Test Results

The differential equation solution (Eq. (3)) showed good agreement with the test performance of single layered glass (Fig 6). The initial fracture occurred always on the tensile surface and, in most cases, within a distance of about three times the glass thickness from the glass edge (Fig 7).
<figure><img src="image_140.png" /><figcaption>Fig 5: Test setup column buckling tests.</figcaption></figure>

<figure><img src="image_141.png" /><figcaption>Fig 6: Test result single layer glass.</figcaption></figure>

b) *Load carrying Behaviour*

The study of the load carrying behaviour showed that the glass thickness *t*, the initial deformation *w*₀ and the load eccentricity *e* have the most important influence on the maximum load. Due to the high compressive strength of glass, the buckling strength of a glass element under compression, with the dimensions applied in building construction (*L* > 300 mm, *t* < 19 mm, initially deformed) is always limited by the maximum tensile strength [Luible 2004].

<figure><img src="image_142.png" /><figcaption>Fig 7: Initial breakage spots of 19 column buckling tests on single layered glass.</figcaption></figure>

The initial breakage can occur on the whole glass surface and depends on the “frozen-in” compressive surface stress in addition to the tensile strength of the annealed float glass as mentioned above. But the study showed that the weakest point of the glass surface can simplistically be assumed to be at the point of the highest tensile stress.

2) *Laminated Safety Glass*

a) *Test Results*

The results of finite element models with linear visco-elastic elements representing the PVB-interlayer showed good agreement with the column buckling test results (Fig 8).

b) *Load carrying Behaviour of Laminated Safety Glass*

Simulations with the visco-elastic model under different temperatures and loading speeds demonstrated the effect of the interlayer on the load carrying capacity of laminated safety glass columns. In practice, the visco-elastic modelling of
the PVB is complicated and may be simplified by an elastic model instead [Booker et al. 1974].

Parametric studies showed that the influence of the shear transfer by the PVB depends on the shear stiffness of the interlayer (which is a function of the temperature and the load duration), but also on the glass geometry (length, thickness of glass and PVB) [Luible 2004]. An improvement of the maximum load, comparing to a similar glass without PVB interlayer, is marginal for long-term loading and temperatures higher than 25°C. From an economical and safety point of view, a shear connection might therefore only be taken into account for short-term loading like wind or impact.

<figure><img src="image_143.png" /><figcaption>Fig 8: Test result laminated safety glass.</figcaption></figure>

## D. Synthesis for Design

### 1) Single Layer Glass

To simplify the design of compression members (e.g. steel columns) column buckling curves are commonly used. The same approach can be applied to compressive glass elements. In contrast to steel, however, the slenderness ratio for glass must be based on the maximum tensile strength, since the compressive strength of glass is not limiting its buckling strength. Unfortunately, simulations of column buckling curves for glass elements based on a slenderness ratio showed large variations for different tensile strength. Therefore, in contrast to steel, it appears unpractical to establish column buckling curves for glass using

this approach. Column buckling curves might still be determined based on the geometric slenderness (Eq. (2)), which results in a family of curves for different tensile strengths. Additional lateral loads and end moments can be taken into account by means of interaction formulas similar to the design of compressive steel members. It was found, however, that for short glass panels these interaction formulas can lead to uneconomic solutions [Luible 2004] [Luible et al. 2004].

A more suitable approach for design might be a direct calculation of the maximum tensile stress in the compressed glass member by means of elastic second order equations (e.g. Eq. (4)) followed by a comparison of the this tensile stress to the design value of the tensile strength of glass. The calculation must be carried out with a reduced glass thickness (e.g. 97.61% of the nominal glass thickness) and a reasonable assumption for the initial deformation (e.g. $w_0 = L_K/400$) [Luible 2004] [Luible et al. 2004].

### 2) Laminated Safety Glass

For the design of a compressed laminated safety glass element, the visco-elastic behaviour might be simplified by the above mentioned elastic approach. The same methods as described for single layered glass can be applied to laminated safety glass elements. The maximum tensile stress for a given load $N$ might be calculated either with a numerical model or an analytical sandwich model that takes into account second order effects (eg. Eq.(4)). For further simplification, the sandwich cross section may be replaced by an effective monolithic cross section with the effective thickness $t_{eff}$ given by:

$$t_{eff} = \sqrt[3]{\frac{12 I_s (1 + \alpha + \pi^2 \alpha \beta)}{b (1 + \pi^2 \beta)}} \quad (12)$$

# IV. LATERAL TORSIONAL BUCKLING

## A. Introduction

Lateral torsional buckling is a limit state of structural stability, where a beam is loaded with pure bending and the deformation is a combination of lateral deflection and twisting. In glass
structures this type of stability failure can occur, for example in beams or swords used as stiffeners in facades.

## B. Lateral Torsional Buckling Models

### 1) Single Layer Glass

The critical torsional buckling moment (bifurcation buckling) of a beam with a rectangular cross section can be calculated with:

$$M_{cr,D} = C_1 \frac{\pi^2 EI_z}{L_D^2} \left[ \sqrt{C_2 z_a + \frac{GK L_D^2}{\pi^2 EI_z}} + C_2 z_a \right] \quad (13)$$

The factors $C_i$ and $z_a$ take into account different boundary conditions, different bending moments and the distance between the centre of gravity and the point where the load is applied [Hirt et al. 1998]. Due to their rectangular cross-section, warping torsion may be neglected in glass beams (Fig 9).

<figure><img src="image_144.png" /><figcaption>Fig 9: Lateral buckling with end moments.</figcaption></figure>

Similar to column buckling the lateral torsional buckling resistance is not limited by the critical torsional buckling moment $M_{cr,D}$. Due to imperfections of the beam the lateral deformation and twisting start already to increase under very small loads and the lateral torsional buckling resistance is reached when the maximum stresses in the beam exceeds the material resistance. Bifurcation buckling (e.g. Eq.(13)) over-estimates the real lateral torsional buckling resistance.

To describe the real load carrying behaviour, analytical and numerical models (finite element method - FEM) were developed. It was found, that the numerical model is more suitable to describe the load carrying behaviour due to the slender geometries of glass beams, than analytical models based on the linear elastic beam theory [Luible 2004].

### 2) Laminated Safety Glass

The critical lateral torsional buckling moment of laminated safety glass may also be calculated using Eq. (13), where the lateral bending stiffness $EI_z$ and the torsional stiffness $GK$ are replaced by a equivalent stiffness, $EI_{z,eff}$ and $GK_{eff}$, to take into account the composite action of the PVB interlayer in laminated safety glass [Luible 2004].

$$EI_{z,eff} = EI_s \left( \frac{\alpha\beta\pi^2 + \alpha + 1}{1 + \pi^2\beta} \right) \quad (14)$$

with

$$I_s = h(t_1 z_1^2 + t_2 z_2^2) \quad (15)$$

For $\alpha$ and $\beta$ see Eq. (6).

$$GK_{eff} = GK_{glass1} + GK_{glass2} + GK_{comp} \quad (16)$$

with

$$GK_{comp} = G I_s \left( 1 - \frac{\tanh \frac{\lambda h}{2}}{\frac{\lambda h}{2}} \right) \quad (17)$$

$$I_s = 4 \left( \frac{t_1 + t_2}{2} + t_{PVB} \right)^2 \frac{t_1 t_2}{t_1 + t_2} h \quad (18)$$

$$\lambda = \sqrt{\frac{G_{PVB} t_1 + t_2}{G t_{PVB} t_1 t_2}} \quad (19)$$

To study the lateral torsional buckling behaviour of a glass beam composed of laminated safety glass with imperfections a finite element model [ANSYS 2002] was developed (Fig 10).
<figure><img src="image_145.png" /><figcaption>Fig 10: FEM-Modell laminated safety glass.</figcaption></figure>

Only half of the glass beam was modelled because of the symmetrical system. The glass layers were modelled with shell elements and the PVB interlayer with volume elements. The applied initial deformation was a scaled shape of the first eigenform of the considered system.

<figure><img src="image_146.png" /><figcaption>Fig 11: Test set up for lateral torsional buckling tests.</figcaption></figure>

## C. Experimental Investigation

### 1) Test Setup

For the test setup a simply supported beam was subjected to a concentrated load at mid-span. The main difficulty was the load application that had to follow the lateral displacement of the upper edge of the glass beam. The hydraulic jack and the load introduction device were therefore fixed on a

carriage (Fig 11).

In Fig 12 a deformed single layered glass beam is shown.

<figure><img src="image_147.png" /><figcaption>Fig 12: Buckled glass beam.</figcaption></figure>

### 2) Main Results

Seventy-nine lateral torsional buckling tests on single layered and laminated safety glass were carried out.

#### a) Single Layer Glass

The results of the tests showed a good agreement with the numerical simulation. Similar to column buckling, the maximum force in the test approaches the critical buckling load $F_{cr,D}$.

<figure><img src="image_148.png" /><figcaption>Fig 13: Tensile stress distribution on the glass surface.</figcaption></figure>

During the test the stress distribution on the glass surface was controlled with strain gauges and compared with the numerical simulation (Fig 13).
It could be seen that the stress distribution is non linear over the beam height *h*. The more the lateral displacement increased this non linear effect became stronger.

The breakages patterns of all tests were analyzed and it was seen that three typical modes for the initial breakage can be identified (Fig 14):

a) initial breakage on the corner of the glass edge

b) initial breakage on the lateral glass surface in a certain distance from the glass edge

c) initial breakage on the glass edge almost in the middle.

<figure><img src="image_149.png" /><figcaption>Fig 14: Three different breakage modes.</figcaption></figure>

<figure><img src="image_150.png" /><figcaption>Fig 15: Distribution of compressive residual stresses on the glass surface near glass edges.</figcaption></figure>

Breakage mode b) and c) can be explained by taking into consideration that the residual stresses of tempered glass near the glass edges can be lower than the residual stresses in the middle of the glass surface [Laufs 2000.2] [Bernard 2001]. These points with minimum residual stresses are located in the middle of the glass edge and at a certain

distance from the glass edge on the lateral glass surface – exactly where the initial breakage occurred (Fig 15).

b) Laminated Safety Glass

The buckling tests confirmed that the load carrying behaviour of a laminated safety glass is characterised by the visco-elastic behaviour of the PVB interlayer (Fig 16). The temperature and the load duration therefore have an important influence on the lateral torsional buckling resistance of a laminated glass beam.

<figure><img src="image_151.png" /><figcaption>Fig 16: Test result laminated safety glass.</figcaption></figure>

D. Synthesis for design

The study of the load carrying behaviour showed that the dispersion glass thickness *t*, the initial deformation of a glass beam, the composite action due to the PVB interlayer and the tensile strength of glass have the most significant influence on the lateral torsional buckling resistance of a glass beam. Due to the high compressive strength of glass the tensile strength of glass is determinant for the buckling resistance. In practice, the visco-elastic behaviour of the PVB interlayer can be simplified by an elastic interlayer with equivalent shear modulus, $G_{PVB}$. Fig 17 shows the influence of an elastic interlayer on the buckling strength. It was found, that for realistic values of $G_{PVB}$ (< 5 N/mm²) monolithic behaviour of the glass beam cannot be achieved.
<figure><img src="image_152.png" /><figcaption>Fig 17: Influence of the shear behaviour.</figcaption></figure>

The buckling resistance is reached when the maximum stress in the glass beam is equal to the tensile strength at one of the three critical points (breakage modes) mentioned above. For a simplification in practice, the tensile strength of the entire cross section may be taken as the minimum of the tensile strength at these three critical points to determine the lateral torsional buckling resistance. The maximum stress in the glass beam has to be calculated with a suitable model that takes into account second order effects and the non linear stress distribution in the cross section.

### 1) Design methods

The lateral torsional buckling resistance of a glass beam may either determined with a suitable model (e.g. FEM) or with buckling curves. The development of buckling curves was studied in this research work [Luible 2004] by means of the developed FEM models.

