22 chapter

CHAPTER 22 - Glass structures 815

LIGHTWEIGHT STRUCTURES IN CIVIL ENGINEERINGPROCEEDINGS OF THE INTERNATIONAL SYMPOSIUM

Warsaw, Poland, 24-28 June , 2002

G E N E R A L L E C T U R E S

DESIGN PRINCIPLES OF GLASS ROOFS

J. SCHLAICH1 and H. SCHOBER

2

1Emeritus Professor University of Stuttgart and Schlaich Bergermann und Partner, Consulting Engineers, Stuttgart 2Schlaich Bergermann und Partner, Consulting Engineers, Stuttgart

ABSTRACT: With his knowledge of spatial geometry, shell theory and manufacturing processes, the structural engineer is able to further develop light and transparent glass roofs. They are attractive from an architectural as well as climatical point of view. Having already been the symbol of the new architecture of the Industrial Revolution during the 18th and 19th century, they experienced a revival during the second half of this century through the work of pioneers like Walther Bauersfeld, Konrad Wachsmann, Buckminster Fuller, Max Mengeringhausen,Frei Otto and others. In the paper the authors will describe some transparent glass roofs designed by their team with emphasis on form finding and manufacture.

GRID SHELLS

The appeal of glass domes grows with their translucence. Double-curved grid shells with triangular mesh offer favourable prerequisites for optimum transparency. Only the triangular mesh enables the membrane forces to basically flow only in the plane with almost no bending stress in the slats, a necessity for single-layer membrane shells.

Surface structures of this type, especially if directly glazed, pose three fundamental problems: - How to resolve the antithesis of favourable load-bearing behaviour

and difficult, double-curved manufacture? - How to cover a triangular structure with the much more favourable

quadrangular panes? - How to cover double-curved surfaces with plane quadrangular

panes?

We have obviously solved the first two problems with our grid shells by fabricating the base grid of the structure from a quadrangular mesh made of slats, square if laid flat on the ground. This plane mesh can assume almost any shape by modifying the original 90° mesh angle. The squares become rhombi.

These quadrangular meshes are diagonally stiffened by pretensioned thin cables, thus forming triangular elements – the essential prerequisite for the favourable structural characteristics of a shell (Fig. 1). The glass panes are directly clamped with the slats.

Fig. 1: Geometric principle of grid shells a) plan without diagonal cables; b) elevation with diagonal cables; c) the grid laid out into a plane (= a plane square net)

This design principle was developed in 1989 and 1990 and built first in Neckarsulm and Hamburg (Figs. 2 and 6). In the meantime it has been applied in various cases for geometrically rather sophisticated glass roofs. The sections "Barrel Vault Domes" and "Domes as Translational Surfaces" include detailed descriptions of some examples.

Fig. 2: The Neckarsulm dome from inside. (Architects: Kohlmeier und Bechler, Heilbronn)

The buckling stability of these filigree ribbed domes is of such importance that it needs to be addressed here. A focal point is how to determine the relevant buckling shape. We developed the subsequent checking procedure for this purpose:

If an ultimate load calculation is conducted using a perfect structure and geometrically non-linear calculation, the load is constantly increased until failure in stability occurs. Depending on the system's stiffness an eigenvalue-analysis is conducted during the final iteration, resulting in natural frequencies and eigenmodes. Fig. 3 shows the resulting buckling shapes for domes with different curvatures.

With the eigenmodes standardized to a maximum value, specified to be the imperfection of the shell, a geometrically non-linear calculation is conducted. It is important to keep in mind that the imperfection consists of a geometrical and a structural part.

If instead of this rather extensive procedure to determine the buckling shape, the deformation shape resulting from snow load, applied to the entire or only half of the shell, is assumed to be the imperfection, the resulting limit load would be on the unsafe side (Fig. 4).

A sound curvature still and forever remains to be the most effective remedy against failure in stability.

PART IV Application traditional and innovative materials 816

Fig. 3: Buckling shapes resulting from ultimate load iterations, left and design loads, right ( g + snow on the entire shell, g + snow on half of the shell)

Fig. 4: Limit loads for different buckling shapes

ROTATIONAL CENTROSYMMETRICAL DOMES

Ribbed domes with ring- and meridional bars only can easily be con-structed because of their rotation-symmetrical structure. However, these domes require correspondingly heavy members, as they rely on frame action resp. on bending stiffness. Diagonal cables stiffening the rectangular glazed meshes transform these ribbed domes into true shell structures with optimum translucence. Examples of such structures are the clinic in Bad Neustadt/Rhön and the shopping mall Grünau in Leipzig (Fig. 5).

Rotation-centrosymmetrical structures can of course always be covered with plane quadrangular panes. The concentration of bars in the zenith - exactly where one would expect maximum transparency - is certainly a basic disadvantage of this approach.

Fig.5a: Glazed dome of Rhönklinikum, Bad Neustadt, (Architect: W. Wilhelm, Bad Neustadt)

Fig.5b: Glazed dome of shopping mall Grünau, Leipzig, (Architects: von Gerkan, Marg + Partner, Leipzig)

BARREL VAULT DOMES

Compared to double-curved domes, barrel domes are "merely" plane or two-dimensional and thus easier to construct - constituting their wide-spread use. In terms of structural behaviour a barrel dome with evenly distributed loads along its lower edges is identical to an arch. If the warp is designed to follow the thrust line, only axial


compression forces will occur, but no bending moments. However, barrel domes must withstand the most diverse loads and therefore require bending-stiff arched girders.

The lattice barrel shell transforms into an efficient cylindrical shell only when its quadrangular mesh is stiffened by diagonal cables and if it is equipped with transverse diaphragms. These diaphragms provide for the double curvature required for shell behaviour and may be made from pretensioned "spoked wheels" with the spacing depending on the barrel's span and curvature (Fig. 6). More about that is provided in [3].

Due to the asymmetric barrel shape of the roof for the spa in Bad Cannstatt, the spoked wheels had to be asymmetrical as well. In the case of the WTC in Dresden pretensioned cable trusses were applied (Figs. 7 and 8).

Fig. 6: Roof for the Museum für Hamburgische Geschichte. (Architect: von Gerkan, Marg + Partner, Hamburg)

Fig. 7: Mineralbad Bad Cannstatt. (Architects: Beck-Erlang + Partner, Stuttgart)


Fig. 8: World Trade Center Dresden. (Architects: Nisch Prasch Sigl, Hamburg)

If - as in case of the courtyard roof in Berlin, Friedrichstrasse – objections are raised towards diagonal bracing of the barrel dome, stronger profiles are necessary because it now predominantly acts in bending. However, it was possible to minimize the size of these profiles by applying more spatial prestressed cable diaphragms, and by using the frame action of the barrel made from welded pipes (Fig. 9).

Fig. 9:

Fig. 9: Atrium roof of Quartier 203, Friedrichstrasse, Berlin. (Architects: von Gerkan, Marg + Partner, Hamburg)

The examples above described continuously supported and diagonally reinforced barrel domes acting like a shell only under asymmetrical loading. In case of symmetrical loading the arch – if following the thrustline – experiences no moments. If the barrel dome cannot be supported continuously at its lower edges but only on columns - like for example the platform roof of the Spandau railroad station in Berlin – the shell transfers the loads longitudinally and transforms into a true cylinder shell, acting like a continuous girder with a depth corresponding to the rise. The barrel's diagonal cables are now indispensable for all loading cases and must be carefully anchored and clamped. A rigid diaphragm at the supports of the shell must not only provide for the transversal stiffening of the barrel dome, but also for the shell-adequate support of the barrel, i. e. the diaphragm must be able to absorb continuous shear forces from the roof. Thus the steel arch functions as transverse diaphragm (Fig. 10).

