exploratory investigation on the cold bending of...

Thierry Mainil

Exploratory investigation on the cold bending of thin glass

Academic year 2014-2015Faculty of Engineering and ArchitectureChairman: Prof. Dr. Ir. Luc TaerweDepartment of Structural Engineering

Master of Science in de ingenieurswetenschappen: architectuurMaster's dissertation submitted in order to obtain the academic degree of

Supervisors: Prof. Dr. Ir.-Arch. Jan Belis, Univ-Prof. Dr. Ing. Geralt Siebert (UniBW)Counselor: Captain Dipl.-Ing. Gordon Nehring

Confidentiality

This master dissertation contains confidential information and/or confidential research results proprietary to Ghent University or third parties. It is strictly forbidden to publish, cite or make public in any way this master dissertation or any part thereof without the express written permission of Ghent University. Under no circumstance this master dissertation may be communicated to or put at the disposal of third parties. Photocopying or duplicating it in any way other is strictly prohibited. Disregarding the confidential nature of this master dissertation may cause irremediable damage to Ghent University. The stipulations above are in force until the embargo date.

Confidential up to and including 01/01/2017 Important

Thierry Mainil


Academic year 2014-2015Faculty of Engineering and ArchitectureChairman: Prof. Dr. Ir. Luc TaerweDepartment of Structural Engineering

Master of Science in de ingenieurswetenschappen: architectuurMaster's dissertation submitted in order to obtain the academic degree of

Supervisors: Prof. Dr. Ir.-Arch. Jan Belis, Univ-Prof. Dr. Ing. Geralt Siebert (UniBW)Counselor: Captain Dipl.-Ing. Gordon Nehring

Foreword and acknowledgements

Glass is a fascinating material full of contradictions. It is very tough and durable, but a small scratch can make it brake. It separates and protects us from the outdoor conditions, but it is transparent, and has been defining the way our buildings’ outlook for multiple decennia. This ambiguity makes the process of designing with glass very interesting and never without surprises. Even though glass has not been the broad subject during the studies of architectural engineering, the material and its specific properties have drawn the attention of both, the architect and the engineer in me. This master thesis has given me the opportunity to research the material extensively with both, an experimental, and a theoretical approach. I would like to thank Prof. Dr. Ir.-Arch. J. Belis and Univ.-Prof. Dr. Ing. G. Siebert, for making it possible for me to do this research abroad and for guiding me in the right direction during this time. This experience has allowed me not only to develop my knowledge about glass, but also to get to know myself better. I would like to thank Gordon for the counseling and motivation until the very end. I would also like to thank Daniel and Robert, as well as the workers in the laboratory, for their advice and help with the experimental study. It would not have been the same without any one of them. I wish to thank my friends from the Oskar von Miller Forum for creating a pleasant working environment. In particular Mariano, Jan, Jaco, Christine and Phillipp for their support during the last weeks. I thank LiSEC for the generous contributions of time, knowhow and products, and also the Universität der Bundeswehr München for providing the space and financial means to perform the experiments. I am very grateful to my parents and brothers for believing in me and being there when I needed them. And last, but not least, I am very grateful for the unconditional and loving support of my girlfriend, who supports my dreams even if it means being apart for so long.

“The author gives permission to make this master dissertation available for consultation and to copy parts of this master dissertation for personal use. In the case of any other use, the copyright terms have to be respected, in particular with regard to the obligation to state expressly the source when quoting results from this master dissertation.”

January 19, 2015

Thierry Mainil

Abstract

In the underlying exploratory investigation, the most important factors for the cold bending of thin glass (t < 3 mm), have been mapped in an attempt to create an affordable and easy alternative for the production of doubly curved anticlastic shapes. In the first chapter, some properties and generalities of the material glass have been presented as well as the production of thin glass. Afterwards the focus shifts to the usage of glass in architecture. The second chapter is about the current bending techniques for glass. The advantages and disadvantages of warm, cold and lamination bending have been explained and two case studies of cold bending were discussed. In the third chapter, numerical analyses have been performed on quadratic (1 m x 1 m) and rectangular (1.1 m x 0.36 m) monolithic thin glass panes with ANSYS Workbench 15.0. This to have a basis to compare the experiments with and to investigate specific factors, which are not perceivable in reality, e.g. the internal stresses. Here, as well as during the experimental research, the shaping of the doubly curved form was integrated in the boundary conditions, being four point supported and two edge linear supported, and achieved by lifting one corner. The experiments were then described in chapter four. For that, the quadratic and rectangular thin glass panes were deformed while measuring the midpoint displacement and the edge displacement with displacement transducers, and the maximum normal stresses with strain gauges. This was done until a corner displacement of 105 mm and that for both the bearing principles and in a vertical and horizontal direction. In that way, the influence of the own weight could be perceived and the horizontal setup allowed loading the pane to observe the form-activation. Before starting the conclusions, a small example design was made to present the possibilities of thin glass. In the interpretation from the numerical analyses and the experimental tests, it was noticed that the most influential factors were the boundary conditions and the width-to-length ratio of the glass pane. It could be concluded that a two edge linear supported pane allows greater corner displacements as a four point supported pane. In addition, a decreasing width-to-length ratio results in an increasing peak midpoint displacement. Next to that, it was also seen that a stiffening effect takes place when twisting the pane. Keywords: Cold bending, thin glass, anticlastic double curvature, numerical analysis, experimental research


Thierry Mainil

Supervisors: Prof. Dr. Ir.-Arch. Jan Belis, Univ.-Prof. Dr.-Ing. Geralt Siebert, Captain Dipl.-Ing. Gordon Nehring

Abstract – This article describes the most important factors for

the cold bending of thin glass (t < 3 mm). By mapping the influential factors it is attempted to develop an affordable and simple alternative for the production of doubly curved anticlastic shapes.

Keywords - Cold bending, thin glass, anticlastic double curvature, numerical analysis, experimental research

I. INTRODUCTION

The development of computer aided design technologies in the past twenty years has led to an increased freedom of forms in contemporary architecture. It makes the use of curved glass in building applications more than ever favored. Additionally, it facilitates the creation of unique free-form facades that are characterized by a combination of aesthetic appeal, transparency and use of natural light within buildings.

Cold bending is an energy efficient method to construct curved glass panes. It is based on the elastic deformation of glass combined with applications of out of pane loads to construct the required shape. The deformations still remain limited though. Nevertheless, the limitations can be diminished by applying thin glass (t < 3 mm). Since internal stresses created by cold bending depend on the pane thickness, thin glass can enable larger deformations. The result is more freedom in architecture [1].

The internal stress is not the only factor that needs to be considered. Staaks has already reported on instability as well as on deformation modes in terms of forcing one corner out of the plane to create a hypar surface. In the first mode, a curved shape characterizes the diagonals and the edges preserve its initial shape. However, if the out of plane displacement of the corner is larger than 16.8 times the pane thickness, the plate buckles. It causes instability at the point where one diagonal straightens and the edges will become curved [2].

In this study, the influential factors of cold twisting thin glass have been investigated for multiple pane sizes and multiple boundary conditions. Experiments were conducted in order to create an affordable and simple alternative for the production of doubly curved shapes.

II. NUMERICAL INVESTIGATION

A geometric non-linear finite element analysis was performed with the ANSYS Workbench 15.0 software. The results for cold twisting normal glass (t > 3 mm) found in literature are compared to the behaviour of thin glass. A parameter study has been executed to evaluate the comparison.

T. Mainil is a student with the Faculty of Engineering and Architecture at

Ghent University (UGent), Gent, Belgium. E-mail: [email protected]. .

Next to that, the behaviour of cold bending thin glass has been investigated for multiple boundary conditions. This kind of investigation was necessary to monitor the factors that cannot be perceived during the tests (e.g. the internal stresses). In addition, the investigation is proficient to validate the experimental research.

A. Parameter study

A four point supported, quadratic shell model was constructed, as it can be seen in Figure 1. A downward imperfection load was added to the model in order to influence the buckling direction and to create consistent data.

Figure 1 Four point supported, quadratic basis model in ANSYS Workbench.

After that, the model was modified to examine the influence of thickness, size and width-to-length ratio of the glass pane and the boundary conditions that support the pane.

B. Thin glass analysis

Since not everything can be observed in the experimental study, a deeper analysis about the behaviour of thin glass during buckling was conducted. For that, the movement of the free edges, the middle axis and the diagonals were observed as well as the changes in the membrane stresses.

C. Results

From the numerical analyses it can be concluded that the critical corner displacement has many influence factors. It cannot be simply summarized in a thickness depending factor of 16.8. Many other factors have to be taken into account.

The boundary conditions as well as the pane’s width-to-length ratio are two factors that are essential. Both have an impact on the behaviour of the pane and they affect the critical corner displacement. Furthermore, it can be noticed that a two edge linear supported pane allows greater corner displacements as a four point supported pane. Additionally, a decreasing

width-to-length ratio results in an increasing peak midpoint displacement.

Subsequently, it was observed that a pure hyperbolical paraboloid could not be made, since it is an unwindable shape. Therefore, the free edges display a slight S-shaped curvature, even for the slightest corner displacement. In the case of its prevention by the linear support, it results in higher stresses in the pane.

Looking at the effects of the buckling on the shape and the internal stresses, it was noticed that in the supported diagonal BD a plain arose after buckling. The plain can also be perceived in the principal stresses that indicate an increasing compressive zone in the middle of the pane. Next to that, the membrane stresses undergo a shift from a double symmetric distribution around the middle axes to a more diagonally oriented distribution.

III. EXPERIMENTAL STUDY

The stability of cold bent thin glass was also a subject within the experiment. Two test setups were built for that, being a four point supported and a two edge linear supported setup, as it can be seen in figure 2. Both designs allowed the lifting of one corner to create the desired doubly curved anticlastic shape for multiple pane sizes. In addition, they were equipped with displacement transducers to measure the midpoint displacement and displacement of the middle of the supported edge AD.

Figure 2 Boundary conditions of the test setups, four point supported (left) and two edge linear supported (right).

