# Is prestressing control possible for tempered glass structures ?

Fabrice Bernard, Ecole des Mines de Douai – Dpt Génie Civil, Douai, France

Laurent Daudeville, Laboratoire Sols Solides Structures, Grenoble, France

This work deals with the use of glass as a real structural material. An in situ non destructive process is proposed for the control of tempered glass structures, especially in joint zones. A design method is also proposed. It is based on one hand, on a limit state ensuring the structure to be perennial on long duration, and the another hand, on the finite element prediction, on every point of the structure, of both residual stresses due to tempering and stresses due to the mechanical loading. The control method consists, during the life of the structure, in comparing images obtained from photoelastic analyses of the loaded structure, to images obtained from finite element analyses of the structure at the ultimate limit state used for the design.

**Keywords:** tempered glass, photoelasticity, in situ control, finite elements, residual stresses, ultimate limit state

## I. INTRODUCTION

The use of glass for civil engineering structures (Fig. 1) is strongly penalized. French offices that regulate construction require full-scale tests of any glass structure to be realised (Fig. 2) and a relatively large global safety factor (about 7). The global safety factor comes from a coefficient equal to 3.5, classical for brittle materials, that factor includes loading and material uncertainties, multiplied by a coefficient of 2, due to a strength reduction estimated over 50 years. This strength reduction is due to the subcritical crack growth

caused by the possible action of water at crack tips
[Michalske and Freiman, 1983].

<figure><img src="image_1.png" /><figcaption>Fig 1: An example of structural glass : tempered beams supporting a glass roof in the Louvre museum</figcaption></figure>

<figure><img src="image_2.png" /><figcaption>Fig 2: Full scale test of a tempered beam presented in Fig. 1</figcaption></figure>

Thermal tempering is a way of reinforcing the glass surface. It not only allows for an increase in the glass tensile strength, since it is necessary to overcome the surface precompression in order to
break the material, but also a certain immunity
against sub critical cracking, as long as the applied
tension does not exceed, in absolute value, the
surface precompression.

In the goal of a flexible design method, it would
be particularly relevant to have a control method of
structures allowing to check with a portable
equipment that critical loads are not exceeded. This
is particularly important for connection areas
where 3D stress states can not generally be
obtained from simple calculations.

This article presents a possible non-destructive
methodology for in situ control of tempered glass
structures. The method is based on an original use
of photoelasticity. This technique, developed
lengthily thereafter, allows to check if a given
stress state is reached.

## II. PHOTOELASTICITY OF GLASS

The transparency and the birefringence of glass
when it is loaded, are two unique properties in the
field of building materials which are proposed to
be used in an original way. For that, it is necessary
to be able to pass from a photoelastic image to a
stress state, and reciprocally. Thereafter, the
principles of photoelasticity and polariscopy are
explained, then a simulation method of
photoelastic images based on the analysis of finite
element results is presented. The difficulty comes
from the three-dimensional feature of the stress
state in the vicinity of joints which makes unusual

### A. Polariscopy and photoelasticity

A plane polariscope is composed of a white or
monochromatic (one wavelength) light source and
two linear polarizers with crossed axes named the
polarizer and the analyser respectively. Two
quarter-wave plates can be added in order to obtain
a circular polariscope. The glass element to be
analysed is positioned between the two polarizers
or the two quarter-wave plates (Fig. 3). The light
ray, emitted by the source, crosses the first
polarizer. It is then constrained to vibrate in a
plane normal to the direction of propagation
determined by the orientation of the polarizer. The
luminous ray is said to be plane or linearly
polarized.

If a quarter-wave plate is added, the amplitude of
the emergent light vector is constant and the vector
extremity draws a circle when the wave
propagates: the luminous ray is circularly
polarized.

The luminous ray then continues to propagate
and meets the glass plate. Glass, like several other
non-crystalline transparent materials, is optically
isotropic under normal conditions but becomes
birefringent, like a crystal, when it is loaded. This
phenomenon is called accidental birefringence or
photoelasticity [Aben and Guillemet, 1993]. The
optical properties of glass can be represented in
each point by an ellipsoid of indices whose
principal axes coincide with the principal
directions of the stresses. So when the luminous

<figure><img src="image_3.png" /><figcaption>Fig 3: Circular polariscopy</figcaption></figure>

the photoelastic analysis.

ray reaches glass, it is divided immediately into
two vibrations whose orientations follow the two secondary principal directions, i.e. the two stress principal directions, in the medium, relative to the plane perpendicular to the axis of light propagation. The third direction does not contribute to birefringence since the propagation velocity of a light ray is not governed by matter alignment along this ray, but by the nature of the medium normally to the ray.

The relative phase retardation $\Delta$ which is created between the two vibrations follows the Neumann equations [Aben and Guillemet, 1993]:

$$ \begin{cases} \frac{d\Delta}{dx_3} = C(\sigma_1 - \sigma_2) + 2\frac{d\varphi}{dx_3}\sin\Delta\cot\kappa \\ \frac{d\kappa}{dx_3} = -\frac{d\varphi}{dx_3}\cos\Delta \end{cases} \quad (1) $$

$\kappa$ is such that $\tan \kappa = E_2/E_1$ where $E_1$ and $E_2$ are the amplitudes of the light vibration along the secondary principal directions ($E_1 > E_2$). $\sigma_1$ and $\sigma_2$ are the secondary principal stresses (in the plane of the plate, i.e. without taking into account the third direction which does not contribute to birefringence), $\varphi$ is the angle between the secondary principal directions and the polarizer axes and C is the glass photoelastic constant.

In the case of circular polariscopy, after having crossed glass, the luminous ray propagates through the second quarter-wave plate, whose fast and slow axes are opposed to those of the first quarter-wave plate. The analyzer brings back the two vibrations in the same plane and makes them interfere. The intensity $I$ of the transmitted light is given by:

$$ \begin{cases} I = I_0 \sin^2(2\varphi) \sin^2(\Delta/2) & \text{if plane polariscopy} \\ I = I_0 \sin^2(\Delta/2) & \text{if circular polariscopy} \end{cases} \quad (2) $$

For a plane polariscopy, this intensity is nil if $\varphi=0(\pi/2)$ or $\Delta=0(2\pi)$. In the first case, the dark fringes are named the isoclinics; they correspond to the points where the secondary principal directions are parallel to the directions of the polarizer axes. In the second case, the dark fringes are called the isochromatics. With white light, each

phase retardation corresponds to a precise colour. The more numerous the isochromatics the higher the difference between the secondary principal stresses. With circular polarization only the isochromatics appear, preventing the superposition of the two types of dark fringes, which may simplify interpretations of images particularly with monochromatic light.

## B. Simulation of the photoelastic images

### 1) Discussion

The phase retardation $\Delta$ depends on the secondary principal stresses, thus on the stress state by the intermediary of the Neumann equations (1).

The resolution of this complete system allows to determine the phase retardation $\Delta$ and thus the light intensity. The photoelastic images can be simulated by this way. However, the resolution of the Neumann equations is complex since it is a system of non-linear differential equations with non-constant coefficients. The system is simplified if the angle $\varphi$ does not depend on the co-ordinate $x_3$ relative to the direction of propagation of the luminous ray, i.e. if the secondary principal directions remain constant along the optical way. In this latter case, the system (1) becomes:

$$ \frac{d\Delta}{dx_3} = C(\sigma_1 - \sigma_2) \quad (3) $$

It is then equivalent to 2D photoelasticity obtained for example with a non holed glass plate loaded in its plane and placed in a polariscope whose axis is normal to the plate. However, in the connection zones of glass structures, the presence of a hole with a possible complex geometry (chamfers) does not allow such an assumption. It is in general impossible to calculate the phase retardation $\Delta$ with equation (3) because the shear stress $\sigma_{12}$ is not constant along the optical way. Thus, the principal directions rotate along the thickness, and the term $d\varphi/dx$ (eq. (1)) can not be neglected. The Neumann equations (1) have to be entirely considered for the analysis of photoelastic
images in joint areas. It is possible to show that system (1) is equivalent to the matrix system (4), obtained from the Maxwell equations of electromagnetism [Aben and Guillemet, 1993]:

$$ \frac{d[E]}{dx_3} = -\frac{i\pi C}{\lambda} \begin{bmatrix} \sigma_{11}(x_3) - \sigma_{22}(x_3) & 2\sigma_{12}(x_3) \\ 2\sigma_{12}(x_3) & -(\sigma_{11}(x_3) - \sigma_{22}(x_3)) \end{bmatrix} [E] \\ \text{denoted: } \frac{d[E]}{dx_3} = M(x_3)[E] \quad (4) $$

$[E]$ is the electrostatic vector field (denoted $E$ in the following text), $\lambda$ is the wavelength of the light ray, $\sigma_{ij}$ are the components of the stress tensor in the plane perpendicular to the light direction of propagation.

## 2) Principle

The integration of differential equations (4) can be carried out using a Cranck-Nicholson scheme [Bernard et al., 2004]:

$$ \frac{E(x_3 + \Delta x_3) - E(x_3)}{\Delta x_3} = M(x_3) \left( \frac{E(x_3 + \Delta x_3) + E(x_3)}{2} \right) \quad (5) $$

The expression of the vector $E$ at altitude $x_3 + \Delta x_3$ is obtained from its value at altitude $x_3$ ($I_d$ is the identity matrix):

$$ E(x_3 + \Delta x_3) = \left[ I_d - \frac{\Delta x_3}{2} M(x_3) \right]^{-1} \left[ I_d + \frac{\Delta x_3}{2} M(x_3) \right] E(x_3) \quad (6) $$

Thus, the assumption of invariance of the secondary principal directions in the thickness of the plate (invariance of shear in the plane of the plate according to direction 3) can be given up.

The choice of the measurement method (circular or linear polariscopy) gives the initial value of the electrostatic field vector $E_0$ (eq. (6)) and the final analyzer [Bernard et al, 2004]. The light intensity $I$ is equal to the square of the electrostatic field vector $E$ and is linked to the phase retardation $\Delta$ through the previously given expression. A FORTRAN program, developed in [Bernard, 2001], reads the data file generated by the FE software (stress state in each Gauss point) and calculates:

* the phase retardation in order to compare the predicted isovalues with the images obtained during experiments with white light (in the case of the plane polarized light, the angle $\varphi$ is obtained from the diagonalization of the stress matrix $2 \times 2$ relative to the plate plane),

* or the light intensity in order to compare locations of light extinction (where the intensity is equal to zero) with isochromatics obtained with monochromatic light (only one wavelength).

# III. VALIDATION OF THE METHOD

## A. Experimental Campaign

In order to check the validity of the analysis code of photoelastic images, the predicted photoelastic fringes are compared with experimental results. The experimental campaign carried out aims at determining the ultimate loads in joint areas of glass structures. It is indeed in this place that the stress states are the most complex, the validation of the approach will be thus relevant. Figure 4 shows one of the tests carried out.

<figure><img src="image_4.png" /><figcaption>Fig 4: Failure test of a joint</figcaption></figure>
The used MTS testing machine has a loading capacity of 500 kN. Different 350*600 mm² glass plates are tested. The glass plate was glued to two metal plates connected to the testing machine frame by means of a knee joint. The mechanical fastener inserted in the hole of the glass plate was embedded in two other plates fixed by means of one pivot joint to the machine horizontal cross-piece. An upwards movement is imposed with a rate of 0.5mm/min. Far from the hole, the stress state is a pure tension. Close to the hole the stress state is three dimensional. The presence of the knee-joint ensures the non-existence of bending and torsion stresses. The loadings is in the plane of the glass plate, as it is the case for structural elements. Experiments were led until failure on both annealed and tempered 19 mm thick glass plates.

Various hole geometries are studied: small, average or large chamfers, small or large diameters (Fig. 5, Table 1). The connection is carried out by means of a symmetrical bolt with conical aluminium washers (Fig. 6).

<figure><img src="image_5.png" /><figcaption>Fig 5: Chamfered hole in the glass plate</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig 6: Steel connector</figcaption></figure>

<table><caption>TABLE 1: HOLE GEOMETRIES</caption><thead><tr><td>Hole</td><td>Dint (mm)</td><td>Dext (mm)</td></tr></thead><tbody><tr><td>a1</td><td>38</td><td>40</td></tr><tr><td>a2</td><td>54</td><td>56</td></tr><tr><td>b1</td><td>24</td><td>40</td></tr><tr><td>b2</td><td>40</td><td>56</td></tr><tr><td>c1</td><td>30</td><td>40</td></tr></tbody></table>

## B. Validation of the photoelastic images analysis program

The first step of the validation of the photoelastic images analysis program consists in carrying out a FEM simulation of the tests in order to obtain the stress state in the connection zones.

The assumptions and the validation of this modelling are exposed in [Bernard and al, 2004 and Bernard et al, 2002]. Fig. 7 shows the comparison between the photoelastic images visualized during a test for a given loading and a given geometry, with the simulated photoelastic images. The results of simulation are thus similar to the experimental ones.

<figure><img src="image_7.png" /><figcaption>Fig 7: Observed and simulated photo-elastic fringes</figcaption></figure>

## IV. APPLICATION OF THE METHOD : IN SITU CONTROL OF JOINTS

Thus, in analysing photoelastic images obtained on existing structures, the stress state can be predicted in the medium. It is also possible to check if the external applied tension exceeds in absolute value the precompression of surface induced by thermal tempering [Zarzycki, 1982]. An ultimate limit state design with this limit state is relevant because tempered glass, whose surface
is not under tension, is not subject to subcritical
crack growth. In [Bernard, 2001], it is mentioned
that for connections areas comprising a hole with
large 45° chamfer and large diameter (Φ<sub>ext</sub> =
56mm), surface is decompressed starting from a
load equal to 80kN.

Figure 8 shows the simulated photoelastic images obtained with the load leading to the surface decompression (a) – 80 KN –, as well as this load divided by the classical 3.5 safety coefficient (b) – 23 KN – (the partial coefficient 2 can be removed because the surface is ensured to be in compression), and the failure load divided by 7 (c) – 16 KN – corresponding to the present design method.

<figure><img src="image_8.png" /><figcaption>Fig 8: Simulated photo-elastic fringes beyond the connector at three different load levels</figcaption></figure>

V. CONCLUSION

This work deals with the context of "structural
glass", i.e. the use of glass for civil engineering
structures.

The lack of knowledge of the long-term
mechanical behaviour of glass led to penalize its
use for such applications. Full-scale tests and high
safety coefficients are required by the offices of
control for construction.

The answer to the question asked in the title is
surely positive. The use of photoelasticity coupled
with a full finite element analysis [Bernard et al.,
2002 and 2004] is proposed for the in situ control
of glass structures. In addition it is proposed to
design tempered glass structures with the ultimate
limit state corresponding to the surface
decompression. This limit state needs to be

calculated by means of finite element predictions
of residual stresses due to tempering [Bernard et
al., 2002 and 2004]. Such a design method allows
to remove the partial safety factor due to the sub-
critical crack growth.

The simulation of the photoelastic images, which
allows to link stress state and isochromatics, is
presented and developed. The originality of the
presented work is to account for the three-
dimensional feature of the stress state in the zones
of connection. The photoelastic images analysis
program developed in this study allows the
integration of a possible rotation of the secondary
principal directions in the thickness of the plate i.e.
along the optical way of the light. This method will
able to check in situ the stress state of the glass
structure, and particularly to know if the glass
surface is decompressed or not. It will allow to
deliver certificates of guarantee.

REFERENCES

[Aben and Guillemet, 1993].

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

[Bernard, 2001]

BERNARD F., Sur le dimensionnement des structures en verre trempé : étude des zones de connexion, thèse de doctorat, Ecole Normale Supérieure de Cachan, 2001.

[Bernard et al., 2002]

BERNARD F., DAUDEVILLE L., GY R., *Sur le dimensionnement des structures en verre trempé*, Revue Française de Génie Civil, Vol. 6, No 7-8, pp.1359-1380, 2002.

[Bernard et al., 2004]

BERNARD F., DAUDEVILLE L., GY R., *Vers une procédure de contrôle des structures en verre trempé*, Mécanique et Industrie, Vol. 5, No 3, pp. 265-273, 2004

[Michalske and Freiman, 1983]

MICHALSKE T.A., FREIMAN S.W., *A molecular mechanism for stress corrosion in vitreous silica*, *J. American Ceramic Society*, Vol. 66, No 4, pp. 284-288, 1983.

[Zarzycki, 1982]

ZARZYCKI J., Les verres et l'état vitreux, Paris,
Masson, 1982
# Glass in Contact with Different Inserts

Martina Eliášová, Czech Technical University in Prague, Czech Republic
Sébastien Floury, Université Blaire Pascal, Clermont-Ferrand, France
František Wald, Czech Technical University in Prague, Czech Republic

The paper presents experimental observations of glass in contact with inserts of steel, aluminium, polyamide and epoxide resin. Four sets of tests with the different contact materials were carried out in the laboratory of Czech Technical University. Influence of the edge finishing, size and thickness of the glass panel and the corner distance was taken into account. The test results and related FE simulation will allow preparing an analytical prediction model of the contact resistance as well as the bearing resistance of bolted connections.

**Keywords:** Glass panel, compression, experiments, material of inserts, size effect, edge finishing.

## I. INTRODUCTION

Modern trends as well as new technologies in production and materials are used in civil engineering. Glass with its new function is one of the most progressive materials nowadays. It is not only the filling but also the load-carrying element. The glass is used for façade systems, roofs of atriums, railing of staircases and over bearing structures.

The structural glass is usually combined with other materials, mostly with steel. There is a lack of knowledge, design rules and procedures, which strengthen use of this attractive material. Designers have limited coherent approach to these problems at present. One of major questions in glass structures design is the connection between the glass components and the joints to the supporting structures. The knowledge in the glass connection is limited even though it is one of the most important parts of the structure.

Glass does not yield, it is a brittle material and its stress concentrations may not be ignored. Ductile material (steel, aluminium) yields if it is locally overstressed and therefore stress concentrations are limited. For glass is important to give an attention to the details and way of their design. Ultimate load depends on the edge finishing, methods of the drilling, bolt positioning and type of the bolts.

## II. EXPERIMENTS

### A. Scope of Test

The test set up was prepared to investigate influence of the inserts from different materials on the glass panel resistance. The test results are summarised in tables, Appendix A1, A2, A3. The length of the inserts was studied on tests Al 90(60,180)/120-15/p-X, tab. A1. The glass panel's size on tests Al 90/120(150,180)-15/p-X, the thickness of glass panel on tests Al 90/120-15(12,10)/p-X and the edge finishing on tests Al 90/120-15/p(s)-X, tab. A2. The material of insert was studied on tests Pa (Fe,Ep) 90/120-15/p-X, tab. A3. The used numbering is adopted to recognize the plates for an analysis. This numbering is given in the following form:

<figure><img src="image_9.png" /></figure>

where: Al = aluminium, Pa = polyamide, Fe = steel, Ep = epoxide resin; insert length: 90 mm, 60 mm, 180 mm; glass size: 120×120 mm, 150×150 mm, 180×180 mm; glass thickness: 10, 12 and 15 mm; edges finishing: p= polished; s = smoothed.

All tests were carried out for annealed float
glass. Heat-strengthened and fully toughened glass samples will be investigated in the second step. Totally 81 tests were performed.

## B. Material Tests

Four point bending tests of glass, Fig. 1, were performed for determination of the strength based on standard [ČSN EN 1288-3: 2001]. The thickness of the glass panel was $t = 15$ mm, test specimen has dimension 1100 x 360 mm.

<figure><img src="image_10.png" /><figcaption>Fig. 1: Test set-up for the bending test</figcaption></figure>

The test specimens were loaded to the failure of the glass panel. During the experiment forces and deflections were recorded, Fig. 2. Average strength of the single float glass, which was used for next experiments, was $f_{g,t} = 67,5$ MPa, Tab. 1.

<figure><img src="image_11.png" /><figcaption>Fig. 2: Force – deflection relation of test T1-2, T1-4</figcaption></figure>

The standard coupon tests, [ČSN EN 10002-1: 1994], [ČSN EN ISO 527: 1997] were carried out for inserts to obtain their material properties, Tab. 2.

## C. Test Set-Up

Glass panels were placed between the inserts and loaded by a force to the failure, Fig. 3. Two test machines with the load capacity 400 kN and 1000 kN were used for the experiments. The first one allowed recording force – deformation curve. A transparent box protected the observer against the dangerous glass shards was used, see Fig. 4.

<table><caption>TABLE 1:
RESULTS OF FOUR POINT BENDING TEST</caption><thead><tr><td>Number of test</td><td>Stress [MPa]</td><td>Deflection [mm]</td><td>Young's Modulus [MPa]</td></tr></thead><tbody><tr><td>1</td><td>39,08</td><td>7,37</td><td>70 413</td></tr><tr><td>2</td><td>48,39</td><td>8,24</td><td>76 054</td></tr><tr><td>3</td><td>85,35</td><td>15,51</td><td>72 011</td></tr><tr><td>4</td><td>97,20</td><td>18,2</td><td>71 048</td></tr></tbody></table>

<table><caption>TABLE 2:
MECHANICAL PROPERTIES OF MATERIALS</caption><thead><tr><td>Material of inserts</td><td>Young's Modulus [MPa]</td><td>Poisson's ratio</td><td>Tensile strength [MPa]</td></tr></thead><tbody><tr><td>Aluminium</td><td>69000</td><td>0,34</td><td>265</td></tr><tr><td>Polyamide</td><td>3500</td><td>0,39</td><td>76</td></tr><tr><td>Epoxide</td><td>5700</td><td>-</td><td>51,5</td></tr><tr><td>Steel</td><td>210000</td><td>0,32</td><td>400</td></tr></tbody></table>

<figure><img src="image_12.png" /><figcaption>Fig. 3: Geometry of the test set-up for the glass in contact under pressure</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 4: Transparent box for protection</figcaption></figure>

## D. Measurements

The shape and thickness of the glass panels and the contact area of inserts were measured before testing. The measured values are summarized in
Appendix Tab. A1, A2 and A3. The edge finishing of the glass panels is taken into account, Fig. 5.

<figure><img src="image_14.png" /><figcaption>Fig. 5: Edge of the glass plate</figcaption></figure>

During the loading of the test specimen an attention was given to the first crack appearance as well as to the shape of the failure in glass panels. Deformations δ of the upper and bottom inserts were measured after the test, Fig. 6, and they are recorded in Tab. A1, A2 and A3 in Appendix.

<figure><img src="image_15.png" /><figcaption>Fig. 6: Measurements of the contact plates</figcaption></figure>

### III. RESULTS

#### A. Failure Modes

The failure modes of the glass panel in contact with inserts were observed during the loading stage. It is possible to describe different initial damages according to the shape of the first crack in the glass panel. “The corner crack” is a breaking of the glass panel from the corner of the insert. “The inside crack” is a vertical crack inside the glass panel. “The surface flake” is a scaling of the glass panel’s surface. “The edge crack” is the breaking of the glass panel’s edge from one insert to the other, see Fig. 7a).

Different failure modes were observed at the collapse as well. “The fast failure” is a fast fragmentation of the glass panel into very small pieces of glass. Usually this failure mode was without any initial damages. Next one is “fragmentation” of the panel into big pieces of glass after crack propagation, see Fig. 7b). The “cut through the insert” occurred for inserts with Young’s modulus lower compared to the tested glass panel, e.g. epoxide resin and polyamide.

<figure><img src="image_16.png" /><figcaption>Fig. 7a): Edge failure, 7b): Glass with flakes and vertical cracks</figcaption></figure>

#### B. Resistance

The comparison of the test results may be based on the predicted reduction of the resistance in contact, which is included in the form of joint coefficient βj. The ultimate resistance can be calculated as

$$F_{red} = \beta_j f_{c,u} A_i \quad (1)$$

where βj is the joint coefficient, Ai is the contact area of the glass, fc,u is the characteristic strength of glass in compression and is considered as 500 MPa.

