BUCKLING DESIGN OF GLASS ELEMENTS UNDER COMPRESSION Andreas Luible Research Engineer Swiss Federal Institute of Technology (EPFL) Lausanne, Switzerland Born 1971, received his civil engineering degree in 1999 from the Technische Universität München and his PhD in 2004 from EPFL Lausanne.

Michel Crisinel Section manager and lecturer Swiss Federal Institute of Technology (EPFL) Lausanne, Switzerland Born 1945, received his civil engineering degree in 1968 from EPF Lausanne. He has worked for a consulting firm prior to his joining the Steel Structures Lab ICOM.

Summary This contribution describes investigations on the buckling strength of glass elements under compression (buckling tests, analytic and numeric modelling). Based on these investigations, design methods for single layer and laminated glass elements under compressive load are proposed. The load carrying behaviour of glass elements under compression depends mainly on parameters such as initial out-of-straightness, glass thickness and shear stiffness of the PVB foil (also depending strongly on temperature and load duration). Models were developed to simulate the buckling behaviour of columns. A parametric study conducted using these models showed the influence of the various parameters. Failure occurs when the maximum tensile stress due to loading exceeds the “frozen-in” compressive surface stress plus the tensile strength of the annealed float glass. The visco-elastic load carrying behaviour of laminated glass was modelled using an elastic sandwich section and reduced to a monolithic cross-section with an effective thickness. When comparing to design curves for steel structures, the slenderness ratio for glass structures has to be based on the maximum tensile strength. The proposed design rules assume that the maximum compressive strength of single layer glass elements as well as laminated glass elements can be determined either with column curves based on a geometric slenderness or by means of direct second order stress analysis.

Keywords: Stability, column buckling, second order analysis.

1. Introduction The increasing demand in modern architecture for more slender and lighter structures requires the use of new construction materials. Glass, a material that has been used for a long time in windows as a filling material, has much to offer in this regard due to its very high compressive strength and transparency. For this reason, there is a growing trend to extend the use of glass to load carrying elements. Due to their high slenderness and high compressive strength, such elements tend to fail because of instability (i.e. column buckling). At the moment little knowledge exists about the load carrying behaviour of glass structural elements, and existing design methods for other materials (i.e. steel) cannot be directly transferred to glass panels, because influences of the following aspects must be investigated in a different way for glass: • production tolerances (i.e. glass thickness) • initial deformations, • the visco-elastic Poly-Vinyl-Butyral foil interlayer (PVB) used for laminated safety glass, • the ideal elastic material behavior without plastic deformability or strain hardening effect as it

is the case for steel, and

• the ultimate breaking stress in glass, which is not a material property but depends on the embedded compressive surface stress due to the tempering process, the degree of damage of the glass surface and the load duration.

The main objective of the research work being conducted is to develop a design method for stability-critical load carrying glass elements which may fail due to lack of stability, i.e. column buckling, plate buckling and lateral torsional buckling [1]. In this paper only the results for column buckling are presented.

2. Column Buckling of Glass Elements Loss of stability (bifurcation buckling) means the instantaneous failure of a structure after the critical load Ncr,K (Fig 1) is exceeded. In reality the critical load can never be obtained because of the out-of-straightness of the bar and/or the eccentricity e of the applied load. If the load N increases gradually, the additional lateral deformations increase disproportionately to the load as the critical load is approached (imperfect column buckling). The maximum load NK is the point where the maximum stresses in the material due to the lateral deformations are reached. For the design of structural elements under compression, the fundamental study of the difference between the critical load Ncr,K and the maximum load NK is necessary.

N

N

e

w0wKL

w

Ncr,K

perfect bar

imperfect bar withinitial deformation w0

w0

NK

N

w

Ncr,K

perfect bar

imperfect bar withinitial deformation w0

w0

NK

N

Fig 1 Eccentrically loaded bar with initial deformation w0.

