In Plane Loaded Glass Panes in Façades, Temperature Loads in Fixed Bonded Glass Panes

(1)

In Plane Loaded Glass Panes in Façades, Temperature Loads

in Fixed Bonded Glass Panes

Citation for published version (APA):

Huveners, E. M. P., Herwijnen, van, F., Soetens, F., & Hofmeyer, H. (2005). In Plane Loaded Glass Panes in

Façades, Temperature Loads in Fixed Bonded Glass Panes. In J. Vitkala (Ed.), Glass Processing Days, the 8th

International Conference on Architectural and Automotive Glas : Tampere, Finland, June 17 - 20, 2005 (pp.

284-286). Tamglass Ltd. Oy.

Document status and date:

Published: 01/01/2005

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284 GLASS PROCESSING DAYS 2005 - www.gpd.fi

New Product Developments and Applications

In Plane Loaded Glass Panes in Façades

Temperature Loads in Fixed Bonded Glass Panes

Ir. Edwin Huveners, Prof. Ir. F. van Herwijnen, Prof. Ir. F. Soetens, Dr. Ir. H. Hofmeyer Eindhoven University of Technology

Department of Architecture, Building and Planning, Group Structural Design Den Dolech 2, Vertigo 9.08, P.O. Box 513, 5600 MB Eindhoven, The Netherlands Keywords

1=Bond line 2=Temperature load 3= Vertical diaphragm 4=Glass structures

Abstract

The author discusses the use of glass panes as transparent stability elements in vertical façade structures subjected to in-plane loads including temperature loads. In the present façade architecture, glass is normally used non-structural. The only mechanical requirement is to resist transversal loads, which determines the thickness of the glass pane. Thus, glass panes are not used to stabilize the structural system. However, glass panes have excellent in-plane stiffness qualities, and providing appropriate connections between glass and the supporting structure, this in-plane stiffness can possibly attribute to the building’s stability. Temperature loads on the glass avoid making a totally stiff connection between glass and structure, whereas this would be required for the stability aspect.

To investigate this problem, this paper presents two conceptual connection designs between glass and structure. For temperature loads, these connections will be studied using the theory of plates. It is shown that a bond line as connection, having a large stiffness directly leads to a uniformly distribution equal to almost the complete temperature load. If the shear stiffness of the bond line is taken into account, a complex stress pattern in the glass pane occurs, even for a geometrical and physical linear approach. Thus, the impact of the temperature loads is signiﬁcant for all glass panes (plate level) and is

independent on façade size and this in contrast with the collaboration of the glass panes and supporting structure to resist laterally mechanical loads (structure level).

Research will continue with experiments to verify the theoretical derived results and with the

development of ﬁnite element models to study the complex stress pattern in the glass pane, dependent on connection type and load case.

Introduction

Glass has always been a popular building material in architectural practise. The major function of glass panes is the transparent part in the building envelope. The present enclosing systems for fixing glass panes to the supporting structure behind are flexible systems for reducing unfavourably additional tensile or compression stresses, which can lead to collapse of the pane. Rubbery materials, such as neoprene, enclose a four-sided supported glass pane in the rebates. The glass pane is flexibly supported along the edges to transfer transversal load only and can move in plane freely. The only in plane load to be supported by blocks in the rebate is its own weight of the pane. So, no additional stresses in the glass pane can occur due to restricted deformation in plane along the edge of the pane.

Code [1] is used for dimensioning glass panes in building practise in the Netherlands. This code can only

be used for linearly supported glass panes in façade and roof structures, which are loaded with uniformly distributed loads only. Furthermore, the prescribed mechanic models are based on geometrically and physically linear theory. Nevertheless, there are some developments to enlarge the ﬁeld of this kind of glass applications.

For other supporting structures of a pane (such as point ﬁxed), glass structures (such as beams, staircases and the like) and for non-uniformly distributed loads (such as concentrated loads) the common sense of the structural engineer is necessary to interpret the current codes and to make a properly mechanic model for reducing the tensile stresses to allowable values. The ﬁnite element method is a convenient tool for special glass applications.

