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Structural Use of Glass Beams Ruth Kasper, Gerhard SedlacekInstitute of Steel Construction of the University of Aachen (RWTH Aachen),

Keywords

1 = Beam 2 = PVB 3 = Test 4 = Sandwich 5 = Laminated safety glass 6 = Temperature

Abstract

Using glass elements as structural components opens new possibilities for architects and engineers to design transparent structures. By this the use of glass is not restricted to the application as an infi ll panel only, but extended to primary load bearing members as beams or columns.

Due to the brittle behaviour of glass robustness and damage tolerance of the structure is a requirement that can only be achieved by laminated safety glass.

Within the frame of a research project the Institute of Steel Construction of the RWTH Aachen is performing tests and numerical studies to determine the serviceability and ultimate behaviour of laminated glass beams. The study includes the determination of the resistance of the cross-section of the beams and their lateral buckling resistance taking account of the time- and temperature dependent behaviour of the PVB-foil. The results will be used as a basis for a design concept for glass components made of laminated safety glass.

Introduction

Examples for the application of glass beams are shown in fi gure 1 and 2. The glass beams are made of laminated glass which includes fl oat glass and PVB-foils.

Due to their fl at cross-section glass beams have a high slenderness for lateral torsional buckling. The collapse load is infl uenced by the imperfection of the glass beam, but also by the stiffness of the sandwich (fi gure 3) represented by the laminated safety glass. The sandwich consists of two materials glass and PVB glass is an elastic material (E ~ 70 000 MPa) and PVB is a visco-elastic material with time and temperature dependent behaviour. The shear modulus GPVB varies between 0.1 to 1000 MPa.

For such structural components so far no design rules or design standards are available. Though some results for the resistance of the edge under tensile stresses have been published [1, 2 and 3], but no investigations are known concerning the resistance to lateral torsional buckling. This project deals with investigations of the lateral torsional buckling behaviour with tests and numerical studies to draw conclusions for a design concept.

Approach

A common approach for a lateral torsional buckling check for a glass beam would be a calculation based on second order theory. Solutions for such checks are given for a monolithic section. These solutions could be used for laminated glass if the second moment of area and the torsional stiffness could be replaced by an equivalent second moment of area and an equivalent torsional stiffness for the sandwich depending on the stiffness of the PVB-foil. These values have been determined by using the Extended bending and torsion theory [4,5 and 6]. From the stress resultants determined by second order theory as

for a monolithic section the stresses in the glass sandwich can than be calculated depending on the stiffness of the PVB-foil.

Below the following steps of the solutions are presented:1) solution for lateral torsional buckling

based on the second order theory of glass beam consisting of a monolithic section,

2) solution based on the Extended bending and torsion theory to determine the equivalent second moment of area and the equivalent torsional stiffness of the section of the sandwich depending on the stiffness of the PVB-foil [5 and 6] to be used for 1) to obtain stress resultants,

3) solution based on the Extended bending and torsion theory to

Fig 1

Arab Urban Develop-ment Institute, Riad, Saudi Arabia [10]

Fig 2

Museum of city history, Luxembourg [10]

Fig 3

Lateral torsional buckling of a glass beam con-sisting of laminated safety glass loaded by a constant moment

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determine the stresses in the glass sandwich [5 and 6] from the stress resultants,

4) tests with beams consisting of monolithic glass and laminated safety glass to analyse the imperfections of a structure member [7],

5) small scale tests to analyse the torsional behaviour of the laminated safety glass with variation of the temperature and the loading time [8].This paper describes the solution for

the loading of by constant moment My only. In future the results will be further developed to account for lateral and transverse loading.

Global static behaviour of the monolithic glass beam - solution based on second order theory

The following solutions are based on the differential equations for lateral torsional buckling [9]:

(1) (2) (3)

where the notations are given in fi gure 4.

Fig 4

Notations

The equations can be simplifi ed for the case of loading by endmoments My only ( and ).

By assuming sinusoidal deformations of the beam (4)

and also initial geometrical imperfections

(5)

the inclination is equal to:

(6)

This equation can be written as:

(7)

where

(8)

and (9)

is the elastic critical moment for a monolithic section.

The stress resultants due to second order effects (fi gure 5) can be written as:

(10).

(11)

where is the solution given in equation (6).