Fig. 10


Fig. 10: Railroad terminal Berlin-Spandau. (Architects: von Gerkan, Marg + Partner, Hamburg)

Fig. 11

Fig. 11: Protective roof for the ancient Roman hot springs in Badenweiler (Architects: Hochbauamt Freiburg)

The protective roof for the ancient Roman hot springs in Badenweiler (Fig. 11) is also a barrel dome with individual supports at the lower edges in 6 resp. 7 m intervals. It spans 36 resp. 40 m and is 68 m long. A crescent-shaped cable with ascending and descending catenaries provides the transversal stiffening.


SUSPENDED CATENARY CANOPY ROOFS

The inversion of the barrel vault is the catenary, transferring all loads as tension forces without any bending. Since each loading case results in a different catenary, the occuring deformations have to be carefully tracked during the design. Without the application of extensive suction-prevention, wind suction can only be counteracted by the structure's deadload.The price for the structure's simplicity and translucence are the anchorages for the cable forces depending on the degree of the sag. The canopy roof for the railroad station in Ulm is an example of a single-glazed suspended roof (Fig. 8). The edge beam (truss) distributes the suspension forces on to strut-and-tie supports.

Fig. 12: The Ulm railroad station canopy. (Architect: H. Gaupp)

In Baden near Vienna a thermopane suspended roof spans 28 m across an indoor swimming pool (Fig. 13). At the high point the suspension forces are diverted across inclined forked supports to the inclined supports of the facade. At the low point a truss positioned in the roof distributes the suspension forces to the strut-and-tie supports.

Extreme attention has to be paid to the increased suction along the edge of the roof. These suction forces are usually determined in wind-tunnel-tests because only the deadloads counteract the suction and the system reacts sensitive if this counterweight should become insufficient. In two fields each at both sides of the roof triangular panes had to be applied instead of square ones to provide for increased deformations there.

Fig. 13: Indoor swimming pool in Baden near Vienna (Architects: Dipl.-Ing. R. Nemetz, Baden/Vienna)

Fig. 14: Glass suspension roof in front of the railway station Heilbronn (Architects: Auer und Weber, Stuttgart)

In front of the railway station in Heilbronn, a preserved monument, a lightweight glass roof was to span across the tracks and the platforms without either obstructing the view of the station’s facade or visually block the station square. Due to the overhead cables of the city railroad the roof had to be constructed with a clearance of 7.35 m.


The result is a 1.000 m² glass mat consisting of large panes which meet the strict demands for overhead glazing. Stainless steel mounting brackets attach the mat to stainless steel cables which hang in a tubular steel frame by cast-iron joints. The glass is also used for the load transfer with the panes transferring the horizontal loads [17].

In the case of the roof in Heilbronn a suspension roof using a minimum of material, just like the roof in Ulm, seemed to be a good approach [1]. However, inclined stays as in Ulm or in Baden (Figs. 12 and 13) were impossible, since the tracks run just along the roof edge. Therefore, the horizontal forces from the suspension roof were short-circuited directly with horizontal compression rods (Fig. 14). And in Heilbronn the glass is separated from the supporting steel strip to avoid any water running off from the center of the roof on to the tracks. This resulted in thin high-strength cables being used to suspend the glass instead of the comparatively wide steel strips necessary in the case of directly applied panes. Raising one side of the slightly curved glass roof facilitates the one-sided water run-off even more and allows an unobstructed view of the station (Fig. 14). Water accummulating at the longitudinal side of the roof is being gathered in a channel and drains through the supports.

CABLE NET FACADES

Our cable net facades consist of a single-layer, plane, prestressed cable net with the glass panes directly attached to its nodes. Wind loads hitting the net at a 90° angle cause large deformations which can be controlled by prestress of the cables. The surrounding structure has to be able to absorb the net forces. The support of the panes and the cable anchorages have to be designed to allow for large deformations of the net. Therefore, the meshes and the glass panes are no longer plane under load. Tests confirmed the sufficient loadbearing capacity of the glass panes. Deformations occuring in the net plane have to be tracked carefully during the design process. A 40 x 25 m2 cable net facade was first designed for the Hotel Kempinski at Munich Airport. The mesh width and the size of the panes are 1.5 x 1.5 m (Fig. 15). The lateral concrete structures together with the arch of the roof form a stiff frame as bracing for the cable net. A similar facade, but with a mesh width of 2.07 x 1.56 m, was designed for the WTC in Dresden (Fig. 16).

Fig. 15: Tensioned grid facade of the Hotel Kempinski Munich, (Architects: Murphy Jahn, Chicago)


Fig. 16: Tensioned grid facade of the WTC Dresden, (Architects: Nietz Prasch Sigl, Hamburg)

Fig. 17: Badenweiler

The canopy covering the ruins of the ancient Roman public bath in Badenweiler called for lightweight structures. The front of this

historical monument was closed off with a tensioned facade. Here, the glazing joints were not sealed with silicone but remained open instead. A gap of about 50 cm remains between the glass membrane, the end cable and the roof to ensure sufficient aeration. The mesh width is max. 1.20 x 1.80 m. For the facade of the Foreign Ministry in Berlin the horizontal cables are recessed for about 40 cm compared to the vertical cables and both net planes are connected by spacers for attaching the dichroic glass strips. The mesh width and the pane size are 1.80 x 2.70 m (Fig. 18).

Fig. 18: Foreign Ministry Berlin (Architects: Müller Reimann, Berlin; Glass art: James Carpenter, New York)


FLAT ROOFS WITH SLIGHT ARCHING

Frequently, roofs over rectangular courtyards are supposed to be flat and pillow-shaped to prevent obstructing the view from the floors above the roof. In this case the pillow-shaped surface may be covered with plane glass, if the rise resp. the warp is small. However, for roofs with insignificant rise single-layer structures are ruled out due to stability reasons calling for dual-layer structures. The glass roof of the Deutsche Bank in Berlin has a grid of slats in compression vaulting 0.60 m, and a cable net in tension sagging 1.40 m. The grid is directly covered with rectangular thermopane glazing and the diagonal cable net only supports every other node to prevent a maze of stakes (Fig. 19).

For stress optimization a formfinding analysis was conducted for the pillow-shaped solution.

The curtain facade of the same building is to connect the newly added storey to the existing structure. The objective of the design was to reflect the old punctuated facade through steel frames covered by glass and to obtain maximum transparency for the loadbearing structure. Therefore, tension rods hold the entire deadload of the 5-storey glass wall at the top and the wind load is transfered to the building at each grid point by thin poles (Fig. 20).

Fig. 20: Facade of the Deutsche Bank, Berlin (Architects: B. Tonon, Berlin; Novotny, Mähner + Assoziierte, Offenbach)

If a flat glass roof surrounded by a concrete slab is chosen as in the case of the Katharinenhospital in Stuttgart, the trusses can be braced between this solid concrete structure which acts like a frame. Depending on the degree of prestress the compression flange of the fish-belly cable truss can be minimized until the prestress finally exceeds the forces of the compression flange due to loading. The compression flange could have even be a cable, but this was not the chosen solution in this case (Fig. 21).