All performed tests concerning the research were executed in the laboratory of the Universität der Bundeswehr München in a controlled environment with a constant temperature of 20.5°C.

A. Monolithic experiments

In these experiments, two pane sizes were tested: a quadratic 1 m x 1 m and rectangular 1.1 m x 0.36 m (w x l) shape, both with a nominal thickness of 2 mm. This was carried out to monitor the effect of the width-to-length ratio.

After the physical and laser-optical measurement of the entire test specimens, strain gauges were applied on one of each kind to examine the normal stresses generated by the cold bending.

At that point, the testing sequence can be started for both setups and both pane sizes. This meaning the lifting of one corner for a total of 105 mm, divided in steps of 5 mm. After each step, the measured data was noted to be compared with the numerical analysis. The tests were conducted in a vertical and a horizontal direction not only to be capable of examining the influence of the own weight but also to apply a known load onto the structure. Based on the deflections it can then be defined if the cold bending had an effect on the pane stiffness or not.

B. Laminated experiments

After defining all pane properties, the identic tests have been performed for the laminated thin glass panes as for the

monolithic panes. It includes also the quadratic and the rectangular panes. Two differences have to be mentioned though. First, no strain gauges were applied on the panes. Because of that, the normal stresses could not be measured during the sequence. Second, two different PVB thicknesses (0.76 mm and 1.52 mm) were available for the rectangular laminates. It enabled a brief analysis of the interlayer thickness during cold bending and more interesting during the loading when bent.

C. Results

For the experiments, the same conclusions can be made as for the numerical study, which validates both. Firstly, linearly supported edges instead of solely point fixed corners allowed a greater corner displacement before buckling. Secondly, the influence of the width-to-length ratio can be seen. The rectangular panes permit greater corner displacements than the quadratic ones. In addition, it can be noticed that the maximum normal stresses are lower for the four point supported panes as for the two edge linear supported. Further on, the horizontal two edge linear supported tests presented that a form-activation was achieved and the deflections for known loads decreased for increasing corner displacements, although the shape was never a perfect hypar surface. The same results may be expected for the vertical setups as long as the buckling point is not reached, even though it was not explicitly tested. From this point on, the pane was not stable anymore and hence it was not capable of carrying loads. Simply focusing on the shape, the tests with laminated thin glass demonstrated the best results. In terms of the vertical position and linearly supported at two edges, a shape that was as close as possible to a hypar surface, was formed for both the quadratic and the rectangular panes.

IV. CONCLUSIONS

The application of cold twisted thin glass is able to form an attractive alternative for warm bending in the field of doubly curved architecture. The experimental and numerical research resulted in a valid model to predict the pattern of deformation, which can be used for design purposes. The key influence factors that were detected in both the studies were the boundary conditions and the width-to-length ratio of the glass pane.

The most promising result for shape forming was achieved with the laminated panes. There, the primary deformation created an almost perfect hypar surface. If these panes are loaded with a constant load in a horizontal setup decreasing deflections were perceived for increasing corner displacements. This proves the form-activation of the shape.

ACKNOWLEDGEMENTS

The author would like to acknowledge the suggestions of Prof. Dr. Ing.-Arch. J. Belis, Univ.-Prof. Dr.-Ing. G. Siebert and Captain Dipl.-Ing. G. Nehring during the research project. A lot of gratitude is also shown to the company LiSEC for providing the test specimens.

REFERENCES [1] Arend, S., Untersuchung zum Tragverhalten von Schalen aus Dünnglas,

Master thesis, Universität der Bundeswehr München, 2014. [2] Staaks, Koud torderen van glaspanelen in blobs, Master thesis,

Technische Universiteit Eindhoven, 2003.

Table of content

1. Glass as a building material ...................................................................................... 1

A ]Overview ............................................................................................................ 1

B ]Composition and structure ................................................................................ 3

C ]Production of flat glass ...................................................................................... 4

D ]Structural application of glass ........................................................................... 6

E ]Free-form design ............................................................................................... 7

2. Bending of glass ....................................................................................................... 9

A ]Overview ............................................................................................................ 9

B ]Current technics................................................................................................. 9

B.1 ] Warm bending ......................................................................................... 9

B.2 ] Cold bending ........................................................................................ 11

B.3 ] Lamination bending .............................................................................. 13

C ]Existing designs ............................................................................................... 14

C.1 ] Single curvature .................................................................................... 14

C.2 ] Double curvature ................................................................................... 17

3. Numerical investigation ........................................................................................... 19

A ]Overview .......................................................................................................... 19

B ]Fundamentals of the finite element method .................................................... 20

C ]ANSYS Workbench .......................................................................................... 21

D ]Models in ANSYS Workbench ......................................................................... 22

D.1 ] Basic model .......................................................................................... 22

D.2 ] Convergence study ............................................................................... 25

E ]Parameter study .............................................................................................. 27

E.1 ] Thickness .............................................................................................. 27

E.2 ] Boundary conditions ............................................................................. 29

E.3 ] Size ........................................................................................................ 31

E.4 ] Shape ratio ............................................................................................ 32

F ]Analyses for experiments ................................................................................. 34

F.1 ] Free edges ............................................................................................ 34

F.2 ] Diagonals .............................................................................................. 35

F.3 ] Middle axis ............................................................................................ 36

F.4 ] Membrane stresses ............................................................................... 37

G ]Summary.......................................................................................................... 40

4. Experimental study .................................................................................................. 41

A ]Overview .......................................................................................................... 41

B ]Preliminary tests............................................................................................... 41

C ]Method ............................................................................................................. 43

C.1 ] Pane properties ..................................................................................... 43

C.2 ] Test setup properties ............................................................................ 48

C.3 ] Test setup adjustments......................................................................... 52

D ]Square experiments ......................................................................................... 54

D.1 ] Four point support ................................................................................ 54

D.2 ] Two edge linear support ....................................................................... 59

E ]Rectangle experiments .................................................................................... 63

E.1 ] Four point support ................................................................................. 63

E.2 ] Two edge linear support ....................................................................... 66

F ]Laminated glass experiments .......................................................................... 69

F.1 ] Pane properties ..................................................................................... 70

F.2 ] Square experiments .............................................................................. 71

F.3 ] Rectangle experiments .......................................................................... 74

G ]Summary.......................................................................................................... 78

5. Application of thin glass .......................................................................................... 80

A ]Overview .......................................................................................................... 80

B ]Concept ........................................................................................................... 80

C ]Design .............................................................................................................. 81

6. Conclusions and recommendations ....................................................................... 84

A ]Conclusions ..................................................................................................... 84

B ]Recommendations .......................................................................................... 85

Bibliography .................................................................................................................... 86

Appendix ......................................................................................................................... 88

A ]Deformation methods ...................................................................................... 89

B ]Laser-optical measurements ........................................................................... 90

C ]Point fixing details ............................................................................................ 91

D ]Displacements transducer details ................................................................... 92

Abbreviations and symbols

APDL ANSYS parametric design language

CAD computer aided design

CNC computer numerical control

CSG chemically strengthened glass

DIN German institute for standardization

E Young's modulus

EPDM ethylene propylene diene monomer

FEA finite element analysis

FEM finite element method

FTG fully tempered glass

GUI graphical user interface

HO horizontal

HP hyperbolical paraboloid

HSG heat strengthened glass

I moment of inertia

IGU insulated glass unit

Ncr,K critical buckling load

OSB oriented strand board

PV photovoltaic

PVB polyvinyl butyral

PVC polyvinyl chloride

RE rectangle

SCALP scattered light polarimeter

SQ square

t thickness

TVG Teilvorgespanntes Glas (see HSG)

VE vertical

w/l-ratio width-to-length ratio

x, y, z Cartesian coordinate system

ΔZBuckling critical corner displacement

ν Poisson’s coefficient

σn normal stress

1

1.

Glass as a building material

A ] Overview

Glass is a material, which is commonly known as a transparent separating material. It

has achieved (the last decennia) an ever more growing position in the construction

industry. The introduction of safety concepts such as lamination and thermal hardening

insured the stability after fracture. They allowed the construction of complete glass

structures that have already proven their capabilities over the last years. At that moment,

full transparency changed from a marvellous dream to an unbelievable reality. Due to the

improving technologies in CAD, architects got more freedom in their design. Powerful

software allows to design, visualize and plan complex free forms. However, where the

classical building materials such as concrete and timber have adjusted to those shapes,

glass still lags behind.

The large amount of curved glass in our daily life as for example in automotive industry

or shop windows demonstrates that free-forms are also possible with glass. Although

aesthetic and constructive advantages could come forward, the main technique of

creating curved glass facades is by approximating the structure, e.g. by triangulation.

That way, straight panes can easily be installed onto the supporting structure, which has

advantages both on production costs and on installation time. An example of this is the

Mercedes Museum in Stuttgart, where the facade exists of large flat panes, although the

overall shape feels like a curvature (see Figure 1.1).

Glass as a building material 2

Figure 1.1: Mercedes Museum, UN Studio, Stuttgart.

Figure 1.2: ILEK glass dome, L. Blandini, Stuttgart.1

1

Looking at the way how curved glass could behave, this does not have to be. By creating

a curved shape, the bending stresses in the glass can be reduced. Knowing that glass

is stronger in compression as in tension due to the flaws on the surface, the reduction of

the bending stresses and therefore also the tensile stresses results in a positive influence

on the structural strength of the glass construction. This allows the underlying bearing

structure to be lighter. This can go as far as to the point that no substructure is needed

anymore and the glass carries itself, as has been shown with L. Blandini's glass dome

(see Figure 1.2). From an aesthetic point of view, a real curve, with a lighter substructure

is more interesting not only for the overall view on the building, but also for the reflections

that appear more natural.

The current production technologies for curved glass are already on a high level and

makes it possible to bend glass panes without derogation to constructive or aesthetic

qualities. Current techniques, are warm bending, cold bending and lamination bending,

and will be explained in the next chapter. Although, they allow more freely shaped forms,

they are often limited to a single curvature and can correspond with high production

costs.