Fig 5

Stress resultants based on second order theory

Equivalent stiffness of laminated glass for bending and torsion

To use the above equations for the beam made of laminated glass, formulas for the second moment of area and torsional stiffness are needed. For a monolithic section the formulas are known:

(12)

(13)

By using the Extended bending and torsion theory [4, 5 and 6] an equivalent stiffness for bending and torsion of the sandwich can be determined depending on the shear modulus G of the PVB-foil. To this end the basic degree of freedoms for monolithic sections are augmented by additional degrees of freedom due to the sandwich characteristics. Figure 6 shows the degree of freedoms of a monolithic section and the additional degree of freedoms of the sandwich for bending and torsion which have been used to solve the equations.

Fig 6

Degrees of freedom of a sandwich for bending and torsion

The equivalent bending stiffness (second moment of area) is depending on the type of loading and the length of the beam. The equivalent torsional stiffness is not depending on the type of loading and the length of the beam. The analytic formulas give the results in fi gure 7. Figure 7 shows the elastic critical moment depending on the shear modulus G of the PVB-foil both determined with the analytic method and with fi nite-element calculations. The results show that the global behaviour of a glass beam consisting of a laminated glass can be modelled with an equivalent stiffness for bending und torsion using the formulas for a monolithic section.

Fig 7

Comparison of elastic critical moments from an analytical solution and from fi nite-element-cal-culations (the curves are almost identical)

Stress distribution in the sandwich

For the design of the section the tensile stress at the edges must be determined. The stresses due to My

II are equal to:

with

(14)

The method to determine the tensile stress due Mz

II to is also given in [2

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and 5] by the Extended bending and torsion theory. Figure 8 shows the stress distribution in the cross section of the sandwich caused by My and Mz

II . The relevant stress is the sum of and .

Fig 8

Stress distribution in the section of the sandwich

Experimental und numerical investigations

To confi rm the analytical results experimental investigations have been carried out for lateral-torsional-buckling of monolithic glass beams and for torsional effects of laminated glass. A further test-series will be carried out in the current year 2003 to investigate the lateral-torsional-buckling behaviour of members made of laminated safety glass.

By using the Finite-Element-Program ABAQUS the tests have been simulated. For all calculations volume elements with 8 nodes have been used. C3D8I-elements has been used for the glass and C3D8H-elements for the PVB-foil.

The simulation of the lateral-torsional-buckling tests with the monolithic glass has been carried out in two steps:

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

Step 2: The displacement of the eigenform determined in the fi rst step are scaled (e.g. L/1000) to the beam. A geometric non-linear calculation has been carried out to determine the collapse load controlled by the measured strain strains at the edges and the measured deformations.

The simulation of the behaviour of the PVB-foil can also be carried out with ABAQUS. The approach and the use of input data for the material PVB follows [11]. In the following the tests, the result of the tests and the simulation are described.

Lateral-torsional-buckling-tests of monolithic glass beams

The tests have been carried out with monolithic glass panels with a length of 3600 mm, a height of 360 mm and thickness of 8, 10 and 12 mm [7]. For this tests thermally strengthened and thermaaly toughened glass have been chosen. Because of the elastic behaviour

and the high strength of the material no failure occurs when testing high slendernesses. Therefore with a single glass beam several tests with different spans could be executed. Testing beams with small slenderness results in failure of the beam before buckling takes place. Figure 9 and fi gure 10 show the test set-up, fi gure 11 and fi gure 12 present the test data of displacements and strains. Figure 12 illustrates the infl uence of the second order effects on the glass beam.

Fig 9

Test set-up: 3-point-bending-test [13]

Fig 10

Test set-up and measurement devices for lateral-torsional-buckling-tests

Fig 11

Test results: Displacement characteristics

Fig 12

Test results: strain measurements at midspan

To conserve the crack pattern the glass was sticked to a bonding sheet. Figure 13 and fi gure 14 show the crack pattern of a thermally strengthened glass beam. The crack pattern adjacent to the supports shows that there is no crack in the non-loaded cantilever of the glass beam. Cracks are only located in the part of the beam with stress resultants caused by torsion or bending. The crack initiation could be caused by the concentrated load at midspan or the torsional moment at the supports.

Fig 13

Crack pattern of a thermally strengthened glass beam at midspan

Fig 14

Crack pattern of a thermally strengthened glass beam at supports

The test results and the numerical studies give information on the imperfections and the collapse loads of a monolithic glass beam. Further tests with laminated safety glass will complete the results to develop a design concept.