The theoretical resistance Ftheor (βj = 1) for insert from aluminium is compared to the experimental one Fexp in the Fig. 8.

<figure><img src="image_17.png" /><figcaption>Fig. 8: Comparison of the theoretical Ftheor and experimental Fexp resistances</figcaption></figure>

#### C. Material of Insert

The inserts of steel, aluminium, polyamide and epoxide were tested. The force at the failure is compared for each material at Fig. 9. The tests with polyamide and epoxide inserts exhibit similar results due to the similar modulus of elasticity.
<figure><img src="image_18.png" /><figcaption>Fig. 9: Comparison of the inserts</figcaption></figure>

The evaluation of the joint coefficient $β_j$ for the different material of inserts is shown at Fig. 10. Suggested values of joint coefficients, which were determined for glass panel with smoothed edges, are summarized at Tab. 3.

<figure><img src="image_19.png" /><figcaption>Fig. 10: $F_{exp}/F_{theor}$ ratio for different material of inserts</figcaption></figure>

<table><caption>TABLE 3:
JOINT COEFFICIENT FOR DIFFERENT MATERIAL OF INSERTS</caption><thead><tr><th>Material</th><th>Al</th><th>Fe</th><th>Pa</th><th>Ep</th></tr></thead><tbody><tr><td>Coefficient<br>β<sub>j</sub></td><td>0,50</td><td>0,55</td><td>0,25</td><td>0,25</td></tr></tbody></table>

The plastic deformation was developed in the insert under the pressure, see Fig. 11.

<figure><img src="image_20.png" /><figcaption>Fig. 11: Residual deformation of the aluminium plate</figcaption></figure>

## D. Size and Thickness Effect

The glass panels of thickness 10, 12 and 15 mm were tested. The results are compared in Fig. 12. Three size of glass panels were examined:

120 x 120, 150 x 150 and 180 x 180 mm.
Comparison of results is given in [Floury 2004] and Fig. 13.

Resistance was observed by tests in range from 400 MPa to 500 MPa. No influence of glass panel thickness was observed, Fig. 12. The maximal forces in compression were at the same level for different size of the glass, joint coefficient $β_j$ varies from 0,65 till 0,75, Fig. 13.

<figure><img src="image_21.png" /><figcaption>Fig. 12: $F_{exp}/F_{theor}$ ratio for different thickness of glass panels</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>Fig. 13: $F_{exp}/F_{theor}$ ratio for different size of glass panels</figcaption></figure>

## E. Length of the Insert

Three different lengths of insert from aluminium were tested (60, 90 and 180 mm). The glass panels had the same size and thickness. Maximal reached forces at failure of the glass panels are compared in the graph. Fig. 14 shows relation between the maximal forces and measured deformation $δ$ of the inserts. For better understanding the force was recalculated according to the contact area to the stress, see Fig. 15.

Behaviour of the 60 mm long inserts was similar to the 90 mm long inserts behaviour. First crack appeared at the corner, then the edge failed and finally the glass panel failed completely. For the 180 mm long inserts, the failure of the glass panel was different. The scales of glass flaked off from the surface of the panel.
<figure><img src="image_23.png" /><figcaption>Fig. 14: Force – deformation relation for the different length of the contact plate</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>Fig. 15: Stress – deformation relation for the different length of the contact plate</figcaption></figure>

### F. Corner Distance

The size of the insert may be demonstrated also from the point of view of distance between the specimen edge and insert edge $L_c$, see Fig. 3. Results are presented on the Fig. 16.

<figure><img src="image_25.png" /><figcaption>Fig. 16: $F_{exp}/F_{theor}$ ratio for different corner distance $L_c$</figcaption></figure>

### G. Edge Finishing

Two types of edge finishing were tested: smoothed and polished. Influence of the edge finishing was investigated for aluminium and polyamide inserts, Fig. 17.

<figure><img src="image_26.png" /><figcaption>Fig. 17: Comparison of the edge finishing</figcaption></figure>

## IV. CONCLUSIONS

Four sets of tests were performed to investigate the behaviour of the of float glass in contact with different material. The type of contact material has an impact to the joint resistance. The tests results indicate an influence of the corner distance $L_c$ as well. The influence of the edge finishing was not observed.

The experimentally obtained joint coefficient $\beta_j$ varies from 0,25 for insert from polyamide up to 0,55 for insert from steel.

Further experiments for fully toughened glass panels and FE method allow to precise contact resistance of glass panel for design. Contact resistance will be used as the first step for the study of bolted connections.

## V. ACKNOWLEDGMENT

This research has been supported by grant ME689 KONTAKT of the Czech Ministry of Education and by grant COST-STSM-C13-00176.

## REFERENCES

* [ČSN EN 1288-3: 2001]

ČSN EN 1288-3: Glass in building – Determination of the bending strength of glass. ČNI, 2001.

* [ČSN EN 10002-1: 1994]

ČSN EN 10002-1: Metals : Tensile test. ČNI, 1994.

* [ČSN EN ISO 527: 1997]

ČSN EN ISO 527: Plastics - Determination of tensile properties. ČNI, 1997.

* [ELIÁŠOVÁ et al. 2003]

ELIÁŠOVÁ, M., WALD, F., SOKOL, Z.: Bolted connections of glass structures, in Czech, In: Ocelové konstrukce a mosty 2003, ČVUT Praha, 2003, ISBN 80-01-02747-3.

* [FLOURY 2004]

FLOURY, S.: Glass in Contact, diploma work Université Blaire Pascal, Clermont-Ferrand, 2004.
# APPENDIX

## A 1: Measurements of the glass and contact plates, failure force –material of inserts from aluminium with different length

<table><thead><tr><th rowspan="2">Test number</th><th colspan="2">Glass</th><th colspan="2">Bottom plate</th><th colspan="2">Upper plate</th><th rowspan="2">First cracking [kN]</th><th rowspan="2">Failure [kN]</th><th rowspan="2">Deformation [mm]</th><th rowspan="2">1) Initial, 2) failure modes</th></tr><tr><th>t<sub>g</sub> [mm]</th><th>b [mm]</th><th>t<sub>pb</sub> [mm]</th><th>L<sub>pb</sub> [mm]</th><th>t<sub>pu</sub> [mm]</th><th>L<sub>pu</sub> [mm]</th></tr></thead><tbody><tr><td colspan="11"><strong>Contact material – aluminium, length 90 mm</strong></td></tr><tr><td colspan="11"><strong>Glass – size 120 x 120 mm, thickness 15 mm</strong></td></tr><tr><td colspan="11"><strong>Edge manufacturing of glass - polished</strong></td></tr><tr><td>Al 90/120-15/p-1</td><td>15.2</td><td>120.3</td><td>10.1</td><td>90.2</td><td>10.0</td><td>90.2</td><td>480</td><td>510</td><td>2.47</td><td>CC, EC</td></tr><tr><td>Al 90/120-15/p-2</td><td>15.1</td><td>120.2</td><td>10.0</td><td>90.1</td><td>10.2</td><td>90.2</td><td>520</td><td>540</td><td>2.68</td><td>CC, FF</td></tr><tr><td>Al 90/120-15/p-3</td><td>15.1</td><td>119.2</td><td>10.1</td><td>90.4</td><td>10.0</td><td>90.3</td><td>477</td><td>477</td><td>1.77</td><td>CC, FF</td></tr><tr><td>Al 90/120-15/p-4</td><td>15.1</td><td>119.8</td><td>10.1</td><td>90.2</td><td>10.1</td><td>90.3</td><td>495</td><td>495</td><td>1.70</td><td>NO, FF</td></tr><tr><td>Al 90/120-15/p-5</td><td>15.1</td><td>120.0</td><td>10.0</td><td>90.1</td><td>10.0</td><td>90.2</td><td>120</td><td>126</td><td>0.00</td><td>CC, I</td></tr><tr><td>Al 90/120-15/p-6</td><td>15.2</td><td>119.6</td><td>10.0</td><td>90.1</td><td>10.0</td><td>90.2</td><td>490</td><td>540</td><td>3.09</td><td>PF, PF</td></tr><tr><td>Al 90/120-15/p-7</td><td>15.2</td><td>119.8</td><td>9.9</td><td>90.1</td><td>10.0</td><td>90.2</td><td>495</td><td>549</td><td>1.61</td><td>NO, FF</td></tr><tr><td>Al 90/120-15/p-8</td><td>15.1</td><td>120.5</td><td>10.2</td><td>90.2</td><td>10.1</td><td>90.2</td><td>42</td><td>42</td><td>0.00</td><td>CC, I</td></tr><tr><td>Al 90/120-15/p-9</td><td>15.1</td><td>119.8</td><td>10.0</td><td>90.2</td><td>10.1</td><td>90.2</td><td>530</td><td>573</td><td>2.28</td><td>EC, PF</td></tr><tr><td>Al 90/120-15/p-10</td><td>15.1</td><td>120.3</td><td>10.0</td><td>90.2</td><td>10.1</td><td>90.2</td><td>540</td><td>560</td><td>2.11</td><td>CC, PF</td></tr><tr><td>Al 90/120-15/p-11</td><td>15.2</td><td>120.4</td><td>10.0</td><td>90.3</td><td>9.9</td><td>90.2</td><td>540</td><td>573</td><td>2.95</td><td>EC, PF</td></tr><tr><td>Al 90/120-15/p-12</td><td>15.2</td><td>120.2</td><td>10.1</td><td>90.2</td><td>9.9</td><td>90.2</td><td>600</td><td>603</td><td>2.73</td><td>EC, PF</td></tr><tr><td>Al 90/120-15/p-13</td><td>15.1</td><td>120.5</td><td>10.0</td><td>90.2</td><td>10.1</td><td>90.2</td><td>570</td><td>600</td><td>2.66</td><td>CC, FF</td></tr><tr><td>Al 90/120-15/p-14</td><td>15.2</td><td>119.8</td><td>10.0</td><td>90.2</td><td>10.2</td><td>90.3</td><td>450</td><td>485</td><td>1.77</td><td>CC, FF</td></tr><tr><td>Al 90/120-15/p-15</td><td>15.2</td><td>120.4</td><td>10.0</td><td>90.2</td><td>10.1</td><td>90.3</td><td>525</td><td>540</td><td>2.42</td><td>CC, FF</td></tr><tr><td>Al 90/120-15/p-16</td><td>15.1</td><td>119.7</td><td>10.1</td><td>90.4</td><td>10.0</td><td>90.4</td><td>575</td><td>585</td><td>2.37</td><td>CC, FF</td></tr><tr><td>Al 90/120-15/p-17</td><td>15.0</td><td>120.3</td><td>10.0</td><td>90.4</td><td>10.0</td><td>90.4</td><td>450</td><td>450</td><td>0.76</td><td>NO, FF</td></tr><tr><td>Al 90/120-15/p-18</td><td>15.2</td><td>120.3</td><td>10.1</td><td>90.3</td><td>9.9</td><td>90.3</td><td>620</td><td>630</td><td>2.70</td><td>EC, PF</td></tr><tr><td>Al 90/120-15/p-19</td><td>15.1</td><td>121.0</td><td>10.0</td><td>90.3</td><td>10.1</td><td>90.3</td><td>600</td><td>615</td><td>2.69</td><td>EC, FF</td></tr><tr><td>Al 90/120-15/p-20</td><td>15.1</td><td>119.6</td><td>10.1</td><td>90.4</td><td>10.0</td><td>90.3</td><td>650</td><td>655</td><td>3.41</td><td>CC, FF</td></tr><tr><td colspan="11"><strong>Contact material – aluminium, length 60 mm</strong></td></tr><tr><td colspan="11"><strong>Glass – size 120 x 120 mm, thickness 15 mm</strong></td></tr><tr><td colspan="11"><strong>Edge manufacturing of glass - polished</strong></td></tr><tr><td>Al 60/120-15/p-1</td><td>15.2</td><td>120.5</td><td>10.1</td><td>60.2</td><td>10.0</td><td>60.3</td><td>190</td><td>215</td><td>0.01</td><td>PF, IC</td></tr><tr><td>Al 60/120-15/p-2</td><td>15.1</td><td>119.7</td><td>10.1</td><td>60.4</td><td>9.9</td><td>60.1</td><td>405</td><td>420</td><td>2.67</td><td>CC, PF</td></tr><tr><td>Al 60/120-15/p-3</td><td>15.1</td><td>120.1</td><td>10.0</td><td>60.1</td><td>9.9</td><td>60.2</td><td>350</td><td>360</td><td>2.14</td><td>CC, FF</td></tr><tr><td>Al 60/120-15/p-4</td><td>15.1</td><td>120.1</td><td>10.0</td><td>60.0</td><td>10.0</td><td>60.0</td><td>280</td><td>335</td><td>1.60</td><td>CC, FF</td></tr><tr><td>Al 60/120-15/p-5</td><td>15.2</td><td>120.1</td><td>9.9</td><td>60.0</td><td>9.9</td><td>60.2</td><td>320</td><td>450</td><td>2.84</td><td>CC, FF</td></tr><tr><td>Al 60/120-15/p-6</td><td>15.1</td><td>120.1</td><td>10.2</td><td>60.0</td><td>10.0</td><td>60.1</td><td>360</td><td>375</td><td>2.81</td><td>CC, FF</td></tr><tr><td>Al 60/120-15/p-7</td><td>15.0</td><td>119.7</td><td>10.0</td><td>60.1</td><td>9.8</td><td>60.2</td><td>328</td><td>336</td><td>4.50</td><td>CC, I</td></tr><tr><td>Al 60/120-15/p-8</td><td>15.0</td><td>119.7</td><td>9.9</td><td>60.2</td><td>9.9</td><td>60.2</td><td>-</td><td>290</td><td>4.40</td><td>I</td></tr><tr><td colspan="11"><strong>Contact material – aluminium, length 180 mm</strong></td></tr><tr><td colspan="11"><strong>Glass – size 120 x 120 mm, thickness 15 mm</strong></td></tr><tr><td colspan="11"><strong>Edge manufacturing of glass - polished</strong></td></tr><tr><td>Al 180/120-15/p-1</td><td>15.1</td><td>120.1</td><td>10.0</td><td>180.0</td><td>10.0</td><td>180.0</td><td>-</td><td>570</td><td>0.57</td><td>NO, IC</td></tr><tr><td>Al 180/120-15/p-2</td><td>15.1</td><td>120.2</td><td>10.1</td><td>180.0</td><td>10.0</td><td>180.0</td><td>-</td><td>630</td><td>0.35</td><td>NO, IC</td></tr><tr><td>Al 180/120-15/p-3</td><td>15.1</td><td>120.2</td><td>10.2</td><td>180.0</td><td>10.0</td><td>180.0</td><td>-</td><td>475</td><td>0.44</td><td>NO, IC</td></tr><tr><td>Al 180/120-15/p-4</td><td>15.1</td><td>119.6</td><td>10.1</td><td>180.0</td><td>10.0</td><td>180.0</td><td>-</td><td>635</td><td>0.29</td><td>NO, FF</td></tr><tr><td>Al 180/120-15/p-5</td><td>15.1</td><td>119.8</td><td>10.1</td><td>180.0</td><td>10.1</td><td>180.0</td><td>-</td><td>640</td><td>0.43</td><td>PF, FF</td></tr><tr><td>Al 180/120-15/p-6</td><td>15.1</td><td>120.1</td><td>9.9</td><td>180.0</td><td>10.0</td><td>180.0</td><td>-</td><td>530</td><td>0.67</td><td>PF, PF</td></tr></tbody></table>

Notes: 1) Initial damages: EC – edge crack

CC – corner crack

PF – panel flakes

2) Failure modes: PF – panel fragmentation

FF – fast fragmentation

IC – internal crack

CU – insert cut
A 2: Measurements of the glass and contact plates, failure force – material of inserts from aluminium, glass panel with different size and thickness

<table><thead><tr><th rowspan="2">Test number</th><th colspan="2">Glass</th><th colspan="2">Bottom plate</th><th colspan="2">Upper plate</th><th rowspan="2">First cracking [kN]</th><th rowspan="2">Failure [kN]</th><th rowspan="2">Deformation [mm]</th><th rowspan="2">1) Initial, 2) failure modes</th></tr><tr><th>t<sub>g</sub> [mm]</th><th>b [mm]</th><th>t<sub>pb</sub> [mm]</th><th>L<sub>pb</sub> [mm]</th><th>t<sub>pu</sub> [mm]</th><th>L<sub>pu</sub> [mm]</th></tr></thead><tbody><tr><td colspan="11"><strong>Contact material – aluminium, length 90 mm</strong></td></tr><tr><td colspan="11"><strong>Glass – size 150 x 150 mm, thickness 15 mm</strong></td></tr><tr><td colspan="11"><strong>Edge finishing of glass - polished</strong></td></tr><tr><td>Al 90/150-15/p-1</td><td>15.1</td><td>149.6</td><td>9.8</td><td>90.3</td><td>9.8</td><td>90.3</td><td>480</td><td>495</td><td>1.80</td><td>CC, IC</td></tr><tr><td>Al 90/150-15/p-2</td><td>15.0</td><td>150.1</td><td>9.8</td><td>90.0</td><td>9.9</td><td>90.4</td><td>435</td><td>468</td><td>2.15</td><td>CC, FF</td></tr><tr><td>Al 90/150-15/p-3</td><td>15.1</td><td>150.5</td><td>9.9</td><td>90.2</td><td>10.0</td><td>90.3</td><td>375</td><td>546</td><td>2.90</td><td>CC, PF</td></tr><tr><td>Al 90/150-15/p-4</td><td>15.1</td><td>150.2</td><td>10.0</td><td>90.2</td><td>9.8</td><td>90.2</td><td>525</td><td>576</td><td>2.90</td><td>CC, PF</td></tr><tr><td>Al 90/150-15/p-5</td><td>15.1</td><td>150.5</td><td>9.8</td><td>90.3</td><td>9.9</td><td>90.2</td><td>435</td><td>492</td><td>2.10</td><td>CC, PF</td></tr><tr><td>Al 90/150-15/p-6</td><td>15.1</td><td>150.4</td><td>9.8</td><td>90.2</td><td>9.8</td><td>90.0</td><td>445</td><td>555</td><td>2.15</td><td>CC, IC</td></tr><tr><td colspan="11"><strong>Contact material – aluminium, length 90 mm</strong></td></tr><tr><td colspan="11"><strong>Glass – size 180 x 180 mm, thickness 15 mm</strong></td></tr><tr><td colspan="11"><strong>Edge finishing of glass - polished</strong></td></tr><tr><td>Al 90/180-15/p-1</td><td>15.1</td><td>180.3</td><td>9.9</td><td>90.1</td><td>9.9</td><td>90.1</td><td>500</td><td>535</td><td>3.05</td><td>CC, IC</td></tr><tr><td>Al 90/180-15/p-2</td><td>15.0</td><td>180.4</td><td>10.0</td><td>90.0</td><td>9.8</td><td>90.1</td><td>350</td><td>625</td><td>2.80</td><td>CC, PF</td></tr><tr><td>Al 90/180-15/p-3</td><td>15.0</td><td>180.4</td><td>9.9</td><td>90.2</td><td>9.9</td><td>90.0</td><td>500</td><td>545</td><td>2.60</td><td>CC</td></tr><tr><td>Al 90/180-15/p-4</td><td>15.0</td><td>180.5</td><td>9.9</td><td>90.0</td><td>9.8</td><td>90.0</td><td>530</td><td>630</td><td>3.40</td><td>CC</td></tr><tr><td>Al 90/180-15/p-5</td><td>15.1</td><td>180.2</td><td>9.9</td><td>90.2</td><td>9.8</td><td>90.0</td><td>640</td><td>680</td><td>3.10</td><td>CC</td></tr><tr><td>Al 90/180-15/p-6</td><td>15.0</td><td>180.3</td><td>10</td><td>90.1</td><td>10.0</td><td>90.2</td><td>475</td><td>615</td><td>3.30</td><td>CC, PF</td></tr><tr><td colspan="11"><strong>Contact material – aluminium, length 90 mm</strong></td></tr><tr><td colspan="11"><strong>Glass – size 120 x 120 mm, thickness 15 mm</strong></td></tr><tr><td colspan="11"><strong>Edge finishing of glass - smoothed</strong></td></tr><tr><td>Al 90/120-15/s-01</td><td>14.6</td><td>120.1</td><td>10.0</td><td>90.2</td><td>10.0</td><td>90.1</td><td>-</td><td>208</td><td>-</td><td>NO, IC</td></tr><tr><td>Al 90/120-15/s-02</td><td>14.8</td><td>120.2</td><td>10.0</td><td>90.2</td><td>9.9</td><td>90.2</td><td>-</td><td>350</td><td>-</td><td>FF</td></tr><tr><td>Al 90/120-15/s-03</td><td>14.9</td><td>120.0</td><td>10.0</td><td>90.2</td><td>10.0</td><td>90.0</td><td>210</td><td>403</td><td>-</td><td>PF, PF</td></tr><tr><td>Al 90/120-15/s-04</td><td>14.8</td><td>120.0</td><td>10.1</td><td>90.1</td><td>10.0</td><td>90.2</td><td>240</td><td>331</td><td>-</td><td>CC, PF</td></tr><tr><td>Al 90/120-15/s-05</td><td>14.8</td><td>120.3</td><td>10.0</td><td>90.3</td><td>10.0</td><td>90.1</td><td>180</td><td>339</td><td>-</td><td>CC, PF</td></tr><tr><td colspan="11"><strong>Contact material – aluminium, length 90 mm</strong></td></tr><tr><td colspan="11"><strong>Glass – size 120 x 120 mm, thickness 12 mm</strong></td></tr><tr><td colspan="11"><strong>Edge finishing of glass - smoothed</strong></td></tr><tr><td>Al 90/120-12/s-1</td><td>12.2</td><td>120.0</td><td>9.9</td><td>90.2</td><td>10.0</td><td>90.3</td><td>50</td><td>75</td><td>0</td><td>EC, PF</td></tr><tr><td>Al 90/120-12/s-2</td><td>12.2</td><td>120.1</td><td>10.1</td><td>90.2</td><td>9.9</td><td>90.3</td><td>450</td><td>460</td><td>1.590</td><td>EC, FF</td></tr><tr><td>Al 90/120-12/s-3</td><td>12.2</td><td>120.0</td><td>10.0</td><td>90.2</td><td>10.0</td><td>90.2</td><td>430</td><td>450</td><td>1.550</td><td>CC, FF</td></tr><tr><td>Al 90/120-12/s-4</td><td>12.2</td><td>119.8</td><td>10.0</td><td>90.3</td><td>10.0</td><td>90.2</td><td>430</td><td>450</td><td>1.550</td><td>NO, FF</td></tr><tr><td>Al 90/120-12/s-5</td><td>12.2</td><td>119.8</td><td>9.9</td><td>90.4</td><td>10.1</td><td>90.4</td><td>450</td><td>465</td><td>2.065</td><td>CC, IC</td></tr><tr><td>Al 90/120-12/s-6</td><td>12.2</td><td>119.8</td><td>10.0</td><td>90.4</td><td>9.9</td><td>90.3</td><td>465</td><td>465</td><td>1.915</td><td>NO, FF</td></tr><tr><td colspan="11"><strong>Contact material – aluminium, length 90 mm</strong></td></tr><tr><td colspan="11"><strong>Glass – size 120 x 120 mm, thickness 10 mm</strong></td></tr><tr><td colspan="11"><strong>Edge finishing of glass - smoothed</strong></td></tr><tr><td>Al 90/120-10/s-1</td><td>10.0</td><td>119.6</td><td>9.9</td><td>90.2</td><td>10.0</td><td>90.2</td><td>-</td><td>50</td><td>0</td><td>PF, IC</td></tr><tr><td>Al 90/120-10/s-2</td><td>10.0</td><td>119.1</td><td>10.1</td><td>90.2</td><td>9.9</td><td>90.2</td><td>-</td><td>45</td><td>0</td><td>PF, IC</td></tr><tr><td>Al 90/120-10/s-3</td><td>10.0</td><td>118.8</td><td>10.1</td><td>90.2</td><td>10.1</td><td>90.2</td><td>420</td><td>446</td><td>4.230</td><td>CC, PF</td></tr><tr><td>Al 90/120-10/s-4</td><td>10.0</td><td>119.3</td><td>10.0</td><td>90.1</td><td>9.9</td><td>90.1</td><td>375</td><td>384</td><td>2.005</td><td>CC, PF</td></tr><tr><td>Al 90/120-10/s-5</td><td>10.0</td><td>119.1</td><td>10.1</td><td>90.1</td><td>10.0</td><td>90.1</td><td>390</td><td>417</td><td>2.960</td><td>PF, IC</td></tr><tr><td>Al 90/120-10/s-6</td><td>10.0</td><td>119.3</td><td>10.0</td><td>90.3</td><td>9.9</td><td>90.2</td><td>390</td><td>402</td><td>2.590</td><td>CC, FF</td></tr></tbody></table>