In the underlying master thesis a numerical and experimental research is made on the

behaviour of thin glass (t < 3 mm) during cold bending. It aims to give a possible answer

to an easy and relatively inexpensive way of producing double curved surfaces. Thin

glass, already known as the middle pane for triple glazing and the cover glass for solar

modules, has the advantage of creating less internal stresses during cold bending

1 Schittch, C., Staib, G., Balkow, D., Schuler, M., Sobek, W., Detail: Glass Construction Manual, Basel, Birkhäuser Verlag AG, 2007, p. 94.


compared to standard glass thicknesses, and therefore appears to be perfect for curved

applications [1].

The goal of this research is to determine the architectural, constructive and production-

technical regularities of shaping heat strengthened thin glass during cold bending or cold

twisting. Based on the results, practical guidelines, rules and boundary conditions are

composed for its structural application. Starting with a flat pane, an investigation about

the possibilities for curving differently shaped panes with various supporting ways is

explored. Additionally, an example of the practical application and installation is

suggested. The crucial points are the size and the shape of the structure and how it could

be formed in reality.

In the following, some overall properties and generalities on the material glass are

described. Afterwards some examples of the use of bent glass in current architecture are

shown.

B ] Composition and structure [14]

In its most principal form, glass purely consists out of silicon oxide and is formed by the

cooling down of the molten substance. The high melting temperature of the material

(2300°C) unfortunately comes along with high production costs, which restricts the scope

of applications. A better processability along with a lower melting point (1500°C) is

created through the addition of sodium oxide (Na2CO3). But instead it loses a part of its

chemical resistance. Therefore, lime (CaO) is added to the mixture, which compensates

the effect of the soda and helps to create a hard and indissoluble material. These three

elements form the main substances for soda-lime-silica glass, which will be the subject

of this research. It is also the most commonly used glass in the construction industry.

Next to these, some other substances can additionally be associated to influence

specific properties and/or change the colour of the panes. The different components and

their ratios are mentioned in Table 1.1.


Component Soda-Lime-Silica Glass SiO2 69-74% Na2O 12-16% CaO 5-12% MgO 0-6% Al2O3 0-3%

Table 1.1: Glass composition.

Figure 1.3: Molecular structure of glass. 2

2

In contrast to other solid materials that solidify in a perfectly structured arrangement of

molecules, glass cooled down from the fluid state does not change to a crystalline phase,

as it can be seen in Figure 1.3. The molecules of glass, called SiO4-tetrahedra, remain

loose elements that share an oxygen atom. Due to the Si-O-Si bridges only a three-

dimensional network structure is formed instead of a logical molecule grid. The achieved

amorphous state, which is then reached, is characteristic for liquids. It is responsible for

the transparency, but also makes it very vulnerable. Flaws are easily formed on the glass

surface, which reduces the theoretically strong tensile strength of the material.

C ] Production of flat glass

People have been fascinated by glass already for centuries. The application of this brittle

transparent material in architecture generated a search for transparency, which was

powered by technical progress, but even more by technical limitations. The limitations

related to the production (e.g. limited pane sizes) still determine the development in the

glass sector.

When glass was first discovered many centuries ago, it had always been linked to a

certain status. Because of its unknown combination of transparency and hardness it was

a very sought-after material, which was most commonly used for jewelry in the beginning.

For the first application in architecture we have to go back to the Roman times, where

Vitruvius wrote that this material possessed the three necessary qualities for good

architecture. Small glass windows combined the properties of functionality (utilitas),

durability (firmitas) and beauty (venustas), although they were still of bad quality. As the

2 Wurm, J., Glas als Tragwerk: Entwurf und Konstruktion selbsttragender Hüllen, Basel, Birkhäuser Verlag, 2007, p. 36.


workability improved, the optical quality of glass also improved and when eventually

ovens were used for the production, the transparency became better and better.

The development of glass blowing led to the creation of new technics that resulted in the

production of larger glass plates. What started from blowing spheres, that were twisted

into circular glass panes, went onto blown cylinders that were cut and rolled open to

create larger flat panes so that bigger windows could be made. This had a significant

effect on architecture at that time, which was less and less defined by mass. But it led,

combined with the increase of knowledge in other sectors, to a more transparent

architecture where the openings now defined the buildings’ outlook. This transparency

created a new contact to the outside world and enabled a lot more daylight to come in.

Of course, today flat glass is not produced like this anymore. The float glass production,

also known as the Pilkington process was invented around the 1950s and is now still the

current standard for glass production. In this process, multiple phases are connected

within a production line that can reach a length of over 500 meter (see Figure 1.4).

Figure 1.4: Principal steps in the manufacture of float glass.3

Generally, continuously operating ovens melt the raw materials to glass. Afterwards, the

viscous mass flows over a bath of molten tin, which gives the glass a perfectly flat

surface. The ribbon comes out of the bath with a natural flow thickness of around

6.8 mm. Pulling or pushing the borders allows it to adjust the thickness to the required

size. Subsequently, it is cut to the desired size up to a maximum of 6.0 m x 3.2 m (l x w),

which are the so called jumbo panes.

This is not the way thin glass is produced though. Although thicknesses as low as

2 mm can be produced with the float glass procedure, the glass that will be used in the

3 Schittch, C., Staib, G., Balkow, D., Schuler, M., Sobek, W., Detail: Glass Construction Manual, Basel, Birkhäuser Verlag AG, 2007, p. 61.


master thesis is produced in a different way, called the down draw process. In this

process, the molten glass is pulled down vertically as it can be seen in Figure 1.5. The

key advantages are thickness control, that is by norm the same as float glass (± 0.2 mm

for thicknesses up to 4 mm, but in reality it is only ± 0.15 mm), and the fact that the glass

has two equal surfaces. This is not the case for float glass, where a tin side and an airside

surface are distinguishable. Within this process, (thin) glass panes with a maximum width

of 1.4 m and a length of 10 m can be produced [15].

Figure 1.5: Principle of the down draw process.4

D ] Structural application of glass

Since the 1950s glass has been used as a structural material itself. The invention of safety

concepts, that assure the stability after fracture, enabled the development of innovative

load-bearing glass structures. By thermal hardening of glass panes the tensile strength

improved and the glass shards got smaller and less sharp, compared to the fractured

conventional glass.

Another safety concept consisted of the connecting of multiple panes, together with an

adhesive synthetic foil. This process of lamination allows the construction to be stable

after fracture and avoids the falling down of splinters. After that, glass beams and

columns were no concept anymore. Nevertheless, they formed the structure as for

example for the New York Apple Store on Fifth Avenue, where full transparency was

achieved (see Figure 1.6).

4 Schott AG, http://www.schott.com/xensation/english/products/look/Production.html, (accessed Oktober 2014).


Although glass has evolved into a universal building material by its transparency, this is

not always desired. How can a total transparent building be architecture? How does

glass architecture relate to more traditional construction methods? These are questions

that are inseparable from designing with glass, a material that can add an interesting

contribution to architecture just by its limitations and contradictions.

An example for this interaction with the transparency of glass could be the glass house

by P. van der Erve in Leerdam. Here the entire house was made out of glass panes. This

creates a surprising effect because of its combination of transparent elements for

separating components (see Figure 1.7). While moving, the glass walls change from

translucent to almost see-through and vice versa. It also creates an interesting shadow

play during the day.

Figure 1.6: Fifth Avenue Apple Store, Bohlin Cywinski Jackson, New York.5

Figure 1.7: Glass house,

P. van der Erve, Leerdam.6

56

E ] Free-form design

When looking at the current practice of the application of glass it can be seen that most

of the structures are made out of connected flat glass panes. For a free-form design,

triangulation is the most common construction method. It shows that the material still has

certain limitations. Mainly high production costs, technical limitations from the production

and a lack of appropriate safety concepts are responsible for that. Many examples can

5 Apple, https://www.apple.com/retail/fifthavenue/, (accessed July 2014). 6 Schleifer, R., Architecture materials: glass, Köln, Evergreen, 2008, pp. 156-167.


be found for this, but the roof of the British Museum designed by N. Foster, as can be

seen in Figure 1.8, might be a well-known one.

Figure 1.8: British museum, Foster + Partners, London.7

It can be clearly seen that a curved form is constructed with glass triangles. What is very

remarkable is that although the bearing structure is designed rather light, the main image

is still determined by it.

Compared to other materials, glass is still a bit left behind if talking about double curved

surfaces. There are already some examples, but it has not yet reached the level of timber

or concrete. Purely from a geometric point of view, the constructions, known from Oscar

Niemeyer for example, are far ahead on what is capable with glass nowadays (see Figure

1.9).

Figure 1.9: National congress building, Oscar Niemeyer, Brasilia.8

There are some exceptions though; it exists more than a handful of structures that are

reaching the limits of glass and define complex, curved geometries, as will be

demonstrated in the case studies in the following chapter.

7 Foster & Partners, http://www.fosterandpartners.com/projects/great-court-at-the-british-museum/, (accessed July 2014). 8 BBC, http://www.bbc.com/news/in-pictures-20266899, (accessed Oktober 2014).

9

2.

Bending of glass

A ] Overview

Bending glass is not a novelty anymore. It can be noticed, that the industry produces

curved glass and establishes standards for it. By giving flat plates a specific (well thought

over) curvature, geometries with greater stiffness and aesthetic qualities are formed.

Additionally, in the context of free-form architecture, curved glass has an ever more

increasing value.

The production of the curved forms can be executed in multiple ways that have their

specific advantages and restraints. For instance the bending radii will always reach

limitations. In this chapter the different production methods, together with their

advantages and disadvantages are scrutinized and explained.

B ] Current technics [16]

A brief focus on the existing production methods is made in the following. The described

processes are those that are common and the most used by now. Later on, the

advantages and disadvantages of cold bending will be explained more deeply by case

studies of existing buildings.