3. Column Buckling Models for Glass

3.1 Single layered Glass The load carrying behaviour of single layered glass can be described using the second order differential equation [2] for a bar with a length LK under axial compression N with pinned ends, an initial sinusoidal deformation w0 and load N applied with an eccentricity e

2

02

( ) sin ( ) 0K

d w x xEI N w e w xdx L

π⎡ ⎤+ + +⎢

⎣ ⎦=⎥ (1)

The solution for the elastic critical buckling load is given by 2

, 2cr KK

EINLπ

= (2)

with: 12

3tbI = (3)

and the geometrical slenderness is defined as

crcrK

ENEA

σππλ == (4)

The maximum deformation is given by

( )0

1 /cos / 2 / crK cr

wewN NL N N

= +−

(5)

and the maximum surface stress can be determined as

0

,1cos 2 cr KK

wN N eA W N NL N EI

σ⎡ ⎤

= ± +⎢−⎢ ⎥⎣ ⎦

⎥ (6)

Thus for a given maximum tensile or compressive stress σ the maximum load N can be determined.

3.2 Laminated Glass The PVB interlayer in laminated glass behaves like a shear connection between the glass layers. The load carrying behaviour can be described using elastic “sandwich” theory [3] or by means of a finite element model [4]. The latter has the advantage that the PVB interlayer can either be represented by elastic or visco-elastic elements [5]. Also different boundary conditions (e.g. end restraints) can be easily adapted. The numerical model is presented in [1]. The critical buckling load of a two layer “sandwich” (Fig 2) with a width b is given as [6]

2 2

, 2

(1 )1

Scr K

K2

EINL

π α π αβπ β

+ +=

+ (7)

For a laminated safety glass with two glass layers

1 2

S

I II

α += (8)

21 2( )

=+

PVB S

PVB K

tG b z z L

β 2

EI (9)

3

12i

ibtI = (10)

21 1 2 2(SI b t z t z= + 2 ) (11)

t1

t2

tPVB

z1

z2

glass

glass

PVB t1

t2

tPVB

z1

z2

glass

glass

PVB

Fig 2 Laminated safety glass with two glass layers.

4. Experimental Investigation

4.1 Test setup The test arrangement represents the ideal column buckling case with two pinned column ends. To achieve this, two steel supports were fabricated, with ball bearings to minimize friction (Fig 3). A total of 80 displacement- and force-controlled tests were carried out. The displacement speed and loading speed of the machine, as well as the temperature, were varied to study the influence of the PVB-foil on buckling behaviour. The test specimens consisted of single layered or laminated safety glass elements with thicknesses t of 8 mm, 10 mm, 8/1.52/8 mm and 10/1.52/10 mm that had been fully toughened and heat-strengthened. The specimen width b was 200 mm and the length LK varied from 350 mm to 1600 mm.

Fig 3 Column buckling test.

4.2 Test Results

4.2.1 Single Layered Glass The second order differential equation solution showed good agreement with the performance of single layered glass in the tests. The load carrying behaviour was independent of the embedded compressive surface stresses but fully toughened glass showed higher deformations and stresses at breakage and a finer breakage pattern, as expected.

0

3

6

9

12

15

18

0 5 10 15 20 25 30w [mm]

N [kN]

test

model (Eq.(5))

N cr,K = 14.9 [kN]

L K = 600mm, t = 8mm

a)

breakage

N

w

L K

N

w

L K

0

2

4

6

8

10

0 6 12 18 24 30 36w [mm]

N [kN]

testmodel (Eq.(5))

b)

N cr,K = 7.5 [kN] breakage

L K = 1200mm, t = 10mm

Fig 4 Buckling test with single layered glass: a) LK = 600 mm, t = 8 mm, b) LK = 1200 mm, t = 10 mm. The initial breakage Fig 5 always occurred on the glass surface under tensile stress and in most cases within a distance of about two to three times the glass thickness from the glass edge. In this region close to the edge the residual stresses are lower than in the centre of the glass surface [7].

L

200 2-

3 t

L /3K

K

initial breakage

Fig 5 Initial breakage spots of 19 column buckling tests on single layered glass.

4.2.2 Laminated Safety Glass In the buckling tests a remarkable composite action, depending on the loading rate and the temperature, due to the PVB-interlayer between the glass sheets could be observed. The results of finite element models with linear visco-elastic elements for the PVB-interlayer showed good agreement with the column buckling test results (Fig 6).

0

10

20

30

40

50

60

0 5 10 15 20 25 30 35w [mm]

N [kN]

test 1 (T=18.0°C, 30mm/h)simulation test 1test 2 (T=16.3°C, 3mm/h)simulation test 2

glass breakage

a) b)

0

40

80

120

160

0 1 2 3 4 5w [mm]

N [kN]

6

test (T=19.3°C, 20 kN/min)

simulation

glass breakage

Fig 6 Buckling test of laminated safety glass: a) L = 800 mm, t = 8/1.52/8 mm (displacement controlled), b) L = 600 mm, t = 10/1.52/10 mm (force controlled).