The codes do not use the in plane stiffness of the glass pane. Unfortunately, the glass pane has good properties for transferring in plane loads. To design a building, which is stabilised with the help of glass panes,

connected to roof (diaphragm)

connected to foundation connected to roof (diaphragm)

internal hinges stiff bearing structure

connected to foundation z (+w) x (+u) y (+v) transom mullion (adherent) z (+w) x (+u) y (+v) transom mullion roof F y (+v) stiff bearing structure

(adherent) internal hinges z (+w) x (+u) z p Txy pTzxy xy T pz Txyz p Froof glass panes Figure 1

Kew garden, London, UK.

Figure 2

(3)

285

GLASS PROCESSING DAYS 2005 - www.gpd.fi

is a bridge to far for building practise and there is much of a chance that the local authorities, such as building inspection departments, reject it. But from the other hand, this is not a new application of glass. In former days, some buildings were stabilised with the help of glass, such as green houses in the nineteenth century in UK (ﬁgure 1). The comparatively small glass panes were structurally bonded to slender iron proﬁles with putty, in favour of transparency.

This leads to the investigation of the modern version of in plane loaded glass structures. To stabilise a façade structure (vertical diaphragm) an interaction has to occur between the glass panes and the linear supporting structure behind (see ﬁgure 2). The glass pane is continuously bonded to a beadwork with the help of modern adhesive technology and then this unit is connected to the supporting structure with mechanical fasteners. So, the structure can stabilize itself without annoying braces.

Joint design

Figure 3 shows two joint types that will be investigated. Joint type 1 is a comparatively simple joint, but the position of the bond line is not correct for connecting a glass pane to an adherent. This location is very sensitive for growing tensile stresses and can lead to prematurely failure. The investigation was started with this model for gathering information about the interaction between bond line and glass pane quickly with the help of analytical formulas before making a ﬁnite element model. Some results of this joint will be discussed in this paper. The location of the bond lines in joint type 2 is a more logical one, because it has two bond lines for transferring predominately

shear stresses, it is symmetrical, it suppresses transversal tensile and compression stresses respectively by the beadwork, there is enough distance to the edge.

The joint has to fulﬁl more requirements than mechanical properties only. As stated before, production and construction are also very important design parameters. It is preferable to make the glass unit in a workshop under constant circumstances. The glass pane and beadwork undergoes the inevitable preparations for making a proper bond line. After curing, the beadwork protects the glass pane along the edges during transport and construction.

In the case of safety, the in plane loaded glass panes have to be laminated. As known, the stiffness of laminated glass strongly depends on the load duration and temperature, which leads to complex plate behaviour. This leads to one of the restrictions of the research, namely only single glass will be used.

An important issue for designing a joint are the minimum and maximum measuring deviations of glass pane and beadwork. The glass pane slightly varies in thickness and normally the thickness is lesser than the nominal thickness of the pane. The measuring deviations of the glass pane and the beadwork are important to know, because the thickness of the bond line is strongly inﬂuenced by small measuring deviations for adhesive like epoxy in particular. The adhesive polyurethane (PU) has normally a larger bond line thickness and is able to adjust to measuring deviations.

Load case temperature

Beside transversal loads and in-plane loads due to mechanical load cases,

the load case temperature also causes in plane stresses, if the edges are prevented to move freely. Figure 4 shows a model with joint type 1. The glass pane is four-sided supported and enclosed by a bond line along the edges. The bond line has two kinds of stiffness (see ﬁgure 4), namely stiffness (k₁) and shear stiffness (k₂). The stiffness is determined by (1) and by (2). In which:

k₁ is the normal stiffness in N/mm3 k₂ is the shear stiffness in N/mm3 E_a is the Young’s modulus of the adhesive in N/mm2

G_a is the shear modulus of the adhesive in N/mm2

t_b is the thickness of the bond line in mm

The stiffness k₃ (shear stiffness along the thickness of the pane) is assumed to be zero in this consideration.