B.1 ] Warm bending

This process enables the biggest geometrical freedom. Warm bending can be described

as the way of shaping glass by heating. Flat panes are placed on a mold of steel or a

ceramic material that will be placed in an oven and heated. The temperatures for the

process vary between 600 – 650 °C. Within this range, glass loses its brittle character

and turns in a rubbery state, where deformations can easily be applied to the panes.

When the glass is cooled down it returns to its amorphous solid state and keeps its now

permanent deformation.

Bending of glass 10

The deformation is typically done through gravity, i.e. the own weight is responsible for

the curving around the mould. This technique makes it possible to receive good results

for smaller deformations on a big scale. For bigger deformations it is not always sufficient

though. For a single cylindrical deformation, for example, minimal bending radii of 2 m

are attainable.

The application of weight on top as well as a vacuum under the glass can be used to

help creating smaller radii. Another technique is the mechanical forcing of the pane into

the mould. It enables larger deformations that are only depending on the thickness of the

glass pane.

However, the big formability has some economical drawbacks. The energy needed for

the heating and the production of the mould makes the process more expensive than

other production ways. Furthermore, the transportation of the panes is rather inefficient,

because the panes are not flat anymore and hence shipping requires a lot more effort.

Next to that, extra costs for producing replacement panes after breakage arise easily.

A single cylindrical curvature is already currently applied. The moulds are easy to

produce and can be reused, which decreases the costs, when repetitive elements are

being produced. For more complex geometries CNC-machines are used to create

custom made moulds out of heat resistant material. The additional costs for this are

disadvantageous for the economic feasibility and make the process very expensive and

time-consuming.

There are already experimental applications where reusable moulds are designed. The

shape-forming surface here is made from parallel bars that can be adjusted in height at

the ends (see Figure 2.1).

Bending of glass 11

Figure 2.1: Bending and tempering line at a pivoting roller bending plant without forms.9

B.2 ] Cold bending

As the name already suggests, this process is based on cold bending panes. Straight

panes are temporarily deformed and are fixed to a bearing structure, so it remains in the

predefined shape. Because the pane is forced into a shape, this deformation is coupled

with permanent internal stresses.

While glass has a brittle character and is quasi linear-elastically deformable with limited

maximal tensile strength, it is impossible to deform standard annealed glass panes with

large deformations without fracture. Tempering is necessary, to be capable of deforming

flat glass to a useful extent. This can be achieved in two ways: thermally or chemically,

and allows bigger deformations due to the higher resistance to tensile stresses. The

biggest deformations can be obtained with thermally tempered glass. Here the glass is

heated first to around 600°C and is then quenched with cool air (see Figure 2.2). The

outcome is that the outside cools down faster than the inside, which has as a final result

that compressive stresses are created in the outer layer (and tensile stresses in the

center).

Figure 2.2: Manufacturing steps for tempering flat glass.10

9 Wurm, J., Glas als Tragwerk: Entwurf und Konstruktion selbsttragender Hüllen, Basel, Birkhäuser Verlag, 2007, p. 36. 10 Wurm, J., Glas als Tragwerk: Entwurf und Konstruktion selbsttragender Hüllen, Basel, Birkhäuser Verlag, 2007, p. 54.

Bending of glass 12

This quenching can be done at different speeds to get a different level of tempering.

Typically, only two specific speeds are used to produce the so called heat strengthened

and fully tempered glass (respectively HSG and FTG) (see Figure 2.3). If then the panes

are bend, the created compressive stresses will be compensated by the introduced

tensile stresses.

Figure 2.3: Typical stress diagrams for different tempering processes, from left to right: FTG, HSG and CSG.10

Another method consists of inserting the glass pane into a hot salt bath. This is called

chemically strengthened glass (CSG). There, the pressure zones are significantly smaller,

because the process relies on the exchange of the sodium ions of the glass surface with

larger potassium ions from the salt bath. This leads to compressive stresses at the

surface. The glass demonstrates a high resistance to mechanical and thermal loads [19].

Because during cold bending the panes are often already deformed in such a way that

60 percent of the allowed bearing strength is reached, the overall capacity of these panes

is by definition lower than warm bent glass, but because of shape advantages this does

not have to be true for the absolute capacity.

The permanent nature of this way of deforming requires the usage of tempered glass of

high quality and long durability. Furthermore, it is also necessary to have a stronger

carrying structure to keep the glass under tension and thereby in shape. Most typically,

the installation takes place in multiple stages. First of all, the deformation is mechanically

applied. This can be achieved in the factory or on site. The panes are then mounted on

the bearing structure and after that they are permanently connected to the specific

substructure.

Bending of glass 13

A big advantage of this technique is that the glass panes can be delivered flat to the site.

It is even more interesting that it is not necessary to create moulds and that there is no

need for high temperatures. However, there is a need for a stronger bearing structure to

fix the glass on.

In general, the applied deformation is almost always a single curvature, whereby the

maximum radius of the deformation is linked to the thickness of the glass and the amount

of tempering. But also the type of interlayer plays a big role in the bending radius, as are

the pane dimensions and the bearing conditions. Single curvature is however, not the

only possibility. There exist already some cases with a double curvature, e.g. the railway

station in Strasbourg, designed by J.-M. Duthilleul (see Figure 2.4 and 2.5).

Figure 2.4: Railway station, J.-M. Duthilleul, Strasbourg.11

Figure 2.5: Interior view of the railway station.11

11

Structurally it is very interesting, but it has also some drawbacks. For example, the

amount of precision necessary for creating normal reflections is very high. Distorted

mirroring is a typical problem for double curved glass facades.

B.3 ] Lamination bending

Lamination bending can be described as a combination of cold bending and lamination.

This kind of bending is performed by forcing multiple glass panes with interlayers into a

fixed form and beginning the process for constructing laminated glass. This means

taking the bent bundle into the autoclave, letting it undergo a pressure and temperature

cycle, and afterwards cooling it down before the shape forcing element is relieved.

11RFR group, http://www.rfr-group.com/en/projects/location/project-singleview, (accessed July 2014).

Bending of glass 14

After the substructure is released, the bundle will try to straighten itself again, but due to

the shear resistance of the interlayer, it will be prevented. The result is that the interlayer

is set under a great shear stress and is now responsible for retaining the shape of the

panes. For this reason it is very important that the interlayer is sufficiently stiff and durable,

as e.g. SentryGlas® interlayers, so that the spring-back is as small as possible and the

shape will remain over time. This in combination with a good bearing system results in

the closest fit to the desired shape.

The major advantage of this method in relation to cold bending is the form freedom it

has. Since the panes are bent as thinner monolithic single panes and not as a laminated

group of panes, there is a bigger bending potential. The disadvantages are quite similar

to the warm bending method. This means the process is rather expensive, time

consuming and the transportation is relatively inefficient. In addition, it is also difficult to

create the exact shape due to the spring-back.

C ] Existing designs

This sections deepens the application of cold bent glass by analyzing existing buildings.

Although this technique is not yet widely used, several interesting constructions can still

be found. It can be indicated that in the current practice glass defines the outlook of the

building, not only within the facade, but also by generating reflections.

C.1 ] Single curvature

Although bent glass is still rather rare, if the most common shape has to be defined it

would have to be a single curvature. The starting point for this are the arcs and shells that

are known from concrete structures. They are capable to carry loads more efficient by

their shape optimization. In these geometries the goal is to develop such a form that the

forces are transferred by normal stresses and to reduce the bending stresses to an

absolute minimum [10]. Because of the shell effect, the structure only has to resist to

compressive normal stresses, where concrete as well as glass have advantages.

In reality, creating a specific shape in which no bending stresses are created is rather

uncommon and mostly the glass is curved in a cylindrical shape. Here the bending radius

is constant over the whole pane. This implies that the structure will always have to cope

with unavoidable tensile stresses.

Bending of glass 15

The stress distribution is rather different compared to flat glass panes. For flat glass

panes, the bending stresses generate peak values for tensile stresses at the free edges.

Already from the manufacturing of the glass panes many imperfections are created on

these edges, which make them sensitive for tensile stresses and can generate crack

initiations. Compared to a cylindrical glass pane, the bending stresses are lower for a

similar load, because the shape provides the structure with a bigger moment of inertia.

Simplified, it means that a smaller glass thickness would be capable of carrying the

identical load.

Esthetically the single curvature has a big influence, too. Where curved facades are used

to be approached with fragmented straight elements, the reflections are not fragmented

anymore, but even.

The technics involving safety and production for single curved glass are also on a high

level. This allows architects to implement them easier and it can be displayed that the

scope of application is broadening evermore. The application of thin glass implies that

the curvature can be stronger as before, since the internal stresses created by cold

bending in a single curvature are depending on the thickness as is proven in the following

[1].

Figure 2.6: Lisec four point bending test for 2 mm HSG.12

12 LiSEC, http://www.lisec.com/Innovation/Flexibles-D%C3%BCnnglas/, (accessed Oktober 2014).

Bending of glass 16

Calculating with an unknown thickness t and a known width of 1 m, the internal bending

stresses can easily be defined for different bending radii R. For that the bending moment

M is defined in equation 2.1:

M E ∙ I ∙ κwithκ 1R

with:

- M Bending moment

- E Young's modulus

- Iy Moment of inertia

- R Bending radius.

With the formulas, the maximal normal stress σn can be defined as following:

σ MW

E ∙ I

R ∙ W

E ∙b ∙ t12

R ∙b ∙ t6

E ∙ t2 ∙ R

with:

- Wy Section modulus.

By filling in the maximum stress we want in the glass in formula 2.2, it can be shown that

for thinner glass thicknesses, the bending radius can be smaller for a similar allowable

stress, as is indicated in Figure 2.7.

Figure 2.7: Bending radius vs. maximum normal stressesσn for multiple glass thicknesses.

0

50

100

150

200

250

300

1000 5000 9000 13000 17000

No

rma

l ste

ss [M

Pa

]

Bending radius R [mm]

2.0 mm 3.0 mm 4.0 mm 6.0 mm 8.0 mm

(2.1)

(2.2)

Bending of glass 17

Case Study: Loggia Aalen-Wasseralfingen [4]

The loggia in Aalen-Wasseralfingen from Freie Planungsgruppe 7 is an excellent example

of a cold bent, single curved glass structure. The design demanded a freestanding

pavilion, which permitted multiple purposes and did not interrupt the view on the church

nearby. With the light structure and the glass roof, this was made possible.