Fig. 21: Courtyard roof, Katharinenhospital, Stuttgart (Architects: Heinle Wischer und Partner, Stuttgart)

The architect’s idea for covering the 32.4 x 40.5 m courtyard of the Foreign Ministry in Berlin was a horizontal glass roof. Therefore, steel main girders with parallel chords were chosen spanning 32.4 m with slightly curved fish-belly girders inbetween at 2.70 m intervals (Fig. 22).

Fig. 19: Unter den Linden, Berlin. (Architects: B. Tonon, Berlin; Novotny, Mähner + Assoziierte, Offenbach)


Fig. 22: Glassroof Foreign Ministry, Berlin (Architects: Müller Reimann, Berlin)

Fig. 23: Roof of the company restaurant Audi, Ingolstadt (Architects: Ahlheim-Nebe-Schoofs, Darmstadt)

In the case of the flat roof over the company restaurant in building A50 at the Audi-plant in Ingolstadt (completed in 2000) the cables were placed on the outside. The result is a suspension-bridge type roof. Its barrel-vault-type steel grid is back-anchored across nine spatially curved cable systems and masts (Fig. 23).

TRIANGULAR MESHED DOMES

We often face the challenge of designing freely shaped glass domes, because they either span across irregular plans, form the transition between different geometric areas or they are intended as sculptures. Since such areas usually can no longer be covered by plane quadrangular panes, the less favourable triangular panes have to be used. Especially when using thermopane, the grid will also be designed as a directly glazed triangular net. Even despite the crooked quadrangular meshes the transition between the two barrel vaults of the roof in Hamburg could be covered in vast areas by quadrangular single glazing. However, the quadrangular panes had to be dissolved into triangles in areas of extreme warping (Fig. 24).

The irregular shaped courtyard of the Palais Bernheimer in Munich and that of the Flemish Council in Brussels were to be spanned by a thermopane pillow-shaped glass roof. The rise as given allowed for a single-layer grid shell formed according to an inverted suspended shape. Since, in both cases, quadrangular meshes would have caused too much warp only a directly glazed triangular grid was applicable. Despite the triangular structure with 6 bars crossing at each node, a filigree and transparent roof was the result (Figs. 25 and 26).

Fig. 24: Museum für Hamburgische Geschichte, transition area (Architects: von Gerkan, Marg und Partner, Hamburg)

Fig. 25: Courtyard at the Palais Bernheimer, Munich. (Architects: A. Freiherr von Branca, E. Freiin von Branca)

Fig. 26: Courtyard roof of Flemish Council, Brussels. (Architects: Arrow, Verstraete, Gent; J. Puyen, Antwerpen)


The architect Frank O. Gehry designed a three-dimensional barrel-shaped roof as a sculpture intervening with the interior for the roof over the atrium at the Pariser Platz in Berlin (Fig. 27). The central courtyard abandons the conventional architecture of roofs in favour of a crimped barrelshaped roof as a sculpture.

Free shapes like this can only be glazed with triangles and made from a triangular grid. The shell structure is not continuously supported at its edges, but only at 16 m intervals. Due to the small longitudinal curvature, the shell had to be additionally stiffened by spoked wheels (Fig. 27). The entire structure is made of stainless steel. The nodes were milled three-dimensionally.

Fig. 27: Atrium roof of DG Bank, Pariser Platz 3, Berlin (Architect: Frank O. Gehry, Santa Monica)

Fig. 27 (cont'd): Atrium roof of DG Bank, Pariser Platz 3, Berlin (Architect: Frank O. Gehry, Santa Monica)

DOMES AS TRANSLATIONAL SURFACES

The previous examples showed that freely shaped, double-curved surfaces can always be formed using triangles. However, they will never attain the same translucence as a structure with quadrangular glazing.

For double-curved surface structures with favourable quadrangular glazed mesh, the glass panes must either be able to absorb the mesh warp or have the same curvature as the load bearing structure, or the net shape must be chosen in such a way that the individual quadrangular meshes can remain plane. The thermopanes of the spherical dome in Neckarsulm are spherically curved, resulting in warped quadrangular meshes and an ideal spherical shape (Fig. 28).

Fig. 28: Grid dome Neckarsulm (Architects: Kohlmeier und Bechler, Heilbronn)

This architecturally sophisticated, but also very expensive type of glazing would definitely restrict the construction of double-curved shells with quadrangular meshes. But there is a geometrical technique for designing almost any shape with plane quadrangles. H. Schober demonstrated that translational surfaces allow for a vast variety of shapes of grid domes with evenly meshed nets consisting of quadrangular meshes [2], [16]. For example, a parabola (generatrix) translating across another parabola (directrix) perpendicular to it, results in a elliptic paraboloid with an elliptic layout curve, which can be covered with an evenly-meshed net consisting of plane quadrangular meshes. This principle was first realized with the roof over the courtyard of the Rostocker Hof in Rostock (Fig. 29).


Fig. 29: Gallery Rostocker Hof, Rostock, grid shell as translational surface. (Architects: Schweger + Partner, Hamburg)

A directrix curving opposite to the generatrix creates a hyperbolic paraboloid, which can also be formed by two systems of linear generatrices (Fig. 30). This allows for the creation of hypar-surfaces with straight edges.

Fig. 30: The hyperbolic paraboloid, a translational surface with plane quadrangular meshes.

The courtyard roof in Leipzig is an example for a translational surface covered with plane rectangular glass panes, spanning the trapezoidal courtyard (Fig. 31).

Fig. 31: Courtyard roof of Industriepalast, Leipzig (Architects: M. Frishman und D. Düttman, Berlin)

However, directrix and generatrix must not necessarily consist of geometrically simple curves, but can also be defined as random spatial curves, and thus offer a huge variety of shapes. A most recent example is the hippopotamus-house in the Berlin Zoo. Here, two parabolas with a freely defined transition curves were chosen as a directrix for the roof of two circular ponds (Figs. 32, 33).

Fig. 32: Glass roof for the Hippo House at the Berlin Zoo. A translational surface with plane quadrangular meshes. (Architect: J. Griebl, München)


Parabolas were also selected as generatrix and had to be identical with the corresponding parabolas of the directrices, in order to yield circular layout curves. The different generatrices of the small and large pond merge in the transition area. The facade of the visitor's hall intersecting with the roof surface is defined by a circular cone standing on its tip and inclined by 8 degrees, covered with standard glazing without any warp, blending with the dome in a freely sweeping edge.

Fig. 33: Glass roof for the Hippo House at the Berlin Zoo. (Architect: J. Griebl, München)

Another example of designing rather difficult geometries as translational surfaces is the roof over the courtyard of the Bosch Areal in Stuttgart. Although in this case there are several adjoining irregular courtyards, it was possible to create a continuously curved transition area consisting solely of plane and evenly meshed quadrangles (Fig. 34). Additions are made to the generatrix in the transition according to the requirements of the entering courtyards. The directrix is freely formed as to allow for sufficient curvature for the transition area and the entering roofs.

Fig. 34: Roof over the courtyard of the Bosch Areal. Grid shell as translational surface with plane meshes. (Architect: Prof. Ostertag, Stuttgart).

This is a simple but by far not the only method of creating grid shells with plane meshes. In [16] further procedures for covering double curved surfaces with plane quadrangular mesh-elements are described. One example is given in Fig. 35.