4.3 Load carrying behaviour and buckling strength

4.3.1 Single Layered Glass The study of the load carrying behaviour showed that the glass thickness t, the initial deformation w0 and the load eccentricity e have the most important influence on the maximum load. Due to the high compressive strength of glass (> 600 N/mm2) the initial breakage of a glass element under compression, with the dimensions applied in building construction (LK > 300 mm, t < 19 mm, initially deformed), always occurs on the surface under tensile stress. The buckling strength of glass

is therefore limited by the maximum tensile strength of the glass surface [1]. The initial breakage can occur on the whole glass surface and depends on the “frozen-in” compressive surface stress in addition to the tensile strength of the annealed float glass as mentioned above. But the study also demonstrated that the weakest point of the glass surface (minimum embedded compressive surface stress and/or most severe surface damage) can simplistically be assumed to be at the point of the highest tensile stress. This assumption underestimates the real buckling strength only by 2.8 % for an initial deformation of w0 = LK/300. [1].

4.3.2 Laminated Safety Glass Simulations with the visco-elastic numerical model under different temperatures and loading speeds demonstrate the effect of the PVB-interlayer on the load carrying capacity of laminated safety glass columns. In practice, the visco-elastic modelling of the PVB is complicated and may be simplified by an elastic model instead [8]. Parametric studies showed that the influence of the shear transfer by the PVB depends not only on the temperature and the load duration but also on the thickness of the PVB foil and the buckling length. For low temperatures and very short loading the maximum load can reach almost the maximum load of a monolithic section. But an improvement of the maximum load, comparing to a similar glass without PVB-interlayer, is marginal for long-term loading and temperatures higher than 25°C. From an economical and safety point of view, a shear connection might therefore only be taken into account for short-term loading like wind or impact. Fig 7 shows the improvement of the buckling load of a laminated safety glass by comparing the critical load Ncr with an elastic shear connection interlayer to the critical load of two independent glass layers (Ncr,without PVB). The buckling strength depends not only on the shear modulus G of the PVB but also on the glass geometry (length, thickness of glass and PVB). That means, a reduction of the buckling length of a laminated glass by means of intermediate supports leads to a smaller buckling strength comparing to the same reduction with a monolithic section.

a)

1

2

3

4

5

6

0.010.11101001000G PVB [N/mm2]

N cr,K /N cr,K(without PVB)

L K = 600mm, VSG 10/1.52L K = 1200mm, VSG 10/0.76

L K = 1200mm, VSG 10/1.52L K = 1200mm, VSG 8/1.52L K = 1600mm, VSG 10/1.52

two glass layers

b)

1

3

5

7

9

11

13

0.010.11101001000G PVB [N/mm2]

N cr,K /N cr,K(without PVB)

L K = 600mm, VSG 10/1.52L K = 1200mm, VSG 10/0.76

L K = 1200mm, VSG 10/1.52L K = 1200mm, VSG 8/1.52L K = 1600mm, VSG 10/1.52

three glass layers

Fig 7 Influence of the shear modulus GPVB on the critical buckling load of a laminated glass with: a) two identical glass layers, b) three identical glass layers, (tPVB = 1.52 mm).

5. Design of load carrying glass elements in compression

5.1 Single Layer Glass To simplify the design of compressive members (i.e. steel columns) column curves were developed

and are commonly used. The same approach can be applied to compressive glass elements. In steel construction, column curves are based on a slenderness ratio. This facilitates the design of members with different steel grades using the same curve. However in contrast to steel, the slenderness ratio for glass must be based on the maximum tensile strength, as the compressive strength is not limiting the buckling strength. A first reasonable approach for glass column buckling curves seemed to be the development of one single curve for all glass strengths. But simulations of column curves which are based on a slenderness ratio showed a big difference for different tensile strengths. Therefore it is not practical - in contrast to steel - to establish a column curve for glass using this approach. However column curves might still be determined based on the geometric slenderness (Eq.(4)). This results in a family of curves for different tensile strengths. Another approach for design might be to directly calculate the maximum tensile stress in the compressed glass member by means of elastic second order equations (eg. Eq.(6)). In contrast to steel construction this is relatively simple to carry out because of the ideal elastic behaviour of glass. The calculated maximum tensile stress has to be smaller than the tensile surface strength of the glass. The calculation has to be carried out with a reduced glass thickness (e.g. 97.61% of the nominal glass thickness) and a reasonable assumption for the initial deformation. Measurements on 200 test specimen and the statistical evaluation gave a value of w0 = LK/400 for the 5% percentile.