Originally, the roof was designed as a plane, but due to the inevitable glass thickness

needed for that it was labeled as uneconomical. After some calculations, a glass roof of

cold bent panes with a tension rod led to a lower and more even stress distribution in the

glass for the uniformly distributed load (own weight, snow and wind load), as it can be

seen in Figure 2.9. This allowed a smaller thickness, which made it possible to create this

slender structure in an economical way.

The laminated glass panes that have a dimension of 5.4 m x 2 m, are constructed out of

two 12 mm FTG panes and a 1.9 mm PVB interlayer. Each pane is held in the curved

shape by two stainless steel, round bars (diameter 14 mm), with an arc rise of 300 mm.

Figure 2.8: Loggia Aalen-Wasseralfingen.13

Figure 2.9: Comparison of structural behaviour

a) beam b) arched structure. 13

13

C.2 ] Double curvature

Double curved surfaces are surfaces that are defined by a minimum of two radii of

curvature and can be divided in synclastic and anticlastic surfaces. The former meaning

that the axial and radial curves are oriented in the same direction, either concave or

convex, and the latter meaning that the axial and radial curves are directed in opposite

directions.

13 Breuninger, U., Stumpf, M., Fahlbusch, M., ‘Tragstruktur der Loggia in Wasseralfingen’, Bautechnik, vol. 80, no. 6, June 2003, pp. 355-361.

Bending of glass 18

The goal is similar to that of the single curvature. It defines a shape in which the bending

stresses are minimal and the loads can be transferred to the substructure by normal

pressure forces. This is often also the most difficult point, the connection between the

structure and the substructure, transferring these normal stresses without generating

peak values in the glass.

Case Study : Victoria & Albert Museum, London [6]

MUMA architects, together with Dewhurst Macfarlane Engineers and Octatube, designed

a glass roof for the new medieval and renaissance gallery of the Victoria & Albert

Museum. The design provided several challenges, including the geometry.

Because of the designed shape, the insulated glass units (IGU's) had to be bent. The

selected process for this project was cold bending and it was the preferred option against

the alternative of hot bending. The first reason for that were the high costs for hot bending

because each panel would need a unique shape, in total nearly 200 different moulds.

The second reason was the visual quality. During the hot bending process, the surface

of the glass does not emerge as completely smooth.

Since the cold bending process creates compression zones in the surface of the glass

panel, the stability of the shape had to be checked early in the process. Octatube

calibrated their FE-models with several physical tests in order to determine the snap-

through point by numerical analysis. Eventually, the surfaces of the IGU‘s became a

perfect hypar shape, as envisioned by the architects and is a great feature for the

museum. 14

Figure 2.10: Victoria & Albert Museum, MUMA , London.14

14 Detail, http://www.detail-online.com/inspiration/glazed-roof-at-the-victoria-and-albert-museum, (accessed July 2014)

Figure 2.11: Detailed view of the curved glass roof.14

19

3.

Numerical investigation

A ] Overview

As a basis to perform the experiments, an elemental numerical analysis has been

executed, with as a starting point the investigations made for normal glass (t > 3mm) by

Staaks and Beer [17] [3].

The experiments performed by Staaks were done on four point supported square panes

where one corner was lifted with a specific displacement to create a double curved,

anticlastic shape. An interesting conclusion he made was that the corner displacement

at which the pane buckles was only dependent on the thickness of the pane and the

translational limitations of the supports. He took these results into account within the

formula: ΔZBuckling = 16.8 · t, which is not material, nor scale dependent. Further on, he

also states that if the width-to-length ratio differs from 1:1 an increasing corner

displacement can be allowed before buckling, which he bases on his numerical model.

The results of Beer demonstrated that the boundary conditions also have an influence on

the buckling effect and the occurrence of it at a specific corner displacement. It can be

seen that line supported glass panes allowed a larger corner displacement before

buckling. Therefore, a larger midpoint displacement can be achieved, which then again

improves the chances of form-activating the glass.

The goal of this chapter is to find out if these assumptions can be analyzed by ourselves

and what the real effects are for thin glass. Additionally, it is important to find out, which

amount of stresses are created in the glass during the cold bending so it can be worked

on in a safe environment and safety measurements can be taken when expecting the

glass to break. Furthermore, a research is carried out on what exactly happens with the

pane during buckling, since it is easier to have a detailed view of the pane behaviour in

the numerical model.

Numerical investigation 20

B ] Fundamentals of the finite element method [8]

Nowadays, the constructions are increasingly complex in terms of special geometries,

materials or connections and a simple calculation by hand is getting too difficult. Hence,

the finite element method (FEM) became one of the most important calculation methods

in the construction industry. Its basics have been established by R. Clough, O.

Zienkiewicz and J. Argyris between 1960 and 1970. Other important contributions on the

subject of applied mechanics were contributed by E. Stein.

With FEM, not only structural problems can be solved, but also it is often applied for heat

conduction, hydro- and aerodynamics, acoustics, magnetism and in many other areas.

Easy problems, such as the deflection of a beam, can be calculated by solving simple

differential equations. FEM is a numerical process for the solving of these differential

equations and is applied, when the to be calculated systems become more complicated.

To calculate a more challenging construction, it will be decomposed into a finite number

of elements that are connected to the nodes by force and displacement boundary

conditions. The breaking down of a structure in small objects with specific properties can

also be described as meshing. An equilibrium state is searched during the solving

process. If that is not possible, an approximate solution is made. Afterwards, all these

subsolutions are composed into a general solution. The result of an approximated

solution depends significantly on the characteristics of the basis function, to which the

approximation is made. These basis functions should essentially already reflect the

expected trend of the solution. For the first approximation Hermite polynomials of the

second or a higher order are suitable. If those solutions are not sufficiently accurate, the

user has two options. Firstly, the number of elements can be increased (finer mesh, h-

adaptive FEM) and secondly, the amount of basis functions could be increased (p-

adaptive FEM) to achieve better results.

In addition to internal parameters, external initial parameters and boundary conditions

need to be defined. They are added to the system of differential equations. With

increasing complexity of the problem, the system of equations also grows. To solve those

problems computer-based programs as for example Maple, ANSYS, MARC and

LS-DYNA can be used.


In FEM, the principle of minimum total potential energy is used. For this two common

variants exist: the principle of virtual work and the principle of virtual displacements.

Generally, the second is more common because it is less complicated.

C ] ANSYS Workbench [9]

For the simulation of thin glass panes into a hyperbolic paraboloid and for further

investigations the software ANSYS Workbench version 15.0 is used. It is a computer-

based FEM program, originally developed by Dr. John Swanson and it is nowadays

distributed by Ansys Inc. It can be applied for calculations of linear and non-linear

problems of various disciplines and it permits a wide range of element types for one- to

three-dimensional problems. ANSYS is the abbreviation for ANalysing SYStem.

There are two versions of ANSYS. In ANSYS Classic, the calculations are entered in a

specific script language and are only presented in a graphical way by the pre- and

postprocessor. However, ANSYS Workbench has an extensive graphic user interface

(GUI), improved algorithms for contact modeling and meshing and an interface towards

multiple CAD-programs. Not all functions are integrated in the GUI though, but with the

APDL (ANSYS Parametric Design Language) all the commands from ANSYS Classic can

be entered in the script language and used in Workbench.

Figure 3.1: Sequence of the task-solving process with FEM.

From the structural mechanical task to a finished FE-model, the sequence as presented

in Figure 3.1 has to be run. In the first step, an appropriate analysis type must be selected.

In this case it is a static structural analysis. Subsequently, the material properties need to


be defined. For that an extensive library can be used or it can be user defined. Afterwards,

a CAD geometry can be implemented or it can also be designed in the included Design

Modeler.

In the next step: Model, all the settings between the geometry designing and the load

definitions are made. The geometry is assigned to a previously selected material,

boundary conditions are added and the requested parameters for the evaluation are

entered. In addition, the mesh size and shape are selected, as can the coordinate system

be defined by the user. Finally, the boundary conditions and the loads have to be

designated. It is also possible to add commands in the APDL. For the output, various

result types can be selected in the Solution branch. In that way, the deformations and

stresses of the object or specifically selected points can be considered after the solution

is completed.

D ] Models in ANSYS Workbench

In this section, the basic numerical model is explained, which will later be subjected to

several variations for the investigation. After explaining the principal settings concerning

material properties, pane geometries and boundary conditions, a convergence study is

made for the mesh, which will be used for all remaining studies.

D.1 ] Basic model

Material

As material properties of glass, an isotropic, linear-elastic material behaviour is entered

into ANSYS with a Young’s modulus of E = 70 000 MPa, a Poisson's coefficient of

υ = 0.23 and a density of ρ = 2 500 kg/m3, as it is prescribed in DIN 18008-1 (12/2010).

Geometry

As geometry of the thin glass panes two sizes were designed, being 1000 mm x 1000

mm x 2 mm ( w x l x t ) and 360 mm x 1100 mm x 2 mm ( w x l x t ). Those are the sizes

and nominal thicknesses of the panes that were available for the experimental

investigation. To make the calculations easier, they were divided in four equal parts and

assembled back into one body so the midpoint could easily be selected. This also


permitted to analyze paths through the midpoint, which allows us to recognize the

deformation figure and its warping modes more easily (see Figure 3.2). Furthermore, this

point is also used as an origin for the imperfection load. This is necessary since in this

part a vertical setup is focused on, and it leaves the own weight out of the calculation to

generalize more on the phenomena itself. Later on when the FEM results are compared

to the experiments, the own weight will also be included, as will the thickness be adjusted

to the actual pane thickness and the deformation will also be analyzed in a horizontal

position.

Figure 3.2: Pane geometries with dimensions.