These examples prove that grid shells can be economically constructed in almost any shape using the technique of translational surfaces and constructing the entire dome from a evenly-meshed net with plane quadrangular panes, an optimum solution for translucence and economy.

Fig. 35: Double curved surfaces with plane quadrangular meshes by centric expansion and translation

REFERENCES

[1] Schlaich, J., Schober, H.: Verglaste Netzkuppeln, Bautechnik 69 (1992), S. 3-10.

[2] Schober, H.: Die Masche mit der Glaskuppel. Netztragwerke mit ebenen Maschen,

Deutsche Bauzeitung 128 (1994), S. 152-163. [3] Knippers, J., Bulenda, T., Stein, M.: Zum Entwurf und zur

Berechnung von Stabschalen, Stahlbau 66 (1997), H. 1, S. 31-37. [4] Das Mineralbad Cannstatt in Stuttgart, Glas 4 (1995), S. 42-47. [5] World Trade Center in Dresden, Glas 1 (1997), S. 34-40. [6] Fassaden und Glasdächer der Deutschen Bank in Berlin, Glas 5

(1998), S. 19-26. [7] Überdachung Vorplatz Hauptbahnhof Ulm, Glasforum 1 (1994), S.

26-28. [8] Der Vlaamse Raad in Brüssel, Glas 1 (1996), S. 18-24. [9] Glaskonstruktion Hotel Kempinski, München, Glas 2 (1995), S. 30-

37. [10] Schlaich, J., Schober, H.: Glaskuppel für die Flußpferde im Zoo

Berlin, Stahlbau 67 (1998), H. 4, S. 3-8.

[11] Schober, H.. Netzkuppeln mit ebenen Maschen, Beratende Ingenieure, September 1998, S. 15-19.

[12] Bahnhof Berlin Spandau, Glas 3 (1999), S. 12-18. [13] Schober, H.: Zum Tragwerk des Funktionsneubaus

Katharinenhospital Stuttgart, Stahlbau 63 (1994), H. 5, S. 129-133. [14] Schlaich, J., Schober, H., Helbig, T.: Eine verglaste Netzschale: Dach

und Skulptur, DG Bank am Pariser Platz in Berlin. Bautechnik 78(2001), H. 7, S. 457-463

[15] Empfehlungen für die Bemessung und Konstruktion von Glas im Bauwesen, Entwurf 2/01, Der Prüfingenieur, April 2001

[16] Schober, H.: Geometrie-Prinzipien für wirtschaftliche und effiziente Schalentragwerke , Bautechnik (2002), H. 1, S.

[17] Schlaich, M., Golenhofen, D., Das hängende Glasdach vor dem Hauptbahnhof Heilbronn, Stahlbau 70 (2001) Heft 11, S. 821 – 826

_______________________________________________________1) J. Schlaich, Emeritus Professor University of Stuttgart and partner of

Schlaich Bergermann und Partner, Consulting Engineers, Hohenzollernstr. 1, D-70178 Stuttgart, Germany

2) H. Schober, partner of Schlaich Bergermann und Partner, Consulting Engineers, Hohenzollernstr. 1, D-70178 Stuttgart, Germany




THE DESIGN OF A FULL GLASS DOME,

USING A NEW GLASS/POLYMER COMPOSITE MATERIAL

G.J. HOBBELMAN1, B. TIMM2, F.A. VEER3, P.M.J. VAN SWIETEN4

1Associate professor, Faculty of Architecture, Delft University of Technology2Graduated at the Faculty of Architecture, Delft University of Technology

3Assistant professor, Faculty of Architecture, Delft University of Technology4Assistant professor, Faculty of Architecture, Delft University of Technology

ABSTRACT: A current trend in Architecture is towards maximum transparency. This counts not only for the cladding and glazing of building but, if possible, also for the structure. In order to obtain transparant building materials a research program [1] has been started in Delft towards the design of a new laminated glass/polymer sandwich composite material. Optimal combination of the ductile polymer and the strengthened glass which is strong but brittle leads to a composite that is strong and ductile. In this research project several graduation students participate, each with a different subject of interest. One of them has investigated the possibilities of the new composite to construct a dome. Fig 1. Load bearing elements of glass are allways combined with steel components in order to provide stability and ductile behaviour of the structure. In this study it was tried to design a dome without any steel added to it. The double curvature of the shell combined with the ductile behaviour of the glass/polymer sandwich should do the job.

Key words: glass, polymer, shell, dome, structural glass,

1. THE PROJECT As a graduating project, Beatrice Timm designed a new opera house for Oslo, Norway and she investigated the possibilities of constructing a full glass dome for the roofing of the main room for 800 spectators. The roof should be fully transparant in order to give the people a view at the sky during breaks. The dome has a span of 22 meters and a height of 7 meters. The shell thickness is 36 mm. It is built on an irregularly shaped ground surface. At first several possible shapes of the crosssection were investigated. This led to the conclusion that a parabolic shape gave the best results. The surface then was divided into components of the sandwich material. The influence of the size of these components was determined by computer calculations. Steel was only used for the joints of the glass elements. Microscale calculations were carried out to find the right form and size of the joints.

Architectural and acoustical considerations also played an important role in the choice of the right shape of the dome. For acoustic reasons the volume of a hall may not be too big compared with the surface. On the other hand from an architectural point of view the dome had to get an outspoken form, that could be achieved by increasing the height. For these two contradictary demands a compromise was found. It was concluded that it was possible to design a dome of glass/polymer of the given size and that it could withstand the loads according to the dutch regulations, with some adaptions for norwegian cicumstances. Second order behaviour was not investigated, however the stresses found were so extremely low that this is not likely to occur.

2. THE USE OF GLASS Glass is an interesting material to investigate as a structural material. It has sufficient strength and can be treated in a way that the strength is maintained over a period of time. It also has several drawbacks. One of them is the brittleness, which can be compensated by laminating the glass. Standard soda lime glass seems to be the best option on the market because it is cheap and easy to laminate. The glass has to be coated to avoid microcracks. Laminating also is necessary because the glass is used overhead with the danger for people that it falls down on the spectators. Thermal strengthening is an effective way of increasing the strength of glass, but unfortunately it is not applicable on double curved, laminated elements. The production of doubly curved laminated elements requires heating of the components wich counteracts the effect of the hardening. Hardening after laminating is impossible because the heat would damage the plastic lamination part. Chemical strengthening could be a possibility but is expensive.

3. THE CHOICE OF THE FORM In order to choose a suitable form for the shell structure, four types of cyclic shells were investigated, Fig 2: a paraboloïd, a sphere, an ellipsoïd and a cycloïd. The radius of curvature was varied: 10, 15, 20 and 25 meters. This resulted in different hights of the shells: the greater the radius, the lower the shell.

Fig. 1, groundplan and crossection of the shell


The radius at the bottom was kept at a constant value of 10 meter, due to the design of the concertroom.

The influence of the variation in radius at the top and near the bottom of the four forms was a criterion. Possibilities for production and construction were also observed.

Comparison of the stresses led to the following conclusions: - Stresses in the cycloïd were allways higher coMPared to a sphere at

own weight and tension occurs at an angle of 42o with the vertical.- ellipsoïds gave much tension at the top due to the small curvature of

the shell, which is not desirable when constructing with glass. - paraboloïds performed better whith smaller radiusses and thus

higher tops. Only when the top was comparably low performed the spheres better then paraboloïds.