### 2) Buckling curves

To determine buckling curves a slenderness ratio $\bar{\lambda}_D$ and a reduction factor were defined. In contrast to steel, these are based on the tensile strength, since the compressive strength does not limit the buckling strength:

$$ \bar{\lambda}_D = \sqrt{\frac{\sigma_{p,t}}{\sigma_{cr,D}}} = \sqrt{\frac{2\sigma_{p,t}I_y}{M_{cr,D}h}} \qquad (20) $$

Where $\sigma_{p,t}$ the is the tensile strength of the glass and $\sigma_{cr,D}$ is the critical lateral torsional buckling

The critical lateral torsional buckling moment $M_{cr,D}$ may be calculated with Eq. (13). For the design of a laminated safety glass the equivalent lateral bending stiffness $EI_{z,eff}$ (Eq.(14)) and the equivalent torsional stiffness $GK_{eff}$ (Eq. (16)) might be used. The reduction factor $\chi_D$ in a buckling diagram is a function of the slenderness ratio:

$$ \chi_D = f(\bar{\lambda}_D) \qquad (21) $$

Hence the maximum bending moment $M$ is:

$$ M = \sigma \frac{2I_y}{z} = \chi_D \sigma_{p,t} \frac{2I_y}{z} \qquad (22) $$

For different types of loading, glass geometries, shear modulus of the PVB interlayer, and initial deformation, $v_0$, reduction factors were generated and plotted in buckling diagrams (i.e. Fig 18).

These diagrams may serve as a preliminary orientation in determining lateral torsional buckling curves for glass. The main results are:

* It is possible to define lateral torsional buckling curves for glass based on the tensile strength.

* Existing buckling curves for example for steel structures can not be transferred to glass.

<figure><img src="image_153.png" /><figcaption>Fig 18: Simulated reduction factors for a concentrated load at mid span compared to buckling tests.</figcaption></figure>

* It might be useful to determine several buckling curves, depending on the composition of the glass (single layer, laminated safety
glass), the type of loading and the initial deformation.

* Further studies with additional systems are necessary (i.e. systems with intermediate lateral support or with partial restraint of the beam by structural silicon joints).

* Buckling curve (c) in Eurocode 3 [EN 1993-1-1:1993] may be used as a conservative approach for design of glass elements since all simulations results lay above this curve.

# V. PLATE BUCKLING

## A. Introduction

When a plate simply supported along its edges is subjected to compression or shear forces in its middle plane the stability phenomenon of plate buckling can occur. The load carrying behaviour is a deformation w perpendicular to the middle plane that comes along with a distortion of the cross section of the plate (Fig 19).

The main difference compared to other stability problems is, that the critical buckling load $N_{cr,P}$ is not necessarily the ultimate load of the plate. The buckled element may sustain greater loads than the critical buckling load due to additional membrane stresses in the plate. Therefore initial out of plane deformation are less important. In glass structures this type of stability failure can occur in glass walls or shear panels.

<figure><img src="image_154.png" /><figcaption>Fig 19: Plate buckling.</figcaption></figure>

## B. Plate Buckling Models for Glass

### 1) Single Layer Glass

The critical buckling load may be calculated with analytical models based on linear elastic bending theory. However, due to post-critical buckling behaviour, the critical buckling load

$N_{x,cr,P}$ is not a criterion for the ultimate strength and thus most of these analytical models are not suitable for design.

$$N_{x,cr,P} = \left( \frac{mb}{a} + \frac{a}{bm} \right)^2 \frac{\pi^2 Et}{12(1-\nu^2)} \left( \frac{t}{b} \right)^2 \quad (23)$$

To study the load carrying behaviour of a buckled glass plate in a more realistic manner (including post-critical buckling) a FE model with shell elements was created. [Luible 2004].

### 2) Laminated Safety Glass

The critical buckling load $N_{x,cr,P,VSG}$ of laminated safety glass may be determined using linear elastic sandwich theory [Zenkert 1997].

$$N_{x,cr,P,VSG} = \left( \frac{mb}{a} + \frac{a}{mb} \right)^2 \frac{\pi^2 D}{b^2} \frac{\frac{(D_1 + D_2)}{D} \left[ \left( \frac{mb}{a} \right)^2 + 1 \right] + \frac{Ab^2}{\pi^2 D_s}}{\left[ \left( \frac{mb}{a} \right)^2 + 1 \right] + \frac{A}{\pi^2 D_s}} \quad (24)$$

with

$$D = D_1 + D_2 + D_s \quad (25)$$

$$D_i = \frac{Et_i^3}{12(1-\nu^2)} \quad (26)$$

$$D_s = \frac{Et_1 z_1^2 + Et_2 z_2^2}{1 - \nu^2} \quad (27)$$

$$A = \frac{G_{PVB}(z_1 + z_2)^2}{t_{PVB}} \quad (28)$$

where

$z_1, z_2$: distance between the centre of gravity of the total cross section to the centre of gravity of the glass layer (Fig 3)

$t_i$: thickness of the corresponding glass layer

$N_x$: pressure force per unit length ($N_x = \sigma_x t$)

$D$: plate stiffness applied to a unit width $b$

For the investigations of the plate buckling behaviour a FE model was developed instead since analytical models are not suitable to describe the load carrying behaviour of a laminated safety glass. The cross section in this model consisted of shell elements for the glass layers and volume elements for the PVB interlayer similar to the FEM for lateral torsional buckling. The load introduction and boundary conditions were applied by means of additional nodes, which were coupled with the element nodes. For symmetrical deformations of
the plate only one fourth was modelled with the
corresponding boundary conditions in the
symmetric axis (Fig 20).

<figure><img src="image_155.png" /><figcaption>Fig 20: Finite element model for plate buckling (reduced model for symmetrical deformations).</figcaption></figure>

C. *Experimental Investigation*

1) *Test setup*

The test setup corresponds to a square plate with four hinged edges. The load introduction in the two horizontal glass edge was achieved with aluminium supports that allow a free rotation of the edge and include a reinforced PTFE interlayer to reduce friction.

The vertical glass edges were supported with
neoprene profiles (Fig 21).

<figure><img src="image_156.png" /><figcaption>Fig 21: Plate buckling test setup.</figcaption></figure>

2) Main results

A total of 9 plate buckling tests on single layered
glass and 6 tests on laminated safety glass were
carried out.

a) *Single Layer Glass*

The load carrying behaviour of the tested glass
plates demonstrated the typical plate buckling
behaviour with a load capacity higher than the
critical buckling load, Ncr,P. (Fig 22). The stiffness
of the tested plates corresponds well with the
model where the load is introduced by a constant
edge displacement, du. Due to plastification of the
aluminium interlayer close to the glass edges,
however the measured deformation, w, in the
centre of the plate is higher than in the model. Below the critical buckling load the curves differ more than in the post buckling region where they are almost identical.

<figure><img src="image_157.png" /><figcaption>Fig 22: Plate buckling test on heat strengthened glass.</figcaption></figure>

In all tests the initial breakage occurred on the glass surface under tensile stress and in a region close to the glass corners (Fig 23).

<figure><img src="image_158.png" /><figcaption>Fig 23: Buckled glass plate.</figcaption></figure>

This is due to the maximum principle tensile stress which is moving as a function of the applied

load in the corners of the buckled glass plate as it can be seen in the numerical simulation (Fig 24) [Luible 2004].

<figure><img src="image_159.png" /><figcaption>Fig 24: First principle stress on the glass surface.</figcaption></figure>

The study of the load carrying behaviour demonstrated that the dispersion of the glass thickness, the boundary conditions, the tensile strength of the glass surface and the initial deformation have the most significant influence on the buckling strength. For loads lower than the critical buckling load the initial deformation has an influence, on the buckling strength.

<figure><img src="image_160.png" /><figcaption>Fig 25: Influence of the buckling shape.</figcaption></figure>

Studies with different initial deformations (applied as a multiple of the eigenform (EF1, EF2)) showed, that the shape of the buckled glass plate may have an influence on the buckling resistance (Fig 25) as well since the maximum tensile stress
determines the buckling strength and not the stiffness of the plate. This important criterion has to be taken into account for glass when for example, the initial deformation does not correspond to the first eigenform (EF1) or when a so called “snap through” of the plate may occur [Luible 2004].

b) Laminated Safety Glass

Tests on laminated safety glass elements demonstrated the influence of the PVB interlayer on the buckling strength. The comparison with the simulations confirmed that a composite action can be activated, but the shear modulus of the PVB has to be relatively high to create a noticeable increase in buckling strength. Even for short time loading and low temperatures the influence is not as high as it is for column buckling or lateral torsional buckling. Therefore the assumption of a monolithic behaviour of a laminated safety glass in buckling overestimates the real buckling resistance. The shear stiffness of the PVB interlayer has to be taken into account. In all tests with laminated safety glass both glass layers broke at the same time. Due to the high stored energy, the breakage pattern of heat strengthened glasses was almost as fine as it normally appears for toughened glass.

D. Synthesis for design

1) Design methods

Investigations of possible design methods for buckling of glass panels showed that it is important to know the distribution of the maximum tensile stress on the glass surface. Therefore, analytical models are not precise enough. A more precise approach may be by means of a FEM calculations presented in [Luible 2004]. In practice, glass panels are usually supported by soft interlayer materials, hence the assumption of restrained edges as it is often the case for steel plates overestimates the real buckling strength. A conservative simplification for design of glass panels are non constrained vertical edges (y-direction, Fig 19) and a load application by a constant compressive force on the horizontal glass edge. For loads lower than the critical buckling load a reasonable assumption for the initial deformation has to be made. FEM calculations are often too fastidious in practice;

therefore the possibility of developing design aids (e.g. buckling curves) was studied.

2) Buckling curves

The slenderness ratio and the reduction factor for plate buckling of glass panels were both based on the maximum tensile strength σp,t and defined as:

$$
\bar{\lambda}_p = \sqrt{\frac{\sigma_{p,t} t}{N_{cr,p}}} \qquad (29)
$$

$$
\rho = \frac{\sigma_p}{\sigma_{p,t}} = \frac{N_{x,p}}{\sigma_{p,t} t} \qquad (30)
$$

Reduction factors for different types of loading, glass geometries, initial deformations w₀ and boundary conditions were generated with FE model and plotted in buckling diagrams (e.g. Fig Fig 26).

<figure><img src="image_161.png" /><figcaption>Fig 26: Simulation of reduction factors.</figcaption></figure>

These simulation results are a first step towards a future definition of plate buckling curves for glass panels. The main results are:

* It is possible to define plate buckling curves for glass panels based on the tensile strength.

* For a slenderness ratio > 1.5 the curves are independent of the initial deformation.

* For a slenderness ratio < 1.5 the initial deformation has an influence on the buckling
strength and therefore the curves have to be
defined as a function of the initial deformation.

* The buckling curves for glass are below existing design curves in steel construction due to the different boundary conditions assumed. Buckling curves for steel are therefore not suitable for glass.

## VI. CONCLUSIONS

Stability is an important design criterion for load carrying glass elements. It was shown in the studies of this research work that the main influences on the buckling resistance of a load carrying glass element are the dispersion of the glass thickness, the initial deformation, the tensile resistance of the glass surface and the composite action due to the PVB interlayer in laminated safety glass. Residual stresses near the glass edges have to be known exactly in order to determine the realistic tensile resistance. Near the glass edges the residual stress of tempered glass is normally less than in the centre of the glass panel. The assumption of a tensile strength that is based on breakage tests carried out in the middle of the glass surface overestimates therefore the real load bearing capacity.

One of the main differences as compared with other materials is that the tensile strength limits the buckling strength.

A suitable design method for column buckling of glass elements seems a second order stress analysis. On the contrary, this approach is not suitable for lateral torsional buckling and plate buckling for which buckling curves are more appropriate. One of the main differences as compared with buckling curves in steel structures is that the slenderness ratio the reduction factors must be based on the maximum tensile strength of glass.

## VII. ACKNOWLEDGMENT

The work in this paper was primarily conducted with the support of the Swiss National Science Foundation (SNF) and the industry partners Glas Trösch (Bützberg, Switzerland) and Verre Industriels SA (Moutier, Switzerland).