Notes: 1) Initial damages: EC – edge crack
CC – corner crack
PF – panel flakes

2) Failure modes: PF – panel fragmentation
FF – fast fragmentation
IC – internal crack
CU – insert cut
<table><caption>A 3: Measurements of the glass and contact plates, failure force – material of inserts from polyamide, steel, epoxide resin</caption><thead><tr><th rowspan="2">Test number</th><th colspan="2">Glass</th><th colspan="2">Bottom plate</th><th colspan="2">Upper plate</th><th rowspan="2">First cracking [kN]</th><th rowspan="2">Failure [kN]</th><th rowspan="2">Deformation [mm]</th><th rowspan="2">1) Initial, 2) failure modes</th></tr><tr><th>t<sub>g</sub> [mm]</th><th>b [mm]</th><th>t<sub>pb</sub> [mm]</th><th>L<sub>pb</sub> [mm]</th><th>t<sub>pu</sub> [mm]</th><th>L<sub>pu</sub> [mm]</th></tr></thead><tbody><tr><td colspan="11"><strong>Contact material – polyamide, length 90 mm</strong></td></tr><tr><td colspan="11"><strong>Glass – size 120 x 120 mm, thickness 15 mm</strong></td></tr><tr><td colspan="11"><strong>Edge finishing of glass – polished, smoothed</strong></td></tr><tr><td>Pa 90/120-15/s-01</td><td>14.7</td><td>120.0</td><td>10.8</td><td>90.1</td><td>10.8</td><td>90.1</td><td>-</td><td>130</td><td>-</td><td>PF, I</td></tr><tr><td>Pa 90/120-15/s-02</td><td>14.7</td><td>120.0</td><td>10.8</td><td>90.2</td><td>10.8</td><td>90.2</td><td>-</td><td>183</td><td>8.2</td><td>EC, I</td></tr><tr><td>Pa 90/120-15/s-03</td><td>14.8</td><td>120.0</td><td>10.8</td><td>90.2</td><td>10.8</td><td>90.2</td><td>-</td><td>180</td><td>6.3</td><td>PF, I</td></tr><tr><td>Pa 90/120-15/s-04</td><td>14.8</td><td>120.0</td><td>10.9</td><td>90.0</td><td>10.9</td><td>90.0</td><td>-</td><td>187</td><td>8.5</td><td>PF, I</td></tr><tr><td colspan="11"><strong>Contact material – polyamide, length 90 mm</strong></td></tr><tr><td colspan="11"><strong>Glass – size 120 x 120 mm, thickness 15 mm</strong></td></tr><tr><td colspan="11"><strong>Edge finishing of glass - polished</strong></td></tr><tr><td>Pa 90/120-15/p-1</td><td>15.0</td><td>119.7</td><td>10.5</td><td>90.1</td><td>10.5</td><td>89.9</td><td>192</td><td>198</td><td>11</td><td>I</td></tr><tr><td>Pa 90/120-15/p-2</td><td>15.0</td><td>120.0</td><td>10.5</td><td>89.9</td><td>10.5</td><td>90.0</td><td>-</td><td>192</td><td>7</td><td>I</td></tr><tr><td>Pa 90/120-15/p-3</td><td>15.0</td><td>120.5</td><td>10.5</td><td>90.0</td><td>10.5</td><td>90.0</td><td>-</td><td>210</td><td>-</td><td>PF, I</td></tr><tr><td>Pa 90/120-15/p-4</td><td>15.0</td><td>120.0</td><td>10.5</td><td>90.2</td><td>10.5</td><td>90.0</td><td>-</td><td>210</td><td>-</td><td>I</td></tr><tr><td colspan="11"><strong>Contact material – steel, length 90 mm</strong></td></tr><tr><td colspan="11"><strong>Glass – size 120 x 120 mm, thickness 15 mm</strong></td></tr><tr><td colspan="11"><strong>Edge finishing of glass - smoothed</strong></td></tr><tr><td>Fe 90/120-15/s-01</td><td>14.8</td><td>120.0</td><td>10.2</td><td>90.0</td><td>10.0</td><td>90.0</td><td>120</td><td>378</td><td>-</td><td>EC, PF</td></tr><tr><td>Fe 90/120-15/s-02</td><td>14.7</td><td>120.4</td><td>10.0</td><td>90.0</td><td>10.0</td><td>89.9</td><td>-</td><td>438</td><td>-</td><td>NO, FF</td></tr><tr><td>Fe 90/120-15/s-03</td><td>14.8</td><td>120.3</td><td>10.0</td><td>90.0</td><td>10.0</td><td>98.4</td><td>-</td><td>480</td><td>-</td><td>NO, FF</td></tr><tr><td>Fe 90/120-15/s-04</td><td>14.7</td><td>119.9</td><td>10.0</td><td>89.2</td><td>10.0</td><td>89.6</td><td>90</td><td>153</td><td>-</td><td>CC, PF</td></tr><tr><td>Fe 90/120-15/s-05</td><td>14.8</td><td>119.9</td><td>10.0</td><td>89.2</td><td>10.1</td><td>89.6</td><td>470</td><td>582</td><td>-</td><td>CC, FF</td></tr><tr><td>Fe 90/120-15/s-06</td><td>14.8</td><td>120.3</td><td>8.5</td><td>89.1</td><td>8.6</td><td>89.2</td><td>-</td><td>495</td><td>-</td><td>NO, FF</td></tr><tr><td colspan="11"><strong>Contact material – epoxide resin, length 90 mm</strong></td></tr><tr><td colspan="11"><strong>Glass – size 120 x 120 mm, thickness 15 mm</strong></td></tr><tr><td colspan="11"><strong>Edge finishing of glass - smoothed</strong></td></tr><tr><td>Ep 90/120-15/s-01</td><td>14.9</td><td>120.2</td><td>10.0</td><td>89.9</td><td>10.1</td><td>89.9</td><td>-</td><td>200</td><td>2.7</td><td>CU</td></tr><tr><td>Ep 90/120-15/s-02</td><td>14.9</td><td>120.0</td><td>10.2</td><td>89.9</td><td>10.0</td><td>89.7</td><td>-</td><td>204</td><td>5.0</td><td>CU</td></tr><tr><td>Ep 90/120-15/s-03</td><td>14.9</td><td>120.2</td><td>10.0</td><td>89.7</td><td>10.1</td><td>89.7</td><td>-</td><td>176</td><td>3.4</td><td>PF, IC</td></tr><tr><td>Ep 90/120-15/s-04</td><td>14.9</td><td>120.1</td><td>10.0</td><td>90.0</td><td>10.0</td><td>89.8</td><td>-</td><td>205</td><td>2.6</td><td>NO, PF</td></tr></tbody></table>

Notes: 1) Initial damages: EC – edge crack
CC – corner crack
PF – panel flakes

2) Failure modes: PF – panel fragmentation
FF – fast fragmentation
IC – internal crack
CU – insert cut
# Design of Glass Members –
A Critical Review of the Present Knowledge

Matthias Haldimann

Ecole Polytechnique Fédérale de Lausanne (EPFL),

Laboratoire de la construction métallique (ICOM), CH-1015 Lausanne, Switzerland

Engineers in Europe and elsewhere are currently using various concepts for the design and analysis of structural elements made of glass. There is no general agreement on a certain design concept yet and all concepts suffer from more or less severe drawbacks.

This paper discusses the present knowledge in structural glass design focusing on difficulties and limitations. It aims at providing an overview for anyone interested in the topic as well as at being a basis for discussion on desirable improvements to design concepts and for research work on the topic.

**Keywords:** structural glass, design concept, code, failure prediction, state of knowledge

## I. INTRODUCTION

In recent years, many research projects have been dedicated to the structural use of glass. Nevertheless, engineers designing load carrying glass elements are in a rather uncomfortable situation when it comes to design concepts:

* In Europe, there is no general agreement on a particular design concept; past code drafts faced stiff opposition.

* Some existing concepts are widely used but are of limited applicability and suffer from significant limitations and drawbacks. Either they are very simplistic and therefore unsuited for sophisticated and innovative designs or they are complex, cumbersome, error prone, non-

transparent and lack flexibility despite their complexity.

* No comprehensive safety concept is available to take the particularities associated with the brittle material glass sufficiently into account.

This leads not only to the frequent need for time-consuming and expensive laboratory testing during the design process for advanced glass structures, but also to many load-carrying glass elements being either oversized and thus unsatisfactory from an economical and architectural point of view or representing a significant safety risk due to incorrect assumptions made in their design.

## II. STATE OF KNOWLEDGE

### A. Overview

The methods for the analysis of structural glass elements available today can be divided into three groups:

a) Simple design rules based on allowable stress;

b) Concepts that allow for the calculation of a design resistance based on Weibull theory [Weibull 1951], the failure probability found in laboratory tests and correction factors for consideration of influences like load duration, panel surface area in tension, stress distribution and environmental conditions;

c) Concepts that are based on the direct simulation of subcritical crack growth during the lifetime of the glass element.

Concepts of group a) (e. g. [TRLV 1998],
[TRAV 2003]) are simple and straightforward but of limited applicability and accuracy. This is mainly due to the inherent problems of allowable stress concepts that are even more in evidence when applied to glass than for other materials.

The failure probability based concepts of group b) ([Beason et al. 1984] [Blank 1993] [Sedlacek et al. 1995] [Güsgen 1998] [prEN 13474-2:2000] and others) are widely used and lead to good results for many standard applications and quasi-static out of plane loads. Both the inherent glass strength and the effects of prestress due to thermal tempering are considered. The drawbacks of these concepts include their limited applicability and flexibility despite their complexity, their lack of transparency for the user and the fact that not all assumptions and concepts used are consistent and scientifically justifiable.

In recent years, efforts are made to develop glass failure prediction models that are based on the direct simulation of subcritical crack growth during the lifetime of the glass element. Reid proposes a so called ‘crack growth model’ for the prediction of the bending strength of annealed glass [Reid 1991], Porter proposes the ‘crack size design method’ for in-plane loaded, annealed glass beams [Porter 2001]. Though further research work is clearly required, the crack size approach is very promising.

## B. Terminology

Each section in this chapter uses the notation as it is introduced in the original literature on the concepts discussed. This does inevitably lead to inconsistencies between different sections.

For clarity, some technical terms and concepts that will be repeatedly used in the further text shall be defined here:

### Residual stress due to heat treatment ($\sigma_{res}$):

The surface compression stress introduced in glass plates by heat treatment (tempering). The term ‘prestress’ is – although widely used – somewhat misleading and shall not be used in

this article.

### Inherent strength:

The (macroscopic) tensile material strength of glass. This is the tensile strength of annealed glass or the part of the tensile strength of heat treated glass that is not due to residual stress (see also III.C).

### Decompressed glass surface:

The part of the element surface where the tensile stress due to loading is bigger than the residual compression stress due to thermal treatment ($\sigma_{tot} = \sigma_{load} + \sigma_{residual} > 0$).

### Weibull distribution:

This probability distribution with the cumulative distribution function given in eq. (1) was developed in [Weibull 1951] and is often used to model the inherent resistance of glass.

$$F(\sigma_{bB,A_0}) = 1 - \exp\left[-\left(\frac{\sigma_{bB,A_0}}{\theta_{A_0}}\right)^\beta\right] \quad (1)$$

$F(\sigma_{bB,A_0})$ represents the failure probability of a glass plate of size $A_0$ under uniform tensile surface stress $\sigma_{bB,A_0}$. $\beta$ is the shape parameter of the distribution, $\theta_{A_0}$ the scale parameter. While the shape parameter is independent from $A_0$, the scale parameter is not.

## C. A brief overview of today's important design concepts

As a basis for the discussion in chapter III, the present chapter aims at very briefly presenting some concepts currently in use. Detailed information can be found in [Haas et al. 2004] (in German).

### 1) RWTH Aachen

This concept is mainly based on [Blank 1993] and [Güsgen 1998] and is outlined in detail in [Sedlacek et al. 1999]. It compares the maximum design stress $\sigma_{max,d}$ with an equivalent resistance as follows:
$$ \sigma_{\max,d} \le \frac{\sigma_{bB,A_0,k}}{\alpha_{\sigma}(p,\sigma_v) \cdot \alpha(A_{red}) \cdot \alpha(t) \cdot \alpha(S_v) \cdot \gamma_{M,E}} \quad (2) \\ + \frac{\sigma_{V,k}}{\gamma_{M,V}} $$

$\alpha_{\sigma}(p, \sigma_v)$ factor to consider the stress distribution within the decompressed glass surface; $p$ = uniformly distributed load perpendicular to the glass surface; $\sigma_v$ = surface compression stress due to tempering

$\alpha(A_{red})$ factor to account for the size of the decompressed surface area (entire surface area for annealed glass)

$\alpha(t)$ factor to account for the load duration

$\alpha(S_v)$ factor to consider load combination and environmental conditions

$\sigma_{\max,d}$ design value of the max. principal tensile stress

$\sigma_{bB,A_0,k}$ characteristic value of inherent bending breaking strength in laboratory tests with surface area $A_0$ (5% fractile at 0.95 confidence interval)

$\sigma_{V,k}$ characteristic value of the residual stress (compression = positive)

$\gamma_{M,E}$ partial factor for the inherent resistance

$\gamma_{M,V}$ partial factor for tempering

Determination of some of the factors is complex, see [Sedlacek et al. 1999] or [Haas et al. 2004] for details. A modified form of eq. (2) is proposed for glass beams. The derivation of all influence factors is based on the assumption that the failure probability of a glass plate follows a Weibull distribution according to eq. (1).

Further assumptions include a fixed design point (see III.J), given load durations and overlapping probabilities as well as material parameters for winter and summer conditions.

## 2) prEN 13474

The European code drafts [prEN 13474-1:1999] and [prEN 13474-2:2000] faced opposition and are still under revision. Nevertheless, their basic concepts shall be outlined here.

The verification of structural safety is based on

the comparison of an effective stress $\sigma_{ef,d}$ with an allowable stress $f_{g,d}$:

$$ \sigma_{ef,d} \le f_{g,d} \quad (3) $$

It is interesting to note that the effective stress

$$ \sigma_{ef,d} = \sigma_{\max,d} \cdot \alpha_{\sigma}(q) \quad (4) $$

is defined *independently* from residual stress. For common geometries and support conditions there are tables giving $\sigma_{ef,d}$ in function of the applied load $q$ and the plate dimensions.

The allowable stress is defined as:

$$ f_{g,d} = \left( k_{\text{mod}} \frac{f_{g,c}}{\gamma_M \cdot k_A} + \frac{f_{b,c} - f_{g,c}}{\gamma_V} \right) \cdot \gamma_n \quad (5) $$

$f_{b,c}$ characteristic value of breaking strength (5% fractile); $f_{g,c}$ for annealed glass, 70 MPa for heat strengthened glass and 120 MPa for fully tempered glass (soda lime silicate glass)

$f_{g,c}$ characteristic value of inherent breaking strength (5% fractile); $f_{g,c} = 45$ MPa

$f_{b,c} - f_{g,c}$ residual stress (compression = positive)

$\gamma_V$ partial factor for tempering (= 2.3 for soda lime silicate glass)

$\gamma_M$ partial factor for the inherent resistance (= 1.8 for soda lime silicate glass)

$\gamma_n$ national calibration factor (mostly = 1.0)

$k_A$ factor to consider surface area, defined independently from the residual stress;

$$ k_A = A^{0.04} $$

$k_{\text{mod}}$ modification factor to consider load duration, load combination and environmental conditions; many assumptions are made to simplify this to only three values: 0.27 for permanent loads, 0.36 for loads of medium duration (snow, climate loads), 0.72 for loads with short durations (wind)

## 3) Allowable stress concept

Despite its numerous drawbacks and oversimplistic approach, it is the extreme ease of use that keeps this concept interesting and still widely used. The verification format is

$$ \sigma_E \le \sigma_{adm} = \frac{\sigma_R}{\gamma} \quad (6) $$
with the global safety factor γ including all uncertainties associated with actions, resistance and modelling. The German technical rules [TRLV 1998] and [TRAV 2003] are well known and widely used examples of design guides using allowable stresses. They give allowable stress values for annealed, heat-strengthened and fully tempered glass.

### 4) Shen

This concept is first presented in [Shen 1997]. In [Wörner et al. 2001] it is adopted to the verification format of [EN 1990:2002]. It is to a large extent a simplification of the concept presented in paragraph II.C.1). For tempered glass however, the approach and numeric values from the Canadian Standard (see II.C.6)) are used. The ULS verification format is

$$ \sigma_{\max,d} \le \sigma_k \cdot \frac{\eta_F \cdot \eta_D}{\gamma_R} \quad (7) $$

* $\sigma_{\max,d}$ design value of the max. principal stress

* $\sigma_k$ characteristic value of the inherent strength

* $\eta_F$ coefficient for the area under traction and the stress distribution within this area

* $\eta_D$ coefficient considering load duration

* $\gamma_R$ partial resistance factor

The verification must be done separately for actions with different load durations. As can be seen from eq. (7), this concept introduces large simplifications. Residual stresses are accounted for indirectly by giving different values of $\eta_D$ and $\eta_F$ for annealed and fully tempered glass respectively.

### 5) Siebert

Building upon the concepts of RWTH Aachen and Shen, [Siebert 1999] proposes a concept that incorporates three important changes:

* It considers the influence of the second principal stress.

* It treats residual stress as an action;

* It uses a less conservative safety factor for the residual stress level.

The following verification format is proposed:

$$ \sigma_{tot,d,max} \cdot f_A \cdot f_\sigma \cdot f_{tS} \le \frac{\theta}{f_P} \quad (8) $$

$$ \text{with } \sigma_{tot,d,max} = \sigma_{d,max} + \sigma_E \quad (9) $$

* $\sigma_{tot,d,max}$ max. principal stress at the surface

* $\sigma_{d,max}$ max. principal stress due to actions

* $\sigma_E$ residual stress (compression = negative)

* $f_A$ coefficient considering the different size of the area under traction in experiment and application

* $f_\sigma$ coefficient considering the different stress distribution in experiment and application

* $f_{tS}$ coefficient for load durations and combination (dead load, snow, wind)

* $\theta$ Weibull scale parameter of the inherent breaking strength; see eq. (1)

* $f_P$ coefficient for the required safety level

As residual stress is considered as an action, $f_\sigma$ becomes dependent from it. $f_A$ and $f_{tS}$ are directly taken from [Güsgen 1998], and therefore equivalent to the coefficients in the RWTH Aachen concept: $f_A = \alpha(A)$, $f_{tS} = \alpha(t) \cdot \alpha(S_v)$.

### 6) Canadian Standard

The Canadian Standard 'Structural Design of Glass for Buildings' [CAN/CGSB 12.20-M89] treats exclusively soda-lime glass under evenly distributed out-of plane loads. It follows a ULS verification format that is extremely similar to what is known from steel design and is very easy to use. For the purpose of this paper, only the resistance part will be discussed (see [Haas et al. 2004] for more details). Glass resistance R is given as

$$ R = c_1 \cdot c_2 \cdot c_3 \cdot c_4 \cdot R_{ref} \quad (10) $$

with strength coefficients as follows:

* $c_1$ glass type
  (value: 1.0 for flat glass and laminated glass)

* $c_2$ heat treatment
  (values: 1.0 for annealed, 2.0 for heat strengthened, 4.0 for tempered glass)

* $c_3$ load duration (simple table; depends also on heat treatment)
$c_4$ load sharing
(values: 1.0 for monolithic glass, 1.7 for double-glazed sealed insulating glass units)

$R_{ref}$ design resistance of reference glasses
(given in a large series of tables and graphs in function of thickness, size and aspect ratio)

Glass resistance is determined using the failure prediction model developed by Beason, Morgan and Vallabhan ([Vallabhan 1983], [Beason et al. 1984]) and targeting an expected failure probability of 0.8 %.

Full shear transmission by the PVB interlayer may be taken into account if the load duration is < 1 minute at a temperature < 70 °C or if it is < 1 week at < 20 °C. For all other cases, shear transmission by PVB must be neglected.

For situations not contained in the tables, the panel size effect shall be considered according to eq. (13), but with $\beta=7$. The resistance for any load duration $t$ in minutes is given by

$$R_t = R_{ref} \cdot t^{(-1/n)} \quad (11)$$

with $n$ being 15 for annealed, 30 for heat strengthened, and 70 for fully tempered glass. Eq. (11) simply makes the resistance of tempered glass less time dependent than that of annealed glass, irrespective of the stress level. As in reality, it is not time dependent at all as long as surface tensile stress due to loading is smaller than the compression stress due to tempering (residual stress), this can lead to obviously wrong results.

### 7) US Standard

The ASTM Code [ASTM E 1300-03] is very similar to the Canadian Standard and uses the same scientific bases. It's only the verification format that is slightly different. Dozens of diagrams for various thicknesses, aspect ratios and support conditions give the nonfactored load (NFL) that an annealed glass panel could support for 3-seconds<fnref n="1" />.

These serve as reference and are multiplied by glass type factors (GTF) including also the load duration. To combine loads of different duration, they are all converted to equivalent 3-s duration loads as follows:

$$q_3 = \sum_i q_i \left[ \frac{d_i}{3} \right]^{\frac{1}{n}} \quad (12)$$

$q_3$ magnitude of the 3-s duration uniform load

$q_i$ magnitude of the load having duration $d_i$

$n$ crack growth parameter
(value for annealed glass: 16)

## III. DISCUSSION: DRAWBACKS OF CURRENT CONCEPTS

### A. Does size matter?

Where the plate size effect is explicitly considered, it is assumed to be given by

$$\frac{\sigma_t(A_1)}{\sigma_t(A_2)} = \left[ \frac{A_2}{A_1} \right]^{\frac{1}{\beta}} \quad (13)$$

$\sigma_t(A_1)$ tensile strength of a structural member with area $A_1$ under tension

$\sigma_t(A_2)$ tensile strength of a structural member with area $A_2$ under tension

$\beta$ shape parameter of the Weibull distribution used to describe the tensile breaking strength; see eq. (1)

Eq. (13) is based on the assumption that the tensile breaking strength follows a Weibull distribution according to eq. (1). The size effect is quite significant for small values of $\beta$ (e. g. 7 as used in North American concepts) while it becomes almost negligible in comparison with other influences and uncertainties for commonly used panel sizes if $\beta = 25$ (according to European concepts for glass under real-life conditions and out-of-plane) is used (Fig. 1).