- The ratio between the top and the radius determined the choice of the form of the shell.

It is very important that the stresses in a full glass shell be kept as low as possible and also equally divided. Therefore the choice for an paraboloid with a height-radius ratio of about 3 was made. The radius will be 22 meter, the height 7 meter.

4. THICKNESS OF THE SHELL The most important feature that determines the thickness of a shell surface is buckling. To avoid buckling the thickness has been approximated with the method of [1]. Using conservative estimations for the loading and safetyfactor a thickness of 36 mm was found. Two layers of glass were used of 18 mm each, connected by a layer of polymer to provide ductility. Fig 3. This value of 36 mm shall be used in following calculations.

For protection of the structural glass shell a second layer of glass was added with the same thickness. This second layer was taken into account in the structural analysis only as loading through its own weight.

Fig. 3 four layers of glass (18 mm) and three layers of epoxy (9 mm)

5. FORMS AND SHAPE OF THE GLASS ELEMENTS

Fig. 4 layouts of glass elements

Several layouts of the glass elements were investigated. Fig 4. The elements were all quadrilateral, only the size varied from 0.4 – 1 - 2 - 4 meter. The shells with the four different elements were loaded with gravity, snow and windloads. The stresses in the four types were allmost the same with small variations. Only in the shells with the largest elements rather big bending stresses occurred. This was due to the fact that the elements in the model were flat, which introduced quite substantial bending stresses, wereas the real elements are doubly curved, following the curvature of the shell. The overall stresses in all the four shells corresponded well with the analytical results.The optimal thickness and the number of layers much be established. First of all the overall stresses in the shell must be calculated.

6. LOADINGS

To determine the correct loads several codes were compared. The snow load in Oslo turned out to be much higher than that in the Netherlands, so the practice of calculation must be adapted to the circumstances in Norway. Windload turned out to be quite the same.

The snowload found was 1.9 kN/m2.

The dead load of the glass was 1.7 kN/ m2 .

The windload 1.1 kN m2.

b

d

c

a

Fig. 2 Four types of shells, with the typical stresses under vertical load. a: sphere, b: ellipsoid, c: cycloid, d: paraboloid


7. RESULTS

Fig 5a stresses self weight, large elements (4m).

Fig. 5b, stresses self weight, small (1m) elements

For the comparison of the sizes of the elements a shell with a circular groundplan was considered. The stresses are equaly divided over the shell. The contours of the (flat) elements are to be seen. The peak stresses at the middle of each element are mainly bending stresses due to the fact that the elements, used for the calculation, were flat. Fig 5a. In the next figure, the peak stresses for self weight are much smaller, the total appearance of the stress figure is much smoother. Fig 5b. This is because the elements are smaller and thus the bending stresses in them. The average stresses coincided well with the theoretical values.The choice has been made to an element of 2 x 2 meters. The compression stresses found under several loadings were maximal -5 MPa wereas the tensional stresses due to asymmetric loading were not higher than 1.9 MPa. This was the case for snowload only on the lower parts of the shell.So the stresses in the shell remain very low, which proves that the shape and curvature of the shell were well chosen and that the thickness of the shell is sufficient.In a former graduate project, Ting [3] found out that a composite of glass and polycarbonate could perform under stresses up to 40 N/mm2

Fig. 6. Glass elements removed.

The problem of progressive collapse was also investigated. This was done on the final design of the shell with an irregular shaped groundplan. On several places an element was removed. The black spots are removed elements. The stresses around the openings were somewhat greater, but the influence was only local. The overall stress distribution of the shell was hardly disturbed. Fig 6. It is however extremely unlikely that these number of elements would fail totally at the same time.

8. JOINTS The joints were designed using the stresses due to asymmetric loading, with the top of the shell unloaded and the lower parts of the shell loaded with snow. The resultant stresses were applied on four elements of the shell held together by one joint. So the stresses in the joint could be calculated.For the design of the joint there are several principles, Fig 7.

1. Line joint

2. Point joint

3. Fully fixed

4. Hinged

5. Joint in between the plates6. Joint over the plates

7. Joint under the plates

8. Joint on both sides of the plates

Fig. 7. priciples of connection

Comparison of the above cases led to the conclusion that a joint on both sides of the shell that can transfer small bending moments gave the best results. So a combination of 3 and 8 was chosen. There are six variables that determine the functioning of the joint: - the height - the width - the area - the material - the thickness of the glass - the width of the joint

All variables were varied under the loads mentioned bofore. The results led to the following conclusion: the joint should be as low as possible, the width should be three to four times the thickness of the glass. A flexible material such as nylon gives lower stresses but bigger


displacements than steel, so steel was considered to be better.The jointsbetween the glass elements should be as small as possible.

To find the right dimensions of the joint three forms of joints were investigated, Fig 8.

Fig. 8 Three forms of joints, ring, square, cross

The ring and square type of joints transfer the forces from one element to the adjacent element, wereas the cross type of joint transfers the forces to all three other elements. So the choice fell to the cross type joint, Fig 9. The dimensions of the joint were determined by the strength, the case of on site of assembly of the connection, and the filling of the joints between two elements. The contact surface between the connection and the glass was determined by the forces that should be transferred and the strength of the adhesive. The width of the joint was set to 25 mm, taking into account the tolerances caused by the the bending and cutting proces of the glass elements and by the construction. For the glue an epoxy resin was chosen, because the ability of filling the gap between the flat surface of the connection foot and the curved surface of the glass element. Also the good performance under maritimal conditions and the cold conditions in Oslo of epoxy resin, made it a good choice.

Fig 9 the cross shaped connection

Fig 10 stresses in connection and glass

The connection then was calculated under different loads, using a diameter of the foot of 50 mm and 75 mm and by varying the dimension of the branches of the connection from 25 to 35 mm The results showed that enlarging the diameter of the foot decreased the stresses in the glass and the adhesive. Enlarging the dimension of the branches increased the stresses in glass and adhesive due to the bending stiffness of the branches. In the end a diameter of the foot of 75 mm and a dimension of the branch of 25 mm was chosen. The stresses in glass and adhesive were –10 MPa to +7.5 MPa, Fig 10.

9. SUPPORTS The supports of the shell structure were designed using a maximum vertical displacement of 1 mm. This dictated the stiffness of the edge beam and the number of supports for it. The value of 1 mm was derived from calculations in which the displacements of the shell were coMPared with the acceptable stresses. The design of the opera house led to a column distance of 3.5 meter for the edge beam. For the edge beam an ordinary HE beam was chosen: HE1000A. If the form of the edge beam should be designed other than an I shape the bending stiffness of that beam should be the same. To support the circular shaped hollow section of the shell an arch was designed in the shape of a sickle with a tension rod. The supports are shown in Fig 11.

Fig 11 The shell and its supports

10. CONCLUSIONS - A full glass dome with a span of 23 meter could be realised. The

dimensions of the shell elements and of the steelconnections between them are realistic. The stresses and strains are acceptable for the materials used.

- In the course of the graduating process a choice for epoxy resin as adhesive to be used for laminating the glass elements and for the connection between them. Further investigation towards a better material is necessary.

- The building of a prototype could not be performed due to lack of time but should be carried out by an other graduate student. This will give more insight in the behaviour of the shell as a whole and of its components.

- The problem of progressive collapse should be examined in more detail, also using scale models.