## REFERENCES

[ANSYS 2002] ANSYS, Release 6.1, SAS IP, Inc, 2002.

[Bennison et al. 2002] BENNISON, S. J., SMITH, C. A., VAN DUSER, A., JAGOTA, A., Structural Performance of Laminated Glass Made with a "Stiff" Interlayer, Glass in Buildings, ASTM STP 1434, V. Block, ed., ASTM International, West Conshohocken, PA, 2002.

[Booker et al. 1974] BOOKER, J.R., FRANKHAM, B.S., TRAHAIR, N.S., Stability of Visco-Elastic Structural Members, Civil Engineering Transactions, pp. 45-51, The Institution of Engineers, Australia, 1974.

[Hirt et al. 1998] HIRT, M. A., BEZ, R., Stahlbau: Grundbegriffe und Bemessungsverfahren, Ernst & Sohn, Berlin, 1998.

[Laufs 2000.2] LAUFS, W., Ein Bemessungs- konzept zur Festigkeit thermisch vorgespannter Gläser, Schriftenreihe Stahlbau RWTH Aachen, Shaker Verlag, Aachen, 2000.

[Luible 2004] LUIBLE, A., Stabilität von Tragelementen aus Glas, Thèse EPFL 3014, Ecole polytechnique fédérale de Lausanne (EPFL), Lausanne, 2004.

[Luible et al. 2004] Luible, A., Crisinel, M., Buckling Strength of Glass Elements in Compression, Structural Engineering International, Vol. 14, No. 2, IABSE, Mai 2004.

[Stamm et al. 1974] STAMM, K., WITTE, H., Sandwichkonstruktionen - Berechnung, Fertigung, Ausführung, Springer Verlag, Wien, 1974.

[Van Duser et al. 1999] VAN DUSER, A., JAGOTA, A., BENNISON, S., J., Analysis of Glass/Polyvinyl Butyral Laminates Subjected to Uniform Pressure, Journal of engineering mechanics, Vol. 125, pp. 435-442, 1999.

[Zenkert 1997] ZENKERT, D., The Handbook of Sandwich Construction, Engineering Materials Advisory Service Ltd., United Kingdom, 1997.
# Fastening of Glass Panes with Undercut Anchors – FEA and experimental investigations

Peter Hof, Darmstadt University of Technology, Statics and Dynamics of Structures,
Darmstadt, Germany

Jens Schneider, Schlaich, Bergermann und Partner, Stuttgart, Germany

Undercut anchors were so far seldom used in glass due to brittleness and the risk of subcritical crack growth. The use of tempered glass in combination with a special drilling-process and -geometry of the undercut holes now made it possible to develop a system that allows this method of fastening glass panes. In a parameter study of 3-D numerical tempering simulations with Narayanaswamy's structural relaxation model, the amount of residual stress in the area of the undercut hole was estimated. Temper stresses were compared to surface stresses in the “infinite” area that were measured by photoelastic methods. Pull-out tests and shear-off tests of anchors from annealed and tempered glass were performed and results compared to the numerical results. Both prove that a significant residual compression stress from tempering exists in the undercut area. Finally load bearing tests showed the performance of the system.

**Keywords:** Glass panes, Undercut Anchors, Pull-out test, heat-transfer coefficient

## I. INTRODUCTION

Glass is one of the oldest materials used by mankind and the applications in the field of civil engineering increase successfully since many years. Nevertheless, the understanding of the non-crystalline solid as a state of matter is still poor.

Undercut anchors are a relatively young but successful method of fastening. Today, they are mainly used in concrete, stone or ceramics. The idea is to drill an “undercut” borehole and fix the anchor in the undercut area with a special mechanism (Fig 1, 2 and 3). In concrete, the anchors are drilled after the setting of the concrete, usually to fix steel elements. They can sustain very high loads. Applications for undercut anchors in combination with stone or ceramics are usually facade panels. Here, they guarantee a simple but powerful way of fastening with one main advantage: invisibility of the anchor from the outside of a facade pane. For these applications, the undercut holes are drilled in each pane automatically with CNC-drilling machines at a very high accuracy. In glass, the use of undercut anchors was so far problematic due to the extreme brittleness, subcritical crack growth in glasses and the stress concentration in the undercut area.

<figure><img src="image_162.png" /><figcaption>Fig 1: Geometry of undercut holes in glass (glass thickness 10 mm and 12 mm)</figcaption></figure>
<figure><img src="image_163.png" /><figcaption>Fig 2: Anchor system</figcaption></figure>

<figure><img src="image_164.png" /><figcaption>Fig 3: Details of the anchor system: a) plastic cap, b) curved steel part (steel snake), c) screw with conical head, d) plastic washer, e) nut</figcaption></figure>

After some years of research, Fischerwerke in Waldachtal, Germany, developed a special drilling process, geometry for the undercut holes and plastic interlayer between steel anchor and glass [Fischer 2000]. In combination with tempered or heat strengthened glass it is now possible to fasten glass panes with undercut anchors. 3-D numerical tempering simulations and pull-out and shear-off tests of anchors from annealed and tempered glass show that a significant residual compression stress from tempering exists in the undercut area. For the numerical tempering models with the finite-element-code ANSYS 5.6 [Ansys 1999], material data and relaxation constants from [Carré 1999] were used to perform a parameter study with different heat transfer coefficients in the undercut area whereas the apparent heat transfer coefficient in the “infinite” (surface) area of the tempered samples was approximated from photoelastic measurements of the surface stress of 10 mm and 12 mm tempered glass samples. These samples and annealed samples were used in pull-out tests of anchors. These tests were simulated in 3-D numerical models with glass, anchor and plastic interlayer to calculate failure stresses. Finally, from the glass strength of the annealed samples and the tempered samples, the apparent heat transfer coefficient in the undercut area could be approximated from a parameter study.

## II. NUMERICAL TEMPERING SIMULATION

### A. Finite-Element modelling

The thermal stresses for glass plates with undercut holes were calculated by simulating the tempering process with the finite-element program ANSYS 5.6. The theories of viscoelasticity and structural relaxation as described by Narayanaswamy [Narayanaswamy 1971, 1978] are implemented in ANSYS. Glass plates of the thickness 10 mm and 12 mm were examined. The glass plates are cooled from an initial temperature of 670 °C to 20 °C by forced convection on both sides.

A 3-D FE-model of the glass plate was constructed using the ANSYS FE code. Due to the rotational symmetry only a part of the glass plate is modelled. ANSYS SOLID90-elements were used to calculate the distribution of temperature over the duration of cooling. Afterwards these elements were replaced by ANSYS SOLID95-elements to calculate stresses. The test specimens consist of 6 mm thick glass plates. Every plate is 1 m long and 0.1 m wide. Altogether nine specimens have been investigated in this preliminary study. Six of the specimens are made of tempered glass and three of heat strengthened glass.

### B. Cooling parameters and Material properties

The cooling parameters and the material properties of the glass, e.g. the viscous and structural relaxation times, were obtained from Carré [Carré 1999]. The thermal conductivity and the specific heat of the glass were considered to be constant and not to vary with the temperature as suggested by Carré. The thermal conductivity was set to $1 = 1,05 \text{ W/(mK)}$ and the specific heat to $cp = 980,4 \text{ J/(kgK)}$.

The only unknown parameter of the tempering process simulation is the apparent heat transfer coefficient. The heat transfer coefficient is
apparent, as the heat radiation influence was
neglected in this study. With photoelastic
measurements (Laser-Gasp, Strainoptics, U.S.A) it
is possible to obtain the tempering stresses at the
free surface of glass samples. The surface
compression stress of the 10 mm and 12 mm glass
plates was found to be approx. 100 MPa for all
tempered samples. Calculations were carried out to
identify the apparent heat transfer coefficient that
leads to a surface compression stress of 100 MPa.
Good results could be obtained with a heat transfer
coefficient of 135 W/(m²K) at the free surface. It is
difficult to measure the tempering stresses within
the area of the undercut hole. To get qualified
results for the stress distribution therefore the heat
transfer coefficient in the area of the undercut hole
was varied between 20 W/m²K and 135 W/m²K.

### C. *Parameter study*

The stresses resulting from thermal tempering
were calculated for an apparent heat transfer
coefficient at the free surface of 135 W/(m²K) and
20, 40, 60, 80, 100, 120, 135 W/(m²K) in the
undercut area. The distribution of the principal
compressive stresses for heat transfer coefficients
of 20, 60 and 135 W/(m²K) in the area of the hole
are shown in figure 4. Figure 4 also shows the path
along which the stresses were calculated. The
principal compressive stresses along this path are
plotted in figure 5.

<figure><img src="image_165.png" /><figcaption>Fig 4: Principle compression stress for heat transfer coefficient of 20, 60 and 135 W/(m²K) in the borehole area</figcaption></figure>

(135 W/(m²K) at free surface), nodes of a path along the
borehole

<figure><img src="image_166.png" /><figcaption>Fig 5: Computed principal stresses (compression) for different heat transfer coefficients in the borehole area (135 W/(m²K) at the free surface) along the path in Fig 4.</figcaption></figure>

## III. DETERMINATION OF THE HEAT-TRANSFER COEFFICIENT IN THE UNDERCUT AREA

### A. *Pull-out tests*

Pull-out tests with annealed (float)-glass and tempered glass were performed (Fig 6). Specimens of 300 mm x 300 mm for each glass type and thickness were tested. In a Finite Element Analysis (FEA) (fig 7) these tests were simulated to calculate failure stresses in the undercut area. Special contact elements between the glass surface and plastic material were used to simulate a realistic load distribution. From the results and the crack origins of annealed glass, the failure stress could be identified for each specimen. It was proved to be almost the same as the glass strength in the borehole area of annealed glass with drilled holes [Schneider 2001]. It was difficult to locate the exact crack origin for tempered glasses but due to an almost linear behaviour, the stress distribution in tempered glass could be calculated from failure loads with the FE-model. Presuming that the glass strength of tempered glass results in an addition of the inherent (annealed) glass strength and the temper stresses (no crack healing), a vector addition of the failure stresses of annealed glasses and the temper stresses for different heat
transfer coefficients along the path of figure 4 could be made to identify the temper stresses and heat transfer coefficient that fit the failure load results of tempered glass (Fig 8, approx. 60 W/(m²K)).

<figure><img src="image_167.png" /><figcaption>Fig 6: Section of test set-up (pull-out tests), FE-simulation of pull-out tests (glass, anchor and plastic interlayer)</figcaption></figure>

<figure><img src="image_168.png" /><figcaption>Fig 7: Crack origins (A to E) for tests with annealed (float) glass, stress distribution in the undercut area (principal tension stress) for failure load [Block 2001]</figcaption></figure>

<figure><img src="image_169.png" /><figcaption>Fig 8. Resulting principle tensile stresses on the surface of a 10 mm tempered glass for different heat transfer coefficients, grey area: failure range of annealed glass, pull-out test</figcaption></figure>

Note that stresses at the top edge of the chamfer (IV) equal to almost zero whereas significant tension stresses cause failure in tempered glasses in the oblique area (II-III) due to lower temper stresses there.

## B. Shear-off tests

Additionally to the pull-out tests shear-off tests were made. The specimens were the same as described for the pull-out tests. The test set-up is given in figure 9. Because of the steel snake as shown in figure 3 it was necessary to divide the simulation of the test in two models. The models differ in the angle between the load-direction and the direction of the steel snake (Fig 10).