<footnote n="1">¹ Using the failure prediction model by Beason, Morgan and Vallabhan assuming that the surface flaw parameters are $m = 7$ and $k = 2.86 \cdot 10^{-53}$</footnote>

<footnote>N⁻⁷m¹². These parameters are considered representing weathered window glass that has undergone in-service conditions for approx. 20 years.</footnote>
<figure><img src="image_27.png" /><figcaption>Fig. 1: The panel size effect's dependency on the Weibull shape factor (calculated using eq. (13))</figcaption></figure>

Both theoretical and experimental data give reasons to question the use of the Weibull distribution (see III.B). Sometimes (see e.g. [Fink 2000.1]) a 'pragmatic' approach which consists in combining a log-normal distribution for the breaking strength with the Weibull-based size effect equation is used.

Failure of glass elements is caused by tensile stress even for compression loaded elements. Thus only the decompressed surface area contributes to the size effect and for heat treated glass only the part of the resistance due to inherent strength is affected by the size effect. The size of the decompressed surface area depends – for given geometry and support conditions – on the loads and is therefore different for each load case. The 'correct' consideration of this increases the complexity of the calculation process considerably, which is why most current concepts choose to consider the total area of the glass element and thus define the size effect independently from the actions (see also III.F).

Interestingly, recent experiments show little or no relationship nor between the total nor the most stressed panel area and the breakage stress [Caldone 2001].

## B. Is the Weibull distribution applicable?

Both theoretical considerations and experimental data suggest that the Weibull distribution may not be very well suited for the prediction of the failure probability of glass panels. The log-normal distribution

seems to often better fit experimental data especially at the lower end. The impressive variation in the Weibull parameters found in experiments and used for design concepts today raises the question of them depending on too many factors other than glass strength to be suitable for design.

On the other hand, the weakest-link model, which is at the basis of the Weibull distribution, remains undoubtedly the most obvious model for the failure behaviour of glass. There have not yet been any proposals on how a non-Weibull distribution type could be consistently explained and integrated into the physical, mathematical and statistical concepts involved in glass testing and failure prediction.

## C. Consideration of residual stress due to heat treatment

Several concepts 'include' residual stress due to heat treatment in the resistance of the glass. It is however crucial to distinguish residual stress clearly from inherent strength for several reasons, the most important being:

* Subcritical crack growth and consequently all effects associated with it do only occur on decompressed surfaces. The factors influencing inherent resistance (the most important being load duration) do *not* influence residual stress.

* The uncertainties and consequently the safety factors are different for residual stress and inherent strength.

In design concepts that consider residual stress due to heat treatment explicitly, it is superposed over the inherent strength of annealed glass (see equations in chapter II.C), thus assuming that the inherent strength is not affected by the heat treatment. There is evidence that the tempering process actually causes a certain amount of 'crack healing' [Hand 2000] [Bernard 2001], this assumption can therefore be considered safe (conservative).

From an engineering point of view and to be consistent with modern codes and design concepts
for other materials, residual stress should be considered as a beneficial action. It is, however, a property of the material which means that the manufacturer has to monitor and guarantee that its specifications (e. g. a minimal surface compression) are met. From this perspective, the residual stress level is part of the product and testing codes, while actions would be part of the design codes.

## D. Flexibility

A design concept should be easily applicable to other situations than evenly distributed out-of plain load. In particular concentrated or line loads, stability problems and in-plane loads are fundamental for the structural use of glass. The user should be able to easily analyse any load combination and to evaluate non-standard transient or accidental load cases.

## E. Accuracy

Accuracy is crucial for economic design. It should however be noted that accurate but complex resistance models are only justifiable in a safety concept where accuracy on the action side is on a comparable level.

## F. Consistency and usability

The problematic situation in terms of consistency and usability shall be illustrated by taking a closer look at the concept for the determination of glass plate resistance given in prEN 13474 (see II.C.2)).

The characteristic value for the inherent strength of glass is said to be $f_{g,d} = 45$ MPa. This value has originally been defined in [DIN 1249-10:1990] and was found by carrying out double-ring bending tests<fnref n="2" /> on new annealed glass specimens. The surface area $A_0$ under uniform tensile stress was $0.24 \text{ m}^2$. To model the failure probability of the test specimens, the Weibull distribution in eq. (1)

was used, the parameters have been determined as $\theta_{A_0} = 74$ MPa and $\beta = 6$. The characteristic value is defined as the stress leading to failure for 5% of the specimens (5% fractile value; eq. (14); 0.95 confidence interval), which gives the 45 MPa mentioned above.

$$F(f_{g,d}) = 1 - \exp\left[-\left(\frac{f_{g,d}}{\theta_{A_0}}\right)^{\beta}\right] = 0.05 \quad (14)$$

As shown in eq. (5), $f_{g,d}$ is divided by a size factor $k_A$ to account for the size effect. $k_A$ is given as

$$k_A = A^{0.04}, \quad (15)$$

with A being the total surface area of the glass plate. Knowing that the ‘real’ size factor assuming Weibull-distributed resistance would be (see III.A)

$$k_A = \left(\frac{A}{A_0}\right)^{\frac{1}{\beta}} \quad (16)$$

$A$ decompressed surface area of the element

$A_0$ decompressed surface area in the tests used to determine the breaking strength

we discover that:

* A resistance value determined using $\beta = 6$ is combined with a correction factor based on $\beta = 25$ (exponent $0.04 = 1/25$).

* $k_A$ becomes 1 for $A = 1 \text{ m}^2$, thus assuming that the surface area in the tests leading to $f_{g,d}$ was 4 times bigger than it really was. (The effect of this is 6% for $\beta = 25$ and 27% for $\beta = 6$.)

* It is assumed that the whole surface of the element is decompressed.

## G. Action history and combination

The inherent glass strength is determined by subcritical growth of surface flaws (often called ‘static fatigue’) and therefore heavily dependent on the action history (Fig. 2). An ultimate limit state verification can not be done based on the maximum stress in an element. The effect of a combination of actions can not be determined by simple superposition.

<footnote n="2">² [DIN 52292-2:1986], replaced by [EN 1288-2:2000]</footnote>
<figure><img src="image_28.png" /><figcaption>Fig. 2: Time-dependence of the constant stress orthogonal to a surface crack of depth $a_i$ that leads to failure</figcaption></figure>

Current European concepts define equivalent load durations $t_{eq}$ for design loads $F$ given in codes such that a glass element under constant load $F$ during $t_{eq}$ undergoes the same strength reduction as it would under the ‘real’ load history. While the background and details of this procedure are not within the scope of this paper, it should be noted that the determination of equivalent load durations requires assumptions on

* action histories and their combination,

* subcritical crack growth model,

* environmental conditions,

* element geometry (!).

The equivalent load durations and a subcritical crack growth model are then used to define ‘allowable’ stresses for the common load types like dead load, snow and wind. This simplified approach is convenient, direct, efficient and extremely similar to the concept used for structural timber that most engineers are familiar with.

On the other hand, this approach does not make the damage accumulation phenomenon transparent to the user and – what is more important – it can lead to very inaccurate results for non-standard load conditions and in general for all situations where the many assumptions made to define the ‘allowable’ stresses are not satisfied.

## H. Considering structural redundancy

In general, verification of the reliability against failure is done on an element level. This means that

a structure is assumed to fail when one individual load-carrying element fails and in consequence, each individual element has to meet the structural safety requirements. Due to the brittle nature of glass and the high importance of hazard scenarios such as impact or vandalism, this can be problematic for glass structures. The required safety level for the isolated elements is often only achieved at excessive costs. A safety concept that takes structural redundancy explicitly into account and offers quantitative information on such redundancy’s influence on required safety levels of components would be of economic and aesthetic interest.

## I. Proof loading, quality assurance and non-structural measures

Similar to wood, the structural efficiency of glass elements suffers from the large uncertainties associated with the material resistance. The large coefficient of variation requires high safety margins for design values.

It is therefore important that a design concept can take advantage, e. g. by modified safety factors, of measures taken to reduce the coefficient of variation of parameters influencing resistance or to restrain them within boundaries. The following measures are of primary interest:

* Proof loading of elements or parts of the structure after fabrication or even after erection;

* Quality assurance (e. g. direct or indirect (fracture pattern) measurement of the residual stress level; visual check for surface damages);

* Non-structural measures that prevent or limit glass damage

## J. Design Point Considerations

To define design values for glass resistance, current design concepts generally use

$$P_f = P(R \le R_d) = \Phi(-0.8 \cdot \beta) \quad (17)$$

from [EN 1990:2002] where $R$ is the resistance, $R_d$ the design resistance, $\beta$ the reliability index and $\Phi$ the cumulative distribution function of the standard
normal distribution. This use of a fixed design
point has two undesirable consequences: (a) the
use of a predefined influence factor (here –0.8) that
may or may not be appropriate and (b) the separa-
tion of resistance from actions assuming their in-
dependence. The reason for the popularity of this
simplification is its convenience: it allows the defi-
nition of the design resistance independently from
the actions. Taking e. g. a target reliability index of
$\beta$= 3.8 (according to [EN 1990:2002] for standard
buildings and a 50 years service life), eq. (17)
gives $P_f$ = 0.0012. Assuming that the glass
strength follows the cumulative distribution func-
tion $P_f(\sigma)$ known from experiments, the design
resistance is simply the 0.12% fractile value of this
distribution function.

A consistent approach that is able to take the par-
ticularities of glass into account should use the
‘real’ design point. Using the reliability index defi-
nition introduced by Hasofer and Lind [Hasofer et
al. 1974], the design point is the point on the fail-
ure surface with the shortest distance to the origin
in the space of normalised variables. As long as all
relevant action and resistance variables are time-
independent and uncorrelated, the determination of
the design point is quite straightforward, e. g. using
the first order reliability method (FORM). For
glass however, the resistance at a given point in
time is heavily influenced by the stress history.
The stress distribution being in general non-
uniform within the element, a time-variant, space-
dependent reliability assessment is required. The
drawback of such an approach is –besides its com-
plexity– the need for appropriate time-dependent
probabilistic models of resistance and action vari-
ables alike.

IV. CONCLUSION

The main difficulties of current glass design
have been identified and discussed. It has been
shown that further research work is required to
provide a modified glass design concept providing

in particular

* extended applicability and flexibility;

* improved consistency;

* better transparency for the user;

* an improved load combination concept;

* the possibility to allow for less conservative
design if data from proof loading or quality as-
surance is available;

* a closer relation to the physical behaviour;

* a more homogenous reliability level;

* facilitated integration of new knowledge
(e. g. on actions or material behaviour).

V. ACKNOWLEDGEMENTS

The author would like to thank Mr. Christoph
Haas of Ernst Basler + Partner AG, Zürich, for
having been an excellent co-author for the state-of-
the-art report [Haas et al. 2004] which is the basis
of paragraph II.C. Thanks are also due to Prof. Dr
Manfred A. Hirt and Mr. Michel Crisinel of EPFL-
ICOM for their ongoing support.

The research on structural glass at EPFL-ICOM
is financially supported by the Swiss Federal Of-
fice for Education and Science as a part of the
European Action COST C13 and by the Swiss Na-
tional Science Foundation. Both contributions are
highly appreciated.

REFERENCES

[ASTM E 1300-03]

Standard Practice for Determining Load Resistance of
Glass in Buildings, ASTM E 1300-03, American
Society for Testing Materials, 2003.

[Beason et al. 1984]

BEASON, W. L., MORGAN, J. R., *Glass Failure Prediction Model*, Journal of Structural Engineering, Vol. 110, No. 2, pp. 197-212, 1984.

[Bernard 2001]

BERNARD, F., *Sur le dimensionnement des structures en verre trempé: étude des zones de connexion*, No. 17, LMT-Cachan, Cachan, France, 2001.

[Blank 1993]

BLANK, K., *Dickenbemessung von vierseitig*
gelagerten rechteckigen Glasscheiben unter
gleichförmiger Flächenlast, Institut für Konstruktiven
Glasbau, Gelsenkirchen, 1993.

[Calderone 2001]

CALDERONE, I. J., *The Fallacy of the Weibull Distribution for Window Glass Design*, Glass Processing Days, 18-21 June 2001, pp. 293-297, Tampere, Finland, 2001.

[CAN/CGSB 12.20-M89]

*Structural Design of Glass for Buildings*, CAN/CGSB 12.20-M89, Canadian General Standards Board, Canada, December 1989.

[DIN 1249-10:1990]

*Flachglas im Bauwesen - Teil 10: Chemische und physikalische Eigenschaften*, DIN 1249-10, Ausgabe 1990-08, DIN, 1990.

[EN 1990:2002]

*Grundlagen der Tragwerksplanung*, DIN EN 1990, Ausgabe:2002-10, DIN, 2002.

[Fink 2000.1]

FINK, A., *Ein Beitrag zum Einsatz von Floatglas als dauerhaft tragender Konstruktionswerkstoff im Bauwesen*, Dissertation, Technische Hochschule Darmstadt, 2000.

[Güsgen 1998]

GÜSGEN, J., *Bemessung tragender Bauteile aus Glas*, Schriftenreihe Stahlbau RWTH Aachen, Shaker Verlag, Aachen, 1998.

[Haas et al. 2004]

HAAS, CH., HALDIMANN, M., *Entwurf und Bemessung von Tragelementen aus Glas – Wissensstandsbericht*, Rapport ICOM 493, Ecole polytechnique fédérale de Lausanne (EPFL), Lausanne, 2004.

[Hand 2000]

HAND, R. J., *Stress intensity factors for surface flaws in toughened glasses*, Fatigue and Fracture of Engineering Materials and Structures, Vol. 23, No. 1, pp. 73-80, 2000.

[Hasofer et al. 1974]

HASOFER, A. M., LIND, N. C., *An Exact and Invariant First Order Reliability Format*, Journal of Engineering Mechanics, Vol. 100, pp. 111-121, ASCE, 1974.

[Porter 2001]

PORTER, M., *Aspects of Structural Design with Glass*, PhD Thesis, The University of Oxford, 2001.

[prEN 13474-1:1999]

*Glas im Bauwesen - Bemessung von Glasscheiben - Teil 1: Allgemeine Grundlagen für Entwurf, Berechnung*

und Bemessung, DIN EN 13474-1, Entwurf 1999-04, DIN, 1999.

[prEN 13474-2:2000]

*Glas im Bauwesen - Bemessung von Glasscheiben - Teil 2: Bemessung für gleichmässig verteilte Belastungen*, DIN EN 13474-2, Entwurf 2000-05, DIN, 2000.

[Reid 1991]

REID, S. G., *Flaws in the failure prediction model of glass strength*, Proceedings of the 6th International Conference on Applications of Statistics and Probability in Civil Engineering, pp. 111-117, Mexico City, 1991.

[Sedlacek et al. 1995]

SEDLACEK, G., BLANK, K., GÜSGEN, J., *Glass in Structural Engineering*, The Structural Engineer, Vol. 73, No. 2, pp. 17-22, January 1995.

[Sedlacek et al. 1999]

SEDLACEK, G., BLANK, K., LAUFS, W., GÜSGEN, J., *Glas im Konstruktiven Ingenieurbau*, Ernst & Sohn, Berlin, 1999.

[Shen 1997]

SHEN, X., *Entwicklung eines Bemessungs- und Sicherheitskonzeptes für den Glasbau*, Dissertation, Technische Hochschule Darmstadt, 1997.

[Siebert 1999]

SIEBERT, G., *Beitrag zum Einsatz von Glas als tragendes Bauteil im konstruktiven Ingenieurbau*, Berichte aus dem konstruktiven Ingenieurbau, 5/99, Technische Universität München, München, 1999.

[TRAV 2003]

*Technische Regeln für die Verwendung von absturzsichernden Verglasungen (TRAV)*, Fassung Januar 2003, Deutsches Institut für Bautechnik, 2003.

[TRLV 1998]

DIBT, *Technische Regeln für die Verwendung von linienförmig gelagerten Verglasungen (TRLV)*, Mitteilungen des Deutschen Instituts für Bautechnik, Berlin, September 1998.

[Vallabhan 1983]

VALLABHAN, C. V. G., *Iterative Analysis of Nonlinear Glass Plates*, Journal of Structural Engineering, Vol. 109, No. 2, pp. 489-502, 1983.

[Weibull 1951]

WEIBULL, W., *A Statistical Distribution Function of Wide Applicability*, Journal of Applied Mechanics, Vol. 18, pp. 293-297, Plenum Press, September 1951.

[Wörner et al. 2001]

WÖRNER, J.-D., SCHNEIDER, J., FINK, A.,
*Glasbau: Grundlagen, Berechnung, Konstruktion*,
Springer Verlag, Berlin Heidelberg New York, 2001.
# Structural Behaviour of Broken Laminated Safety Glass

Alexander Kott, IBK, ETH Zurich, Switzerland

Prof. Thomas Vogel, IBK, ETH Zurich, Switzerland

This research paper reports the experimental and theoretical results of the project "Remaining Structural Capacity of Laminated Safety Glass" at ETH Zurich.

The aim of the project is to develop new mechanical models and to control the post breakage behaviour of laminated safety glass (LSG). At first the different stages of failure of LSG and the corresponding definitions are given to explain the different capacities of the structure. In broken LSG the poly-vinyl-butyral (PVB) foil works as tension reinforcement and the upper broken glass layer of the sandwich structure carries the compression forces. Therefore the mechanical properties of the foil were determined using tensile tests. Bending tests combined with impact tests demonstrate that the different glass types, bearings, the type of initial failure and dimensions of the specimen determine the post breakage behaviour. As observed in these tests the fracture behaviour affects the remaining structural capacity (RSC). Different types of yield lines can evolve from initial cracks. The yield line patterns influence the ultimate load decisively.

**Keywords:** Failure modes, sandwich structure, viscoelastic stress-strain behaviour, yield line mechanism.

## I. INTRODUCTION

With the revolution of the manufacturing process laminated safety glass (LSG) is available consisting of panes of float glass (FG), toughened glass (TG) or heat strengthened glass (HG). The multi-layer glass can be used as a structural material, combining even these different glass types. The laminates have to be composed in such a way that

after the first crack the interlayer, a Poly-Vinyl-Butyral (PVB) foil holds the pieces together and the whole structure resists. In this way the remaining structural capacity (RSC) can be provided beside the load carrying capacity (LCC). Until now structural engineers conduct costly destructive glass experiments to assess the structural safety.

To avoid these costs a mechanical model has to be developed. Definitions of RSC according to the stage of failure allow to distinguish the different capacities. The stages consider not only the ultimate loads but also the corresponding external works. In the broken LSG the PVB foil works as tension reinforcement and the upper broken glass layers carry the compression forces. The visco elasto plastic stress-strain behaviour of the PVB foil was determined using tensile tests under stationary temperature conditions in a climate chamber. The ultimate failure, which leads to the collapse of the structure, occurs, when the compression zone in the yield line fails reaching the compression strength of glass. Temperature, strain level and strain rates are significant parameters that affect the tensile stresses in the thermoplastic polymer. The different yield line mechanisms depend on the actions and the structural supports of the glass specimens. These results were confirmed by experimental tests.

## II. THREE STAGES OF LSG

Figure 1 shows a typical LSG consisting of two glass layers and a PVB foil. The glass layers are partially damaged but the PVB foil is intact. Three different stages can be identified due to the type and location of the fracture. The pane is divided into sections classified by these different stages as shown in Figure 1.
<figure><img src="image_29.png" /><figcaption>Fig. 1: Longitudinal section of a LSG pane divided in sections according to the stages reached.</figcaption></figure>

## A. Stage I

In section 3 as long as the pane is unbroken for each uncracked glass layer the hypotheses of Bernoulli can be adopted. Tensional and compression stresses exist only in the glass layers. The modulus of elasticity of the PVB foil can be neglected and the interlayer, as an adhesive joint holds the glass panes together. This structural behaviour can be interpreted with the sandwich theory of thick sheets [1]. Considering the membrane effects due to large deflections, finite element methods with volume elements or multi layer elements have to be used [2].

<figure><img src="image_30.png" /><figcaption>Fig. 2: Stress distribution in cross section of equal and unequal glass layers keeping the overall thickness constant.</figcaption></figure>

The problem of all these calculations is the unknown shear stiffness, respectively the shear modulus *G* of the PVB foil.

Time dependency of the loads as well as the influence of temperature can modify the shear modulus [3]. A parametric study of stress distribution depending on the shear modulus in a glass pane explains the location of the first failure.

<figure><img src="image_31.png" /><figcaption>Fig. 3: Ultimate loads depending on the shear modulus for two side supported LSG calculated by the sandwich theory.</figcaption></figure>

The failure appears immediately when the ultimate strength $\sigma_{I,u}^T$ of the glass is reached. As shown in Figure 3 the ultimate loads $F_{I,u}^T$ were calculated with different shear moduli for two side supported glass pane charged by two loads. The static system can be seen in Figure 1. The overall thickness of the LSG is fixed; the partial thicknesses of the glass layers are varied. As shown in Figure 2 the bond can be expressed with the distance $s(G)$ between the neutral axes.

Two extremes can be considered: For the unbonded cross sections the shear modulus of the PVB foil is zero and the distance can be assumed $s(G = 0) = H/2$, independent of the glass layer thicknesses. Unbonded cross sections with equal glass layers have the same stress distributions in
both glass layer, therefore the theoretical failure
can occur in the lower surface of the upper and
lower glass layer, respectively. Because, the lower
side of the upper layer is protected by the PVB foil
the lower glass layer will always break first. For
unbonded cross sections with unequal glass layers
the thicker layer, which is stiffer, breaks first on
the lower side. The other extreme is full bonded
cross sections; the distances of the neutral axes in
full bonded cross sections reach the minimum of
s(G = ∞) = 0. In this case the thicknesses of the
layers does not influence the location of the initial
failure. Indeed, always the lower layer will break
first. Other cases lie between these two extremes.
The graphs in Figure 3 for unequal cross sections
are characterised by two shapes. With small shear
moduli G the thicker layer breaks first, otherwise
with higher shear moduli the lower glass layer
fails. The choice of unequal glass layers depending
on the shear modulus rises the ultimate load F<sup>T</sup><sub>I,u</sub>
from 15% to 40%.

For specimens hit by an impact, two aspects have
to be taken into account. The pane acts as fully
bonded and therefore the lower layer breaks first.
Even so the upper layer can be destroyed first if the
object, which hits directly the upper layer, is stiff
enough or if its shape is irregular. Peaks and edges
cause singularities.

B. *Stage II*

To explain this stage II also the pane, divided in
sections as shown in Figure I, has to be considered.
In stage II the whole load has to be carried by the
unbroken layer. The PVB-interlayer serves for two
purposes: Firstly the high adhesion of PVB to glass
ensures the glass fragments to adhere firmly and
secondly the static function of the interlayer re-
mains carrying the bond stresses. As long as the
distance between two cracks is sufficiently large,
bond stresses can be activated and the sandwich
theory can be adopted at least partially, as shown
in section 3 of the LSG pane. If one glass layer of a
LSG pane is broken, it is important to know the
number and size of such stage I sections to predict
the ultimate load F<sub>II,u</sub>. The tests demonstrate,
however, that in LSG panes with TG sheets the
distance of adjacent cracks in stage II does not
allow intermediate stage I sections. In section 4 it
is obvious that the distance between the cracks is

too small and the stresses cannot disperse into both
glass layers.