11. LITERATURE [1] F.A. Veer, M.A.C. van Liebergen, S.M. Benedictus-De Vries:

Designing and engineering transparent building components with high residual strength. 5th glass processing days, Finland 1997

[2] C.B.Wilby: Concrete Dome Roofs, Longman Scientific & Technical, 1993 [3] G.J. Hobbelman, G.P.A.G. van Zijl, C.N. Ting: A new structural

material by architectural demand, Structural Engineering, Mechanics and Computation, ed. A. Zingoni, Elsevier, 455-462.

[4] C.J.J. Vreedenburgh, J.G. Bouwkamp: Axiaalsymmetrische Schalen [5] T.H.Hsu, Volume 4: Shells, Gulf publishing Company, 1991. [6] P.Csonka, Theory and Practice of Membrane Shells, VDI Verlag, 1987. 1) G.J. Hobbelman, Delft University of Technology, Faculty of Architecture,

Berlageweg 1, 2628 CR, Delft, The Netherlands 2) B. Timm, Delft University of Technology, Faculty of Architecture,

Berlageweg 1, 2628 CR, Delft, The Netherlands 3) F.A. Veer, Delft University of Technology, Faculty of Architecture,

Berlageweg 1, 2628 CR, Delft, The Netherlands 4 ) P.M.J. van Swieten, Delft University of Technology, Faculty of

Architecture, Berlageweg 1, 2628 CR, Delft, The Netherlands




THE EFFECTS OF TORSION ON THE LOAD-BEARING CAPACITY OF

SINGLE-PANE GLASS BEAMS

J.BELIS1, R.VAN IMPE2, F.VERNAILLEN3, G.LAGAE4 AND W.VANLAERE5

1,5Doctoral Research Assistant, 2,4Professor, 3Graduate Student, Ghent University, BELGIUM

Key words: glass, beams, transparency, lateral torsional buckling, experiment, analysis

1. INTRODUCTION

Transparency has become an important characteristic of contemporary architecture and building design. Not only infill and cladding elements, but also the primary load-bearing building structure can be made of glass. Considerable know-how exists for the first type of applications, but for the latter several years of further scientific research will be needed for optimisation of load-bearing capacity, material use, safety and construction methods.

The lack of national or international codes or standards makes it difficult for designers to bring forth economic load-bearing glass designs. Two main causes are responsible for exaggerated costs: the used safety factors are very well on the conservative side (resulting in uneconomical use of material), and expensive full-scale experimental tests which are required for every individual project.

However, this relatively new application is looked at with great interest by designers and the glass industry. Though a load-bearing glass structure will probably never be used on the same scale as e.g. steel or concrete, its relevance is shown by a growing number of realisations. The connecting pedestrian bridge between two office buildings in Rotterdam, The Netherlands is a speaking example (Fig 1). The Laboratory for Research on Structural Models is active in research on load-bearing glass beams since a few years.

Fig.1 D.J.Postel: Kraaijvanger-Urbis architects,The Netherlands [7]

2. GLASS AS A BUILDING MATERIAL

Glass has specific properties of thermal and acoustical insulation, chemical resistance, durability and recycling, all of which are beyond the scope of this article. Moreover, glass is an interesting structural engineering material. Some typical properties of annealed glass can be found in literature [2, 7, 9] (Table 1).

Density 2500 kg/m3

Young’s Modulus 70-74 kN/mm2

Poisson’s ratio 0.22 Tensile strength Theoretical value 3600-5000

N/mm2 (cf. §4) Compressive strength > 1000 N/mm2 but

complimentary tensile stresses will govern

Hardness 6 MoH Maximum service temperature Approx. 280° C (beware

temperature differences!) Coefficient of thermal expansion 7.7-8.8 x 10-6 /K

Table 1. Some typical properties of annealed glass [5]

The combination of a very good transparency with a high strength (cf. §3.) makes glass a unique building material. The main disadvantage, however, is its total lack of tensile ductility: glass is a brittle material.

In order to build in some residual strength in the concept, glass beams will usually be drafted as laminates: a number of individual glass panes held together by means of a soft transparent interlayer. Polyvinyl butyral (PVB) or resin are chosen in most cases. If a glass pane breaks, pieces stay adhered to the interlayer, and people below are saved from injuries.

Even with composed glass beams, the geometry of the structural elements is limited to a rectangular cross section. The reason is that there are no connection systems that are suitable to deal with the shear forces between a glass web and flanges. Glues are too much deformable or simply not strong enough, and glass welding induces complicated residual stresses. Moreover, this last technique is not very well known.

Before examining the sandwiched beam, it is necessary to look at the structural behaviour of single-pane glass beams first.

3. THE GLASS STRENGTH CONCEPT

3.1. Micro-macro scales

Glass strengths as given in Table 1 are to be situated at a microscopic scale, where the material strength depends above all on the quality of the molecular bonding. The scale of building components, on the other hand, is called “macro-scale”. Due to the production process, transport

ABSTRACT: Contemporary architects seem to strive constantly for more transparency in their designs. Even the primary load-bearingbuilding structure can be made of glass. Because of the slenderness of their cross-section, glass beams tend to collapse due to lateral torsionalbuckling. The critical buckling load is far below the theoretical critical load under simple in-plane bending. We examined the effectivenessof horizontal supports of the upper “flange” in order to avoid the negative effects of torsion in the beam. The dependency of the load-bearingcapacity on the lateral restraint of the supported upper “flange” is examined numerically. Both continuous and point-like, elastic lateralsupports along the upper fibres are examined. It is shown that the load-bearing capacity can be considerably increased by preventing theupper fibres of the beam from moving out of the beam’s plane. Conclusions are drawn on the efficiency of the different flange supportingconcepts and on the strength requirements needed for realising efficient connections, able to prevent torsion. The results of this study provideuseful data for a conceptual optimisation of the load-bearing capacity of glass beams and for the design of “flange” supporting connections.

Detail


and handling, glass at macro-scale shows unavoidable surface imperfections and invisible or nearly visible micro-cracks, known as Griffith flaws. [4, 5]

3.2. Statistics

If a glass surface is subjected to tension, crack tips act as stress concentrators. The value of the concentrated stress depends on the crack length, the radius of curvature of the discontinuity, and the initiated tensile stress. Crack propagation will occur if the energy released upon crack growth is sufficient to provide all the energy that is required for crack growth. Griffith flaws are randomly distributed across the surface of a piece of glass. The probability that a crack will propagate and glass will break equals the probability that a flaw or crack is situated at a critical stress location. The statistical distribution, which is mostly used to express this probability, has been developed by Weibull [13], although the relevance of this theory has been questioned by Calderdone [3]. In the literature we find values for the characteristic tensile strength of glass between 10 and 100 MPa, but laboratory experiments showed that values for the tensile strength could go up to 200 MPa.

3.3. Reinforcements

An improved resistance against tensile stresses can be realised by artificial prestressing. High compressive stresses can be induced on the glass surface by a thermal or chemical treatment [4, 7]. Depending on the degree of stress induced, the glass is called heat-strengthened, tempered or toughened (= fully strengthened).

3.4. Stress concentrations

It is clear that stress concentrations on the glass surface can cause it to break. For this reason, a direct contact of glass with hard materials such as steel or concrete is usually avoided by a soft interlayer (e.g. rubber).