<figure><img src="image_170.png" /><figcaption>Fig 10: Model 1 (left) and Model 2 (right) for Shear-off test</figcaption></figure>

The screw and the steel snake were modelled as a quarter for each part of the steel snake. So the resulting tensile stresses differ in value and distribution. The stresses for the two models are shown in figure 11 for a glass thickness of 10 mm. As shown only half of the drilling and the anchor system is simulated because of symmetric reasons. The results of the principle tensile stresses are given in figure 13 for the model 1. The results are calculated separately for two paths (Fig 12). One path called edge, at the edge of the model and one called middle, which is set in the area of the maximum stresses beside the screw.
<figure><img src="image_171.png" /><figcaption>Fig 11: Tensile stresses for the shear-off test for the two models.</figcaption></figure>

<figure><img src="image_172.png" /><figcaption>Fig 12: Paths edge and middle</figcaption></figure>

<figure><img src="image_173.png" /><figcaption>Fig 13 a: Resulting principle tensile stress on the surface of a 10 mm tempered glass for different heat transfer coefficients, grey area: failure range of annealed glass, shear-off test, model1, path edge</figcaption></figure>

<figure><img src="image_174.png" /><figcaption>Fig 13b: Resulting principle tensile stress on the surface of a 10 mm tempered glass for different heat transfer coefficients, grey area: failure range of annealed glass, shear-off test, model 1, path middle</figcaption></figure>

### C. Results

With consideration of the pull-out test and the shear-off test for both examined thicknesses, the apparent heat transfer coefficient could be identified to be always in the range of 60 ± 20 W/(m²K) for different commercial glasses and tempering processes. The results of the pull-out test deviate less than the results from the shear-off tests. For this heat transfer coefficient, the whole undercut area is under compression (Fig 5) to prevent subcritical crack growth.

## IV. FULL SCALE TESTS

As further investigations full scale test were examined. The specimens were tested in the dimension from 1200 x 2200 mm² up to 1200 x 4000 mm². The glass types varied from tempered glass with 10 mm and 12 mm, to laminated safety glass made of tempered and heat strengthened glass. Different numbers of fittings for the different dimensions of the glass specimens were taken. The tests were done by loading the glass specimen using an airbag with air compression. The load was kept constant for a certain span of time at defined load steps. For each load step the displacement of several points of the specimens were taken. The test set-up is shown in figure 14. The results for the compression load test were satisfying and lead to the possibility to use the tested specimens up to 100 m high buildings with a safety factor of 3,0 (for this load case). The suction test gave inferior results. Especially the high loads at the edges of buildings from wind suction allow the use only up to 8 m at the moment.
<figure><img src="image_175.png" /><figcaption>Fig 14: Full scale load bearing tests</figcaption></figure>

## V. CONCLUSION AND FURTHER RESEARCH

Fastening of glass panes with undercut anchors is a possibility for tempered glass. A special geometry of hole and anchor, a plastic interlayer and very accurate CNC-drilling are required. Numerical results show that apparent heat transfer coefficients (neglecting the heat radiation influence) of 60 ± 20 W/(m²K) in the undercut area are realistic for commercial glasses and tempering processes preventing subcritical crack growth due to surface compression stresses in the whole undercut area. Photoelastic measurements in the undercut area should be made to validate the results. Moreover, full scale load capacity tests proofed the performance of the anchor system. Further researches deal with a new circlip instead of the curved steel for a higher load bearing capacity.

## REFERENCES

**[Aben, 1993]**
ABEN, H., GUILLEMET, C., *Photoelasticity of Glass*, Springer, Berlin 1993

**[Ansys 1999]**
ANSYS 5.5.2, *Theory Reference 4.6: Viscoelasticity*, Eigth Edition, 1999

**[Bathe 1990]**
BATHE, K.J., *Finite-Elemente-Methoden*, Springer, Berlin 1990

**[Block 2001]**
BLOCK, C., *Bestimmung der Festigkeit von Glas im Bereich einer Hinterschnittbohrung*. Studienarbeit am Institut für Statik, TU Darmstadt, 2001, unpublished

**[Carré 1999]**
CARRÉ, H.; DAUDEVILLE, L., "Load-bearing capacity of tempered structural glass", Journal of Engineering Mechanics, Vol. 125 (1999), No. 8, pp 914-921

**[Fischer 2000]**
ARTUR FISCHER GMBH & CO. KG, *Informationsblatt über Glasbefestigung mit Hinterschnittanker FZP-G*, Edition 4/2000

**[Laufs 2000]**
LAUFS, W., *Ein Bemessungskonzept zur Festigkeit thermisch vorgespannter Gläser*. Dissertation (PhD-thesis), Lehrstuhl für Stahlbau der RWTH Aachen, Shaker-Verlag, 2000.

**[Narayanaswamy 1971]**
NARAYANASWAMY, O.S., *A Model for Structural Relaxation in Glass*, J. Am. Ceram. Soc., Vol. 54 (1971) No. 10, pp 491-498

**[Narayanaswamy 1978]**
NARAYANASWAMY, O.S.: *Stress and Structural Relaxation in Tempering Glass*, J. Am. Ceram. Soc., Vol. 61 (1978) No. 3, pp 146-152

**[Scherer 1986]**
SCHERER, G.W., *Relaxation in Glass and Composites*. Wiley, New York 1986.

**[Schneider 2001]**
SCHNEIDER, J.: *Festigkeit und Bemessung von Glas: Punktgelagerte Verglasungen, Dynamisch beanspruchte Verglasungen bei weichem Stoß*. Dissertation (PhD-thesis), Institut für Statik TU Darmstadt, 2001.

**[Soules 1987]**
SOULES, T.F., BUSBEY, R.F., REKHSON, S.M., MARKOVSKY, A., BURKE, M.A., *Finite Element Calculation of Stresses in Glass Parts Undergoing Viscous Relaxation*, J. Am. Ceram. Soc., Vol. 70 (1987) No.2, pp 90-95
# Point bearing elements - Research investigations

Ruth Kasper and Gerhard Sedlacek

RWTH Aachen – Institute of Steel Construction, Germany

Point bearings are a popular element of modern glass constructions. On the one hand point bearings are elements that can be used to design transparent buildings, on the other hand point fixings allow for a good failure resistance after breakage of a laminated glass panel.

Due to the brittle behaviour of glass the knowledge about the rupture process in the glass hole and the influence of the point bearing geometry is important.

The article gives an overview on the investigated research projects of the RWTH Aachen, the results as well as the proposed methods to solve the design problem “point bearing”.

**Keywords:** glass, design, point-fixings

## I. INTRODUCTION

Figure 1 shows an transparent roof structure made of point supported glazing.

<figure><img src="image_176.png" /><figcaption>Fig 1: Example</figcaption></figure>

Point fixings possess different geometries and materials to avoid contact between glass and steel. The geometry of the hole can be cylindrical or conical (figure 2). Besides the general static system

these parameters have an influence on the stress concentration around the glass holes caused by loading.

<figure><img src="image_177.png" /><figcaption>Fig 2: Types of point-fixings</figcaption></figure>

The quality of the stress concentrations can be determined with finite element methods by modeling the hole, the geometry of the point-fixing and the stiffness of the separating materials. The results can be manipulated by variation of several parameters. Following this, finite element analysis are not useful for this application without any verifying tests. To ensure a safe design of point-supported glazing a method needs to be developed that takes into account the individual behaviour of the types of point-fixings.

Generally, the material used for point-supported glazing is thermally toughened or heat strengthened glass. Thermally treated glasses
possess a higher material resistance and can bear the concentrated stresses near to the holes. The material resistance near to the hole has been analyzed by [Laufs 2000, Carre 1996, Schneider 2001]. The investigations included breakage tests (figure 3), numerical modelling of the distribution of the pre-stressing and measuring of the pre-stressing, both near to the hole. The latter has been done by using optical measurement devices. Therefore the material resistance of the hole of tempered glass is known. Figure 4 shows the distribution of the pre-stressing of a conical hole.

<figure><img src="image_178.png" /><figcaption>Fig 3: Breakage tests at glass holes</figcaption></figure>

<figure><img src="image_179.png" /><figcaption>Fig 4: Distribution of the pre-stressing near to the hole of tempered glass</figcaption></figure>

In the frame of actual research investigations of the University of Aachen three different methods

have been analyzed and evaluated. All design concepts are based on small-scale tests. With the aid of the small-scale tests force-stress-characteristics for each type of point-fixing could be developed.

**Design concept 1** uses the developed force-stress-characteristics for the calibration of simplified finite element models. The functionality of this method has been proofed [Wolf 2004, Kasper 2004].

**Design concept 2** contains the proposition that the quantity of the stress concentration is depending on the reaction forces at the point-fixings, but the numerical investigation showed that this proposition is not valid [Wolf 2004].

**Design concept 3** is based on the hot-spot method used in the field of steel constructions. The adaptability on point-fixings has been analyzed and the evaluation showed that further investigations are necessary for the validation [Wolf 2004].

In the following the execution of the tests for the development of the force-stress-characteristics are described. Furthermore the results of the examination of the different design methods are summarized.

## II. DEVELOPMENT OF FORCE-STRESS-CHARACTERISTICS FOR POINT-FIXINGS

The force-stress-characteristic of point fixings depends on the geometry of the test specimen and the geometry of the point-fixings. Here, a standard point-fixing is used exemplary to show the usability of the concept.

The force-stress-characteristic of point-fixings has been developed for test specimen with the size of 400 mm x 400 mm. The thickness of the test specimen was 10 mm. The tests specimens were line-supported on two opposite sides (figure 6). The specimens were loaded under 4 different angles. Figure 5 shows the test set-up and the different loading directions. During the tests strain gauges measured the strain at several points (figure 6).
<figure><img src="image_180.png" /><figcaption>Fig 5: Test set-up and loading directions of the small-scale tests</figcaption></figure>

<figure><img src="image_181.png" /><figcaption>Fig 6: Strain measurements and static system</figcaption></figure>

The evaluation of the test results (force-strain-curves) has been based on finite element calculations. For the finite element model volume elements and contact approaches were used. Figure 7 shows details of the finite element model.

<figure><img src="image_182.png" /><figcaption>Fig 7: Details of the finite-element-model</figcaption></figure>

The model's calibration has been done by variation of the E-Modulus of the separating materials and by comparing the measured as well as the calculated strains. For each loading direction a data sheet has been developed. The data sheet shows the comparison between the measured and the calculated values on the different paths, the distribution of the principal stresses and the maximum stress concentration next to the hole. Table 1 shows an abstract of the data sheet for the loading direction of 0°. Analogous data sheets for each loading angles have also been developed.
<figure><img src="image_183.png" /><figcaption>TABLE 1:
DATA SHEET WITH STRESS-STRAIN-CHARACTERISTIC FOR THE
LOADING DIRECTION OF 0°</figcaption></figure>

### III. TEST OF DESIGN CONCEPT 1 - USING FORCE-STRESS-CHARACTERISTICS FOR THE CALIBRATION OF SIMPLIFIED FINITE ELEMENT MODELS

For the verification of this design method further original size tests were executed. The proceeding was the same as for the small-scale tests. The tests were made for two different geometry: 1000 mm x 1000 mm with 4 point-fixings and 2000 mm x 1000 mm with 6 point-fixings. The test specimens were loaded with a single load at midspan.

The original size tests were also evaluated with finite element calculations: the results show that the discrepancies between the measured and the calculated strain values are higher than for the small-scale tests. Reasons for this can be imperfections of the test set-up.

The next step was the use of simplified models calibrated with the aid of the data sheets including the force-stress-characteristics. If the model confirms the results or gives higher stress-concentrations next to the hole, the model can be used for the design of the point-supported glass panel. After the results of the original size tests were confirmed with the simplified model.

Table 2 and table 3 compare the results of the stress concentration of the calibrated and a simplified model. For the simplified model also volume elements were used but the contact approach was neglected. The results show that it is possible to determine the stress concentrations of large scale components with a model which has been calibrated on the basis of small scale tests.