C. *Stage III*

The next crack appears in the uppermost sheet in
a cross section with already broken lower layers.
Therefore a LSG pane switches to stage III and the
structure resists although all glass sheets are bro-
ken. The PVB foil takes over now a decisive part
of the structural function and works as a tension
reinforcement. The broken pieces of glass of the
upper layer, which continue to be effective in re-
sisting the internal forces, are subjected mainly to
compression as shown in Figure 1. Although sec-
tion 7 is in stage III the cracks are too close to each
other and the upper small glass pieces do not
firmly adhere to the foil. Therefore the tensile
force of the foil cannot be transmitted by bond
stresses to the glass. As a consequence, the glass
pane acts like a cloth without bending stiffness.
For the pane it forms a hinge and if this is simply
supported, it falls down. This typical phenomenon
can be seen in all simply supported LSG panes of
TG sheets, as shown in Figure 4.

<figure><img src="image_32.png" /><figcaption>Fig. 4: TG pane consisting of two broken glass layers, slipped from the bearings.</figcaption></figure>

The bending stresses of section 5 are distributed
into glass layers of section 6, which remains in
stage I. Therefore a yield line can be developed in
section 5. The calculation of the forces and stresses
in the foil as well as in the broken upper layer of
this section, satisfying the equilibrium and the ma-
terial laws, shows that the yield moment increases
the distance between the inner forces (called the
lever arm of inner forces) [4]. This can also be
achieved with unequal glass layers. Another possi-
bility to increase the moment is to extend the thick-
ness of the interlayer, as shown in Figure 5. A sys-
tem of yield lines forming a collapse mechanism is known as a yield line pattern. It is to be noted that a yield line is, in fact, an idealization for a band of intensive cracking. Perpendicular to these lines the PVB foil yields. The yield-line method of the theory of plasticity gives an upper bound approach of the ultimate load.

<figure><img src="image_33.png" /><figcaption>Fig. 5: Yield line mechanism in different cross sections; the internal forces as well as vertical distribution of stresses and strains.</figcaption></figure>

The developed mechanism depends on the initial fracture pattern evolved from stage II or I. For the purpose of analysis, the band of intensive cracking is concentrated to a single yield line and all plastic rotation is considered to occur along that line. For estimating a required RSC, energy approach is used, assuming that the total external work W is equated to the total internal work U, as shown in Equation 1.

$$W = U \qquad (1)$$

### III. REMAINING STRUCTURAL CAPACITY

The remaining structural capacity can be expressed as a ratio of stage II and stage III properties to the according stage I value, either considering load or physical work. According to Equation 2, $RSCR_{II,F}^T$ is based on the failure load of the intact sheet and only ultimate loads of the three stages are taken into account. $RSCR_{II,W}^T$ in

Equation 3, however, is a ratio of the total works. Not only the ultimate forces but also the corresponding deformations are considered.

$$RSCR_{II,F}^T = \frac{F_{II,u}^T + F_{III,u}^T}{F_{I,u}^T} \qquad (2)$$

$$RSCR_{II,W}^T = \frac{W_{II,u}^T + W_{III,u}^T}{W_{I,u}^T} \qquad (3)$$

According to Equation 4 and 5 the RSC in stage III can also be formulated with similar attributes in the ratios of $RSCR_{III,F}^T$ and $RSCR_{III,W}^T$ respectively.

$$RSCR_{III,W}^T = \frac{W_{III,u}^T}{W_{I,u}^T} \qquad (4)$$

$$RSCR_{III,F}^T = \frac{F_{III,u}^T}{F_{I,u}^T} \qquad (5)$$

### IV. ACTIONS AND THE RESULTING FRACTURES

The pattern of fracture influences the RSC in stage II and III. In this study four different causes of breakage are explained. Depending upon the type of action LSG can break in completely different ways. The first cause of breakage can be assumed as an impact, which hits the upper side of the upper glass layer. Depending on the dropping height one or more glass layers can break. If the impact is stiff enough or if its shape is irregular the surface of the hit glass layer breaks first, as mentioned in chapter II(a). In all other cases the lower glass layer breaks at first. Glass panes hit by an impact act as full bonded sandwiches. In full bonded sandwiches always the lower layer breaks first, as shown in Figure 6.

The second cause of breakage that generates a fracture is the exceeding of the ultimate stress of one glass layer by permanent loads, self weight or snow. As shown in Figure 2, depending on the bond and cross section of the pane the lower or upper layer breaks first.

The third cause, the inclusion of nickel sulphide is widely prevented by the heat soak test. Although a spontaneous failure of one glass layer has to be considered. The initial crack evolved not from the location of the maximum stresses caused by the actions but from the location of the inclusion.
The last cause of breakage, which can be found in particular conditions, is a prefabricated and artificial crack introduced by a glass cutter or another mechanical instrument.

<figure><img src="image_34.png" /><figcaption>Fig. 6: Four side simply supported glass slab damaged by a steel sphere.</figcaption></figure>

# V. STRESS-STRAIN BEHAVIOR OF PVB FOILS

As shown in Figure 1, a RSCR in stage III can be achieved, if tension is transmitted by the foil. PVB is a thermoplastic, semi crystalline polymer with a visco elastic-plastic stress strain behaviour. Therefore the stress in PVB depends on three variables as shown in Equation 6:

$$ \sigma_{PVB} = f(\varepsilon, \dot{\varepsilon}, T) \quad (6) $$

## A. Tensile tests

In order to be able to simulate this behaviour and to apply the constitutive laws for the mechanical model, tensile tests were executed in a climate chamber under stationary temperature conditions as shown in Figure 7(a). The details of the experimental set-up are reported in [5]. At first, displacement-controlled tests with different temperatures were carried out to determine the material properties taking into account that the glass transition temperature of PVB ranges from 10°C to 15°C. As a consequence, there are in general two types of curves to describe the stress-strain behaviour as shown in Figures 8 and 9. If the experimental temperature is below the glass transition temperature of the thermoplastic, the behaviour is linear elastic until the material reaches the yield point. Due to the plastic deformation, micro cracks can be seen as a white discoloration as shown in Figure 7(b). Finally, before the foil tears apart, the material hardens. If the experimental temperature is above the glass transition temperature, the foil shows no linear

elastic behaviour. Instead, the higher the temperature, the larger are the plastic deformations.

<figure><img src="image_35.png" /><figcaption>Fig. 7: (a) Tensile test of a PVB specimen, (b) PVB specimen with white discoloration in creep test</figcaption></figure>

The influence of the strain rate was investigated at constant temperature. In that tests, the ultimate strength of the foil decreases with lower strain rates. In fact, with higher strain rates the ultimate strain diminishes, too.

<figure><img src="image_36.png" /><figcaption>Fig. 8: Stress-strain curves, experienced under constant strain rate and different temperatures.</figcaption></figure>

<figure><img src="image_37.png" /><figcaption>Fig. 9: Stress-strain curves, experienced above and below the glass transition temperature depending on the load rate.</figcaption></figure>

## B. Tri-linear approach

To determine a computable mechanical behaviour a tri-linear curve is proposed. As shown in Figure 10, the tensile stresses can be calculated with one
function and the strain is the only unknown variable. Based on the results of the tensile tests the three gradients and the characteristic stresses can be taken from the created database. The necessary equations are defined in [5].

<figure><img src="image_38.png" /><figcaption>Fig. 10: Proposed tri-linear approximation.</figcaption></figure>

### C. Creep of the PVB foil

For the verification of the RSC different requirements for the remaining lifetime can be formulated. The lifetime of a VSG pane in stage III is affected decisively by the creep of the foil. In Figure 11(b) the deformation depending on time under constant load and ambient temperature is shown. The constitutive equations most commonly used to describe the creep effects in polymers are based on the Newtonian model (Voigt model) as shown in Equation 7.

$$ \varepsilon(t) = \frac{\sigma_0}{E} \left(1 - e^{-\frac{t}{\lambda}}\right) \qquad \lambda = E/\eta \quad (7) $$

<figure><img src="image_39.png" /><figcaption>Fig. 11: (a) Voigt model for mechanical behaviour of linear visco elasticity [6], (b) creep curve of a PVB specimen under constant tension and constant ambient temperature.</figcaption></figure>

The creep response, i.e. the strain depending on time under constant stress can be modelled by dashpots and springs as shown in Figure 11(a). A more precise model consists of an arbitrary number of Maxwell models connected in parallel called

Maxwell-Wiechert model. In this case the Kelvin Model is sufficient under constant loads without strain recovery, because the stress does not relax.

## VI. BENDING TESTS

### A. Four-point bending tests

Four-point bending tests were executed with LSG panes of HG, TG and FG as shown in Figure 12(a). The details of the experimental setup are reported in [5]. The first part of the specimens was tested conventionally beginning with stage I. In the other part, to investigate the effects of the actions and of resulting crack patterns, the specimens were damaged by a steel sphere, dropped from a height of two meters. The upper layer broke and the initial crack patterns exhibited the typical form of an impact. To examine stages II and III, damaged specimens were installed in the experimental setup of the four-point bending test.

<figure><img src="image_40.png" /><figcaption>Fig. 12: (a) Experimental setup for four-point bending test (b) and for point loaded simply supported, square slab.</figcaption></figure>

Varied thicknesses of glass layers and PVB foils were investigated keeping the overall thickness of the cross section constant. The intact LSG panes of equal glass layers (each glass layer 8 mm thick and a PVB foil of 1.52 mm) were loaded in stage I. The velocity of the displacement-controlled test was $v = 0.2$ mm/s. The load-deflection curve is linear elastic. Therefore the creep effect of the foil can be neglected. The same procedure was repeated with glass panes of unequal glass layers (upper layer of 12 mm, lower layer of 4 mm and a PVB foil of 1.52 mm). As shown in Figure 13, LCC increases with unequal glass layers. The ultimate load $F_{I,u}^T$
of the pane with unequal glass layers is 23% higher than the load of panes with equal layers. In sections of equal layers the first failure occurred in the lower sheet. In cross sections of unequal layers, however, the initial crack appears in the thicker upper glass layer.

<figure><img src="image_41.png" /><figcaption>Fig. 13: Load-deflection diagram in stage I for specimen with equal and unequal glass sheets.</figcaption></figure>

To avoid the break of the next sheet, immediately after the first break had occurred, the system was unloaded and a plastic deflection of the pane was observed in stage II. The remaining deformations $w_{II,u}^T$ at the beginning of stage II were evaluated for all glass types. The average elongation of the panes was 19.5 mm in TG, 13.7 mm in HG and 5.5 mm in FG, respectively the associated strains were 1.8% in TG, 1.2% HG and 0.5% in FG. The reason for the elongation could be attributed to differences in form and number of the small fragments of TG or HG as shown in Figure 15(a). Fragments of FG panes were of longitudinal shape as shown in Figure 15(b). Then the specimens were reloaded till the ultimate load $F_{II,u}^T$ was reached. Specimens hit by an impact were also loaded till the ultimate load $F_{II,u}^T$ was reached. The diagram in Figure 14 shows the load-deflection behaviour in stage II for various specimens with glass sheets of 8 mm and a PVB-foil of 1.52 mm thickness. A graph with linear elastic shape for a theoretical monolithic one-glass-pane of 8 mm is also shown in the diagram. $F_{II,u}^{TG}$ is 167% higher than $F_{II,u}^{FG}$ and 71% higher than $F_{II,u}^{HG}$. All failure points lie on a straight line. The curve for the LSG pane of TG sheets is also linear elastic and the ultimate load $F_{II,u}^{TG}$ is more or less the same than the theoretical one. This can be explained by the fracture patterns of the TG sheet. As shown in Figure 15(a) the pane consisting of TG sheets exhibits in stage II a narrow pattern of cracks not leaving relevant stage I sections. Only the upper layer is intact and the

whole bending moment has to be carried by the lower glass sheet.

<figure><img src="image_42.png" /><figcaption>Fig. 14: Load-deflection diagram for two specimens in stage II tested in four-point bending test.</figcaption></figure>

a)

<figure><img src="image_43.png" /></figure>

b)

<figure><img src="image_44.png" /><figcaption>Fig. 15: (a) cracks of TG on the lower layer, (b) longitudinal cracks of FG.</figcaption></figure>

Panes of HG or FG, damaged by an impact, behave differently than those, which were damaged by increasing loads. The panes previously damaged by an impact, however, had more crack patterns with stage I regions. At the beginning the graphs show a linear elastic load deflection behaviour. Even so, a phase of crack initiation began accompanied by loud sounds of cracking. When this phase is completed the strain and the load rises up linearly until $F_{II,u}^T$ is reached. The same behaviour was observed with panes of FG sheets, which were also previously damaged by an impact. This phenomenon can be explained as tension-stiffening of glass. A similar behaviour of tension-stiffening can be found in concrete structures. Panes of HG or FG sheets subjected to increasing loads do not show this phase of crack initiation and therefore the load deflection curve is linear. TG and HG panes collapsed immediately by reaching stage III. The whole pane was in section 7, as shown in Figure 1 and mentioned in chapter II(C). Also FG panes of equal glass layers exhibited localized section 7 and the panes collapsed immediately. FG panes of unequal glass layers or with thicker foil
resist after the second failure. After the force $F_{II,u}^T$ was reached and the last intact glass layer was broken a yield line mechanism depending on the initial crack patterns began to develop. Figures 17(a,b,c) show three different yield line mechanisms in two side supported LSG. The ultimate load $F_{III,u}^T$ of panes with all yield lines perpendicular to the principle stresses is the highest as shown in Figure 16. The panes damaged by the sphere do not have the possibility to transfer the compression forces in the centre of the pane, where the sphere hit the pane.

<figure><img src="image_45.png" /><figcaption>Fig. 16: Load-deflection diagram for simply two side supported LSG panes with three different yield line mechanisms in stage III.</figcaption></figure>

The surface of the hit layer is completely destroyed and a compression zone cannot be found. It is to be noted that to develop a perfect perpendicular yield line a notch was introduced by a glass cutter. The artificial crack appears also

<figure><img src="image_46.png" /><figcaption>Fig. 17: Three different yield line patterns in two side supported panes; (a) yield line perpendicular to the principle stresses; (b) half yield line perpendicular to the principle stresses, (c) four yield lines crossing the centre.</figcaption></figure>

perpendicular to the principle stresses after hitting the pane with a rubber hammer. Also here the panes were installed into the set-up. Considering the cross-section, tests showed that the ultimate load $F_{III,u}^T$ increases by enforcing a larger lever arm of the internal forces by unequal glass thicknesses and by the extension of the thickness of the foil.

<figure><img src="image_47.png" /><figcaption>Fig. 18: Load-deflection diagrams for LSG panes with similar yield line but different cross-sections.</figcaption></figure>

Finally, all four-point bending tests showed that yield line mechanisms were necessary to reach stage III.

## B. Tests with simply supported slabs

The bearings have a decisive part in a structural system to carry the loads. Therefore in yield line theory the bearings influence the yield line patterns. In addition to the four-point bending tests other bending tests with simply supported square slabs were conducted. Both spans were 0.90 m long and the cross sections of the FG slabs consist of two 8 mm glass layers and a 1.52 mm PVB foil.
The span/thickness ratio can be calculated as 513. The specimens were loaded in the centre of the slab by a concentrated load $F_{II}^{FG}$ until the lower layer broke. The graph in the corresponding load-deflection diagram in Figure 20 is not linear. Then the specimens were un- and reloaded until the ultimate load $F_{II,u}^{FG}$ was reached. The loads increase nonlinear due to the high values of the deflections and the span/thickness ratio. This effect can be observed in stage I as well as in stage II. A remaining deformations $w_{II,r}^{T}$ at the beginning of stage II was observed, as shown in Figure 20. Finally in stage III all specimens developed yield line patterns as shown in Figures 19(b,c). Not all slabs were tested in this way. A specific number of specimens were first of all hit by an impact, as shown in Figure 6. Depending on the dropping height one or both glass layers broke. Also here the damaged specimens were installed in the experimental set-up to test them in stage II and/or III.

In the following three different modes show the different yield line patterns of a square slab. As shown in Figure 19(a), the slab in mode 1 resists the induced load developing yield line patterns in form of a circular fan. Specimens of collapse mode 2 with yield lines in form of fans in the corners have the smallest ultimate load $F_{III,u}^{FG}$ as shown in Figure 21. The collapse mode with the minimum

ultimate load $F_{III,u}^{FG}$ is the proper value for the calculation of RSC.

<figure><img src="image_48.png" /><figcaption>Fig. 20: Load-deflection diagrams for four side simply supported LSG slab in stage I and II</figcaption></figure>

<figure><img src="image_49.png" /><figcaption>Fig. 21: Load-deflection diagrams for four side simply supported LSG slab with three different yield line mechanisms in stage III.</figcaption></figure>

Therefore collapse mode 3, which is also possible, is not the proper one. The smallest ultimate load of mode 2 is not the exact solution but this approximation is quite accurate.

<figure><img src="image_50.png" /><figcaption>Fig. 19: Three different yield line patterns in simply supported square slabs, (a) mode 1; circular fan, (b) mode 2; corner levers in form of fans, (c) mode 3; yield lines from the slab centre to the four corners.</figcaption></figure>
In general in stage III the yield line theory is applicable to glass slabs of LSG that are reinforced by a PVB foil.

## VII. CONCLUSIONS

Three stages were proposed to determine the structural safety of a glass structure. With the corresponding ultimate forces and virtual works it is possible to give statements about the remaining structural capacity according to the degree of damage. The four-point bending tests as well as tests with simply supported glass slabs combined with impact tests have shown that with glass sheets of unequal thickness the load carrying capacity in stage I and the remaining structural capacity in stage III can be increased. Remaining structural capacity in stage III exists in all glass structures of laminated safety glass only if the initial cracks form yield line patterns. Therefore laminated safety glass of toughened glass simply supported in two sides as well as in four sides have no RSC in stage III. The ultimate collapse occurs when the glass on the surface of the upper layer fails by compression. Then the foil tears or the pane slide from the bearing. The ultimate moment of resistance at the yield line can be determined taking into account the material laws of the glass and the PVB foil. Finally the total collapse in stage III of a glass structure can be determined from the yield line patterns using the equations of equilibrium. As a result the ultimate load in stage III can be found as an estimation of the RSC. In future, bending tests with point fixed glass plates have to be carried out. The resulting types of yield lines have to be discussed. Also other enhancements like interlayers with other material laws or interlayers with a matrix of glass fibre have to be investigated.

## VIII. ACKNOWLEDGMENT

The presented research project is part of the European COST action C13 and supported by the Swiss Federal Office for Education and Science. All test specimens were delivered by the Swiss branch of Pilkington AG free of charge. Both contributions are highly appreciated.

## REFERENCES

[1] Stamm, K., Witte, H., *Sandwichkonstruktionen*; Springer-Verlag; Wien, New York 1974.

[2] Bohmann, D., *Ein numerisches Verfahren zur Berechnung von Verbundglasscheiben*; Shaker Verlag; Aachen 1999.

[3] Sobek, W., Kutterer, M., Messmer, R.; *Untersuchungen zum Schubverbund bei Verbundsicherheitsglas*; Bauingenieur Bd. 75, Heft 1, Januar 2000; pp. 41-46.

[4] Kott, A., Vogel, T., *Safety of laminated glass structures after initial failure*; Structural Engineering International 2/04, pp. 134-138.

[5] Kott, A., Vogel, T., *Remaining Structural Capacity of Broken Laminated Safety Glass*; Glass Processing Days 2003; Tampere (Finland) 15. – 18.06.2003; pp. 403-407.

[6] Osswald, T., Menges, G., *Materials Science of Polymers for Engineers*; Carl Hanser Verlag; München, Wien, New York 1996.

## NOTATIONS

### Upper case letters:

$F_{i,u}^T$ Ultimate load in stage *i* for glass type *T* (*i*=I, II, III)

$K_{III,y}^T$ Internal forces in the yield line in stage III

$M_{III,y}^T$ Ultimate moment of resistance in stage III

$RSCR_{i,W}^T$ Remaining structural capacity ratio of external works in stage *i* for glass type *T* (*i*=II, III)

$RSCR_{i,F}^T$ Remaining structural capacity ratio of ultimate forces in stage *i* for glass type *T* (*i*=II, III)

$W_{i,u}^T$ Physical work in stage *i* for glass type *T* (*i*=I, II, III)

$U$ Internal work in the yield line

### Lower case letters:

$w_{II,r}^T$ Plastic deflection in stage II for glass type *T*

### Greek letters:

$\sigma_{III}^{PVB}$ Tensile stress in the PVB foil in stage III

$\sigma_{III,u}^{PVB}$ Ultimate tensile stress in the PVB foil in stage III

$\sigma_I^T$ Tensile stress for glass type *T* in stage I

$\epsilon_{III,y}^T$ Strains in the yield line in stage III

$\sigma_{I,u}^T$ Flexural strength for glass type *T* in stage I

$\sigma_{II}^T$ Tensile stress for glass type *T* in stage I

$\sigma_{II,u}^T$ Flexural strength for glass type *T* in stage II

$\sigma_{II}^T$ Tensile stress for glass type *T* in stage II

$\sigma_{III}^T$ Compressive stress for glass type *T* in stage III

$\sigma_{III,u}^T$ Ultimate compressive stress for glass type *T* in stage III

### Superscripts:

$FG$ Float glass

$HG$ Heat strengthened glass

$TG$ Toughened glass

$T$ Glass types (FG, HG or TG)

### Subscripts:

$F$ Load

$r$ Residual

$u$ Ultimate

$W$ External work

$I,II,III$ Stage I, II or III
# Proposal for a code calibration procedure

Jørgen Munch-Andersen, Danish Building and Urban Research, Hørsholm, Denmark
Randi Kruse Vestergaard, NIRAS - Consulting Engineers and Planners A/S, Allerød, Denmark

A method for estimating the strength parame-
ters of the Weibull distribution and the power
in Brown's integral from full-scale experiments
with rectangular glass panes loaded uniformly
at different load rates is proposed.

Knowing the parameters, it is possible to es-
timate the load capacity of a glass pane sub-
jected to any load history. Provided accurate
load histories (load as a function of time during
the service life) for e.g. wind and snow loads are
available this enables fairly simple and very ac-
curate estimates for the load capacity of a glass
pane for different load types.

The method is applied to permanent load and
Danish snow load in order to demonstrate the
consequences.

**Keywords:** Weibull distribution, parameter esti-
mation, code calibration, duration of load

I. INTRODUCTION

The use of glass for structural purposes suffers
from insufficient knowledge of the properties of
glass and the nature of the loads. It is therefore dif-
ficult to formulate simple and accurate code re-
quirements for load bearing glass structures.

A measure of the failure probability can be ob-
tained if loads and strength are sufficiently de-
scribed by statistical distributions. Two properties
inherent in glass make it difficult to deal with the
safety level. One is the damage accumulation that
causes the strength to become severely dependent
on the load duration, the other a significant size
dependency. The established models for these phe-
nomena, Brown's integral and the Weibull distri-
bution, requires estimates for several parameters.

However, different methods to estimate these pa-
rameters lead to very different values.

The large deflections of glass panes add to these
difficulties, because calculation of damage accu-
mulation for variable load has to be made on stress
level.

This paper deals with determining the parameters
in Brown's integral and the Weibull distribution
using results from full-scale experiments with glass
panes simply supported along the four edges and
loaded at different rates. The advantage of this
method is that uncertainties about the models for
size effect and damage accumulation becomes less
important because the experimental results used for
calibration of the models originates from speci-
mens of the same size, and with the same support
and loading conditions as the real glass panes.

If the parameters are determined from idealised
test set-ups, e.g. double ring tests, possible weak-
nesses of the theoretical description will signifi-
cantly increase the uncertainty for real glass panes.