The adhesive epoxy has an average Young’s modulus in the order of 10,000 N/mm2_{for room temperature. If the}

epoxy behaves like a solid, the relation between Young’s modulus and shear modulus is known. The relation is given by equation 3.

(3) The Poisson’s ratio (νa) of epoxy is

approximately 0.35 and the shear stiffness is in the order of 3700 N/mm2_.

A bond line of epoxy has a thickness in the order of 0.5 mm. So the stiffness k1 = 20,000 N/mm3_{and the shear stiffness}

k2 = 7400 N/mm3_{respectively.}

The relationship between modulus and thickness of the bond line is clear. A greater modulus and a smaller thickness give a greater stiffness. The adhesive epoxy has a greater value for the stiffness (greater modulus and a very small thickness of the bond line) than for instance polyurethane (smaller modulus and a comparatively large thickness of the bond line). The stiffness

k₁ is based on the linear relation between stress and displacement of the bond line and this is a restriction of its application, because the bond line

1 2 tb t b t t b b w t = b w

bond line bond line

Figure 3

Two different types of investigated bond lines.

1 2 tb t b t t b b w t = b w

bond line bond line

Figure 3 Two different types of investigated bond lines.

The joint has to fulfil more requirements than mechanical properties only. As stated before, production and construction are also very important design parameters. It is preferable to make the glass unit in a workshop under constant circumstances. The glass pane and beadwork undergoes the inevitable preparations for making a proper bond line. After curing, the beadwork protects the glass pane along the edges during transport and construction. In the case of safety, the in plane loaded glass panes have to be laminated. As known, the stiffness of laminated glass strongly depends on the load duration and temperature, which leads to complex plate behaviour. This leads to one of the restrictions of the research, namely only single glass will be used.

An important issue for designing a joint are the minimum and maximum measuring deviations of glass pane and beadwork. The glass pane slightly varies in thickness and normally the thickness is lesser than the nominal thickness of the pane. The measuring deviations of the glass pane and the beadwork are important to know, because the thickness of the bond line is strongly influenced by small measuring deviations for adhesive like epoxy in particular. The adhesive polyurethane (PU) has normally a larger bond line thickness and is able to adjust to measuring deviations.

Load case temperature

Beside transversal loads and in-plane loads due to mechanical load cases, the load case temperature also causes in plane stresses, if the edges are prevented to move freely. Figure 4 shows a model with joint type 1. The glass pane is four-sided supported and enclosed by a bond line along the edges. The bond line has two kinds of stiffness (see figure 4), namely stiffness (k1) and shear stiffness (k2). The stiffness is determined by

b a

t

E

k �

1 (1) and by b a

t

G

k �

2 (2). In which:

k1 is the normal stiffness in N/mm3 k2 is the shear stiffness in N/mm3

Ea is the Young’s modulus of the adhesive in N/mm2 Ga is the shear modulus of the adhesive in N/mm2 tb is the thickness of the bond line in mm

The stiffness k3 (shear stiffness along the thickness of the pane) is assumed to be zero in this consideration.

The adhesive epoxy has an average Young’s modulus in the order of 10,000 N/mm2_{for room} temperature. If the epoxy behaves like a solid, the relation between Young’s modulus and shear modulus is known. The relation is given by equation 3.

)

1 (

2

a a a

E

G

�

(3) 1 2 tb t b t t b b w t = b w

bond line bond line

b a

t

E

k �

1 (1) and by b a

t

G

k �

2 (2). In which:

)

1 (

2

a a a

E

G

�

(3) 1 2 tb t b t t b b w t = b w

bond line bond line

b a

t

E

k �

1 (1) and by b a

t

G

k �

2 (2). In which:

)

1 (

2

a a a

E

G

_�

�

(3) Figure 4

Flexible enclosed glass pane with stiffness k₁ based on joint type 1.

bond line

k

b

w

k

b

l

1 3

k

2

t

b

t

k

h

1 1

k

y (+v) z (+w) x (+u)

T

xy

w

1

k

1

E,

_t

_k

k

1

w

1 1

k

b

t

k

1

(4)

286 GLASS PROCESSING DAYS 2005 - www.gpd.fi

strongly depends on force-deformation course and inﬂuences the stresses in the pane signiﬁcantly. Thus, the compressed bond line shows another force

deformation course than the bond line under tensile. Beside path-dependency, temperature and time are other adhesive properties, which inﬂuence the behaviour of the bond line and also the glass pane.