<table><caption>TABLE 2: COMPARISON BETWEEN THE STRESS CONCENTRATIONS (CALIBRATED MODEL AND SIMPLIFIED MODEL) - SMALL SCALE TESTS</caption><thead><tr><th></th><th>Calibrated model</th><th>Simplified model</th><th>Ratio</th></tr></thead><tbody><tr><td>E-Modulus elastomer ring</td><td>E = 25 N/mm²</td><td>E = 10 N/mm²</td><td>[-]</td></tr><tr><td>0°</td><td>92,4 N/mm²</td><td>95,9 N/mm²</td><td>1,04</td></tr><tr><td>22,5°</td><td>89,7 N/mm²</td><td>90,2 N/mm²</td><td>1,01</td></tr><tr><td>45°</td><td>75,9 N/mm²</td><td>75,1 N/mm²</td><td>0,99</td></tr><tr><td>90°</td><td>46,5 N/mm²</td><td>55,2 N/mm²</td><td>1,19</td></tr></tbody></table>

<caption>TABLE 3: COMPARISON BETWEEN THE STRESS CONCENTRATIONS (CALIBRATED MODEL AND SIMPLIFIED MODEL) - LARGE SCALE TESTS</caption>
<table><thead><tr><th></th><th>Calibrated model</th><th>Simplified model</th><th>Ratio</th></tr></thead><tbody><tr><td>E-Modulus elastomer ring</td><td>E = 25 N/mm²</td><td>E = 10 N/mm²</td><td>[-]</td></tr><tr><td>1000 mm x 1000 mm with 4 point supports</td><td>53,5 N/mm²</td><td>62,5 N/mm²</td><td>1,17</td></tr><tr><td>2000 mm x 1000 mm with 6 point supports</td><td></td><td></td><td></td></tr><tr><td>Hole at the edge</td><td>8,8</td><td>9,98</td><td>1,134</td></tr><tr><td>Hole in the corner</td><td>88,4</td><td>93,5</td><td>1,058</td></tr></tbody></table>

<table><thead><tr><th rowspan="2">System</th><th rowspan="2">Geometry</th><th rowspan="2">Span</th><th rowspan="2">q</th><th>A<sub>x</sub></th><th>A<sub>y</sub></th><th rowspan="2">σ<sub>max,hole</sub></th></tr><tr><th>an<br>d<br>A<sub>y</sub></th><th>[N]</th></tr><tr><th></th><th>[m]</th><th>[mm]</th><th>[N/mm²]</th><th>[N]</th><th>[N]</th><th>[N/mm²]</th></tr></thead><tbody><tr><td>1</td><td>1,0 x 1,0</td><td>800</td><td>0,005</td><td>0</td><td>125</td><td>31,4</td></tr><tr><td>2</td><td>1,25x1,25</td><td>1050</td><td>0,0032</td><td>0</td><td>125</td><td>35,3</td></tr><tr><td>3</td><td>1,5 x 1,5</td><td>1300</td><td>0,0022</td><td>0</td><td>125</td><td>37,7</td></tr><tr><td>4</td><td>2,0 x 1,0</td><td>900</td><td>0,0028</td><td>0</td><td>125</td><td>46,5</td></tr></tbody></table>

#### IV. TEST OF DESIGN CONCEPT 2 - “THE STRESS CONCENTRATION NEXT TO THE HOLE DEPENDS ON THE REACTION FORCES AT THE POINT-FIXING”

It can be easily shown that this thesis is not valid. For the demonstration the calibrated finite element model was used. The system is statically determined and the eccentricity of the point-fixing is equal to zero, which means that only vertical reaction forces exist. The geometry of the system is varied and the system is loaded with a uniformly distributed loading. The quantity of the loading is determined in that way that the vertical reaction forces are equal to 1250 N for each system.

The results are shown in Table 4. The maximum principal stresses vary between 31,4 N/mm² and 46,5 N/mm². That means that the stress concentration does not depend only on the quantity of the reaction force but also on the global proportions of the plate. For comparison the stress concentration for the centrical loaded (0°C, F = 1250 N) small scale test specimen is 46,2 N/mm².

The results show that the thesis is not useful and the quantity of the stress concentration is influenced by further parameters.

#### V. TEST OF DESIGN CONCEPT 3 - “HOT SPOT METHOD”

The hot-spot method is used, for example, for designing details of steel constructions. The stress concentrations due to welds or details are taken into account by the aid of stress concentration factors. It is sufficient to determine the global stresses near to the detail and to multiply the stress with the stress concentration factor.

The first question to answer is: Is it possible to determine a reasonable stress distribution over the plate without a detailed point-fixing? The plate is modeled without holes and one node in the geometrical center of the hole is restrained in vertical direction. Depending on the number of elements the global stress distribution is equal to the stress distribution of the model with a detailed point fixing (figure 8 and 9). Consequently it is possible to verify a geometrical stress that is more or less independent of the modeling of the point fixing.

Figure 10 shows the influence of the plate size on the stress distribution in the area of the point fixing. The scaled illustration makes clear that the form of the stress concentration is not only depends on the geometry and the materials of the point fixing but also on the geometry of the plate.

<caption>TABLE 4 RESULTS OF NUMERICAL STUDY FOR DESIGN CONCEPT 2</caption>
<figure><img src="image_184.png" /><figcaption>Fig 8: Stress distribution over the diagonal of a 1,0 m x 1,0 m plate depending on the number of elements</figcaption></figure>

<figure><img src="image_185.png" /><figcaption>Fig 9: Comparison between stress distribution of the simplified model and the detailed model next to the hole (1,0 m x 1,0</figcaption></figure>

<figure><img src="image_186.png" /><figcaption>Fig 10: Comparison between the stress distribution next to the hole for different sizes of the plate (non-scaled and scaled illustration)</figcaption></figure>

Furthermore the investigations have shown that the stress concentration factor not only depends on the size and the geometry of the plate but also on the kind of loading (existent distributed loading near to the point fixing).

## VI. SUMMARY AND PROSPECTS

The paper shows different possibilities for the design of point-supported glazing. The design based on force-stress-characteristic is already a proved possibility for the specification of the behaviour of point-fixings. With further research investigations the possibility of the application of the “hot-spot” needs to be analyzed to define the different parameters.

## VII. ACKNOWLEDGMENT

The investigations could be done by the support of the “Fachverband Konstruktiver Glasbau (FKG) e.V.”.

## REFERENCES

[Carre 1996] Carre, H: Etude du comportement a la rupture d'un materiau fragile precontraint : le verre trempe. These 1996.

[Kasper 2004] Kasper, R; Sedlacek G: Bemessungskonzept für punktförmig gelagerte Scheiben. FKG - Arbeitskreis Punktgestützte Gläser. 8. Forschungsvorhaben 2004.

[Laufs 2000] Laufs, W: Ein Bemessungskonzept zur Festigkeit thermisch vorgespannter Gläser. Promotion 2000.

[Schneider 2001] SCHNEIDER, J.: Festigkeit und Bemessung von Glas: Punktgelagerte Verglasungen, Dynamisch beanspruchte Verglasungen bei weichem Stoß. Dissertation (PhD-thesis), Institut für Statik TU Darmstadt, 2001.

[Wolf 2004] Wolf, H: Experimentelle und numerische Untersuchung von punktförmig gehaltenen Glasscheiben im Hinblick auf ein Bemessungskonzept. Diplomarbeit 2004 Lehrstuhl für Stahlbau RWTH Aachen (not published)
# Pre-design of discretely supported glass under uniform loading with the help of interpolation

Jürgen Neugebauer

Graz University of Technology – Institute of Structural design, Austria

In the planning process it is helpful that rectangular glass panels under uniform loading can be designed very fast. Furthermore it is important to check the results of a FE calculation.

A calculation of many different discretely supported glass panels under an uniform load with the value $q=1,0 \text{ kN/m}^2$ with the help of the FE method results in values of stresses and deformations. The insertion of these values in a 3D-diagramm, results in a surface of results of stresses or deformations. For a pre-design this surface of the shape of a hyperbolic paraboloidal shell is exact enough. A linear interpolation in this surface is very easy.

With a short number of input data, such as the distance in direction length, the distance in direction across or the thickness of the glass panel, it is possible to interpolate the results of stress in this shape. The interpolation results must be calibrated with other influences for example with the diameter of the hole and the distance of the hole to the edges and with the shore hardness of the rubber at the support.

**Keywords:** interpolation, discrete support, glass fitting

## I. INTRODUCTION

For a finite element calculation there are many different parameters to be defined, on the one hand geometrical parameters such as the size or the thickness of the glass pane, and on the other hand parameters of the different materials such as the glass, the rubber or the steel of the glass fitting.

For results a lot of time is needed, which in some cases does not exist. A concept to shorten the design time is desirable. Furthermore it is important to check the computed results of a FE method calculation. A possibility is, to check all the input data, and to believe the computed results. Another possibility is, to check the results itself, with calculated results one gets in another way.

## II. STRESSES AND DEFORMATIONS

A finite element calculation results in values for the stresses and values for the deformations of each point of the glass pane. In figure 1 to figure 3 see the results of a finite element calculation. The results are the principal stresses $\sigma_1$ and $\sigma_2$ and the deformation $w$.

<figure><img src="image_187.png" /><figcaption>Fig 1: Results of the principal stresses $\sigma_1$</figcaption></figure>

<figure><img src="image_188.png" /><figcaption>Fig 2: Results of the principal stresses $\sigma_2$</figcaption></figure>
<figure><img src="image_189.png" /><figcaption>Fig 3: Results of the deformation w</figcaption></figure>

One can see that the maximum of the stresses $\sigma_1$ are located in the area around the hole and the maximum of the principal stresses $\sigma_2$ are located in the middle of the glass pane. The maximum deformation is in the middle of the glass pane. In most cases the maximum of principal stresses are located around the hole.

### III. DIFFERENT STATIC SYSTEMS

The most used glass panes are rectangular. For this type it is possible to separate the static systems by the numbers of the glass fittings, see in figure 4. The results of a calculation of the principal stress $\sigma_1$, see in figure 5, shows that there is a big difference between the results of the system with 4 and system with 6 glass fittings. By a further increase of the number of the supports the difference between the results is little. One can reduce the problem to three kinds of rectangular glass panes with 4, 6 and 8 glass fittings, see in figure 4.

<figure><img src="image_190.png" /><figcaption>Fig 4: Different static system of point load supported glass panes</figcaption></figure>

<figure><img src="image_191.png" /><figcaption>Fig 5: Results of the principal stresses $\sigma_1$ with different numbers of glass fittings</figcaption></figure>

### IV. INPUT DATA

The basic idea for the interpolation is to define different geometrical sizes, analyse them, and interpolate in between the different values.

The geometrical input data required for the interpolation are the distance in direction length $d_L$ the distance in direction across $d_C$ and the thickness see in figure 6.

The materials are defined with the elasticity modulus $E = 70000 \text{ N/mm}^2$ and the Poisson's ratio $\nu = 0.23$. These values for the materials are constant values in the interpolation process.

<figure><img src="image_192.png" /><figcaption>Fig 6: Input data of the glass pane</figcaption></figure>

* $d_C$: distance between the glass fitting in direction across [mm]

* $d_L$: distance between the glass fitting in direction length [mm]

* $d_H$: diameter of the glass hole [mm]

* $d_{e1}$: distance to the edge in direction across [mm]

* $d_{e2}$: distance to the edge in direction length [mm]
## V. INTERPOLATION OF PRINCIPAL STRESSES

The results of the principal stresses of different calculations with different geometries are able to be drawn into diagrams. The diagrams for the different static systems are shown in the figures 7 to figure 9. An important fact is that the graphs of the results in the different static systems are nearly straight. With this fact a linear interpolation in between them is possible. Coefficients depending on the numbers of glass fittings are created in table 1 to table 3.

The basis for the calculation of the maximum principal stresses is the equation (1). Depending on the numbers of the glass fittings the coefficient C has to be chosen out of table 1 to table 3. Interim values can be interpolated linear. The interpolation is based on a fixed diameter of the hole, a fixed distance to the edges and a fixed shore hardness of the rubber at the supports.