The paper also discusses how simple rules for
designing glass panes can be achieved. As a part of
this an estimated load history for snow loads have
been applied to simulated panes with strength pa-
rameters determined from the full-scale experi-
ments. This enables accurate estimates of the load
capacity for snow load. The method can of course
also be used for other load types.

This paper is based on an idea outlined in
[Munch-Andersen & Ellum, 1995] and developed
in a MSc-thesis [Vestergaard, 2004]. Somewhat
similar methods are described in [Beason & Mor-
gan, 1984] and [Grüters et al., 1990].
## II. DEFINITIONS

The well-known Brown's integral is used to describe the damage in an area of the glass pane due to the load history up to time T.

$$K(T) = \int_{0}^{T} \sigma(t)^{n} dt. \qquad (1)$$

K is somewhat related to the fracture mechanics parameter $K_{f}$, see e.g. [Simiu et al., 1984].

The equivalent 1 second stress is defined as the constant stress that, when it acts for 1 second, causes the same damage as the real stress does during the time T. It can be determined from $K(T)$ as

$$\sigma_{e,1\text{sec}} = (\frac{1}{1\text{sec}} K(T))^{1/n} \quad (2)$$

The Weibull distribution can only yield for the strength of an area with a constant stress field. In particular it deals with the size-effect, which is caused by the fact that the probability of the presence of a severe micro-crack is increased when the size of the area is increased. The distribution of the 1 second strength $f_{1\text{sec}}$ can be written as

$$F(f_{1\text{sec}}) = 1 - \exp\left[-\left(\frac{f_{1\text{sec}} - a}{(A/A_0)^{-1/k} b}\right)^k\right] \quad (3)$$

where a is a minimum strength (which is often assumed to be zero), b is related to the average 1 second strength, k is the size parameter, A is the actual area and $A_0$ is a reference area for which b is determined.

k is closely related to the variance for which reason k can be estimated from tests with only one size of test specimens. A small value means large size-effect and variance. The influence of the parameters is illustrated in Figure 1.

It should be noted that the Weibull distribution presupposes that the strength of two neighbouring areas is uncorrelated.

The 'weakest link of a chain' analogy is often used to explain the Weibull distribution. The ratio $A/A_0$ replaces the number of links in the chain.

<figure><img src="image_51.png" /><figcaption>Figure 1. Influence of the parameters a, b and k on the distribution function of the Weibull distribution for $A/A_0 = 1$ and a, b and x in kPa.</figcaption></figure>

## III. OUTLINE OF METHOD

The method is based on physical and simulated experiments with rectangular glass panes, which are exposed to uniformly distributed load increased at a constant rate d until failure. Since glass is linearly elastic, the stress history $\sigma(t)$ at any point of the pane can be quite accurately calculated by means of a FEM programme when the load rate d, the failure load $p_f$ and the time to failure T is known. Also the equivalent $\sigma_{e,1\text{sec}}$ at any point can be determined for an assumed value of n.

In the simulated experiments the pane was divided into a number of sub-areas in which the stress field was assumed to be constant. The strength of each sub-area was assumed to be Weibull distributed. For an assumed set of strength parameters (a, b, k) the 1 second strength of each sub-area was simulated by means of a random number.

Load was applied at a constant rate and $\sigma_{e,1\text{sec}}$ was calculated for each sub-area at each load step. Failure occurs when $\sigma_{e,1\text{sec}} > f_{1\text{sec}}$ in any of the sub-areas. This failure determined the load capacity of the simulated pane.

This procedure was repeated for a large number of simulated panes. Then the distribution function for the failure load was estimated and compared with the results from the experiments. In the present study the strength parameters were determined so that the square of the difference between the
<figure><img src="image_52.png" /><figcaption>Figure 2. Distribution of failure loads from test results and simulated tests. The estimated strength parameters of the Weibull distribution are taken as the values that minimise the square distance between the distributions.</figcaption></figure>

simulated and the measured distribution of the failure loads are minimised, see Figure 2. More sophisticated methods, like maximum likelihood estimation, could also be used.

It should be noted that the resulting distribution of the failure loads will not be a Weibull distribution. A LogNormal distribution might be a good approximation. This is the distribution type usually used to describe material strengths.

In principle the power $n$ in Brown's integral could be estimated from experiments with only one loading rate, because the different failure loads causes a slight time dependency of $a$ and $b$ in the Weibull distribution. However, a reliable estimate requires physical experiments carried out at different loading rates, $d$. Then $n$ can be included in the estimation.

## IV. EXPERIMENTAL RESULTS

This study was based on thorough test series by [Johar, 1981] and [Johar, 1982] with 6 mm thick glass panes 1.5 m × 2.4 m. They were loaded at constant rates ranging from 0.0025 kPa/s up to 25 kPa/s. This caused the time to failure to range from less than 1 second to about 30 minutes. There are about 20 tests for each loading rate. The results used for this study are shown as distribution functions in Figure 3.

<figure><img src="image_53.png" /><figcaption>Figure 3. Tests results from [Johar, 1981 & 1982] plotted as distribution functions for the failure load for each loading rate. Tests are carried out at normal room conditions.</figcaption></figure>

For the slower loading rates it is seen that the failure load decreases with the loading rate as would be expected from Brown's integral. For loading rates above about 1 kPa/s, the experiments indicate that this effect is significantly reduced as the strength distribution is almost independent of the loading rate.

The estimation is therefore divided into two parts, one for loading rates up to 0.25 kPa/s and one for the faster rates.

The reduction is most obvious for the rates 1.5 kPa/s and 15 kPa/s, because the distribution of the results for 2.5 kPa/s and 25 kPa/s is seen to be quite awkward. The results for loading rates 2.5 and 25 kPa/s is not included in the analysis further on in this paper.

Results from experiments where the failure occurs at the edge cannot be used directly as the failure mechanism at the edge is different. Using the information that the strength of the internal part of the pane is larger than the failure value requires use of statistical methods for censored samples. This was not attempted in this study, but it was proved that the estimated parameters became almost the same if the results from panes with edge failure were excluded or were included without correction.
## V. DETAILED PROCEDURE

1. The pane was divided into $m \times m$ sub-areas. The values $m = 5, 10, 15$ and $20$ were used.

2. A set of strength parameters ($a, b, k, n$) was assumed.

*Step 3 was performed for each sub-area.*

3A. The relation between the load $p$ and the largest principal stress $\sigma_1$ at the centre point was established by means of a FEM programme, taking large deflections into account. This was done for 25 load steps, allowing for sufficiently accurate determination of the stresses by linear interpolation.

3B. The ($p, \sigma_1$) relation was transformed to a ($t, \sigma_1$) relation by means of the known loading rate and $t = p/d$.

3C. Equivalent stresses $\sigma_{e,1\text{sec}}$ for each load step were determined by Eq. (2) from the ($t, \sigma_1$) relation and the assumed $n$. Hereby a ($t, \sigma_{e,1\text{sec}}$) relation was established.

*Step 4 was repeated many times (200 - 1000) with new simulated panes and for each relevant loading rate*

4A. The 1 second strength, $f_{1\text{sec}}$, for each sub-area was simulated by generating a random number $x$ between 0 and 1. $f_{1\text{sec}}$ was then calculated from Eq. (3) as $x = F(f_{1\text{sec}})$.

4B. The time to failure for which $\sigma_{e,1\text{sec}} = f_{1\text{sec}}$ was found for each sub-area from the ($t, \sigma_{e,1\text{sec}}$) relation from step 3C.

4C. The sub-area with the shortest time to failure was identified and the failure load $p_{f,i}$ was calculated by multiplying that time by the loading rate $d$.

5. The simulated failure loads for each loading rate was ranked and the square of the distance between all the experimental results and the simulated distribution for the corresponding loading rate was calculated.

6. New sets of parameters were assumed and the procedure from step 3C was repeated until a good approximation was reached, i.e. a small square distance. If $n$ is unchanged, the repetition can start at step 4.

## VI. RESULTS OF ESTIMATION

### A. Low loading rates

The results for the slower loading rates, 0.0025, 0.025 and 0.25 kPa/s, were used for this analysis. The parameters ($a, b, k, n$) were estimated when they were all free, see Table 1, and for $n = 16$, see Table 2. Also the number of sub-areas was varied. All numbers in Tables 1 and 2 are determined for 500 simulated panes for step 4 and $b$ refers to $A_0 = 1 \text{ m}^2$.

<table><caption>TABLE 1: ESTIMATED PARAMETERS FOR LOADING RATES BELOW 1 KPA/S WHEN ALL PARAMETERS ARE FREE.</caption><tr><td><i>m</i>, no of sub-areas = <i>m</i> × <i>m</i></td><td>5</td><td>10</td><td>15</td><td>20</td></tr><tr><td><i>n</i>, power in Brown's integral</td><td>11.7</td><td>11.4</td><td>11.6</td><td>11.3</td></tr><tr><td><i>a</i>, minimum strength, MPa</td><td>26.3</td><td>25.3</td><td>0.0</td><td>3.2</td></tr><tr><td><i>b</i>, strength parameter, MPa</td><td>55</td><td>47</td><td>74</td><td>70</td></tr><tr><td><i>k</i>, size parameter</td><td>3.9</td><td>3.9</td><td>6.2</td><td>6.0</td></tr></table>

<table><caption>TABLE 2: ESTIMATED PARAMETERS FOR LOADING RATES BELOW 1 KPA/S WHEN $n = 16$ IS CHOOSEN.</caption><tr><td><i>m</i>, no of sub-areas = <i>m</i> × <i>m</i></td><td>5</td><td>10</td><td>15</td><td>20</td></tr><tr><td><i>n</i>, power in Brown's integral</td><td>16</td><td>16</td><td>16</td><td>16</td></tr><tr><td><i>a</i>, minimum strength, MPa</td><td>3.5</td><td>0.0</td><td>0.4</td><td>0.3</td></tr><tr><td><i>b</i>, strength parameter, MPa</td><td>72</td><td>68</td><td>68</td><td>67</td></tr><tr><td><i>k</i>, size parameter</td><td>6.2</td><td>6.2</td><td>5.8</td><td>6.4</td></tr></table>

When all parameters were free, it was seen that $n$ became a very stable value around 11.5, significantly smaller than the values usually quoted which range from 16 to 20, depending i.a. on humidity. It was further seen that the other parameters for $m = 5$ and 10 were quite different from those for $m = 15$ and 20. A supplementary investigation showed that if $a$ was assigned the value zero, $k$ and $b$ approached the values found for $m = 15$ and 20, where $a$ was close to zero.

It should also be mentioned that for $m = 10$ the number of simulations affected $a, b$ and $k$ significantly. Using either 200 or 1000 simulations in stead of 500 changes the estimates to numbers close to those quoted for $m = 15$ and 20 in Table 1. The high value for $a$ found in a few cases was likely to be a statistical coincidence.

When $n = 16$ was chosen as a fixed value, it is
<figure><img src="image_54.png" /><figcaption>Figure 4. Estimated distribution functions compared with experimental data for loading rates below 1 kPa/s when all parameters are free (n ~ 11) and m = 15.</figcaption></figure>

<figure><img src="image_55.png" /><figcaption>Figure 5. Estimated distribution functions compared with experimental data for loading rates below 1 kPa/s when n = 16 is chosen and m = 15.</figcaption></figure>

seen from Table 2 that *a* became close to zero and the size parameter equalled the higher values in Table 1. Comparing Figures 4 and 5 shows that the approximation was much less satisfying for *n* = 16 than when *n* was free. The values obtained for *m* = 15 are used henceforth.

## B. High loading rates

The difference in the failure load depending on the load duration was dealt with by estimating a different value of *n* for each loading rate but common values for *a*, *b* and *k*. This was done for *m* = 15 and 500 simulations. The estimated values for *n* are given in Table 3.

The value of *n* increased significantly between 0.25 and 1.5 kPa/s. The high values were so high that the time dependency could be ignored for practical purposes. (For *n* approaching infinity

only the maximum load mattered, the usual assumption for most other materials). The approximation was good as can be seen from Figure 6.

<table><caption>TABLE 3:
ESTIMATED VALUES FOR *n* WHEN PARAMETERS FOR ALL LOADING RATES ARE USED AND *n* IS ALLOWED TO DEPEND ON THE RATE AND ALL OTHER PARAMETERS ARE ASSUMED INDEPENDENT OF THE RATE.</caption><thead><tr><td>Loading rate</td><td>n</td></tr></thead><tbody><tr><td>0.0025 kPa/s</td><td>13.0</td></tr><tr><td>0.025 kPa/s</td><td>12.1</td></tr><tr><td>0.25 kPa/s</td><td>17.7</td></tr><tr><td>1.5 kPa/s</td><td>52.2</td></tr><tr><td>15 kPa/s</td><td>50.0</td></tr></tbody></table>

<figure><img src="image_56.png" /><figcaption>Figure 6. Estimated distribution functions compared with experimental data for all loading rates when *n* is estimated separately for each loading rate.</figcaption></figure>

For the three lower rates *n* was on average higher than 11.6. The reason is that due to statistical uncertainties *a* became significantly larger than zero for this estimation. This also affected the other estimated parameters so they took values somewhat different from the stable values in Table 1. Other estimations indicated that the average would approach 11.6 if *a* was given the value zero.

## VII. APPLICATIONS

Having estimated the parameters of the Weibull distribution and the power *n* in Brown's integral, a stochastic model for the strength of a glass pane was established. It can be used to estimate the failure probability by simulation for any load history, $q(t)$. This can be done very much similar to the Detailed procedure described above.
## A. Constant load

The procedure can be used to estimate a distribution function for the load capacity of a glass pane subjected to constant load $q_T$ acting for the time $T$. These distributions can be used to derive a relation between the characteristic load capacity of a glass pane and the duration of the load.

The simulation procedure for constant load was slightly different from the simulated tests, where the load was increased until failure. The way to handle this was, for each simulated glass pane, to increase $q_T$ gradually until failure occurred after precisely the time $T$. Results from the simulations is shown in Figures 7 and 8 for $n = 11.6$ and 16.

The characteristic 5% fractiles of the load capacity and material strength are given in Table 4.

<figure><img src="image_57.png" /><figcaption>Figure 7. Distribution of load capacity for constant loads $q_T$ for $n = 11.6$ and $T$ equal to 1 sec, 5 min, 16 days, and 50 years. The capacity for estimated Danish snow load is shown, too.</figcaption></figure>

<figure><img src="image_58.png" /><figcaption>Figure 8. Distribution of load capacity for constant loads $q_T$ for $n = 16$ and $T$ equal to 1 sec, 5 min, 16 days, and 50 years. The capacity for Danish snow load is shown, too.</figcaption></figure>

<table><caption>TABLE 4
ESTIMATED CHARACTERISTIC LOAD CAPASITIES AND MATERIAL STRENGTHS FOR DIFFERENT DURATIONS OF LOAD.</caption><thead><tr><th>T</th><th>1 sec</th><th>5 min</th><th>16 days</th><th>50 years</th></tr></thead><tbody><tr><td>n = 11.6</td><td>3.3 kPa<br>45 Mpa</td><td>1.7 kPa<br>25 MPa</td><td>0.6 kPa<br>12 MPa</td><td>0.3 kPa<br>8 Mpa</td></tr><tr><td>n = 16</td><td>2.9 kPa<br>41 Mpa</td><td>1.8 kPa<br>27 MPa</td><td>0.8 kPa<br>14 MPa</td><td>0.4 kPa<br>10 MPa</td></tr></tbody></table>

It is seen that the relative decrease with increasing duration was smaller for the strength than for the load capacity. This should be regarded when using the values to estimate $k_{\text{mod}}$ values. The coefficient of variation was 25% independent of the duration.

## B. Variable load

When the load varies over time, as most loads do, it becomes more complicated to estimate $\sigma_{e,1\text{sec}}$, but if the load history $q(t)$ is known, the principle is almost the same as described above.

When the load history was applied to a sufficient number of simulated panes, the failure probability could be estimated from the number of failed panes. This did not, however, enable estimation of the coefficient of variation of the strength.

An estimate that included the coefficient of variation was achieved by applying a load factor $\alpha$ to the load history and then for each simulated pane, by determining the highest value of $\alpha$ that did not cause failure. The variation of $\alpha$ was then a measure for the variation of the strength.

### 1) Snow load

The load history for snow is the snow load as a function of time during each of the major snow packs during the service life, $q_1(t)$, $q_2(t)$, ..., $q_N(t)$ and 0 the rest of the time. A snow pack is defined as the period from the first day where the ground is covered by snow until it has all melted away again.

The load history was modelled by three stochastic parameters:

1. The peak load $q_{\text{max},i}$ during snow pack $i$.

2. The number of snow packs during service life $T_s$ (50 years).

3. The equivalent duration $t_{e,i}$ of each snow pack, see below.
The distribution of the peak load was taken as the distribution of the yearly maximum ground snow load used for the Danish code for actions [DS 410:1998]:

$$F(q_S) = \exp\left(-\exp\left(-\frac{q_S - 415 \text{ Pa}}{124 \text{ Pa}}\right)\right) \quad (4)$$

The characteristic 98% value was 900 Pa.

The return time for snow packs with this distribution of the maximum load was obviously 1 year. The number of snow packs during the service life $T_s$ was therefore modelled as a Poisson process with the parameter $T_s \times 1 \text{ year}^{-1}$.

The equivalent duration of a snow pack is defined as the time that the peak load $q_{max,i}$ should act in order to cause the same damage K (defined by Eq. (1)) as the real load during the snow pack.

Similar damage means that the contribution to the equivalent stress $\sigma_{e,1sec}$ in all parts of the pane for the actual and the simulated snow pack must be similar. Due to the non-linear load-stress relation this requires that the load level is representative. Because the contribution to the damage from loads below say 90% of $q_{max,i}$ was very small, it was judged that using the peak load to estimate $t_{e,i}$ was sufficiently accurate.

The equivalent time was estimated from an estimation of the ground snow load during all snow packs for 32 years at two Danish locations. The locations were that far apart that major events could be considered independently, so that the data represented 64 years. The distribution of the major peak loads agreed well with the distribution in the code, Eq. (4).

Figure 9 shows the run of the load during a snow pack and the equivalent constant load by which it is represented.

$t_{e,i}$ is determined for all snow packs during the 64 years for which $q_{max,i} > 400 \text{ Pa}$. By inspection it is seen from Figure 10 that $t_{e,i}$ was significantly higher for snow packs with $q_{max,i} > 700 \text{ Pa}$ than for the others. Because correct estimation was most important for the higher loads, the estimate was

<figure><img src="image_59.png" /><figcaption>Figure 9. The load during an (exceptionally long) snow pack and the equivalent load represented by $t_{e,i}$ and $q_{max,i}$.</figcaption></figure>

based on the incidents with $q_{max,i} > 700$. Unfortunately, there were only four of these incidents. The duration was therefore assumed independent of the load and was modelled as a LogNormal distribution with parameters estimated from the four incidents. The mean value was somewhat dependent on the power $n$ in Brown's integral. It is about 11 days for $n = 11.6$ and 8 days for $n = 16$.

<figure><img src="image_60.png" /><figcaption>Figure 10. The relation between peak load $q_{max,i}$ and the logarithm of estimated equivalent duration $t_{e,i}$ in seconds for $n = 11.6$.</figcaption></figure>

Simulated snow load histories could then be applied to simulated glass panes. For each simulation the snow load was multiplied by a load factor $\alpha$ which was increased until failure occurred. Hereby the distribution of the load capacity $\alpha q_{max}$ could be found ($q_{max}$ was the largest peak load during the service life so the distribution of $q_{max}$ was the usual snow load in the code). The distribution of $\alpha q_{max}$ is shown in Figures 7 and 8 for two values of $n$. The 5% fractile represents the characteristic load capacity for Danish snow loads.

The lower part of the distribution was seen to be very similar to the distribution for constant load for 16 days, but the coefficient of variation was smaller. A sound conclusion about $k_{mod}$ for snow requires calculation of partial coefficients with and without duration of load effects. This might be done in line with [Sørensen et al., 2003].
VIII. CONCLUSIONS

A model for estimation of the statistical distribu-
tion of the load capacity of simply supported rec-
tangular glass panes subjected to any load history
is suggested. The input parameters in the models
are estimated from tests with rectangular glass
panes which means that only severe shortcomings
of the assumed models for strength (Weibull) and
damage accumulation (Brown) will affect the load
capacity.

Considering the limited number of test results
available, these models appear to give a satisfying
description of the test results.

The estimated parameters for the Weibull distri-
bution (3) conform quite well with other estimates.
It appears that the minimum strength *a* should be
taken as zero and that the size dependency *k* is
about 6. (The latter is assigned the value 25 in the
proposal for a CEN standard for the design of
glass. This value means that the size dependency
becomes insignificant).

The parameter *n* in Brown's integral (1), which
governs the damage accumulation (or Duration-of-
Load effect) deviates significantly from the ex-
pected *n* ~ 16-20. The estimated *n* = 11.6 means
that the Duration-of-Load (DoL)-effect is much
more severe than usually anticipated. This estimate
is based on tests lasting no more than an hour, thus
there is a crucial need to carry out tests with real
size glass panes subjected to constant load and
with expected time to failure of 1 month - 10 years.

The estimation of *n* also showed that for short
duration loads - like wind - there seem to be no
DoL-effect. Also this should be investigated more
closely.

The reliability of all the estimated parameters
should be improved by carrying out more tests, in-
cluding other glass thicknesses and other areas and
aspect ratios. The model should in principle be ap-
plicable to all shapes of panes and uneven load dis-
tributions. This should also be confirmed be ex-
periments.

A principle for stochastic modelling of snow

load is also described. The principle requires
knowledge of the run of the ground snow load dur-
ing major snow packs. The necessary parameters
are estimated for Danish circumstances. The inter-
pretation to code format need further work.

ACKNOWLEDGEMENT

The study on which this paper is based [Vester-
gaard, 2004] is supervised by L. O. Nielsen, De-
partment of Civil Engineering, Technical Univer-
sity of Denmark, P. Friis Hansen, Department of
Mechanical Engineering, Technical University of
Denmark and the first author.

REFERENCES

[Beason & Morgan, 1984]

BEASON, W. L., MORGAN, J. R. Glass failure predic-
tion model. J. of Structural Engineering, Vol 110, no 2.
1984

[Sørensen et al, 2003]

SØRENSEN, J. D., STANG, B. D., SVENSSON, S. *Ef-
fect of load duration on timber structures in Denmark.*
In ICASP9 - the 9th International Conference on Appli-
cations of Statistics and Probability in Civil Engineer-
ing, July 6-9, 2003, San Francisco, California, USA.

[DS 410:1998]

*Code of Practice for Loads for the Design of Structures.*
DS 410 (4.1). Dansk Standard. 1999.

[Grüters et al, 1990]

GRÜTERS, H.,HACKL, K., WILLMS, H. *Stress and strain analyses of rectangular plates for large deformations, Part 2, Calculation of fracture probabilities in rectangular plates under area load.* Glastech. Ber 63, no 4. 1990.

[Johar, 1981]

JOHAR, S. *Dynamic fatigue of flat glass. Phase II.* Ontario Research Foundation. Department of Metals, Glass and Ceramics. Mississagua. 1981.

[Johar, 1982]

JOHAR, S. *Dynamic fatigue of flat glass. Phase III.*
Ontario Research Foundation. Department of Metals,
Glass and Ceramics. Mississagua. 1982.

[Munch-Andersen & Ellum, 1995]

MUNCH-ANDERSEN, J., ELLUM, J. C. *Glass in the building envelope - a review* (In Danish). Danish Building Research Institute (SBI Bulletin 107). Hørsholm, 1995.