Inﬂuence of stiffness k

₁

The next derivations are based on glass panes, which are ﬂexibly enclosed by stiffness k1 only. If glass panes can

move freely in plane, the strain ε is the product of the linear thermal expansion coefﬁcient and the uniformly distributed temperature (α·ΔT) for both directions and no shear stresses occur. If the glass pane is completely prevented to deform, the strains are suppressed to zero and this leads to stresses in the pane. The stresses in both directions are the same and are independent of the pane size. No shear stresses occur in this case, because the strains along the edges are completely suppressed to zero and the stresses are thickness independence indeed. The equation for uniformly distributed thermal stresses for plates [2,3,4] is

(4).

Substituting the properties of glass, equation (4) can be rewritten as: (5).

In which

σ(x,y) is the thermal stress in N/mm2

α is the linear thermal expansion coefﬁcient of glass

(9·10-6 mm/mmºC)

ν is the Poisson’s ratio of glass (0.23) E is the Young’s modulus of glass (70,000 N/mm2₎

T0 is the initial temperature of the pane

in ºC

Te is the considered end temperature

in ºC

ΔT is the uniformly change of temperature in ºC

A bond line with stiffness k1 can obtain

a more realistic stress distribution of an enclosed glass pane. The following two equations (6,7) give the uniformly distributed thermal stress in the glass pane as well as in the bond line (serially connected).

The maximum not-prevented deformation can also be calculated: (8) (9) In which:

σx, σy are the normal stress in

direction and y-direction respectively in N/mm2

u, v are the maximum deformation in x-direction and y-direction respectively in mm

h is the height of the pane in mm

w is the width of the pane in mm

For determining the strain in the glass pane and bond line in both directions, the stress has to be divided by the Young’s modulus.

Figure 5 shows the relation between the square size of the glass pane and the stresses in case of a temperature change for 4 values of k₁. The chosen temperature changes are -25ºC and 60ºC. The inﬂuence of the stiffness of the bond line on the glass pane is signiﬁcant. The smaller the stiffness and the glass pane, the smaller are the normal stresses in both directions. If the pane size increases with comparatively low stiffness of the bond line, the normal stresses increase rapidly to almost -9_/

11 ΔT, because the bond line

resists more deformation of the pane. Increasing the stiffness leads to larger stresses in the pane. If the bond line is very stiff, the maximum stress (-9_/

11

ΔT) is reached immediately and is even dimension independent.

The linear relation of stiffness k₁ and the geometrically linear behaviour of the glass pane are not physically correct. If the glass pane is subjected to decreasing temperatures, the maximum occurred tensile stress is the utmost tensile stress reduced by factors such as time and the bond line has an allowable deformation belonging to its geometry and property. If the glass pane is subjected to increasing temperature, the bond line will be suppressed and leads to larger values for stiffness k₁. The compression stresses in the glass pane are large

and in combination with its geometry (large size and small thickness) the pane will buckle. Small glass panes, which were applied in green houses, can better resist increasing of temperature, because the glass pane is completely under compression and stiff enough to resist buckling.

Inﬂuence of shear stiffness k

₂ In this consideration, the bond line has a linear stiffness along the edges, defined as k₂. This is the so-called shear stiffness of the bond line. It introduces stresses in the glass pane along the edges, which influences the entire plate. The stiffer the bond line, the larger is the shear stress especially at the corners, because the glass pane deforms from the centre of the plate outwards and the prevented deformations (distortion) is the largest at the corners. Stiffness k₂ also influence the stress distribution in x-direction as well as in y-direction. The centre of the edge is more prevented to deform by stiffness k₂ and leads to great normal stresses along and decrease to the corners. This leads to a complex stress distribution in the plate. The stress criterion for glass panes is the principle stress. The maximum as well as the minimum principle stress must be determined, because the maximum principle stress leads to failure of the utmost tensile stress and the minimum principle stress can lead to buckle of the glass pane.