Analysis settings

In the branch of the analysis settings, the amount of load steps can be selected. But

more important is the consideration of the geometrical nonlinear effects. For that, large

deformations have to be enabled, since the displacements that are later defined in the

boundary conditions, are very large (105 mm) compared to the pane thickness

(2 mm).

Boundary conditions

Two kinds of boundary conditions will be investigated in the analysis. First, one being a

four point bearing system, where all the corners are supported, and second, another one

where two edges are linearly supported, which is also a common bearing technique. For

both techniques, rotations in the possible directions are allowed and only the translations

are limited in specific directions as it can be seen in Figure 3.3, Table 3.1 and Table 3.2.


Figure 3.3: Boundary conditions, four point supported (left) and two edge linear supported (right).

Corner X Y Z A Free Free 0 B 0 Free 0 C Free Free Δ D 0 0 0

Edge X Y Z AD Spring Spring 0 BC Free Spring Function

Table 3.2: Translational limitations of the edges of the two edge linear support.

Table 3.1: Translational limitations of the corners of the four point support.

As it can be derived from the figures, by lifting corner C the desired double curved HP-

shape arose. It is done for multiple reasons, e.g. to have a comparable starting point as

our predecessors. But it must be noticed that it is not sufficient. During the designing

process three alternative possibilities were considered. The first one being a method

where the edges are rotated around an imagined middle axis, the second being lifting

corner A and C and finally, the one presented in the drawings, only lifting corner C.

Comparing the data of the different methods that can be found in Attachment A, it was

observed what a little difference it made. The biggest difference was perceived in the

method where one edge was twisted around the middle axis. Here, the midpoint

displacement could not directly be measured, which is a drawback, since it is one of the

key measuring points. Further on, the decision was made to choose the model where

only one corner was lifted. This is the simplest for the test setup, because it is probably

the way in which the glass pane would be installed in reality. Before the last corner is

displaced in the support, the glass is rested and fixed on three points.


APDL-command

It is tried to create a membrane structure and since it is not standardly implemented in

ANSYS Workbench, the insertion needed to be done manually. This to implement the

function from ANSYS Classic and be capable of calculating with it. To activate the

membrane and bending stiffness the following code is applied:

“/ PREP7

/ keyopt, l, l, 0

/ solve ”.

In the “keyopt”-command following instructions are included: 1 - for all bodies, 1 – for all

elements and third number 0 – activates the membrane and bending stiffness of the

elements. In this way, the elements obtain three translational degrees of freedom

(heaving, swaying and surging) and three rotational degrees of freedom

(pitching, yawing and rolling). If 1 is entered instead of 0, it would mean that only the

membrane stiffness would be activated (the three translational degrees of freedom). This

would not lead to a convergent solution. The reason is that ANSYS uses Newton’s

method to solve nonlinear equations. This is an iterative method for approximating the

solution with a quadratic rate of convergence. The convergence can only be achieved if

the starting value is already in the proximity of the solution, also known as local

convergence.

D.2 ] Convergence study

To determine a suitable mesh fitness at which only a limited computational time is needed

for correct results, a convergence study is made on the four point supported square pane

in a vertical position. Here, the buckling load of 0.5 N in the downward direction was also

added to the model (see Figure 3.4). This allows to influence the buckling direction and

therefore creates consistent data.


Figure 3.4: Mesh, boundary conditions and buckling load for four point supported square 2 mm pane.

Since a shell model is applied, the used elements are also shell elements. Normally two

element types can be investigated, being type 181 and 281. But the study is limited to

type 281 with a planar geometry, since it has more nodes (not just at the corners, but

also in the middle of each edge). Therefore, it is more precise for the calculation of

complex shapes. For the element shape , both the 8-node quadratic (Q8) and 6-node

triangular (T6) elements have been investigated. These elements have six degrees of

freedom at each node, being translations in the x, y, and z directions, and rotations about

the x, y, and z axes. When we would only have enabled the membrane stiffness, these

would of course be limited to the three translational degrees of freedom.

Two factors were observed during the numerical analysis, as it can be seen in Figure 3.5.

The midpoint displacement, which is represented on the first y-axis by full lines, since for

that a clear peak value exists, and the computational time, which is represented on the

second y-axis by dotted lines, so that it can be restricted to an allowable limit. Both were

compared to the element size in the following.


Figure 3.5: Four point supported 1000 mm x 1000 mm x 2 mm glass pane, without own weight.

Midpoint displacement and computational time to element size for multiple element shapes.

From the graph in Figure 3.5 can be derived that the 8-node quadratic elements converge

quicker than the 6-node triangular elements. Next to that, it can be seen that the value

for the midpoint displacement is stable from an element size of 20 mm and the

computational time is still acceptable. This is also the mesh that will be used.

E ] Parameter study

In this part, the goal is to define the influential parameters for the cold bending of (thin)

glass, which cannot be investigated in the experiments. Although many samples are

available, they are still limited to two sizes. Therefore, other sizes have been investigated

numerically, too. Next to that the numerical analysis will later also be compared to the

experimental data to be capable of revising the correctness of both.

E.1 ] Thickness

Since the focus is on thin glass, the first thing to know is what this actually means for cold

bending. It is known that in the experiments performed by Staaks [17] a factor

"ΔZBuckling = 16.8 · t" was concluded, meaning that for a four point supported quadratic

pane, the buckling point is reached if one corner is lifted out of the plane 16.8 times the

thickness. This was only tested for normal glass thicknesses though (t > 3 mm).

0

8000

16000

24000

5,84

5,86

5,88

5,90

0 20 40 60 80 100 120 140 160 180 200

Tim

e [s

]

Dis

plac

emen

t [m

m]

Element size [mm]

Midpoint Q8 Midpoint T6

Computational time Q8 Computational time T6


Here, multiple glass thicknesses, ranging from 2.0 mm to 8.0 mm, were investigated to

check the parameter put forward by Staaks and look if thin glass reacts in an identic way.

For that the graph in Figure 3.6 is made up with the midpoint displacement versus the

corner displacement. This is then the forced and controlled displacement of corner C. In

the next step, a second axis is introduced in the graph (as in most following graphs in

this thesis), which represents the maximum principle stresses for the different glass

thicknesses. Those lines do not have their own legend in the graph, but have exactly the

same colour as for the midpoint displacement and hence are differentiated by being

dotted.

Figure 3.6: Four point supported,1000 mm x 1000 mm glass pane, without own weight.

Midpoint displacement and maximum principal stress to corner displacement for multiple glass thicknesses.

0

11

22

33

44

55

66

77

88

0

5

10

15

20

25

30

35

40

0 20 40 60 80 100 120 140

Max

imum

prin

cipa

l str

ess

[MP

a]

Mid

poin

t dis

plac

emen

t [m

m]

Corner displacement [mm]

2.0 mm 2.5 mm 3.0 mm 4.0 mm 6.0 mm 8.0 mm


Thickness [mm] ΔZBuckling [mm] ΔZBuckling / t [-] Dev. from 16.8 [%]

2.0 32 16.0 5.0 2.5 40 16.0 5.0 3.0 48 16.0 5.0 4.0 66 16.5 1.8 6.0 101 16.8 1.0 8.0 134 16.8 1.0

Table 3.3: Corner displacement at which the pane buckles for multiple glass thicknesses and deviation from Staaks' results.

As it can be seen in the graph by the red dotted line that connects the maxima of all the

different thicknesses, and also in Table 3.3, there seems to be a difference for panes with

a thickness smaller or equal to 4 mm. Therefore, the factor of 16.8 cannot be used

anymore and an alternative of 16.0 could be derived from the results. This means that a

thinner pane buckles quicker compared to relatively greater thicknesses.

An explanation for it can be that small imperfections have a similar scale for glass panes

with different thicknesses. But the influence of for example 0.1 mm will be greater on a

pane with a thickness of only 2 mm than on an 8 mm glass pane.

E.2 ] Boundary conditions

In this study the focus is on two common boundary conditions, being four point corner

supporting and two edge linear supporting. This is an important part of the numerical

research, because here the corner movement, which is responsible for creating the

double curved shape, is implemented. Practically, it means that for the four point support,

one of the corners is pushed up at the bearing. In the case of the two edge linear support

the edge BC will rotate around the corner B as it can be seen in Figure 3.3.

As for most other graphs that will be presented in this work, again two Y-axes are

integrated (see Figure 3.7). The first one is for full lines and shows the midpoint

displacement compared to the corner displacement. The second one displays the

maximum principal stresses for each bearing type with the same colour, but as a dotted

line.


Figure 3.7: Four point and two edge linear supported,1000 mm x 1000 mm x 2 mm and 360 mm x 1100 mm x 2 mm.

Midpoint displacement and maximum principal stress to corner displacement for multiple boundary conditions.

It can be noticed that the maximum principal stresses in the panes are lower in the case

of the four point support than in case of the two edge linear support. This can be

explained by looking at the edges of the four point supported pane after it was buckled.

As marked by the black dotted line in Figure 3.8, it can be seen that the edge follows a

curvature and it is not straight as it would be for a perfect HP-shape. If the edge is forced

to be straight, as is it done within the linear support, it leads to extra stresses in the pane

because of the restriction.

Figure 3.8: Four point supported square pane, corner displacement of 50 mm, 200% enlarged side view.

0

15

30

45

60

75

90

-5

0

5

10

15

20

25

0 20 40 60 80 100

Max

imum

prin

cipa

l str

ess

[MP

a]

Mid

poin

t dis

plac

emen

t [m

m]


Square 4P Square 2L Rectangle 4P Rectanlge 2L


Next to that, it can be noticed that the rectangular pane supports a greater corner

displacement before buckling. For the four point support this results in a 32 mm corner

displacement for the square pane compared to 54 mm for the rectangular pane, which

is around 70% more. This can be explained with the length of the loaded diagonal AC,

which is shorter and can therefore withstand a higher buckling load. In terms of

simplifying the pane and just looking at the buckling of the diagonal AC, the following

equation can be assumed for the critical buckling load [12]:

N ,π ∙ E ∙ I

L

with:

Ncr,K Critical buckling force

E Young's modulus

I Moment of inertia

LK Buckling length.