4. BEAM THEORY

4.1. Geometry

In order to demonstrate our reasoning and to make results practical and easier to interpret, we choose to show our research starting from one example beam geometry. This beam geometry corresponds to the one of real single pane glass beams which have been (and will be) used in the Laboratory for Research on Structural Models of the Ghent University [9]. The beam consists of a single float glass pane with dimensions L=2200 mm, H=400 mm and t=10 mm. It is supported by two fork bearings, which allow rotations around the Y and Z-axis (Fig 2):

4.2. Buckling

The considered beams have a relatively small thickness compared with their height; the cross-section is very slender with an H/t ratio of 40. At the supports, any lateral movement or rotation about the longitudinal axis is prevented by the boundary conditions.

When the horizontal beam is subjected to an in-plane load (e.g. a bending moment at the supports or a vertical load at midspan), bending will appear in its vertical main plane. Bending stresses will occur, so that the outer material fibres at the top and at the bottom of the web will be subjected to compression and tension respectively. An increase of the load might cause the stress in the compressed side of the web to exceed a certain critical value, which causes buckling of the compressed fibres out of the beam’s plane. The cross section of the beam makes an out-of-plane movement combined with a rotation about the longitudinal axis. Since this rotation is restrained towards the ends of the beam, the structural element gets exposed to the effects of torsion. This is structurally unacceptable: even if the laterally deformed beam may still be able to carry its buckling load, lateral deflections and twisting deformations will get so large that they cannot be tolerated anymore. Moreover, additional out-of-plane bending stresses and torsional stresses tend to grow in an uncontrolled manner and may induce complete failure soon.

The whole phenomenon is known as “lateral torsional buckling”. The load at which buckling is initiated is called the critical buckling load.

Several authors have proposed expressions to determine the critical load for lateral torsional buckling of beams [6, 8, 9, 11]. A comparative study has demonstrated that the results of the different expressions show good agreement [8,9]. For that reason, it is sufficient for our purpose to focus on only one solution, for example the one given in Eurocode 3 [6]. The critical bending moment for lateral torsional buckling Mcr is given by:

g

.

g

y

t

y

wy

cr yCyCEI

GIL

I

I

L

EICM 2

50

22

2

3

2

1

4 (1)

The symbol It denotes the torsion constant, Iw is the warping constant, Iy

is the second moment of area about the minor axis, L=2200 mm is the length of the beam between points which have lateral restraints and yg=200 mm is the distance between the point of load application and the shear centre. Values of C1 and C2 are given in Eurocode 3 for various load cases [6]. For a point load P at midspan and for a uniformly distributed load along the total length of the beam, values are respectively C1=1.365 or 1.132 and C2=0.553 or 0.459. The obtained critical load Pcr is 9755 N for a point load at midspan and pcr=7.52 N/mm for a constant load. (The resultant of the constant load p in the given geometry is 16548 N).

4.3. Bending

Since both the glass beam and its load are in a vertical plane, we will discuss here the in-plane bending of the glass element.

From classical beam theory we can easily calculate the bending stresses at the outer fibres of the rectangular beam section for the critical loads as obtained above (cf. §4.2.):

MPa.W

M1220 for the point load (2)

MPa.0717 for the constant load

It is seen that the maximum in-plane bending stresses are very low when the beam is loaded with the critical load for lateral torsional buckling. On the other hand, if we suppose a typical characteristic tensile strength of 60 MPa for the glass (§3.2.), loads could go up to 29061 N for a point load at midspan and to 58182 N for a constant load. Characteristic breaking strengths can easily go up to 100 MPa [9], resulting in even higher allowable loads if only bending is considered.

The example above shows clearly that the load-bearing capacity of glass beams can be improved considerably (critical loads can get up to five times as high) if lateral torsional buckling can be prevented properly and the beam is forced to fail in simple bending. Lateral torsional buckling can be excluded if the compressed rim can be restrained laterally. The effects of reducing torsion in the beam by means of lateral restraints on the load-bearing capacity of glass beams are examined below.

5. NUMERICAL ANALYSIS

5.1. General

At present the main part of our study consists of a numerical analysis. Computer calculations are performed by the computer program Rasta, which is based on finite element techniques and has been developed at the Laboratory for Research on Structural Models by Van Impe [12].

Fig.2

Two main loading types are considered. The first is a constant loading over the full beam’s length with a value p=60 N/mm. The second is a concentrated load at midspan with the same resultant as for the uniformly distributed load case: P=60*2200=132000 N. Since prevention of lateral movement in a real construction can vary from zero (i.e. without restraints) to infinity (i.e. laterally totally immovable) -but usually lies somewhere in between-, we introduced lateral springs to support the compressed rim of the beam model.

Y p

X Loading type 1 Z P fork bearings

Loading type 2

L/4 L/4 L/4 L/4


Springs are placed at the compressed rim along the whole beam length in all simulations. The influence of the spring stiffness k (kN/cm) on the load-bearing capacity of the beam is examined numerically by varying its value during the simulations.

5.2. Load factor

The concept of a load factor is introduced in the following manner: it is the factor by which the basic loads (as determined in the previous paragraph) must be multiplied to determine the total load on the beam. The loads are supposed to stay vertical during the deformation of the beam.

5.3. Uniformly distributed load p=60N/mm

Three main cases with different boundary conditions are used, as illustrate by Fig 3.

The first case represents the beam, supported by a fork bearing at each end and by lateral springs at the compressed rim. It represents for example a simply supported glass beam which has its compressed upper rim silicon glued to a glass roof plate. The roof structure itself is supposed to be very stiff in its plane and flexible in the direction perpendicular to its plane.

Fig.3

The second case has a supplementary restraint, which doesn’t allow any out-of-plane movement of the compressed rim: it has only its Z-direction fixed at midspan. This model simulates the influence of e.g. an inextensional stabilising cable (Fig 3).

The third case has two such restraints, sub-dividing the beam in three equal lengths.

5.4. Observations

In general it is noticed that higher spring stiffnesses result in an important increase of the load factors. Applying lateral “spring” supports can seriously augment the load-bearing capacity of the beams in all cases. The results of the analysis are compiled in Fig 4.

For a beam without any lateral spring restraints (k=0) the load factor increases quickly when an increasing number of laterally immovable supports is added. This corresponds with buckling loads pcr1=7.20 and pcr2=17.98 pcr3=24.06 N/mm for the first, second and third case respectively. For beams without lateral springs, the addition of point-wise lateral supports has a positive effect on their load-bearing capacity.

Higher values of the spring stiffness (k > 0) result in a considerable growth of the load factor, especially for beams without additional lateral restraints (case 1). The graph of case 1 shows a very steep part for 0 < k < 0.2 kN/cm, indicating that an important improvement of the load-bearing capacity can be achieved even with relatively weak springs on a two-fork supported beam. The same effect is noticed for beams with one lateral restraint, but in a less pronounced way. The steeper part of the curve is situated in the region 0 < k < 0.3 kN/cm. For a beam with two or more lateral restraints, the curve is approximately linear and no steep part exists.

The lines of case 1 and case 3 touch and are virtually identical in the interval 0.2 < k < 0.4 kN/cm. For k > 0.4 kN/cm the difference between both cases is not very important either. It seems that the addition of two lateral restraints as in case 3 has a poor effect for values of the spring stiffness, which exceed 0.1. The addition of only one restraint at

midspan, on the contrary, has a positive effect on the load-bearing capacity, except for values of k around 0.15 and above 1.0 kN/cm, where the other curves are intersected.