5.2 Laminated Safety Glass For the design of a compressed laminated safety glass element, the visco-elastic behaviour might be simplified by the above mentioned elastic approach. The same methods as described for single layered glass can be applied also to laminated safety glass elements. A comparative study [1] confirmed that the column buckling curves of single layered glass can also be used for laminated safety glass. The second approach, a maximum stress calculation either with a numerical model or an analytical sandwich model that takes into account second order effects, may be carried out for laminated glass as well. For further simplification, the sandwich cross section can be replaced by an effective monolithic cross section with the effective thickness teff given by

2

32

12 (1 )(1 )

seff

Itb

α π α βπ β

+ +=

+ (12)

a)

10

12

14

16

18

20

22

500 1000 1500 2000L K [mm]

VSG 10/1.52/10mmt eff [mm]

0.1 N/mm2

0.5 N/mm2

2 N/mm2

1 N/mm2

5 N/mm2

b)

10

12

14

16

18

500 1000 1500 2000L K [mm]

VSG 8/1.52/8mmt eff [mm]

0.1 N/mm2

1 N/mm2

0.5 N/mm2

2 N/mm2

5 N/mm2

Fig 8 Effective thickness for laminated glass with two glass layers: a) 10/1.52/10 mm, b) 8/1.52/8 mm.

This expression is derived from Eq. (7). For a given Load N the maximum tensile stress in the compressed laminated glass member can be calculated by means of elastic second order equations (eg. Eq.(6)). A reduced glass thickness and a reasonable assumption of the initial deformation have to be considered (§ 5.1). Fig 8 shows the effective thickness teff for two different laminated glass compositions as a function of the shear modulus (G = 0.1 … 5 N/mm2) and the buckling length LK.

6. Conclusions The column buckling strength of a compressed glass element depends mainly on the initial deformation, the glass thickness and the shear stiffness of the PVB foil interlayer. Failure occurs when the maximum tensile stress due to loading exceeds the embedded compressive surface stress plus the tensile strength of the annealed float glass. The load carrying behavior of laminated safety glass in compression depends strongly on the temperature and the load duration due to the shear connection with the visco-elastic PVB interlayer. Investigations showed that this shear connection, which can be simplified on the safe side with an elastic approach, might only be taken into account for short-term loading like wind and impact. The design of single layered glass and laminated safety glass might be carried out either with column buckling curves based on geometric slenderness or by means of direct second order stress analysis. The latter seems to be the more convenient approach especially when additional bending moments are applied. The dispersion of the glass thickness from the nominal value as well as the initial deformation of the glass members might be taken into account in design by the values given in the paper. For further simplification the cross section of a laminated safety glass structural element can be modeled as a monolithic cross section with an effective thickness.

7. Acknowledgement The work presented in this paper was primarily conducted with the support of the Swiss National Science Foundation (SNF) and the industrie partners Glas Trösch (Bützberg, Switzerland) and Verre Industriels Moutiers - VIM (Moutier, Switzerland).

8. References [1] LUIBLE, A., “Stabilität von Tragelementen aus Glas”, Thèse EPFL 3014, Ecole

polytechnique fédérale de Lausanne (EPFL), Lausanne, 2004. [2] HIRT, M. A., BEZ, R., Stahlbau: Grundbegriffe und Bemessungsverfahren, Ernst & Sohn,

Berlin, 1998. [3] ZENKERT, D., The Handbook of Sandwich Construction, Engineering Materials Advisory

Service Ltd., United Kingdom, 1997. [4] ANSYS, Release 6.1, SAS IP, Inc, 2002. [5] VAN DUSER, A., JAGOTA, A., BENNISON, S., J., Analysis of Glass/Polyvinyl Butyral

Laminates Subjected to Uniform Pressure, Journal of engineering mechanics, Vol. 125, pp. 435-442, 1999.

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

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

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