If it is assumed that all these factors are the same for the two panes the only difference

is the length of the diagonals:

L . √1m ∙ 1m 1.41m . √1.1 ∙ 0.36 1.15 .

From (3.1) and (3.2) it can be concluded that:

N , , . N , , ..

Although, it can also be seen that the buckling point might be a bit higher for the two

edge linear support, it also seems very unstable after having reached the warping point.

For the square panel it is very clear that the behaviour for the four point supported pane

is more fluid. The graph has a smoother layout and does not go back and forth like the

two edge supported square. This was already clear in the calculation of the model. For

some points the mesh had to be made smaller to be capable of calculating the zone

between 40 – 75 mm corner displacement.

E.3 ] Size

Another point, which was not described in the found literature, was the influence of the

pane size. Now knowing the factor for the buckling-displacement for a 1 m x 1 m

quadratic point fixed glass pane, multiple pane sizes were investigated. And although it

(3.1)

(3.2)

(3.3)


was stated that the critical corner displacement is only dependent on the glass thickness,

it seems rather unusual that the pane size would not have an influence. Therefore,

multiple pane sizes with a 1:1 width-to-length ratio are modeled to get an idea of the

differences of varying sizes.

Figure 3.9: Four point supported, 2 mm thick square panes, without own weight.

Midpoint displacement and maximum principal stress to corner displacement for multiple sizes.

From the graph in Figure 3.9 it can be concluded that the size of the pane does not have

an influence on the corner displacement at which the pane buckles. This can be

explained with the maximum principal stresses. The smaller the pane is, the higher the

principal stresses are. Therefore, the pane buckles at the same displacement because

although it has a higher critical force (shorter loaded diagonal), the internal stresses are

also higher. However, a difference in the amount of midpoint displacement can be

noticed at the peak.

E.4 ] Shape ratio

Having dealt with the size of the pane, another interesting factor might be the width-to-

length ratio. It could already be noticed in Figure 3.7, that for a rectangular pane the

0

12

24

36

48

60

0,0

1,5

3,0

4,5

6,0

7,5

0 20 40 60 80 100

Max

imum

prin

cipa

l str

ess

[MP

a]

Mid

poin

t dis

plac

emen

t [m

m]


500 mm x 500 mm 1000 mm x 1000 mm

1500 mm x 1500 mm 2000 mm x 2000 mm


corner displacement can be higher than for a quadratic pane. However, the question is

if it is possible to find a specific function between different ratios, as for the glass

thickness. For this, multiple 2 mm glass panes with an identical length and varying width

were modeled.

Figure 3.10: Four point supported, 2 mm thick glass panes, without own weight.

Midpoint displacement and maximum principal stress to corner displacement for multiple w/l-ratios.

As can be seen in Figure 3.10, the smaller the width-to-length ratio, the higher the corner

displacement at which the pane buckles. This can be explained in the same way as it

was done for the rectangular and the square pane in the part about the boundary

conditions (see section E.2). The length of the loaded diagonal gets shorter and shorter

and because of that the critical force increases, too. In Table 3.4, the exact critical corner

displacements are mentioned in combination with the midpoint displacement at that

point and the amount of deviation of that displacement to the pane thickness compared

to the square pane.

y = 0,0181x² + 0,826x - 0,9038

0

10

20

30

40

50

60

70

-10

-5

0

5

10

15

20

25

0 20 40 60 80 100

Max

imum

prin

cipa

l str

ess

[MP

a]

Mid

poin

t dis

plac

emen

t [m

m]


1000 mm x 1000 mm 750 mm x 1000 mm 500 mm x 1000 mm

333 mm x 1000 mm 250 mm x 1000 mm


W/L-ratio [-] ΔZBuckling [mm] Midp. displ. [mm] ΔZBuckling / t [-] Dev. from 1:1 [%] 1:1 32 5.9 16.0 0.0 3:4 33 6.4 16.5 3.1 1:2 39 8.1 19.5 21.9 1:3 52 11.0 26.0 62.5 1:4 65 14.0 32.5 103.1

Table 3.4: Corner displacement at which the pane buckles for multiple pane ratios and deviation from 1:1-ratio.

Connecting the peak values resulted in a parabolic trend, which is visualized by the red

dotted line in the graph. This means that the rate at which the critical corner displacement

increases slows down towards smaller ratios. But additionally the peak midpoint

displacement increases more and more.

F ] Analyses for experiments

In this part the focus is on examining the behaviour of the available panes at the buckling

point as a basis for the experiments that will be done. It is investigated what changes if a

pane buckles and how the buckling can be perceived.

Although two pane sizes are available and two supporting ways are tested, in the

numerical modeling it can be recognized that the visible phenomena can be generalized

over all and they can be divided over three zones: the free edges, the diagonals and the

middle axis. For each of them the focus will be made on the pane shape and bearing

type, which shows the behaviour in the most explicit way, being the two edge linear

supported square pane.

Of course, with FEM the opportunity exists to have a look at the internal stresses that

cannot be visualized in the experiments. Therefore, some notes are made here.

F.1 ] Free edges

As it was already seen before, the free edges have the tendency to take over an S-like

shape instead of staying straight. The origin for that can possibly be that a HP-shape is

not unwindable. This means that a pure hyperbolical paraboloid (HP) cannot be formed

from a single uniform pane. It is not clear why it results specifically in that shape. But

since it is forced into a HP, the glass pane will try to create a close fit to it.


For the presentation of the shape related factors, both the corners A and C are lifted in

the model to have a more symmetrical representation of the form. Additionally, for the

following figures and graphs it seemed more interesting to have a symmetry in it, to make

them easier to read. This results in comparable data, but the corner displacements of

both corners A and C have to be summed up to have the resulting corner displacement

as it has been used throughout this master thesis.

Figure 3.11: Two edge linear supported square pane. Corner displacement of 40 mm, 500% enlarged view.

Figure 3.12: Two edge linear supported square pane. Corner displacement of 50 mm, 500% enlarged view.

As it can be seen in Figure 3.11 and 3.12, the S-curve is not depending on the buckling

or the critical corner displacement, but it could be caused by the desired, unwindable

shape. Therefore, it does not only appear after the pane buckled, but it can also be

perceived for smaller corner displacements, although very slightly.

F.2 ] Diagonals

When lifting corner A and C at the same time and keeping B and D at their origin, the

created shape ought to be a HP. Since that is an anticlastic double shaped form, both

diagonals (AC and BD) of this shape should have an opposite, but equal curvature. In

Figure 3.13, a graph is made displaying both the diagonals and their displacements over

the full length for specific corner displacements. The displacements are the summation

of both corners A and C, so it can easily be compared to the lifting of only one corner.


Figure 3.13: Two edge linear supported, 1000 mm x 1000 mm x 2 mm glass pane, without own weight.

Displacement of the diagonales before and after buckling.

Of course, the midpoint of both the downside curvature and the upper curvature touch

in the middle, since it is also where the diagonals reach the same value. If looking at the

movement of the midpoint for the different corner displacements, it can be noticed that

it is still rising between the 40 mm and the 45 mm corner displacement. But, it decreases

afterwards when the pane has buckled. Here, the diagonal BD is also not parabolic

anymore, but it shows a plain. Therefore, the shape is not regular anymore.

F.3 ] Middle axis

The middle axis is another criteria to determine, which shape is formed by the corner

displacements. To have a perfect HP, this axis ought to be straight. When looking at

Figure 3.14, something different can be observed. The axis is already slightly (less than

0.5 mm) curved in the stable phase. Then, with a corner displacement of 45 mm the

curvature of the axis changes direction. For the 50 mm corner displacement, when the

pane has buckled, the axis is completely bent.

0

5

10

15

20

25

0 246 491 737 982 1227

Dis

plac

emen

t [m

m]

Position on diagonal [mm]

40 mm 45 mm 50 mm

A C

B D


Figure 3.14: Two edge linear supported, 1000 mm x 1000 mm x 2 mm glass pane, without own weight.

Displacement and principal stress of the middle axis, before and after buckling.

The buckling is not only visible in the displacements. Here, the principal stress is again

presented with the dotted lines and by the second Y-axis. When taking a closer look, it

can be seen that for the 40 mm corner displacement the whole axis is still in tension. But

from the 45 mm corner displacement on, a pressure zone is formed in the middle area,

which gets bigger and bigger when the plain in the middle zone is created.

F.4 ] Membrane stresses

With FEM, the opportunity arises to have a look at what happens inside the glass pane.

Here, the internal stresses, its build-up and the influence of the buckling are investigated.

This part, takes a closer look at the membrane stresses of the four point supported 2 mm

square glass pane. Already knowing from the previous that the buckling happens at a

corner displacement of 32 mm, the moment before and after the buckling can be

compared, respectively being a corner displacement of 30 mm and 35 mm.

-3

-1

1

3

5

7

9

8

9

10

11

12

13

14

0 250 500 750 1000

Prin

cip

al s

tres

s [M

Pa

]

Dis

plac

emen

t [m

m]

Position on middle axis [mm]

40 mm 45 mm 50 mm


Figure 3.15: FE-model of the four point supported square pane, 1000 mm x 1000 mm x 2 mm, without own weight.

As in the other studies, a model is used, where the pane is put in the vertical position and

no own weight is applied to it. Only a small imperfection load of 0.5 N is put in the middle

to force the pane to buckle (see Figure 3.15). Again, in this model only one corner is

lifted, being corner C as it was shown in Figure 3.3.

The differences of both corner displacements are presented in a series of membrane

stress figures 3.16-3.23. For all stresses that are analyzed in the FE-model, the same

kind of shift can be noticed as in the membrane stresses. Generally, the image of the

non-buckled pane is double symmetrical, compared to the buckled one. It appears to

have a shift in its stress distribution. Because of that, a large pressure zone arises in the

middle of the pane.