Summary

A bond line with a great stiffness k₁ (such as epoxy) directly leads to a uniformly distributed stress of approximately -9_/

11 ΔT and is

independent from the size of the pane. A bond line with a small stiffness k₁ gives a lower uniformly distributed stress in the pane, but if the size of the pane increases, the prevented deformation also increases and leads to a stress of approximately -9_/

11 ΔT and is size

dependent.

References

[1] NEN 2608-2, 3rd draft, Glass in building – Glazed installations non-vertical installed – resistance against wind load, snow load, and self-weight – requirements and determination method.

[2] Szilard, Theories and Applications of Plate Analysis, Classical, Numerical and Engineering Methods

[3] S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, 2nd edition, McGraw-Hill, 1959

[4] S. Timoshenko and J.N. Goodier, Theory of Elasticity, 2nd edition, McGraw-Hill, 1951 The Poisson’s ratio (�a) of epoxy is approximately 0.35 and the shear stiffness is in the order

of 3700 N/mm2_{. A bond line of epoxy has a thickness in the order of 0.5 mm. So the stiffness}

k1 = 20,000 N/mm3 and the shear stiffness k2 = 7400 N/mm3 respectively.

bond line k b w k b l 1 3 k2 t b t k k h 1 1 k y (+v) z (+w) x (+u) Txy w 1 k1 E, _t _k k1 w 1 1 k b t k1

Figure 4 Flexible enclosed glass pane with stiffness k1 based on joint type 1.

The relationship between modulus and thickness of the bond line is clear. A greater modulus and a smaller thickness give a greater stiffness. The adhesive epoxy has a greater value for the stiffness (greater modulus and a very small thickness of the bond line) than for instance polyurethane (smaller modulus and a comparatively large thickness of the bond line). The stiffness k1 is based on the linear relation between stress and displacement of the bond line

and this is a restriction of its application, because the bond line strongly depends on force-deformation course and influences the stresses in the pane significantly. Thus, the compressed bond line shows another force deformation course than the bond line under tensile. Beside path-dependency, temperature and time are other adhesive properties, which influence the behaviour of the bond line and also the glass pane.

Influence of stiffness k1

The next derivations are based on glass panes, which are flexibly enclosed by stiffness k1

only. If glass panes can move freely in plane, the strain � is the product of the linear thermal expansion coefficient and the uniformly distributed temperature (�·�T) for both directions and no shear stresses occur. If the glass pane is completely prevented to deform, the strains are suppressed to zero and this leads to stresses in the pane. The stresses in both directions are the same and are independent of the pane size. No shear stresses occur in this case, because the strains along the edges are completely suppressed to zero and the stresses are thickness independence indeed. The equation for uniformly distributed thermal stresses for plates [2,3,4] is ) 1 ( ) 1 ( ) ( 0 ) , ( _� � � � � � � � � � � � � ETe T E T y x (4).

Substituting the properties of glass, equation (4) can be rewritten as: �(x,y)��119�T (5).

In which

�(x,y) is the thermal stress in N/mm2

� is the linear thermal expansion coefficient of glass (9·10-6_mm/mmºC)

� is the Poisson’s ratio of glass (0.23)

E is the Young’s modulus of glass (70,000 N/mm2₎

T0 is the initial temperature of the pane in ºC

Te is the considered end temperature in ºC

�T is the uniformly change of temperature in ºC

A bond line with stiffness k1 can obtain a more realistic stress distribution of an enclosed

glass pane. The following two equations (6,7) give the uniformly distributed thermal stress in the glass pane as well as in the bond line (serially connected).