5.5. Point load P=132000 N

For the point-wise loading type, only boundary condition cases 4 and 5 are examined (Fig 3). In analogy with the previous loading type, a meaningful difference is noticed in the initial load factors of case 1 and case 2 (k = 0). Here too, an additional lateral support increases the load-bearing capacity of the beam considerably: the addition of one lateral support at midspan of the compressed material fibres triples the critical buckling load. The reason is that the renewed boundary condition actually forces the beam into higher buckling modes, which correspond with higher eigenvalues or critical loads (Fig 5).

Fig.5

In contradiction with the constant load, both curves, which correspond to a point load, are very clearly separated and do not show any points where the curves are tangent to one another. Moreover, the difference between the corresponding load factor values of both curves remains very well pronounced for the total range of spring stiffnesses k. Since we wanted to understand why we found such an important influence of the additional lateral support here and not in the previous loading type, we checked some additional loading types, as described below.

5.6. Additional loading types

Two more loading types have been simulated (Fig 6) in combination with boundary condition case 2 (i.e. with one additional lateral restraint at midspan). The point load P in loading type 3 is positioned on a quarter length of the beam. For this position, maximum lateral deflection is expected for the second eigenmode, in which the beam is forced by its boundary conditions (Fig 5). In comparison with a load position at midspan (Loading type 2, Boundary condition case 2) load factors lie now below that previous curve (Fig 4).

L/2 L/2

Case 1,4

Y X Case 2,5

Z Case 3

L/3 L/3 L/3

Mode 1

Mode 2

Mode3

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2spring stiffness k (kN/cm)

loa

d f

act

or

Uniformly distributed load - case 1



Concentrated load - case 1

Concentrated load - case 2

Loading type 4 - case 2

Loading type 3 - case 2

Fig.4


In loading type 4, two times half the initial point load is placed at a quarter span. Spreading the load into two half loads P/2 gives a more favourable situation than concentrating it at one location, as we see the position of the corresponding curve lying far above the previous one.

Fig.6

For type 3, we notice that load factors are found lying in between the curves of one point load and a constant load over the full length. Recall that all loading types represent the same total force of 132000 N!

6. EXPERIMENTS

6.1. Test set-up

In order to simulate the theoretical boundary conditions as good as possible, the beam is supported by steel fork bearings, which mechanically allow translations and rotations as described in §4.1. At the moment of submission of this paper, loading types are limited to point loads. Loads are introduced in the system by placing masses of 1 and 0.5 kg on a balancing yoke, built around the beam. Lateral restraints of the compressed upper rim are realised by steel spheres, which are laterally fixed to the test set-up frame at both sides of the beam (Fig 7). Dynamometers are used to control lateral reaction forces in the restraints. Displacements are measured with electronic devices.

Fig.7

6.2. Aluminium

For the first experiments, aluminium is chosen as material for the scale models, because it is easier to produce specimens, and costs are relatively low. The Young’s moduli of aluminium and glass are very similar, so rectangular aluminium strips initially seem to be a valid alternative to examine instability problems of glass beams [9].

The results of the numerical analysis are confirmed by the experiments. However, not all cases have been experimentally tested at this moment. The research on glass specimens is in progress. For a detailed description, we refer to an oncoming report, which will be published in near future.

7. CONCLUSIONS

Analogous conclusions could be drawn from numerical analysis and preliminary experiments:

7.1. Lateral restraints

The addition of point-wise lateral restraints has a very important positive effect on the load-bearing capacity of a beam subjected to a concentrated load. For constant loads, the effect is significant only for cases with “weak” springs (k < 0.1 kN/cm). If the compressed rim is laterally

constantly supported by stronger springs, point-wise lateral restraints are not very useful. E.g. if glass beams are silicon-glued to the roof structure, stabilising cables could be omitted in the design.

7.2. Spring stiffness

Independently from loading type or boundary condition case, spring stiffeners on the compressed rim increase the beam’s load-bearing capacity. The beneficial effect of the springs on the critical load grows more rapidly at relatively low values of the spring stiffness than at relatively large values of the stiffness. Quite evidently there must be an asymptotic value of the lateral stiffness beyond which no further increase of the critical load factor is possible.

7.3. Loading types

In comparison to a local load, the load-bearing capacity of the beam is higher if the same load is spread out over the length of the beam. This was expected since the maximum bending moments are doubled for

local loads (44

2plPlMP against

8

2plM p ).

8. FUTURE RESEARCH

In near future more laboratory experiments will be executed on single pane glass specimens. Tests on large models (span of 2.2 m) will be performed.

The value of spring constants of the lateral supporting springs has to be interpreted and translated towards realistic supports. These realistic point-wise supports could e.g. be steel cables; realistic continuous supports could be realised by silicones.

A geometrical parameter study on the lateral torsional buckling load for beams with lateral supports is in progress, in order to generalise our conclusions to beams with other geometries.

9. REFERENCES

1. J.Belis: Glass Beams, Proceedings of the Second FTW PhD Symposium, Ghent, 2001 paper 7

2. J.Belis, R.Van Impe, G.Lagae, P.Buffel: Glass and transparent plastics: a structural engineering point of view, Proceedings of the 7th European Conference on Advanced Materials and Processes, Rimini, 2001 paper 777

3. I.Calderdone: The Fallacy of the Weibull Distribution for Window Glass Design, Proceedings of the Glass Processing Days, Tampere, 2001, pp293-297

4. H.Carré: Tempered glass, a new structural material (in French), Cahiers du CSTB, Cahier 3003, 1997

5. A.Griffith: The phenomena of rupture and flow in solids, Phil. Trans. Roy. Soc. of London, vol. 221, 1921

6. NBN-ENV 1993-1-1, Eurocode 3: Design of steel structures – Part 1-1: General rules and rules for buildings, Belgian Institute for Normalisation (BIN) vzw, 1992

7. The Institute of Structural Engineers, Structural use of glass in buildings, SETO, 1999

8. S.Pattheeuws: Calculation of lateral torsional buckling loads with finite elements method, Graduate thesis (in Dutch), Ghent, 2001

9. M.Roelandt: Lateral Torsional Buckling of Glass Beams, Graduate thesis (in Dutch), Ghent, 2000

10. S.Timoshenko and J.Gere: The Theory of Elastic Stability, 2nd edition, McGraw-Hill Book Company Inc., New York/Toronto/London, 1991

11. N.Trahair: Flexural-Torsional Buckling of Structures, E&FN Spon, 1993

12. R.Van Impe: Calculation of constructions with Rasta, manual v 3.0 (in Dutch), 1998

13. W.Weibull: A statistical Distribution Function of Wide Applicability, Journal of Applied Mechanics, September 1951, pp293-297

14. W.Young: Roark’s Formulas for Stress an Strain, 6th edition, McGraw-Hill Book company Inc., 1989

1,2,4,5) J.Belis, R.Van Impe, G.Lagae, W.Vanlaere, Laboratory for Research on Structural Models, Department of Structural Engineering, Ghent University, Technologiepark-Zwijnaarde 9, B-9052 Zwijnaarde, BELGIUM.3) F.Vernaillen, Department of Structural Engineering, Ghent University, Technologiepark-Zwijnaarde 9, B-9052 Zwijnaarde, BELGIUM.

Y P/2 P/2

X Loading type 3 Z P Loading type 4

L/4 L/4 L/4 L/4

22 chapter

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