Small scale torsional-tests

Laminated safety glass panels with two glass panes (1100 mm x 360 mm) and a PVB-foil (0.76 and 1.52 mm) have been loaded by a constant torsional moment by twisting them by a certain angle (fi gure 15) kept constant over time. Figure 16 demonstrates the time- and temperature depending behaviour of the PVB-foil. At a temperature of 40 C there is no visible dependence of the loading-time of the PVB-foil. At lower temperatures - 9C - and for short loading rates the resistance of the system is much higher.

With the aid of the new developed torsional solution, the G-modulus of system can be determined from the measured moment and the measured rotation. Figure 17 shows these results evaluated for 9 C and 40 C. The G-modulus for a temperature of 9 C converge after some time to 7 MPa whereas the G-modulus for a

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temperature of 40 C converge to only 0.5 MPa.

By adapting the input dates of the PVB-foil, the numerical simulation of the test data is possible based on the approach in [11].

Acknowledgement

We thank Saint Gobain Deutschland GmbH in particular Anne Wittenkmper and Matthias Meiner for the material support.

Summary

The fi rst results show that from a study on the lateral tosional buckling of beams made by laminated glass the stiffness of the PVB has a large infl uence on the bending and torsional stiffness of the section depending on the loading time and the temperature. This infl uence can however be modelled by equivalent stiffness values. By this it is possible to take account of the stiffness variation of the PVB-foil depending on the climatic boundary conditions and the loading time.

Prospect

The experimental and numerical investigations will be fi nished by the end of 2003 [12]. The aim of the project is to develop a proposal for the design of glass beams not only loaded with a constant moment but also by biaxial bending and line loads.

References [1] Hess, R.: Glastrger. Forschungsbericht. vdf

Hochschulverlag AG an der ETH Zrich 1999. [2] Gsgens, J.: Bemessung tragender Bauteile aus

Glas, 1998, Shaker Verlag. [3] Holberndt, T.: Querschnittstragfhigkeit von

Glastrgern bei mehrachsiger Beanspruchung. Diplomarbeit TU Berlin, 2001, unverffentlicht.

[4] Roik, K.; Sedlacek; G.: Erweiterung der

Fig 15

Test set-up for torsional tests

Fig 16

Time- and temperature dependent behaviour of a laminated safety glass sandwich (2 x 6 mm glass panes combined with a PVB-foil of d = 1.52 mm)

Fig 17

Time- and temperature dependence of the G-modulus of the PVB-foil (2 x 6 mm glass panes combined with a PVB-foil of d = 1.52 mm loaded by torsion).

technischen Biege- und Verdrehtheorie unter Bercksichtigung von Schubverformungen, 1970, Die Bautechnik 47 Heft 1, S, 20-32.

[5] Vlling, B.: Berechnungsverfahren fr Balken und Platten in Sandwichbauweise nach der erweiterten Biegetheorie. Diplomarbeit RWTH Aachen, 2000, unverffentlicht.

[6] Scarpino, P.: Berechnungsverfahren zur Bestimmung einer quivalenten Torsionssteifi gkeit von Trgern in Sandwichbauweise. Diplomarbeit RWTH Aachen, 2002, unverffentlicht.

[7] Mnnikes, J.: Biegedrillknicken von Trgern aus thermisch vorgespannten Glasscheiben. Diplomarbeit RWTH Aachen, 2002, unverffentlicht.

[8] Schulz, J: Experimentelle und numerische Untersuchung von Verbundglasscheiben mit Bercksichtigung des visko-elastischen Verhaltens des Verbundmittels PVB. Diplomarbeit RWTH Aachen, 2003, unverffentlicht.

[9] Roik, K.; Carl, J.; Lindner, J.: Biegetorsionsprobleme gerader dnnwandiger Stbe. Verlag Ernst & Sohn. Berlin Mnchen Dsseldorf 1992.

[10] Knaack, U.: Konstruktiver Glasbau. Rudolf Mller, 1998.

[11] van Duser, A.; Jagota, A.; Bennison, S.: Analysis of Glass/ Polyvinyl Butyral Laminates subjected to uniform pressure. Journal of engineering mechanics, April 1999.

[12] DFG-Endbericht: Untersuchung des Biegedrillknickverhaltens thermisch vorgespannter Glastrger

[13] Kasper, T.: Analytische und experimentelle Untersuchungen zum Biegedrillknickverhalten thermisch vorgespannter Glasschwerter, 2000, unverffentlicht.