2 1 2 2 1 1 1 2 ₂ ₂1 1 1 4 ) 2 ( whk wh k w Ek h Ek E h k h k E TEwk x _� � � � � � � � � � � � � (6)

The Poisson’s ratio (�a) of epoxy is approximately 0.35 and the shear stiffness is in the order of 3700 N/mm2_{. A bond line of epoxy has a thickness in the order of 0.5 mm. So the stiffness} k1 = 20,000 N/mm3 and the shear stiffness k2 = 7400 N/mm3 respectively.

The next derivations are based on glass panes, which are flexibly enclosed by stiffness k1 only. If glass panes can move freely in plane, the strain � is the product of the linear thermal expansion coefficient and the uniformly distributed temperature (�·�T) for both directions and no shear stresses occur. If the glass pane is completely prevented to deform, the strains are suppressed to zero and this leads to stresses in the pane. The stresses in both directions are the same and are independent of the pane size. No shear stresses occur in this case, because the strains along the edges are completely suppressed to zero and the stresses are thickness independence indeed. The equation for uniformly distributed thermal stresses for plates [2,3,4] is

)

1 (

)

1 (

)

(

0 ) , (

_�

�

E

T

e

T

E

T

y x (4).

Substituting the properties of glass, equation (4) can be rewritten as:

�

(x,y)

�

119

�

T

(5).

In which

� is the linear thermal expansion coefficient of glass (9·10-6_mm/mmºC) � is the Poisson’s ratio of glass (0.23)

E is the Young’s modulus of glass (70,000 N/mm2₎ T0 is the initial temperature of the pane in ºC Te is the considered end temperature in ºC �T is the uniformly change of temperature in ºC

A bond line with stiffness k1 can obtain a more realistic stress distribution of an enclosed glass pane. The following two equations (6,7) give the uniformly distributed thermal stress in the glass pane as well as in the bond line (serially connected).

2 1 2 2 1 1 1 2 1 1 1

2

4 )

2 (

whk

wh

k

w

Ek

h

Ek

E

h

k

h

k

E

TEwk

x

�

_�

�

(6)

The Poisson’s ratio (�a) of epoxy is approximately 0.35 and the shear stiffness is in the order of 3700 N/mm2_{. A bond line of epoxy has a thickness in the order of 0.5 mm. So the stiffness} k1 = 20,000 N/mm3 and the shear stiffness k2 = 7400 N/mm3 respectively.

The next derivations are based on glass panes, which are flexibly enclosed by stiffness k1 only. If glass panes can move freely in plane, the strain � is the product of the linear thermal expansion coefficient and the uniformly distributed temperature (�·�T) for both directions and no shear stresses occur. If the glass pane is completely prevented to deform, the strains are suppressed to zero and this leads to stresses in the pane. The stresses in both directions are the same and are independent of the pane size. No shear stresses occur in this case, because the strains along the edges are completely suppressed to zero and the stresses are thickness independence indeed. The equation for uniformly distributed thermal stresses for plates [2,3,4] is

)

1 (

)

1 (

)

(

0 ) , (

�

_�

�

_�

�

E

T

e

T

E

T

y x (4).

Substituting the properties of glass, equation (4) can be rewritten as:

�

(x,y)

�

119

�

T

(5).

In which

� is the linear thermal expansion coefficient of glass (9·10-6_mm/mmºC) � is the Poisson’s ratio of glass (0.23)

E is the Young’s modulus of glass (70,000 N/mm2₎ T0 is the initial temperature of the pane in ºC Te is the considered end temperature in ºC �T is the uniformly change of temperature in ºC

A bond line with stiffness k1 can obtain a more realistic stress distribution of an enclosed glass pane. The following two equations (6,7) give the uniformly distributed thermal stress in the glass pane as well as in the bond line (serially connected).