For the calculations the thickness of the glass pane was assumed to be $h=10$ [mm] and the diameter of the hole was $d_H=30$ [mm]. The distance to the edges were given with $d_{e1} = 100$ [mm] and $d_{e2} = 110$ [mm]. The load was given with $q=1.0$ [kN/m²]. For the rubber at the supports the shore hardness was assumed to be 20 [-]. The results of the principal stresses are drawn into the diagram shown in figure 7 to figure 9. For all other geometrical input data one can multiply the interpolated result with the coefficients ($c_1, c_2, c_3$). These influences are dealt with later. With the assumption of theory of small deformations one can do a linear static calculation.

$$ \sigma_{\max} = \frac{C}{h_{ef, \sigma i}^2} \cdot q \cdot c_1 \cdot c_2 \cdot c_3 \cdot 1E6 \quad (1) $$

$\sigma_{\max}$ maximum principal stress [N/mm²]

C coefficient dependent on the numbers of glass fittings [mm²]

q uniform load on the glass pane [N/mm²]

$c_1$ coefficient depending on the influence of the diameter of the hole [-]

$c_2$ coefficient depending on the influence of the distance to the edge [-]

$c_3$ coefficient depending on the influence of the shore stiffness of rubber [-]

$h_{ef, \sigma i}$ fictive thickness of the glass pane for the

verification of the stresses [mm]

### A. Point load fixation with 4 glass fittings

<figure><img src="image_193.png" /><figcaption>Fig 7: Principal stress $\sigma_1$ of a glass pane with 4 glass fittings</figcaption></figure>

<table><caption>TABLE 1: COEFFICIENT C FOR 4 GLASS FITTINGS</caption><thead><tr><th>dC/dL</th><th>200</th><th>600</th><th>1000</th><th>1400</th><th>1800</th></tr></thead><tbody><tr><td>200</td><td>0,11</td><td>0,23</td><td>0,37</td><td>0,52</td><td>0,66</td></tr><tr><td>600</td><td>0,23</td><td>0,46</td><td>0,77</td><td>1,08</td><td>1,39</td></tr><tr><td>1000</td><td>0,37</td><td>0,77</td><td>1,28</td><td>1,82</td><td>2,35</td></tr><tr><td>1400</td><td>0,52</td><td>1,08</td><td>1,82</td><td>2,58</td><td>3,36</td></tr><tr><td>1800</td><td>0,66</td><td>1,39</td><td>2,35</td><td>3,36</td><td>4,38</td></tr></tbody></table>

### B. Point load fixation with 6 glass fittings

<figure><img src="image_194.png" /><figcaption>Fig 8: Principal stress $\sigma_1$ of a glass pane with 6 glass fittings</figcaption></figure>
<table><caption>TABLE 2:
COEFFICIENT C FOR 6 GLASS FITTINGS</caption><thead><tr><th>dC/dL</th><th>200</th><th>600</th><th>1000</th><th>1400</th><th>1800</th></tr></thead><tbody><tr><th>200</th><td>0,11</td><td>0,67</td><td>1,73</td><td>3,24</td><td>5,22</td></tr><tr><th>600</th><td>0,26</td><td>1,09</td><td>2,55</td><td>4,46</td><td>6,83</td></tr><tr><th>1000</th><td>0,45</td><td>1,48</td><td>3,51</td><td>6,01</td><td>8,94</td></tr><tr><th>1400</th><td>0,67</td><td>1,78</td><td>4,43</td><td>7,59</td><td>11,21</td></tr><tr><th>1800</th><td>0,89</td><td>2,02</td><td>5,25</td><td>9,12</td><td>13,49</td></tr></tbody></table>

### C. Point load fixation with 8 glass fittings

<figure><img src="image_195.png" /><figcaption>Fig 9: Principal stress $\sigma_1$ of a glass pane with 8 glass fittings</figcaption></figure>

<table><caption>TABLE 3:
COEFFICIENT C FOR 8 GLASS FITTINGS</caption><thead><tr><th>dC/dL</th><th>200</th><th>600</th><th>1000</th><th>1400</th><th>1800</th></tr></thead><tbody><tr><th>200</th><td>0,11</td><td>0,57</td><td>1,42</td><td>2,64</td><td>4,24</td></tr><tr><th>600</th><td>0,26</td><td>0,97</td><td>2,18</td><td>3,75</td><td>5,68</td></tr><tr><th>1000</th><td>0,49</td><td>1,38</td><td>3,10</td><td>5,17</td><td>7,61</td></tr><tr><th>1400</th><td>0,75</td><td>1,75</td><td>4,01</td><td>6,69</td><td>9,72</td></tr><tr><th>1800</th><td>1,03</td><td>2,06</td><td>4,88</td><td>8,20</td><td>11,90</td></tr></tbody></table>

## VI. INTERPOLATION IN 3D-DIAGRAMM

It was shown, in the earlier chapters, that the lines of the results for the principal stresses $\sigma_1$ are nearly straight. It is possible to create a 3D-diagram, like a coordinate system – (x, y, z). The coordinate x is the distance in direction lengthwise and the coordinate y is the distance in direction across. In the direction z the results of the maximum principal stress of the glass pane are

plotted in. The sum of all these values results in a shape of a hyperbolic paraboloidal shell.

With this fact it is possible to make a linear interpolation between the different distances in direction lengthwise and cross.

<figure><img src="image_196.png" /><figcaption>Fig 10: 3D-concept of the interpolation</figcaption></figure>

This concept is easy to implement it into a program such as Excel. With a calculation of results at the corners of the 3D-surface all other geometrical possibilities can be obtained by an interpolation in between them, see in figure 10.

## VII. APPROXIMATION OF THE DEFORMATION

The idea was, to model beams in the line of the supports with the unit width of $b=1$ [mm] and the height of the thickness of the glass pane. It is possible to arrange these beams in a special way to get a good approximation for the deformation. One can see the assembling of these beams for the different number of glass fittings in figure 16, 18 or 20. The load $q$ [$N/mm^2$] of the uniform load of the glass pane is equal to the load $q$ [$N/mm$] of the beam, due to the unit width of $b=1$ [mm].

For a beam there are three basic possibilities for their support behaviour. A free supported beam on both sides is the first possibility. The second is a beam restrained on one side and hinged on the other side. The third beam is fixed on both sides see in figures 11 to 13. For the beams it is also possible to have a projecting end on the hinged supported side, see in figure 14 and 15. [Sträussler et al., 1996]
$$w = \frac{5}{384} \cdot \frac{q \cdot d^4}{E \cdot I} \qquad (2)$$

<figure><img src="image_197.png" /><figcaption>Fig 11: Hinged supported beam</figcaption></figure>

$$w = \frac{2}{369} \cdot \frac{q \cdot d^4}{E \cdot I} \qquad (3)$$

<figure><img src="image_198.png" /><figcaption>Fig 12: One side hinged supported and the other side restrained supported beam</figcaption></figure>

$$w = \frac{1}{384} \cdot \frac{q \cdot d^4}{E \cdot I} \qquad (4)$$

<figure><img src="image_199.png" /><figcaption>Fig 13: Restrained supported beam</figcaption></figure>

$$w = \frac{1}{16} \cdot \frac{q \cdot d^4}{E \cdot I} \left( \frac{5}{24} - \frac{d_e^2}{l^2} \right) \qquad (5)$$

<figure><img src="image_200.png" /><figcaption>Fig 14: Hinged supported beam with projecting ends</figcaption></figure>

$$w = \frac{1}{384} \cdot \frac{q \cdot d^4}{E \cdot I} \qquad (6)$$

<figure><img src="image_201.png" /><figcaption>Fig 15: One side hinged supported and the other side restrained supported beam with a projecting end</figcaption></figure>

With these values one is able to compute each deformation of each beam in both directions. The concept of the approximation is to sum up each deformation of each beam. The possibilities of the assembling for the different numbers of supports will be shown in the following.

## A. Point load fixation with 4 glass fittings

For a glass pane with 4 supports it is the sum of the deformations both directions (along and across) of hinged supported beams with projecting ends, (figure 14).

<figure><img src="image_202.png" /><figcaption>Fig 15: Deformation of a glass pane with 4 glass fittings</figcaption></figure>

<figure><img src="image_203.png" /><figcaption>Fig 16: Approximation concept of a glass pane with 4 glass fittings</figcaption></figure>

$$ \begin{aligned} w &= w_L + w_C \\ w &= \frac{q \cdot d_L^4}{16 \cdot E \cdot I} \left( \frac{5}{24} - \frac{d_e^2}{d_L^2} \right) + \frac{q \cdot d_C^4}{16 \cdot E \cdot I} \left( \frac{5}{24} - \frac{d_e^2}{d_C^2} \right) \end{aligned} \qquad (7) $$

* w deformation [mm]

* q load on the beam [N/mm]

* d_L distances between the glass fittings along [mm]

* d_C distances between the glass fittings across [mm]

* E elasticity modulus [N/mm²]

* I moment of inertia [mm⁴]

* d_e projecting end (distances to the edge) [mm]
## B. Point load fixation with 6 glass fittings

For a glass pane with 6 supports it is the sum of the deformations along of a one side hinged supported beam with projecting end (figure 12) and across a hinged supported beam with projecting ends (figure 14).

<figure><img src="image_204.png" /><figcaption>Fig 17: Deformation of a glass pane with 6 glass fittings</figcaption></figure>

<figure><img src="image_205.png" /><figcaption>Fig 18: Approximation concept of a glass pane with 6 glass fittings</figcaption></figure>

$$ 
\begin{aligned}
w &= w_L + w_C \\
w &= \frac{2 \cdot q \cdot d_L^4}{369 \cdot E \cdot I} + \frac{q \cdot d_C^4}{16 \cdot E \cdot I} \cdot \left( \frac{5}{24} - \frac{d_1^2}{d_C^2} \right)
\end{aligned} 
\quad (8) 
$$

## C. Point load fixation with 8 glass fittings

For a glass pane with 8 supports it is the sum of the deformation along of a one side free supported

<figure><img src="image_206.png" /></figure>

and the other side restrained supported beam (figure 12) and the deformation across of a hinged supported beam (figure 11). The values were modified slightly, see in equation (9).

<figure><img src="image_207.png" /><figcaption>Fig 20: Deformation of a glass pane with 8 glass fittings</figcaption></figure>

$$ 
\begin{aligned}
w &= w_L + w_C \\
w &= \frac{2,5 \cdot q \cdot d_L^4}{369 \cdot E \cdot I} + \frac{4,5 \cdot q \cdot d_C^4}{384 \cdot E \cdot I}
\end{aligned} 
\quad (9) 
$$

# VIII. INFLUENCE OF THE THICKNESS

For glazing systems a single glass or a laminated glass is possible. For a single glass the definition of the thickness is trivial. In the case of a laminated glass one has to consider a composite cross section. It is possible, to arrive at a simple solution using the concept of a virtual thickness. With this trick one can compute laminated glass in the same way as a single glass. How it goes, is shown in the following equations [Bucar et al., 2002].

$$ h_{ef,w} = \sqrt[3]{h_1^3 + h_2^3 + 12 \cdot \Gamma \cdot I_s} \quad (10) $$

$$ h_{1,ef,\sigma} = \sqrt{\frac{h_{ef,w}^3}{h_1 + 2 \cdot \Gamma \cdot h_{s,1}}} \quad (11) $$

$$ h_{2,ef,\sigma} = \sqrt{\frac{h_{ef,w}^3}{h_2 + 2 \cdot \Gamma \cdot h_{s,2}}} \quad (12) $$

with
$$h_s = 0,5 \cdot (h_1 + h_2) + h_v$$

$$h_{s,1} = h_s \cdot h_1 / (h_1 + h_2)$$

$$h_{s,2} = h_s \cdot h_2 / (h_1 + h_2)$$

$$I_s = h_1 \cdot h_{s,1}^2 + h_2 \cdot h_{s,2}^2$$

h thickness of the glass pane [mm]

$h_1$ thickness of the glass pane 1 of a laminated glass with two glass panes [mm]

$h_2$ thickness of the glass pane 2 of a laminated glass with two glass panes [mm]

$h_{ef,w}$ virtual thickness for the verification of the deformation [mm]