[Simiu et al, 1984]

SIMIU, E. et al. *Ring-on-Ring Tests and Load Capacity of Cladding Glass.* U.S. Department of Commerce. National Bureau of Standards. Gaithersburg, 1984.

[Vestergaard, 2004]

VESTERGAARD, R. K. *Strength of glass Construc-
tions* (in Danish). Master Thesis. Dept. of Civil Eng.,
Technical University of Denmark. 2004.
# To increase the residual bearing capacity of glass with a local reinforcement

Jürgen Neugebauer

Institute of Structural Design, University of Technology, Graz, Austria

A requirement of overhead glazing constructions is that they have a minimum residual bearing capacity. Wide spanned broken laminated glass with a continuous or a discrete support has a risk of falling down. The solution is to fix the laminated glass to the substructure. Wire-cloth or other synthetic fabrics are embedded in the PVB-interlayer or the synthetic resin, near the edges were the glass is supported. The fabric can be fixed to the glazing beads or to the glass fittings with special connections.

**Keywords:** reinforcement, residual bearing capacity

## I. INTRODUCTION

Residual bearing capacity is defined as the resistance against a collapse of a broken system. In glass technology this definition is used in connection with the carrying behavior of laminated glass or laminated safety glass, which is already destroyed by load effect or spontaneous break of one or more glass panels. [Wörner et al. 2001]

<figure><img src="image_61.png" /><figcaption>Fig 1: Example of a broken glass [Bauen mit Glas, 2002]</figcaption></figure>

## II. STATIC OF BROKEN GLASS

The principal concept for increasing the residual bearing capacity for continuously supported glass panes is to fix the glass to the supports. In the case of damage the glass panes cannot fall down. Therefore the values of the horizontal (membrane) forces at the support have to be computed, to be able to design the fixation.

In the case of a discrete supported glass pane the correlation between the forces at the support and the membrane forces in the glass in the area around the glass hole are needed.

### A. *Horizontal forces by continuous support*

The initial system is an unbroken laminated glass pane with the length *l* simply supported only on two edges. Figure 2 shows a sketch of the static systems of the glass panel. The sketch is structured into the initial system, the deflected static system, the system with tension cracks and the static system of a cable.

The load *q* is increased from zero. At the supports, only vertical (*A*<sub>v</sub> or *B*<sub>v</sub>) reactions result if small deformations are assumed, see in figure 2 (a). With a further load increasing up to the fracturing load the theory of the small deformations cannot be assumed any longer. It is a change from the theory of small deformations to the theory of large deformations. At the supports, vertical (*A*<sub>v</sub> or *B*<sub>v</sub>) and horizontal (*A*<sub>h</sub> or *B*<sub>h</sub>) reactions result, see in figure 2 (b).

At the moment of the fracture there is a change in carrying behavior. The bending moment decreases and the traction increases, and at the supports larger horizontal (*A*<sub>h</sub> or *B*<sub>h</sub>) reactions
result, see in figure 2 (c) [Sobek et al. 1999].

The PVB (polyvinyl butyral) interlayer stretches near the cracks. Dependent on the load the forces of static system can be in an equilibrium. But with a higher load $q$ a further deformation is possible. With this further deformation of the system, the stretch of the foil can be so large, that the system can transfer no more bending moment to the supports and the static system of a broken toughened glass changes to that of a cable system see in figure 2 (d).

<figure><img src="image_62.png" /><figcaption>Fig 2: Load carrying systems of continuously supported glass</figcaption></figure>

In the case where this deflection of the static system of a cable is the smallest the largest horizontal force results. Further stretching of the foil due to non linear effects causes an enlargement of the deflection $f$ and thus a decrease of the horizontal bearing force ($A_h$ or $B_h$).

These horizontal forces tend to pull the glass out of the supporting system.

## B. Horizontal forces by discrete support

With point load supported glass panes, the stress maxima are concentrated in the range of the glass fittings. The load $q$ over the whole area is increased from zero to the fracturing load. At the moment of the fracture there is a change in carrying behavior, a change from small to large deformations and from pure bending moment to bending moment and traction. The traction membrane force is concentrated near the supports, see figure 3.

These membrane forces result in a ring of traction around the hole, which causes an expansion of the hole. If the deformations at the hole edge are too large, the broken glass panel can slide out of the glass fittings.

<figure><img src="image_63.png" /><figcaption>Fig 3: Load carrying systems of discretely supported glass</figcaption></figure>

# III. CONCEPT OF THE REINFORCEMENT

## A. Continuous support

The principal concept for increasing the residual bearing capacity is the fixation of the glass with a fabric to the substructure. A fabric is embedded into the PVB-interlayer between the glass panes near the edges, and one side of the fabric protrudes of the glass. The projecting fabric is cast into a plastic fastening batten, or is welded to the metal profile. Into that fastening border holes for the screws are drilled, and with these screws the border is fastened to the support profile. In order to handle thermal deformations the holes must be bored with a larger diameter. A squeeze-free storage for the unbroken glass panel must be ensured. The grip is supposed to become effective only in case the glass breaks.

<figure><img src="image_64.png" /></figure>
<caption>Fig 4: Concept for continuously supported glass</caption>

## B. Discrete support

The function of the reinforcement is similar to that in reinforced concrete. The expansion of the glass hole due to the membrane forces is hampered by the reinforcement. A round fabric is embedded into the PVB-interlayer between the glass panes around the glass hole. The fabric is cast into a plastic hollow shaft or welded to a metal hollow shaft. The height of this core is slightly smaller than the total thickness of the structure and the outside diameter is slightly smaller than the hole diameter. The inside diameter results from geometry of the glass holder. In order to handle all movements of the glass cladding a squeeze-free fixation of the glass pane must be ensured.

<figure><img src="image_65.png" /><figcaption>Fig 5: Concept for discretely supported glass</figcaption></figure>

## IV. MATERIALS OF THE REINFORCEMENT

To be able to select the correct material used for the reinforcement, the production conditions must be considered. Furthermore it is important that the materials used are themselves chemically compatible. Only with the knowledge of the boundary conditions a sound connection between the glass and the fabrics on the one hand and between the fabric and the support on the other hand, can be reached.

The production of a laminated safety glass takes place at a temperature of 140°C, and a pressure of 14 bar, therefore the fabric must have a temperature resistance of approx. 140°C. [Leicht und Glasbau, 2002]

The fabrics should possess a high measure of transparency due to a requirement of the architects to the glass.

The thickness of the fabrics cannot be more than 1,0 mm due to the generally used thickness of the PVB-interlayer with thickness of 1,52 mm.

<table><caption>TABLE 1:
REINFORCEMENT MATERIALS</caption><thead><tr><th>MATERIALS</th><th>TENSILE STRENGTH</th><th>FUSING TEMPERATURE</th></tr></thead><tbody><tr><td>fabrics</td><td>N/mm²</td><td>°C</td></tr><tr><td>glass fibre</td><td>3400</td><td>400</td></tr><tr><td>polyamide</td><td>400-700</td><td>225</td></tr><tr><td>polyester</td><td>600</td><td>220</td></tr><tr><td>wire</td><td>450-750</td><td>600-1200</td></tr></tbody></table>

<figure><img src="image_66.png" /><figcaption>Fig 6: Reinforcement materials (fabrics)</figcaption></figure>

## V. TESTS AND RESULTS

### A. Continuous support

<figure><img src="image_67.png" /><figcaption>Fig 7: Testing concept for the traction tests</figcaption></figure>

For the horizontal forces which tend to pull the glass out of the supporting system a sound fixation
has to be developed. To be able to design the fixation it is required to get values for the load resistance of the different materials. It is important to find out which is the weakest member of this composition of glass, fabric and the fastening profile. With simple traction tests it is possible to get the fracturing load of the weakest member and information of the fracturing behaviour of this system

<figure><img src="image_68.png" /><figcaption>Fig 8: Test set up in the laboratory</figcaption></figure>

In the laboratory the traction test series were done with a cylinder press, see in the figure 8. In the test set up a laminated glass with two 6 mm float glass with a width of 360 mm and a height of 500 mm for the test objects were used. In the PVB-interlayer the wire-cloth and the syntactic fabrics were embedded at both sides of the testing object. The upper position was the carrying side with a strong wire fabric and the lower side was the testing side. The size of each fabric was 300 mm in the length and 200 mm in the width. At the testing side fabrics with different materials, different diameters of the threads and different mesh openings were arranged. The fabrics were glued with a synthetic resin to two small steel profiles on both sides of the fabrics. With three bolds the profiles were fixed to two bigger steel plates on both sides, see in figure 8. Two big bolds were the fixation to the hydraulic press. With this symmetrical testing set up only axial forces are ensured.

The load deformation diagram in figure 9 shows the result of a test series with a wire-cloth. The threads had a diameter of 0.5 mm and a mesh opening of 1.6 mm. The result of this test was a fracturing load of approximately 20.0 kN. The crack was in the space between the glass and the fastening profile, it was a fracture of the fabric itself, as shown in figure 10.

<figure><img src="image_69.png" /><figcaption>Fig 9: Load deformation diagram of the traction test with a wire-cloth</figcaption></figure>

<figure><img src="image_70.png" /><figcaption>Fig 10: Broken wire-cloth</figcaption></figure>
In the load deformation diagram, see in figure 11, the result of the test series with a polyamide fabric is shown. The threads had a diameter of 0.3 mm and the mesh opening was 0.5 mm. In the figure 11 is an oscillating curve before the fracturing load with approx. 8.5 kN was reached. The reason is, that the bond between the PVB-interlayer and some threads has gone lost (figure

12).

<figure><img src="image_71.png" /><figcaption>Fig 11: Load deformation diagram of the traction test with a</figcaption></figure>

<figure><img src="image_72.png" /><figcaption>Fig 12: Broken polyamide fabric</figcaption></figure>

After the maximum load was reached the oscillating curve was the breakage of the polyamide fabric thread by thread.

The result of the test series with a polyester fabric see in the load deformation diagram in

figure 13. The threads had a diameter of 0.35 mm and the mesh opening was 1.18 mm. The result of this test was a fracturing load of approximately 8.5 kN, it was a breakage of the polyester fabric itself. The crack was in the space between the glass and the fastening profile, see in figure 14.

<figure><img src="image_73.png" /><figcaption>Fig 13: Load deformation diagram of the traction test with a polyester fabric</figcaption></figure>

<figure><img src="image_74.png" /><figcaption>Fig 14: Broken polyester fabric</figcaption></figure>

With these testing results with the different materials it is possible to say, that the less deformable wire-cloth (approx. 6 mm), is better than synthetic fabrics with deformations in the space between the fastening profile and the glass of more than 10 mm. To prevent a collapse of glazing system, it is a requirement to ensure that the glass panes can not slip out of their supports. With an opening of more than 10 mm, as with synthetic
fabrics, a slip out of the supporting system is
possible.

## B. Discrete support

To get information of the behaviour of the broken laminated glass around the glass hole, pressure test series were prepared. The concept of these tests was, to get the maximum force where the glass fitting is pulled out of the glass. The goal of these series was, to get the difference in the residual bearing capacity between the laminated glass system without and with reinforcement.

<figure><img src="image_75.png" /><figcaption>Fig 15: Testing concept for the pressure tests</figcaption></figure>

In the test set up a laminated glass with two 6, 8 and 10 mm toughened glass panes with a 30 mm hole in the middle were used. The laminated glass with a size of 600 mm in both directions is supported on a frame, as shown in the figure 16. A glass fitting with a diameter of 60 mm was used.

<figure><img src="image_76.png" /><figcaption>Fig 16: Test set up in the laboratory without reinforcement</figcaption></figure>

The reinforcement with a diameter of 200 mm and a hole in the middle with a diameter of 30 mm was embedded in the PVB-interlayer around the

glass hole (see figure 17). For the test series different materials with different diameters of the threads and different mesh opening were used.

<figure><img src="image_77.png" /><figcaption>Fig 17 Test set up in the laboratory with reinforcement</figcaption></figure>

The results of these test series without a reinforcement one can see in the figure 18. As a function of the thickness of the laminated glass the load deformation diagram shows the fracturing load of the glass with approx. 4.0 kN up to approx. 11,5 kN.

<figure><img src="image_78.png" /><figcaption>Fig 18 Load deformation diagram of the system without reinforcement</figcaption></figure>

An important fact is, that the load maxima with which the glass is pulled out of the glass fitting was with approximately 0.5 kN and with a deformation of approx. 90 mm in all cases of the
glass thickness nearly the same. The maximum force for the residual bearing capacity depends on the extensional stiffness of the PVB-interlayer.

<figure><img src="image_79.png" /><figcaption>Fig 19 Result of the system without reinforcement</figcaption></figure>

In the first test set up a laminated glass with reinforcement two 6 mm toughened glass panes with a 30 mm hole in the middle were used.

<figure><img src="image_80.png" /><figcaption>Fig 20: Load deformation diagram of the system with and without a reinforcement of a 2x6 mm laminated glass</figcaption></figure>

The figure 20 shows a comparison of the test result of a 2x6 mm laminated glass without and with reinforcement. The fracturing load of the test with reinforcement was, with approx. 4.5 kN, a little bit higher than the result of the system without a reinforcement. The load where the glass

is pulled out of the glass fitting was with approx. 2.3 kN, and the deformation was approx. 140 mm. The value was more than 4 times higher than the load where the glass is pulled out of the glass fitting of the system without reinforcement. The residual bearing capacity was increased by a factor of 4.

<figure><img src="image_81.png" /><figcaption>Fig 21 Result of the system with reinforcement</figcaption></figure>

The comparison of the two pictures, in figure 19 the system without reinforcement and picture in figure 21 the system with reinforcement, shows the different shapes of the broken glass. With the reinforcement the broken glass gets more bending stiffness around the glass hole and the shape changes from a funnel to a bowl.

<figure><img src="image_82.png" /><figcaption>Fig 22 Fractured fabric around the glass hole</figcaption></figure>

Figure 22 shows the glass hole after the glass fitting was pulled out. One can recognize that the collapse occurred by the fracture of the fabric due to the membrane forces.
# VI. SUMMERY AND PROSPECTS

With the proposed construction it is possible to design wide spanned laminated glass simply supported on two sides. By point load supported glazing it will be possible to use laminated toughened glass with its high bending stress resistance. The goal is that the broken glass (in all qualities - float glass, heat strengthened glass, or toughened glass) remain fixed and is prevented from falling down.

The future research will concentrate on an optimization process of this system.

# VII. 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 and the industry partner Herbert Maierhofer Glastechnik GmbH (St. Marein/Mürztal, Austria).

# REFERENCES

## [Bauen mit Glas, 2002]

Bauen mit Glas, Information für den Bauherrn, Architekten und Ingeniure, Wirtschaftsministerium Baden-Württemberg, .2002

## [Leicht und Glasbau, 2002]

Bemessung von Glaskonstruktionen, Institut für Stahlbau, Holtbau und Flächentragwerke, Technische Universität Graz, 2002.

## [Sobek et al. 1999]

W.SOBEK, F. MAIER, M. KUTTERER, IL
Forschungsbericht 1/99, "Mutifunktionale Glassysteme",
Bewehrtes Verbundsicherheitsglas, 1/99 Universität
Stuttgart, 1999.

## [Wörner et al. 2001]

WÖRNER, SCHNEIDER, FINK, Glasbau, Berechnung,
Konstruktion, Springer Verlag, 2001.
# Inelastic material behaviour of Soda-Lime-Silica Glass

Frank Schneider, Darmstadt University of Technology, Statics and Dynamics of Structures, Darmstadt, Germany

J.-D. Wörner, Darmstadt University of Technology, Statics and Dynamics of Structures, Darmstadt, Germany

The inelastic material behaviour of thermally toughened and heat strengthened Soda-Lime-Silica glass at room temperature was investigated. A four-point bending test configuration was used. The delayed elastic deformations were measured and reveal a significant time-dependent behaviour. The possible origin such as sub-critical crack growth, ion diffusion or absorption of water molecules is discussed briefly. The aim of the study is a better understanding of the nature of glass itself, in order to use this brittle material as a structural member in the field of civil engineering according to appropriate safety requirements.

**Keywords:** inelastic material behaviour, Soda-Lime-Silica glass, creep

## 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. Especially the long term behaviour and the influence of long duration loading on the glass structure and the material strength have to be investigated and verified. Therefore, a brief summary of the constitution of glassy materials will be given in the next paragraph of this paper before the experimental results are discussed.

## II. THE GLASS MATERIAL

### A. *Chemical composition*

Glasses used in structural applications consist mainly of SiO₂. In the crystalline form of SiO₂ each Si-atom is surrounded tetrahedrally by four O-atoms (Fig. 1), whereby each O-atom is connected to two-neighboured tetrahedras and therefore counts to ½ in the chemical formula. The three-dimensional network formed of the crystalline structure exhibits equal bond angles between its composites. In the glassy state these angles vary slightly and the periodicity of the crystalline lattice, i.e. the long-range order, is lost [Greaves, 1997]. Therefore, glassy materials are characterized by a certain short-ranged order of the nuclei, but without long-range order, as it can be found in the regular lattice of a crystal.

<figure><img src="image_83.png" /><figcaption>Fig 1: Si-O₂ tetrahedra [Wörner, 2001]</figcaption></figure>

Pure-fused silica is difficult to produce economically due to a very high softening temperature of around 1200°C. By adding oxides such as NaO₂ and CaO₂ the glass transition range of the modified glass, like the most common Soda-Lime-Silica Glass, is lowered to about 550°C. The added oxides enter the network as cations. The
radical oxygen breaks the covalent bond of the SiO₂ network apart into two groups of Si-O⁻ connected by a weaker polar bonding.

## B. *The structure of glass*

Despite many other solids, glass does not have a crystalline structure on microscopic scale. At room temperature the molecules have a disordered arrangement like a fluid, but they are rigidly bound like in a solid. Structurally, non-crystalline solids are similar to fluids, but the entire network is rigid. This state is called an amorphous solid or a glass. Therefore, glass is rather more a state of matter, than a special material with a certain chemical composition. The glass state does not fit into the classical classification of substances by three states of aggregation. It is an own class halfway between fluids and solids. Because of this, glass is sometimes called a frozen super-cooled liquid. These misnaming has led to the legend that antique windowpanes would very slowly flow downwards under their own weight [Zanotto, 1998].

Theoretically every material independent on its chemical composition can achieve the glassy state, if crystallization can be prevented by cooling the melt very fast. The molecules have no time to move into the ordered crystalline arrangement until the melt solidifies. Due to the increasing viscosity during the cooling process the molecules remain in this metastable frozen-in non-equilibrium condition. In this sense a solid can be defined as a liquid with a very high viscosity [Scholze, 1988]. A glass-forming melt has a tendency to achieve this super-cooled condition especially easy. The temperature Tg, at which the glass transition takes place, and the achieved material properties depend on the chemical composition and also cooling rate. By cooling slowly the material has more time to relax during solidification, the transition occurs at lower temperatures and the glass is more dense. This phenomenon is technically used by the production of thermally toughened glass.

## C. *Strength and material properties*

The material properties are determined by the glassy state. Its characteristic brittle behaviour results from this disordered structure. Plastic deformations, i.e. the slip of molecular layers observed in a crystalline metal, cannot occur. A tensional loading leads to stress concentration at

any surface crack and finally to a spontaneous brittle fracture without plastic deformations at the crack-tip. Therefore the technical strength of the glassy material extremely depends on the condition of the surface in the tension zone and is some orders of magnitudes lower than the expected theoretical strength of the atomic bonds. To overcome this deficit it is possible to prestress the glass by heating it above Tg and cooling it rapidly afterwards. The result is a residual stress state with compression at the surface and tension in the interior. Depending on the surface compression achieved the glass can be classified as toughened glass or heat strengthened glass. Usually, anorganic glass as an amorphous material is treated to be both ideal linear-elastic and isotropic. Nevertheless, small inelastic deformations could be measured in a bending test. After unloading from a long term loading, the glass specimens exhibit a remaining deformation of up to 1% of the instant elastic one. Subsequently, during a recovery phase the inelastic deformations mostly disappeared.

# III. EXPERIMENTAL PROCEDURE

## A. *Specimens*

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 toughened glass and three of heat strengthened glass.

<figure><img src="image_84.png" /><figcaption>Fig 2: Experimental setup during loading</figcaption></figure>
## B. *Experimental procedure*

A four-point bending test configuration with a span of 0.91 m was used to apply a static load P. Tree different loading levels have been applied to the test specimens in order to achieve the maximum tensile stress showed below in Table 1. The increasing inelastic deformation was measured manually each day after temporary unloading each specimen. A mobile rigid frame was installed above the supports and the deflection was measured in the middle of the span by a digital dial gage (Fig 3). Afterwards the loading was applied to the specimen again. Every specimen was loaded at room temperature for a period of seven days at all. During the subsequent 101 days recovery period the decrease of the deformation was measured similarly.

<table><caption>TABLE 1:
LOAD LEVELS AND NUMBER OF SPECIMENS</caption><thead><tr><th>Glass type</th><th>applied tensile stress [N/mm²]</th><th>degree of utilization [%*]</th><th>Number of specimens</th></tr></thead><tbody><tr><td>toughened glass</td><td>120</td><td>100</td><td>2</td></tr><tr><td>toughened glass</td><td>100</td><td>83</td><td>2</td></tr><tr><td>toughened glass</td><td>70</td><td>58</td><td>2</td></tr><tr><td>Heat strengthened glass</td><td>70</td><td>100</td><td>3</td></tr></tbody></table>

<figure><img src="image_85.png" /><figcaption>Fig 3: Schematically experimental setup of four-point bending testResults and Discussion</figcaption></figure>

## C. *Experimental results*

The experimental results reveal a significant time-dependent behaviour. During the loading period an increasing delayed-elastic deformation

was observed. After seven days loading, the inelastic deformation grew up to 1% of the instant elastic one. Subsequently, during the 101 days recovery phase the deformations mostly disappeared. Therefore no viscous flow is observed. For that reason it is more accurate to speak about a delayed elastic behaviour.

<figure><img src="image_86.png" /><figcaption>Fig 4: Measured deflection during a 2600 h time period</figcaption></figure>

## D. *Discussion*

Two out of three of the heat strengthened glass sheets failed during the loading period. The remaining did not exhibit a significant different behaviour to the equally loaded toughened glass sample. The delayed-elastic deformation is proportional to the applied load, i.e. the achieved stress, as illustrated in Figure 4. Therefore, an influence of the load-level is not evident.

Visco-elastic flow of glass, as it is observed in the glass transition range, can be excluded as a possible reason for the observed behaviour at room temperature [Gy, 1994], [Duffrène, 1999]. A probable explanation could be the growth of sub-critical crack inside the initial tension zone, which weakens the material [Gy, 1999] and could give an explanation for the failure of two out of three heat strengthened glass specimens after a certain load period. This static fatigue could also lead to a time dependant decrease in the material strength and should be considerate in the design of load bearing glass members. Even the recovery phase could be explained by cracks-healing after unloading. At the other hand, no significant differences exist between the evolution of the delayed deformations based on the instantaneous elastic one, neither between the different load levels, nor between toughened glass and heat strengthened glass, i.e.

<footnote>* based on the characteristic value of the bending strength</footnote>
the value of the residual stress state (Fig 5). An influence of the overall stress state on the growth of cracks is not identifiable, which could be expected if sub-critical crack growth causes the observed behaviour. Furthermore, the curvature in Figure 5 leads to the assumption of a linear viscoelastic material behaviour.