2 1 2 2 1 1 1 2

₂

1 1 1

4 )

2 (

whk

wh

k

w

Ek

h

Ek

E

h

k

h

k

E

TEwk

x

�

_�

�

(6) (6) (7)

�

wh

k

wh

k

w

Ek

h

Ek

E

w

k

w

k

E

h

TEk

y 2 1 2 2 1 1 1 2 1 1 1

2

4

2 �

�

(7)

The maximum not-prevented deformation can also be calculated:

1

k

u

_�

�

x ₍₈₎ 1

k

v

_�

�

y (9) In which:

�x, �y are the normal stress in x-direction and y-direction respectively in N/mm2 u, v are the maximum deformation in x-direction and y-direction respectively in mm h is the height of the pane in mm

Square pane -60 -40 -20 0 20 40 0 0.5 1 1.5 2 2.5 3 Pane size in m St re ss in N /m m 2 k1 = 100 N/mm³ �T=-25ºC k1 = 300 N/mm³ �T=-25ºC k1 = 1000 N/mm³ �T=-25ºC k1 = 20000 N/mm³ �T=-25ºC k1 = 100 N/mm³ �T= 60ºC k1 = 300 N/mm³ �T= 60ºC k1 = 1000 N/mm³ �T= 60ºC k1 = 20000 N/mm³ �T= 60ºC

Figure 5 Compression and tensile stresses in the pane and bond line by increasing and

decreasing temperature respectively under several kinds of stiffness of the bond line for a square pane.

Figure 5 shows the relation between the square size of the glass pane and the stresses in case of a temperature change for 4 values of k1. The chosen temperature changes are -25ºC and 60ºC. The influence of the stiffness of the bond line on the glass pane is significant. The smaller the stiffness and the glass pane, the smaller are the normal stresses in both directions. If the pane size increases with comparatively low stiffness of the bond line, the normal stresses increase rapidly to almost -9_/

11�T, because the bond line resists more deformation of the pane. Increasing the stiffness leads to larger stresses in the pane. If the bond line is very stiff, the maximum stress (-9_/

11�T) is reached immediately and is even dimension independent.

The linear relation of stiffness k1 and the geometrically linear behaviour of the glass pane are not physically correct. If the glass pane is subjected to decreasing temperatures, the maximum occurred tensile stress is the utmost tensile stress reduced by factors such as time and the bond line has an allowable deformation belonging to its geometry and property. If the glass pane is subjected to increasing temperature, the bond line will be suppressed and leads to larger values for stiffness k1. The compression stresses in the glass pane are large and in combination with its geometry (large size and small thickness) the pane will buckle. Small glass panes, which were applied in green houses, can better resist increasing of temperature, because the glass pane is completely under compression and stiff enough to resist buckling. Influence of shear stiffness k2

In this consideration, the bond line has a linear stiffness along the edges, defined as k2. This

is the so-called shear stiffness of the bond line. It introduces stresses in the glass pane along

�

wh

k

wh

k

w

Ek

h

Ek

E

w

k

w

k

E

h

TEk

y 2 1 2 2 1 1 1 2

₂

1

₂

1 1

4

2 �

�

(7)

1

k

u

_�

�

x₍₈₎ 1

k

v

_�

�

y (9) In which:

is the so-called shear stiffness of the bond line. It introduces stresses in the glass pane along

�

wh

k

wh

k

w

Ek

h

Ek

E

w

k

w

k

E

h

TEk

y 2 1 2 2 1 1 1 2 1 1 1

2

4

2 �

�

(7)

1

k

u

_�

�

x₍₈₎ 1

k

v

_�

�

y (9) In which:

is the so-called shear stiffness of the bond line. It introduces stresses in the glass pane along Figure 5

Compression and tensile stresses in the pane and bond line by increasing and decreasing tempera-ture respectively under several kinds of stiffness of the bond line for a square pane. Square pane -60 -50 -40 -30 -20 -10 0 10 20 30 0 0,5 1 1,5 2 2,5 3 Pane size in m St re ss in N /m m 2 k1 = 100 N/mm³ �T=-25ºC k1 = 300 N/mm³ �T=-25ºC k1 = 1000 N/mm³ �T=-25ºC k1 = 20000 N/mm³ �T=-25ºC k1 = 100 N/mm³ �T= 60ºC k1 = 300 N/mm³ �T= 60ºC k1 = 1000 N/mm³ �T= 60ºC k1 = 20000 N/mm³ �T= 60ºC