$h_{1ef,σ1.}$ virtual thickness of the glass pane 1 for the verification of the stresses [mm]

$h_{2ef,σ2.}$ virtual thickness of the glass pane 2 for the verification of the stresses [mm]

$h_v$ thickness of the PVB-interlayer [mm]

The shear transmission factor $\Gamma$ is given with:

$$\Gamma = \frac{1}{1 + 9,6 \cdot \frac{E}{G} \cdot \frac{l_s \cdot h_v}{h_s^2 \cdot a^2}} \quad 0 \le \Gamma \le 1 \quad (13)$$

with

$$I_s = h_1 \cdot h_{s,1}^2 + h_{s,2}^2$$

$$h_s = 0,5 \cdot (h_1 + h_2) + h_v$$

$$h_{s,1} = h_s \cdot h_1 / (h_1 + h_2)$$

$$h_{s,2} = h_s \cdot h_2 / (h_1 + h_2)$$

E... elasticity modulus [N/mm²]

G... shear modulus in respect to the boundary conditions, see in table 1 [N/mm²]

a... length of the shorter side of the glass pane [mm]

l_s... span length (distance between the glass fittings) [mm]

<table><caption>TABLE 4: SHEAR MODULUS G IN N/mm²</caption><thead><tr><th rowspan="2">temperature in the PVB-interlayer</th><th colspan="3">loading duration</th></tr><tr><th>short-</th><th>mid</th><th>always</th></tr></thead><tbody><tr><td></td><td></td><td></td><td></td></tr></tbody></table>

<table><caption>TABLE 5: LOADING DURATION – LOAD TYPE</caption><thead><tr><th></th><th>time</th><th></th><th></th></tr></thead><tbody><tr><td></td><td></td><td>0,75</td><td>0,5</td><td>0,01</td></tr><tr><td>&lt; 25°C</td><td></td><td></td><td></td><td></td></tr><tr><td>≥ 25°C</td><td></td><td>0,5</td><td>0,25</td><td>0,01</td></tr></tbody></table>

<table><thead><tr><th>loading duration</th><th>load type</th></tr></thead><tbody><tr><td>short-time</td><td>wind</td></tr><tr><td>mid</td><td>climatic load</td></tr><tr><td>always</td><td>dead load</td></tr></tbody></table>

These equations can be simplified with an assumption of two extreme shear transmission behaviours, in a carrying behaviour of full bond between the glass panes for short load durations and without a full bond for long load durations.

If there is a system with a full bond it is the following equation:

$$h_{ef,w} = h_{ef,σ} = \sum h_i \quad (14)$$

If there is a system without bond it will be the following equations:

$$h_{ef,w} = \sqrt[3]{\sum h_i^3} \quad (15)$$

$$h_{ef,σ} = \sqrt{\frac{\sum h_i^3}{h_i}} \quad (16)$$

$h_i$ thickness of each glass pane [mm]

$h_{ef,w}$ virtual thickness for the verification of the deformation [mm]

$h_{1ef,σi.}$ virtual thickness of each glass pane for the verification of the stresses [mm]

## IX. INFLUENCE OF THE DISTANCE OF THE HOLE TO THE EDGE

To get the influence of the distance $d_i$ of the hole to the edge on the principal stresses,
calculations were done. The distance to the edge $d_1$ was varied from 40 mm to 200 mm. With these values of the diagram the interpolated results can be calibrated for another distances to the edge.

<figure><img src="image_208.png" /><figcaption>Fig 21: Coefficient $c_1$ [-]</figcaption></figure>

## X. INFLUENCE OF THE SHORE

To get the influence of the shore hardness of the supports on the principal stresses, calculations were done. The shore hardness was varied from shore hardness 20 to 100. With these values of the diagram the interpolated results can be calibrated for another shore hardness of the rubber at the supports.

<figure><img src="image_209.png" /><figcaption>Fig 22: Coefficient $c_2$ [-]</figcaption></figure>

## XI. INFLUENCE OF THE DIAMETER OF THE HOLE

To get the influence of the diameter $d_H$ of the hole on the principal stresses, calculations were

done. The diameter $d_H$ of the hole was varied from 30 mm to 70 mm. With these values of the diagram the interpolated results can be calibrated for another diameter of the hole.

<figure><img src="image_210.png" /><figcaption>Fig 23: Coefficient $c_3$ [-]</figcaption></figure>

## XII. CONCLUSION AND PROSPECTS

With this concept of the interpolation it is possible, to pre design point load supported glass panes under uniform loading very easily. Further it is possible to check the results of finite element calculations, to be sure that the calculations did what you wanted.

## XIII. ACKNOWLEDGMENT

The work in this paper was primarily conducted with the support of Prof. Gernot Beer from the Institute of Structural Analysis at the Graz University of Technology.

## REFERENCES

[Bucar et al., 2002]
Ö. BUCAR, C. SCHULER, A. MEYER, A. FRIED, Bauen mit Glas, VDI Bericht 1527 zur Berücksichtigung der Verbundtragwirkung bei VSG mit PVB-Zwischenschichten.

[Sträussler et al., 1996]
R.KRAPFENBAUER, E. STRÄUSSLER, Bau Tabellen Verlag Jugend und Volk GmbH Wien 1996
# Glued Joints in Glass Structures

Frank Wellershoff, Gerhard Sedlacek

Institute of Steel Construction, RWTH Aachen University, Germany

**ABSTRACT.** In facade engineering the potential of glued joints of glass members is not exploited except in structural sealant glazing systems. In other technical fields the application of glued joints is already state of the art, even in safety relevant parts. A further development of glued glass joints in facades can therefore be expected in the near future.

To state the potential, but also the risks in using glued joints the chemical and physical background is described. This includes the chemical principles of bonding forces and the physical principles of the deformation behaviour of polymer materials.

To allow the development of new glued joints the behaviour of high modulus adhesives under variable loads and different environmental conditions has to be tested. Test results of point supports with epoxy or acrylic adhesives, tested in compliance with the aging and loading conditions of the European Technical Approval for Structural Sealant Glazing Systems, are shown. Furthermore results of linear overlapping supports with Polyurethanes and Acrylics under short-term loads and long-term loads are presented.

In compliance with the Eurocode, the ultimate limit state as well as the serviceability of glued joints must be verified. For the development of appropriate design concepts, which can be implemented in corresponding standards, suitable computation methods need to be verified. The calculation of the hyper elastic deformation behaviour of polyurethanes with FE-Methods using energetic models is explained and shown exemplarily. The calculation of deformations under long-term loads with logarithmic functions, which have been calibrated by tests, are explained and verified with respect to their accuracy.

**Keywords:** joints, gluing, adhesives, glass

## I. INTRODUCTION

The accredited use of glued joints in glass structures is today restricted to structural sealant glazing facades with linear glued joints, consisting of silicon adhesives [ETAG 002]. These joints are allowed to transmit forces acting to the single façade element only. In mechanical engineering glued connections are already applied to safety relevant parts. In modern cars the windows are used to stiffen the car body. The stiffening forces are transmitted by glued connections with high modulus polyurethanes.

However, that the potential of glued glass joints is not entirely exploited in civil engineering. Possible applications of glued glass joints could be point supports or linear bearings which:

* carry single façade elements

* transmit stiffening forces

* connect elements to plates, columns or beams

Adhesives for the above mentioned applications have to be as stiff as possible, but able to equalize different temperature elongations between the components. Further the adhesive must be resistant to environmental conditions like temperature, humidity and UV-radiation.

## II. BASIC PRINZIPLES

### A. Characterisation of Adhesives

#### 1) General molecular structures

Adhesives are polymer materials that consist of simple monomer units recurrently chained to macromolecules.
romolecules. The atoms in each macromolecule are
chemically bonded and the macromolecules are
physically or chemically bonded to each other and
intertwining is inevitable.

<figure><img src="image_211.png" /><figcaption>Fig. 1: molecular structure of polymers [Ehrenstein 1999]</figcaption></figure>

## 2) Classification

Polymers can be classified according to their
thermo-mechanical properties.

### a) Thermoplastics

Relatively weak intermolecular forces keep
molecules in a thermoplastic together, so that the
material softens when exposed to heat, but returns
to its initial condition when cooled. Thermoplastic
polymers can be repeatedly softened by heating
and then solidified by cooling - a process similar to
the repeated melting and cooling of metals. Most
linear and slightly branched polymers are thermo-
plastics. All the major thermoplastics are produced
by chain polymerisation.

### b) Thermosets

A thermosetting plastic solidifies or "sets" irre-
versibly when heated. Heating cannot reshape
Thermosets. Thermosets usually are three-
dimensional networked polymers with a high de-
gree of cross-linking between polymer chains. The
cross-linking restricts the motion of the chains and
leads to a rigid material.

### c) Elastomers

Elastomers are rubbery polymers that can be
stretched easily to several times their unstretched
length and which rapidly return to their original
dimensions when the applied stress is released.

Elastomers are cross-linked, but have a low
cross-link density. The polymer chains still have
some freedom to move, but are prevented from

permanently moving relative to each other by the
cross-links.

## B. Deformation behaviour of Adhesives:

### 1) Deformation shares

Under external forces three different deforma-
tions, which have to be superimposed, could be
identified:

* A) Spontaneous elastic deformation (spontane- ous reversible) according to changed valence bond angles of atoms in chemical bonding.

* B) Time dependent viscoelastic deformation (time dependent reversible) according to stretched molecular chains.

* C) Time dependent viscous deformation (time depending irreversible) according to movement of molecular chains.

<figure><img src="image_212.png" /><figcaption>Fig. 2: Deformation shares</figcaption></figure>

### 2) Rheological models

To describe the time depending deformation be-
haviour of polymers different rheological models
are developed and can be classified:

#### a) Linear viscoelatic models

Under linear vicoelasticity the time depending
yieldingness depends only on the material tempera-
ture but not on the stress.

Some common used models are explained in the
following:
<figure><img src="image_213.png" /><figcaption>Fig. 3: Burgers-Model (Four-Parameter-Model)</figcaption></figure>

The strain function for the Burgers Model is:
$$
\begin{aligned}
\epsilon = \epsilon_0 + \epsilon_v + \epsilon_{rel} \\
= \frac{\sigma_0}{E_0} + \frac{t \cdot \sigma_0}{\eta_0} + \frac{\sigma_0}{E_{rel}} \cdot (1 - e^{-t/\tau}) \quad \text{with} \quad \tau = \frac{\eta_{rel}}{E_{rel}}
\end{aligned}
\quad (1)
$$

<figure><img src="image_214.png" /><figcaption>Fig. 4: Generalized Kelvin-Voigt-Model</figcaption></figure>

The strain function for the generalized Kelvin-Voigt-Model is:

$$ \epsilon = \left( \sum_{i=1}^{n} \frac{1}{E_i} \cdot (1 - e^{-t/\tau_i}) \right) \cdot \sigma_0 \quad \text{with} \quad \tau_i = \frac{\eta_i}{E_i} \quad (2) $$

<figure><img src="image_215.png" /><figcaption>Fig. 5: Generalized Maxwell-Model</figcaption></figure>

The stress function for the generalized Maxwell-Model is:

$$ \sigma = \left( \sum_{i=1}^{n} E_i \cdot e^{-t/\tau_i} \right) \cdot \epsilon_0 \quad \text{with} \quad \tau_i = \frac{\eta_i}{E_i} \quad (3) $$

### b) Non linear viscoelastic models

In these models the young's modulus E or the shear modulus G depend on the load duration, the temperature of the material and the value of the applied stress.

<figure><img src="image_216.png" /><figcaption>Fig. 6: Deformation Model</figcaption></figure>

For the solution of the differential equation of the deformation model the non-linear function for each damping element must be calculated. Optimised results are verified with 20 parallel Maxwell-Elements. As analytic solutions are nearly impossible, numerical methods are developed [Lewen 1991].