<figure><img src="image_87.png" /><figcaption>Fig 5: Inelastic deformation in percent of the elastic deformation.</figcaption></figure>

In technical literature alternative explanations, such as ion diffusion [Holloway, 1973], [Kahnt, 1996], switches in atomic bonds of the SiO₂ network [Pelletier, 1999] and absorption of water molecules [Franek, 1983] are also frequently discussed. The idea of water molecules entering the tensioned glass surface and after unloading generating the observed deformations by acting like a wedge needs further investigations.

A slow diffusion of cations in the glass structure is the most probable explanation. Due to external stress sodium ions get slowly squeezed out of their initial position to an energetically more favourable place in the lattice. After removing the load the ions move back into their original position, thus an explanation of recovery phase could be given. A time-depending decrease of the material strength is not expected in this case. The determination of the ultimate bending strength has not been performed by now, due to the small amount of similar specimens and load conditions and the expected scattering of the results. Furthermore, a combination of two ore more of the discussed mechanisms could be responsible for the effect. The dominant mechanism can not clearly be identified using the current test setup.

A viable rheological model will be developed to describe the behaviour in the next chapter.

## E. Rheology

The general constitutive equation of a linear viscoelastic material can be written in the form of a hereditary integral

$$ \sigma(t) = \int_{0}^{t} \Phi(t - \bar{t}) \dot{\epsilon}(\bar{t}) d\bar{t} . \quad (1) $$

For any applied strain history $\epsilon(t)$ the achieved stress $\sigma(t)$ can be calculated easily if the relaxation modulus $\Phi(t)$ is known. At the other hand, the analogue equation

$$ \epsilon(t) = \int_{0}^{t} \Psi(t - \bar{t}) \dot{\sigma}(\bar{t}) d\bar{t} \quad (2) $$

can be used to describe the strain for a particular stress. The creep compliance function $\Psi(t)$ can be obtained by fitting the parameters of the assumed rheological model to the results of a performed creep experiment. Therefore, an average curve of the measured delayed elastic deformations illustrated in Figure 5 is calculated as a basis of the fit. Subsequently, an adequate rheological model composed of linear elastic springs ($\sigma=E*\epsilon$) and linear viscous dashpots ($\sigma=\eta*\epsilon$) has to be chosen. A single Maxwell or Kelvin Element does not describe the observed characteristics of the viscoelastic behaviour sufficiently. For that reason, a chain of three Kelvin solids in series is used.

$$ \epsilon(t) = \epsilon_1(t) + \epsilon_2(t) + \epsilon_3(t) \quad (3) $$

<figure><img src="image_88.png" /><figcaption>Fig 6. Kelvin chain</figcaption></figure>

The total strain is the sum of the strains of each Kelvin element (Fig 6), thus the creep compliance $\Psi(t)$ is:

$$ \Psi(t) = \sum_{i=1}^{3} w_i (1 - e^{-\frac{t}{\tau_i}}) \quad (4) $$

Whereby $w_i$ represents weighting factors and $\tau_i=\eta_i/E_i$ is the retardation time of any Kelvin solid. Nevertheless, the elements and its parameters can not directly be related to any physical mechanism.
A similar behaviour could be described by three parallel Maxwell elements and its particular parameters. The equations of the Maxwell model are more convenient if a strain is applied and the stress-relaxation is measured. Nonetheless, both models are equivalent.

An equal approximation can be done by the b-function, which represents a continuous distribution of relaxation times. In this case the creep compliance $\Psi(t)$ has the form

$$ \Psi(t) = (1 - e^{-\left(\frac{t}{\tau}\right)^b}) \quad (5) $$

<figure><img src="image_89.png" /><figcaption>Fig 7: Inelastic deformation, approximation by rheological models and ultimate value</figcaption></figure>

Figure 7 illustrates the average values of the inelastic deformation during the loading period and the approximations by the Kelvin chain and the b-function. Furthermore, the calculated limit of the ultimate deformation $\varepsilon(t=\infty)$ is presented for both models and the increasing inelastic deformation is extrapolate to 600 h.

<table><caption>TABLE 2:
PARAMETERS OF THE RHEOLOGICAL MODELS, LOADING PERIOD</caption><thead><tr><th rowspan="2"></th><th colspan="3">Kelvin Chain</th><th rowspan="2">b-function</th></tr><tr><th>1</th><th>2</th><th>3</th></tr></thead><tbody><tr><th>weighting factor*</th><td>0.389</td><td>0.294</td><td>0.317</td><td>b=0.408</td></tr><tr><th>relaxation time [s]</th><td>10<sup>4</sup></td><td>10<sup>5</sup></td><td>10<sup>6</sup></td><td>87957</td></tr><tr><th>relaxation time [h]</th><td>2.78</td><td>27.78</td><td>277.78</td><td>24.43</td></tr><tr><th>Ultimate strain [%]</th><td colspan="3">1.44</td><td>1.34</td></tr></tbody></table>

<table><caption>TABLE 3:
PARAMETERS OF THE RHEOLOGICAL MODELS, UNLOADING PERIOD</caption><thead><tr><th rowspan="2"></th><th colspan="3">Kelvin Chain</th><th rowspan="2">b-function</th></tr><tr><th>1</th><th>2</th><th>3</th></tr></thead><tbody><tr><th>weighting factor*</th><td>0.637</td><td>0.237</td><td>0.127</td><td>b=0.288</td></tr><tr><th>relaxation time [s]</th><td>10<sup>4</sup></td><td>10<sup>5</sup></td><td>10<sup>6</sup></td><td>7820054</td></tr><tr><th>relaxation time [h]</th><td>2.78</td><td>27.78</td><td>277.78</td><td>5.57</td></tr><tr><th>ultimate strain [%]</th><td colspan="3">1.16</td><td>1.17</td></tr></tbody></table>

The same procedure can be performed for the unloading period. The parameters given in Table 2 and 3 enable someone to describe the deformations in both cases mathematically.

## F. Restrictions

The deflection could not be determined immediately after the removal of the load. The delay between unloading and measuring was approximately one minute, but the delay-time differs slightly, dependent on the quantity of the applied dead load. For that reason, a short term creeping-effect could not be acquired and the time period between unloading and measuring the deflection was not exactly equal.

Sufficient statistical information could not be ensured, due to the small number of test specimens. Nevertheless, a tendency can be deducted. An improved test assembly with an enlarged numbers of specimens and detailed investigations to comprehend the different influences is planed for the future.

## G. Further investigations

A test series of 30 samples of toughened glass is planned to be carried out. Each glass specimen will be investigated on its material behaviour, e.g. Young's Modulus, weight, dimension, surface compression, etc. before and after the long term loading test. In addition the transparency of the specimens will be determined before and after the loading to prove cracks inside the glass directly, as illustrated in Figure 8 schematically. A laser diode will be used as an intense light source and a photo sensitive diode (PSD) will gauge the intensity of the received signal. Eight samples will remain unloaded for the whole test period to exemplify as a reference.

<footnote>* in case of the b-function the value of the parameter b</footnote>
<figure><img src="image_90.png" /><figcaption>Fig 8: Schematic illustration of the transmission detection test setup</figcaption></figure>

An advanced measurement device, namely laser triangulation, has been developed to increase the accuracy of the test method and detect short time creep. The deformation will be determined by a laser-optical device (Fig. 9). A laser beam above the support will be deflected at the surface of the glass. The rotation $\phi$ of the support, which is directly related to the deflection by a simple linear equation, is measured by a position sensitive detector continuously, even during loading and unloading. Finally, the bending strength of every specimen will be determined.

<figure><img src="image_91.png" /><figcaption>Fig 9: Schematic illustration of the advanced test setup</figcaption></figure>

Four specimens will remain in an immersion bath during the whole test period to investigate the influence of the water. Furthermore, the detection of water molecules inside the glass structure, e.g. by infrared-spectroscopy and microwave-irradiation will be investigated. Unfortunately, first tests performed at the Darmstadt University of Technology department of Physics did not show sufficient results. One possible reason for that are the different bonding conditions in physical and chemical bonded water compared to bulk water. The influence of ion-diffusion will be analysed by studying specimens of pure Silica-glass, which does not include oxides in the glass structure. If no significant delayed elastic behaviour can be

observed in pure Silica-glass, the effect in Soda-Lime-Silica glass can be related to that.

In the investigated glass specimens, the inelastic part of the deformation is very small and for that reason difficult to verify. To visualize the inelastic deformations more easily it is projected to study the reaction of glass-fibre bundles, 6 m in length, under static tensile load. Due to its high strength and small cross-section measurable deformations can easily be achieved. Preliminary investigations exhibit a noticeable delayed deformation of about 30% of the instant elastic one after 24 hours loading. Subsequently, during a 24 hour recovery phase the deformations mostly disappeared. A small remaining deformation can be explained by the failure of single fibres in the bundle. Unlike bulk glass, fibres exhibit an axial orientation of the $SiO_2$ network due to the manufacturing process. Nevertheless, the achieved results are promising and will be carried on in the future.

## IV. CONCLUSION

Thermally toughened glass was tested. The increase of the inelastic deformation is measured by a digital dial gauge. More accurate measurements by laser triangulation sensors are planned for the future. The value of the evidenced delayed elastic deformation is small, compared to the instantaneous elastic one. Nevertheless, it could affect the safety of permanently loaded units, e.g. externally prestressed members. The origin of the observed behaviour is not completely understood at present. Time dependent material behaviour at room temperature of glass is frequently attributed to viscous flow. Sub-critical crack growth inside the initial tension zone, ion diffusion, switches in atomic bonds of the $SiO_2$ network or absorption of water molecules are more likely the reason of the observed inelastic deformation. To understand the origin of that behaviour and the consequence for the safety requirements further investigations have to be carried out.

## REFERENCES

[Greaves, 1997]
GREAVES, G.N., *Local structure, microstructure and glass properties*. Journal of Non-Crystalline Solids., Vol. 222, pp. 13-24, 1997

[Wörner, 2001]

WÖRNER, J.-D., *Glasbau-Grundlagen, Berechnung, Konstruktion*, Berlin Heidelberg New York, Springer, 2001

[Zanotto, 1998]

ZANOTTO, E., *Do Cathedral Glasses Flow?* Am. Phys. Vol. 66, pp. 392-396, 1998

[Scholze, 1988]

SCHOLZE, H., *Glas - Natur, Struktur und Eigenschaften*, Berlin Heidelberg New York, Springer, 1988

[Gy, 1994]

GY, R., *New insights into the viscoelasticity off glass*. *Journal of Non-Crystalline Solids*, Vol. 175, pp. 103-117, 1994

[Duffrène, 1999]

DUFFRÈNE, L., *Viscoelastic behaviour of a soda-lime-silica glass: inadequacy of the KWW function*, Journal of Non-Crystalline Solids, Vol. 215: pp. 208-217, 1999

[Gy, 1999]

GY, R., *Creep and Recovery of a Thermally Tempered Glass Plate at Room Temperature*, Glass Processing Days 13-16 June 1999

[Holloway, 1973]

HOLLOWAY, D.G., *The Physical Properties of Glass*, London, Wykeham, 1973

[Kahnt, 1996]

KAHNT, H., *Ionic transport in Glasses*, Journal of Non-Crystalline Solids, Vol. 203, pp. 225-231, 1996

[Pelletier, 1999]

PELLETIER, J.M., *Evidence for a residual elastic modulus in inorganic glasses by mechanical spectroscopy*, Journal of Non-Crystalline Solids, Vol. 258, pp. 119-130, 1999

[Franeck, 1983]

FRANEK, H. J., *Reaktionen zwischen Glasoberfläche und Wasser bzw. wässrigen Lösungen*, Glastechnische Berichte Vol. 56 No. 6/7, pp. 165-175, 1983
<figure><img src="image_92.png" /></figure>
# Glass Strength in the Borehole Area of Annealed Float Glass and Tempered Float Glass

Jens Schneider, Senior Engineer

Schlaich Bergermann und Partner, Stuttgart, Germany

Thermal stresses of tempered glass in the area of typical boreholes were estimated using Narayanaswamy's structural relaxation model implemented in a Finite-Element-Code (ANSYS 5.5.2). Surface stresses were measured for samples with holes of different float glasses (soda-lime-silica-glass, borosilicate glass) from different commercial tempering processes. The bending strength in the borehole area of the samples and of annealed (float) glass samples was determined using a modified coaxial double-ring bending test. Results are compared and evaluated on a statistical basis. It is shown that the characteristic glass strength of tempered glass in the borehole area is not lower than in the "infinite area" given in European standards. From the results it is also assumed that crack healing plays an important role for the bending strength of tempered float glass.

**Keywords:** glass strength, tempered glass, numerical modelling, tempering process, structural glass, photoelastic measurements

## I. INTRODUCTION

In many facade applications, structural glass is fixed in holes with glass fittings. Finite-element calculations show that the tensile stresses in such glass panes mostly have their maximum in the area of the chamfers of holes, for usual dimensions and bending from wind loading [BATHE 1990, SCHNEIDER 2001].

Thus, it is important to determine the glass strength in the hole area for the structural design of the glass. The glass strength very much depends on the surface condition influenced by the drilling process and - for tempered glass - the amount of

temper stress. Because of sub-critical crack growth and the stress concentration in the hole area, annealed float glass is usually inadequate for this applications. For tempered glass, the qualitative distribution of thermal stresses in the borehole area was calculated in [LAUFS 2000, SEDLACEK 1999] with a viscoelastic model but only little quantitative data about their influence on the glass strength was given.

In [CARRE 1997, CARRE 1999] the amount of thermal stresses at the edges of tempered glass was re-calculated by means of photoelastic measurements and their influence on the glass strength was compared with the results of bending tests.

In this investigation the same methodology was used for the borehole area. First, photoelastic methods (Laser-Gasp®, Epibiaskop, [ABEN 1993]) were used to determine the surface stresses in the "infinite" area of tempered samples with holes from different commercial tempering processes. Then, 3-D finite-element calculations for symmetrical and asymmetrical holes in tempered glasses were done to calculate temper stresses in the borehole area using Narayanaswamy's structural relaxation model that is implemented in ANSYS [ANSYS 1999, DUFFRENE 1997, GY 1994, NARAYANASWAMY 1971, NARAYANASWAMY 1978, SCHERER 1986, SOULES 1987]. In a parameter study for different apparent heat transfer coefficients, the amount of temper stresses in the "infinite" and in the borehole area was then re-calculated with the FE-models. Finally, tempered samples and identical annealed test samples
(samples before tempering) were tested in a
modified coaxial bending test [DIN EN 1288,
SCHNEIDER 2001] to determine the bending
strength at the edges or chamfers of the holes.

## II. HOLE GEOMETRY AND TEMPER STRESS DISTRIBUTION IN THE HOLE AREA

For architectural glass, usually two types of
holes are used: cylindrical and conical holes. The
diameter of the holes depends on different types of
glass fittings.

Table 1 shows the geometry of the holes from
the three producers A, B and C, the number of
tested samples and stress rates for the tests (see
chapter III). All samples had a nominal thickness
of 10 mm and were stored in a box outside in the
same conditions for approx. 3 months after
drilling. Figure 1 shows a typical test sample after
the drilling process, figure 2 the breaking pattern
of the glass after the bending test.

Table 2 shows the results of the measured
thickness of all samples from the different test
series. It is interesting to note that the thickness of
all soda-lime-glasses just meet the allowable lower
tolerance given in the standards (10 mm – 0,2 mm
= 9,8 mm). This is important for stress calculations
from bending.

Table 3 shows the measured surface compression
stresses for all tempered samples from producer A
and B (soda-lime-glass) in the “infinite area” in
comparison with numerical results at the edges/
chamfers of the holes. For the numerical
calculations, the apparent heat transfer coefficient
was re-calculated iteratively from the mean value
of the surface stress from photoelastic
measurements in the “infinite area” of the samples.
Apparent means that the effects from heat radiation
were neglected in this study as they are of minor
influence for the given thickness. It resulted in
150 W/m²K for series A and in 200 W/m²K for
series B. In the borehole area, a different heat
transfer coefficient was applied. It was reduced by
50 W/m²K (100 W/m²K and 150 W/m²K,
respectively) considering the results of stress
measurements in the borehole area from [LAUFS
2000]. For both cylindrical and conical holes it was
assumed to be constant in the hole area. This is not

strictly true for conical holes where a division in
three parts (infinite area, cone area, cylindrical
area) could be made. Figure 3 shows typical results
of 3-D tempering simulations for a cylindrical and
a conical hole of test series A. For calculations
with borosilicate glass, the heat transfer coefficient
was assumed to be 500 W/m²K which is the
maximum for air quenching achievable
[SCHERER 1986]. The surface stress of
borosilicate glass could not be measured due to the
different stress optical coefficient of the glass.

<figure><img src="image_93.png" /><figcaption>Figure 1: Typical test sample after drilling process (tempered soda-lime glass)</figcaption></figure>

<figure><img src="image_94.png" /><figcaption>Figure 2: Breaking pattern of test samples after coaxial double ring bending test, left: float glass, right: tempered glass</figcaption></figure>

Results from table 3 and figure 3 show, that the
surface compression stress near the chamfers of the
holes is slightly higher than in the infinite area
(approx. 10% to 15%) but that - for the chosen heat
transfer coefficients - inside the holes in the centre
(half the thickness of the pane) the surface stress is
lower. Also, on the surface, close to the holes, in
an area of about half the thickness of the pane,
compression stresses are slightly lower than in the
infinite area of the pane. Here tangential membrane
stresses at the edges of the holes use some of the
surface stresses to get in equilibrium. Similar
results were obtained in [LAUFS 2000,
SEDLACEK 1999]. Therefore, the glass strength
for bending in the area of the holes should be
approximately equal to the strength of the infinite
plate. For in-pane loading with maximum stresses
in the centre of the holes glass strength should be
lower.

<figure><img src="image_95.png" /><figcaption>Figure 3: Test series A: Principal compression stress at the edge of a hole (glass thickness 10 mm).</figcaption></figure>

Above: cylindrical hole ∅ 46 mm (for symmetry reasons, only one half of the thickness was modelled),

Below: conical hole Ø 57 - 46 mm, [N/m²]
<table><caption>TABLE 1:
TEST SERIES AND GEOMETRY OF HOLES TESTED</caption><thead><tr><th>pr.</th><th>test series / type<br>(F= annealed float glass,<br>E= tempered float<br>glass)</th><th>nominal<br>thickness<br>[mm]</th><th>number<br>of samples</th><th>stress rate<br>[MPa·s<sup>-1</sup>]</th><th>con. (CO) or<br>cyl. (CY)<br>part under<br>tension</th><th>hole<br>diagram<br>[mm]</th><th>hole geometry</th></tr></thead><tbody><tr><td rowspan="4">A</td><td>A.F1 conical hole</td><td rowspan="4">10</td><td>40</td><td rowspan="4">2,0</td><td rowspan="4">CY</td><td rowspan="4">57<br>(con.<br>part)</td><td rowspan="4">□□□</td></tr><tr><td>A.E1 with chamfer,</td><td>39</td></tr><tr><td>A.F2 diamond</td><td>43</td></tr><tr><td>A.E2 drilled,<br>soda-lime-<br>silica-glass</td><td>39</td></tr><tr><td>A.F3 cylindrical<br>hole with<br>chamfer,</td><td rowspan="5">10</td><td>30</td><td rowspan="5">0,2</td><td rowspan="5"></td><td rowspan="5">46</td><td rowspan="5">□□□</td></tr><tr><td>A.E3, chamfer,</td><td>30</td></tr><tr><td>A.F4 diamond</td><td>30</td></tr><tr><td>A.E4 drilled,<br>soda-lime-<br>silica-glass</td><td>30</td></tr><tr><td>A.F5 drilled,<br>soda-lime-<br>silica-glass</td><td>33</td></tr><tr><td>A.E5 drilled,<br>soda-lime-<br>silica-glass</td><td>36</td><td></td><td>20,0</td><td></td><td></td><td></td></tr><tr><td>A.F6 cylindrical<br>hole without<br>chamfer,<br>water-jet<br>drilled, soda-<br>lime-silica-<br>glass</td><td>10</td><td>30</td><td>2,0</td><td></td><td>46</td><td>□□□</td></tr><tr><td rowspan="12">B</td><td>B.F1 conical hole</td><td rowspan="12">10</td><td>35</td><td rowspan="12">0,2</td><td rowspan="12">CY</td><td rowspan="12">32<br>(con.<br>part)</td><td rowspan="12">□□□</td></tr><tr><td>B.E1 with chamfer,</td><td>33</td></tr><tr><td>B.F2 diamond</td><td>33</td></tr><tr><td>B.E2 drilled,<br>soda-lime-<br>silica-glass</td><td>32</td></tr><tr><td>B.F3 diamond</td><td>31</td></tr><tr><td>B.E3 drilled,<br>soda-lime-<br>silica-glass</td><td>30</td></tr><tr><td>B.F4 diamond</td><td>31</td></tr><tr><td>B.E4 drilled,<br>soda-lime-<br>silica-glass</td><td>32</td></tr><tr><td>B.F5 diamond</td><td>32</td></tr><tr><td>B.E5 drilled,<br>soda-lime-<br>silica-glass</td><td>33</td></tr><tr><td>B.F6 diamond</td><td>31</td></tr><tr><td>B.E6 drilled,<br>soda-lime-<br>silica-glass</td><td>31</td></tr><tr><td>C.F1 cylindrical<br>hole with<br>chamfer,</td><td rowspan="5">10</td><td>31</td><td rowspan="5">0,2</td><td rowspan="5"></td><td rowspan="5">40</td><td rowspan="5">□□□</td></tr><tr><td>C.E1 with chamfer,</td><td>29</td></tr><tr><td>C.F2 diamond</td><td>29</td></tr><tr><td>C.E2 diamond</td><td>29</td></tr><tr><td>C.F3 diamond</td><td>29</td></tr><tr><td>C.E3 drilled,<br>borosilicate<br>glass</td><td>29</td><td></td><td>20,0</td><td></td><td></td><td></td></tr></tbody></table>
<table><caption>TABLE 2:
MEASURED THICKNESS OF ALL SAMPLES</caption><thead><tr><th>Test Series</th><th>no. of samples</th><th>maximum value [mm]</th><th>minimum value [mm]</th><th>mean value [mm]</th><th>standard deviation [mm]</th><th>coeff. of variation [-]</th></tr></thead><tbody><tr><td>A</td><td>413</td><td>9,97</td><td>9,78</td><td>9,86</td><td>0,03</td><td>0,004</td></tr><tr><td>B</td><td>384</td><td>10,08</td><td>9,75</td><td>9,93</td><td>0,08</td><td>0,008</td></tr><tr><td>C</td><td>183</td><td>10,25</td><td>9,95</td><td>10,07</td><td>0,06</td><td>0,006</td></tr></tbody></table>

<table><caption>TABLE 3:
SURFACE STRESSES OF TEMPERED GLASSES, MEASURED BY PHOTOELASTIC MEASUREMENTS VS. COMPUTED SURFACE STRESSES AND COMPUTED STRESSES AT THE CHAMFER</caption><thead><tr><th rowspan="2">Test Series</th><th rowspan="2">no. of samples</th><th colspan="3">infinite area</th><th colspan="2">hole area</th></tr><tr><th>measured mean value [MPa]</th><th>measured standard dev. [MPa]</th><th>5%-fractile [MPa]</th><th>computed surface stress [MPa]</th><th>computed stress at edge/ chamfer [MPa]</th></tr></thead><tbody><tr><td>A</td><td>207</td><td>127</td><td>17</td><td>98</td><td>130</td><td>152</td></tr><tr><td>B</td><td>191</td><td>160</td><td>8</td><td>146</td><td>150</td><td>166</td></tr><tr><td>C</td><td>-</td><td>n.a.</td><td>n.a.</td><td>n.a.</td><td>50</td><td>55</td></tr></tbody></table>