<figure><img src="image_217.png" /><figcaption>Fig. 7: Modified Burgers-Model</figcaption></figure>

The strain function for the modified Burgers-Model is:

$$ \begin{gathered}
\epsilon_{ges} = \frac{\sigma_0}{E_0(T)} + t \cdot \frac{\sigma_0}{\eta_0(T, \sigma)} + \frac{\sigma_0}{E_{rel}(T, \sigma)} \cdot \\
\left( 1 - \exp\left( - \frac{E_{rel}(T, \sigma)}{\mu(T, \sigma) \cdot (1 - \nu(T, \sigma))} \cdot t^{(t - \nu(T, \sigma))} \right) \right)
\end{gathered}
\quad (4) $$

### 3) Super positioning

The linear viscoelasticity is the basis for linear accumulation of loads and deformations.
<figure><img src="image_218.png" /><figcaption>Fig. 8: Boltzmann's superposition</figcaption></figure>

Left: If the strain $\varepsilon_{10}$ is causing the stress $\sigma_1(t)$ and the strain $\varepsilon_{20}$ yields to the stress $\sigma_2(t)$, than the strain $\varepsilon_{10} + \varepsilon_{20}$ is leading to the stress $\sigma_1(t) + \sigma_2(t)$.

Right: If the stress $\sigma_{10}$ is causing the strain $\varepsilon_1(t)$ and the stress $\sigma_{20}$ yields to the strain $\varepsilon_2(t)$, than the stress $\sigma_{10} + \varepsilon_{20}$ is leading to the strain $\sigma_1(t) + \varepsilon_2(t)$.

### 4) Corresponding principle

All solutions based on the theory of linear elasticity (plate theory, beam theory) could be used with linear viscoelatic materials. In this case the time and temperature depending behaviour of the materials must be considered.

## C. Glass as an assembly part in glued connections

The characteristics of the glass surface must be considered if higher bonding forces to an adhesive are desired.

<figure><img src="image_219.png" /><figcaption>Fig. 9: Glass surface</figcaption></figure>

The surface consists of silicon atoms saturated with OH-groups and some metal ionic (e.g. Na).

In glued glass joints chemical bonding with silanized bonding agents are very common. On one side the bonding agent owns a reactive group for the glass surface and on the other side a reactive group for the adhesive [Röder 1996].

The hydrolysis of the silane to a silanol is enabled by the humidity on the glass surface.

<figure><img src="image_220.png" /><figcaption>Fig. 10: Hydrolysis</figcaption></figure>

After the hydrolysis the bonding happens in two steps:

* Hydrogen bonds arise between the OH-molecules of the silanol and the glass surface.

* By splitting of water some hydrogen bonds change into chemical SI-O-SI bonds.

<figure><img src="image_221.png" /><figcaption>Fig. 11: Atomic Bonding at the glass surface</figcaption></figure>

# III. GLUED POINT BEARINGS

## A. Test set up

### 1) Test specimen

Stainless steel cylinders are glued to thermally
toughened glass panes. The maximum surface roughness of the cylinders was $R_{max} = 6.9 \ \mu m$ and the thickness of the adhesives was $d = 0.5 \ mm$.

<figure><img src="image_222.png" /><figcaption>Fig. 12: Glued point bearings / test specimen</figcaption></figure>

## 2) Adhesives

For the described purpose of point supports under various environmental conditions a producer of technical adhesives proposed the adhesives to be tested.

* *Epoxi* (Delo Duopox 1895) is a silane modified two-component epoxy adhesive.

* *UV-Acrylate* (Delo Photobond 4436) is a toughened one-component acrylate where the initial polymerisation is applied by UV-radiation.

<table><caption>TABLE 1: ADHESIVE CHARACTERISTICS</caption><thead><tr><th></th><th>Epoxy</th><th>UV-Acylate</th></tr></thead><tbody><tr><td>Young's modulus</td><td>3200 N/mm<sup>2</sup></td><td></td></tr><tr><td>Shore A</td><td></td><td>83</td></tr><tr><td>α<sub>t</sub></td><td>6 x 10<sup>-5</sup></td><td>8 x 10<sup>-5</sup></td></tr><tr><td>Operating Temp.</td><td>-40°C - + 100°C</td><td>-30°C - + 120°C</td></tr></tbody></table>

## 3) Tensile test set up

The glass panes were fixed with a steel ring $D = 70 \ mm$ and tensile forces were applied to the point bearings with a rate of 500N/s.

<figure><img src="image_223.png" /></figure>

<figure><img src="image_224.png" /><figcaption>Fig. 12: Tensile test</figcaption></figure>

## 4) Shear test set up

A steel fork supported the point bearings and the

glass panes were pushed out with a force rate of 500N/s.

<figure><img src="image_225.png" /><figcaption>Fig. 14: Glued point bearings / shear test</figcaption></figure>

# B. Results

## 1) Initial strength

The characteristic strength before aging $F_c$ (5% quantile; significance level 95 %) is a major input variable for fatigue tests. Therefore larger numbers of test specimen are tested without aging.

<table><caption>TABLE 2: INITIAL STRENGTH / TENSILE TEST</caption><thead><tr><th></th><th>Epoxy</th><th>UV-Acylate</th></tr></thead><tbody><tr><td>N</td><td>16</td><td>11</td></tr><tr><td>F<sub>min</sub></td><td>[kN]</td><td>1.074</td><td>10.38</td></tr><tr><td>F<sub>max</sub></td><td>[kN]</td><td>4.38</td><td>31.71</td></tr><tr><td>F<sub>mean</sub></td><td>[kN]</td><td>2.72</td><td>22.07</td></tr><tr><td>Standard deviation</td><td>[kN]</td><td>0.98</td><td>6.37</td></tr></tbody></table>

<table><caption>TABLE 3: INITIAL STRENGTH / SHEAR TEST</caption><thead><tr><th></th><th>Epoxy</th><th>UV-Acylate</th></tr></thead><tbody><tr><td>N</td><td>12</td><td>12</td></tr><tr><td>F<sub>min</sub></td><td>[kN]</td><td>2.77</td><td>19.40</td></tr><tr><td>F<sub>max</sub></td><td>[kN]</td><td>7.43</td><td>23.18</td></tr><tr><td>F<sub>mean</sub></td><td>[kN]</td><td>5.12</td><td>21.55</td></tr><tr><td>Standard deviation</td><td>[kN]</td><td>1.55</td><td>1.02</td></tr></tbody></table>

The mean values $F_{mean}$ of these specimen indicate a higher initial strength of the connection with the *UV-Arylate*. This is most likely due to the better stress balancing behaviour of the *UV-Arylate*. This theory is supported by the breakages pattern of the stiffer *Epoxy*-specimen in the shear tests, where the glass surface is cracked like a shell.

## 2) Tensile Strength after fatigue load

Cycle loads according to [ETAG 002] were applied before the ultimate limit load was tested.

* Phase 1: 100 cycles with $0.1 F_c - 1.0 F_c$

* Phase 2: 250 cycles with $0.1 F_c - 0.8 F_c$
* Phase 3: 5000 cycles with 0.1 F<sub>c</sub> – 0.6 F<sub>c</sub> characteristic tensile strength

<table><caption>TABLE 4: STRENGTH AFTER FATIGUE LOAD / TENSILE TEST</caption><thead><tr><th></th><th>UV-Acylate</th></tr></thead><tbody><tr><th scope="row">N</th><td>4</td></tr><tr><th scope="row">F<sub>min</sub> [kN]</th><td>14.26</td></tr><tr><th scope="row">F<sub>max</sub> [kN]</th><td>23.59</td></tr><tr><th scope="row">F<sub>mean</sub> [kN]</th><td>19.69</td></tr><tr><th scope="row">Standard deviation [kN]</th><td>3.92</td></tr></tbody></table>

The value F<sub>mean</sub> decreases by 12.1%, however with only four specimen tested this effect is not statistically proved.

### 3) Strength after UV-Radiation

The test specimen were stored in an aging chamber and illuminated with UV-A light (320 – 400 nm) for 21 days. The radiated power of the lamp was 162 W/m² at the beginning and 80 W/m² at the end of the aging period and therefore higher than the postulated power of 50 W/m² [ETAG 002].

<table><caption>TABLE 5: STRENGTH AFTER UV-RADIATION / SHEAR TEST</caption><thead><tr><th></th><th>Epoxy</th><th>UV-Acylate</th></tr></thead><tbody><tr><th scope="row">N</th><td>4</td><td>4</td></tr><tr><th scope="row">F<sub>min</sub> [kN]</th><td>8.50</td><td>0.00</td></tr><tr><th scope="row">F<sub>max</sub> [kN]</th><td>21.34</td><td>8.52</td></tr><tr><th scope="row">F<sub>mean</sub> [kN]</th><td>12.29</td><td>5.03</td></tr><tr><th scope="row">Standard deviation [kN]</th><td>6.06</td><td>3.62</td></tr></tbody></table>

Although the *UV-Acylate*-specimen need UV-radiation to initialize the polymerisation the strength of these specimen decrease extremely by UV-aging. On the other hand the *Epoxy*-specimen showed higher strength values after UV-aging. This could be evoked by the temperature in the UV-chamber that might have enabled a degradation of residual stresses which arose during the polymerisation process.

### 4) Strength after Immersion in water

In this aging scenario the specimen were stored in 45°C hot demineralised water for 28 days [1].

<table><caption>TABLE 6: STRENGTH AFTER IMM. IN WATER/TENSILE TEST</caption><thead><tr><th></th><th>Epoxy</th><th>UV-Acylate</th></tr></thead><tbody><tr><th scope="row">N</th><td>5</td><td>5</td></tr><tr><th scope="row">F<sub>min</sub> [kN]</th><td>1.94</td><td>0.00</td></tr><tr><th scope="row">F<sub>max</sub> [kN]</th><td>2.53</td><td>5.82</td></tr><tr><th scope="row">F<sub>mean</sub> [kN]</th><td>2.27</td><td>3.38</td></tr><tr><th scope="row">Standard deviation [kN]</th><td>0.25</td><td>2.11</td></tr></tbody></table>

<table><caption>TABLE 7: STRENGTH AFTER IMM. IN WATER / SHEAR TEST</caption><thead><tr><th></th><th>Epoxy</th><th>UV-Acylate</th></tr></thead><tbody><tr><th scope="row">N</th><td>3</td><td>3</td></tr><tr><th scope="row">F<sub>min</sub> [kN]</th><td>4.18</td><td>0.95</td></tr><tr><th scope="row">F<sub>max</sub> [kN]</th><td>6.29</td><td>2.68</td></tr><tr><th scope="row">F<sub>mean</sub> [kN]</th><td>5.31</td><td>2.09</td></tr><tr><th scope="row">Standard deviation [kN]</th><td>1.07</td><td>0.99</td></tr></tbody></table>

Comparable to the UV-aging the strength of the *UV-Arylate*-specimen decreased and the strength of the *Epoxy*-specimen is not negatively influenced.

### 5) Strength after Immersion in water under high temperature

After immersion in demineralised water (28d, 45°C) four specimen (two with each adhesive) were heated to 80°C for two hours. Only one *Epoxy*-specimen did not delaminate by heating and reached a maximum force of 11.6 kN in the shear test. Higher temperature is apparently the limiting environmental condition for the tested connections. Different temperature elongations between the stainless steel and the adhesives is assumed to be the reason for the delamination.

## C. Conclusions

Both of the two investigated connections are not suitable for the dedicated application in facades. The *UV-Arylate*-specimen showed substantial aging effects due to UV-radiation or immersion in water and both specimen types (*UV-Arylate* and *Epoxy*) delaminate when heated up to 80°C.

Better results could be expected with a toughened *Epoxy*-Adhesive and a larger application thickness which could balance stresses due to temperature elongations.

# IV. LINAR GLUED JOINTS

## A. Tested Adhesives

* PU-1 (Sikaflex 265) and PU-2 (SikaTack HM) are 1-component adhesives on polyurethane basis. Both react with the air humidity by polyaddition to an elastomer.

* PU-3 (Sika 250 HMA-1) is a 1-component adhesive with an hot melting component. Under the application temperature of 80°C this com-