As it can be seen in Figure 3.17 by the curved dotted line, the shape is not regular

anymore like in Figure 3.16. But a plain is created in the middle. Further on, it is noticeable

that the membrane normal stresses get distorted because of the buckling and the

pressure zones are positioned around the diagonals as the shear stresses are. Before,

the stress distributions were focused around the middle axes, the buckling changes the

focus on the diagonals.


30 mm corner displacement

Figure 3.16: Displacements Z-direction.

Figure 3.18: Membrane normal stress, X-direction.

Figure 3.20: Membrane normal stress, Y-direction.

Figure 3.22: Membrane shear stress.

35 mm corner displacement

Figure 3.17: Displacements Z-direction.

Figure 3.19: Membrane normal stress, X-direction.

Figure 3.21: Membrane normal stress, Y-direction.

Figure 3.23: Membrane shear stress.


G ] Summary

To conclude this chapter, a small overview is made on what was revealed about the cold

bending of thin glass and what can be used in the following chapter. Firstly, it can be

stated, that the critical corner displacement has many influence factors and it cannot be

summarized in a thickness depending factor of 16.8 [17]. Many other factors have to be

taken into account, too.

The boundary conditions, as well as the pane’s width-to-length ratio are two factors that

are important. Both have an impact on the behaviour of the pane and influence the critical

corner displacement. From the numerical analysis it was learned that a two edge linear

supported pane allows greater corner displacements compared to a four point supported

pane. Furthermore, it was detected that a decreasing width-to-length ratio results in an

increasing peak midpoint displacement.

Further on, it was noticed that a pure hyperbolical paraboloid cannot be made, since it

is an unwindable shape, and that the free edges therefore show a slight S-curve. If this

is prevented by the linear support, it also results in higher stresses in the pane.

Finally, the effects of the buckling on the shape and the internal stresses were examined.

After the pane has buckled, the supported diagonal BD is flattened and a plain is created

in the middle. This can also be seen in the principal stresses that show an increasing

compressive zone in the middle. Next to that the membrane stresses undergo a shift

from a double symmetric distribution around the middle axes to a more diagonally

oriented distribution.

41

4.

Experimental study

A ] Overview

In this chapter, an experimental investigation about the cold bending of thin glass is

performed. As a first step, some basic tests were executed to get in touch with the (before

unknown) material and its behaviour. Although the results from the numerical analysis

are known, some differences during the handling of the material can still be possible.

After this preliminary check, the real test setups were built. It exists of a four point

supported and a two edge linear supported setup. The glass panes were measured and

tested, the setups were modified if needed and to make a conclusion the results were

compared to the FE-results.

All the tests that were done for this research, were performed in the laboratory of the

Universität der Bundeswehr München in a controlled environment with a constant

temperature of 20.5°C.

B ] Preliminary tests

What was learned from the introductory chapters could be bundled as following: the

thinner the glass is, the less tension is created in it during cold bending. The most

common examples for cold bending are single curved or anticlastically double curved.

In this thesis, the focus is on double curvature. First of all, since good alternatives are

already available for single curvature. But also while single curvature can only stiffen out

the pane in one direction, what would leave the pane sensitive in the other direction,

which is not favorable for small thicknesses.

Since performing tests on glass is not expected to be basic knowledge for an architecture

student, some preliminary tests were performed to get a feeling with the thin glass and

the desired shape. The panes that were used for these preliminary tests are also the ones

that have been investigated in the previous chapter. They are from the same batch as

those for the experiments, being a monolithic rectangular 360 mm x 1100 mm (w x l) and

Experimental study 42

a monolithic square 1000 mm x 1000 mm (w x l) glass pane with a nominal thickness of

2 mm.

Figure 4.1: First test with rectangular pane.

Figure 4.2: Detailed view first test.

As it can be seen in Figure 4.1 and 4.2, a HP-like shape was created with the rectangular

pane by clamping the four edges onto a predefined frame. Since the wooden frame was

made with a computer-controlled mill and designed in such a way that two corners were

lifted by 25 mm, it can be stated that the shape has four straight edges and therefore

should be a HP. The straight red rope, which was spanned over the middle of the pane

shows also that the created shape has to be a close fit to a HP.

Figure 4.3: First test with square pane.

Figure 4.4: Breaking pattern of square pane.

After the rectangular pane, a square pane was added to the test. Again, a wooden frame

was built, but this time only one corner was meant to be displaced and that for

100 mm (see Figure 4.3). The test was less successful, but taught a lot. Looking at the

breaking pattern in Figure 4.4, the origin of the crack can clearly be seen at the right

bottom, which was also the location where the last clamp was put on (too firmly). This is


no coincidence. If looking at the results from Galuppi [7], the tips of the loaded diagonal

AC clearly show a different movement after the pane buckled. By prevent this movement,

extra tensile stresses could have been created (see Figure 4.5). A good detailing of the

bearing should therefore be looked after and the force with which the glass is held should

be controllable.

Figure 4.5: "Double cylindrical" deformed shape.15

C ] Method

In this section, the properties of the glass panes that will be used for testing, are

measured and described as the test setups that have been built. Next to that, the first

tests as well as the necessary modifications to the setups will be mentioned here.

C.1 ] Pane properties

For this part, two pane sizes, 1000 mm x 1000 mm and 360 mm x 1100 mm (w x l) of

heat strength thin glass with a nominal thickness of 2 mm were available for testing. Out

of them, five of each were selected that came out of the same batch.

The first step, before testing them, was to define the exact properties of the available

panes. For that, the length, width and exact thickness was measured for each pane. In

addition, the breaking pattern of one of the panes was analyzed according to DIN EN

12150 (01/2005) to check which kind of glass it could be.

15 Galuppi, L., Massimiani, S., Royer-Carfagni, G., ‘Large deformations and snap-through instability of cold-bent glass’, Challenging Glass 4 & COST Action TU0905 Final Conference,London, Taylor & Francis Group, 2014, pp. 681-690.


Figure 4.6: HSG fracture pattern of one of the test panes.16

Based on the fracture pattern it can be concluded that the glass had to be heat

strengthened glass (HSG), although the standard norm (DIN EN 12150) does not

describe 2 mm thin glass yet [13]. This can also be seen in Figure 4.6. The pane did not

break into small pieces (dice), but instead into larger fragments, which is typical for heat

strengthened glass. It was informed though that the breaking pattern of the fully tempered

thin glass looks really similar, what is normally not the fact. This is because the tempering

process is still in the developing phase. The machine (produced by LiSEC), used to

temper the thin glass, does not let the glass roll on from the oven to the quencher.

Instead, it has the glass panes floating on so called air pillows that enables it to develop

tempered glass without the usual roller waves and therefore a clearer view [11]. It is still

in a developing phase, with the result that they are still not capable of creating the

standard breaking pattern for the fully tempered glass (small dice), although the internal

stresses are stated to be correct. Because of that, the panes were also investigated by

laser-optical measurements so that the exact amount of pre-stressing can be known,

(this is done with a SCAttered Light Polarimeter SCALP-05 and the accompanied

software GlasStress 5.7.0.8.)

Physical measurements

As it can be seen in the following tables and figures, all the panes were carefully

measured. For the physical measurements the widths and lengths were measured at the

two edges and in the middle, and the thicknesses at the four corners (see Figure 7).

Afterwards, the average was taken of all these measurements and it was used for the

numerical model (see Table 4.1 and 4.2).

16 Nehring, G., Beitrag zur Bemessung von kaltverformten Strukturen aus Dünnglas, ongoing PhD research (internal), Universität der Bundeswehr München, 2015.


Table 4.1: Square pane measurements.

Square LAvg. [mm] WAvg [mm] tAvg [mm] TVG SQ 1 999 1000 2,06 TVG SQ 2 1000 1001 2,08 TVG SQ 3 1000 1001 2,08 TVG SQ 4 1000 1000 2,09 TVG SQ 5 1000 1000 2,09

Table 4.2: Rectangle pane measurements.

Rectangle LAvg. [mm] WAvg [mm] tAvg [mm] TVG RE 1 1100 360 2,08 TVG RE 2 1100 360 2,09 TVG RE 3 1100 360 2,08 TVG RE 4 1101 360 2,07 TVG RE 5 1100 360 2,08

Laser-optical measurements

After the physical measurements, the pre-tensioning of the panes was determined. For

that, multiple points (S#) were investigated as it can be seen in Figure 4.7. At each point,

both directions were measured. Since the values can vary slightly, three measurements

were done at each position in both directions.

Figure 4.7: SCALP measurement points.

In table 4.3 and 4.4, an overview of the most important data concerning both sets of

panes is provided. As it can be seen by the amount of pre-stressing, the glass is indeed

HSG and not FTG. This is also not necessary for the research, because the stresses

caused by cold bending (, which the panes will undergo,) are not so high due to the small

thickness.


Table 4.3: Square laser-optical measurements.

Square U [MPa] M [MPa] L [MPa] Avg. X Y

-70.8 -73.3

33.2 32.5

-70.5 -72.8

Std. Dev. X Y

3.03 2.29

1.39 1.71

3.25 2.40

Var. X Y

9.16 5.25

1.94 2.93

10.53 5.77

95% C.I. X Y

-69.94 -72.59

32.83 32.00

-69.52 -72.11

Table 4.4: Rectangle Laser-optical measurements.

Rectangle U [MPa] M [MPa] L [MPa] Avg. X Y

-69.6 -74.5

32.9 32.8

-69.4 -74.2

Std. Dev. X Y

4.23 7.55

2.32 2.74

4.03 7.94

Var. X Y

17.91 57.01

5.40 7.51

16.25 62.99

95% C.I. X Y

-68.13 -71.77

32.08 31.86

-67.91 -71.37

Avg. = Average; Std. Dev. = Standard Deviation; Var. = Variance; C.I. = Confidence Interval U = Upper Surface; M = Middle; L = Lower Surface.

In Table 4.4 it can be noticed that the variance for the rectangular pane is very high. This

is because the measurements at the corner and in the middle show large differences

(see Attachment B). At the corner, the stresses in the Y-direction seem to be higher (5 –

15 MPa). This is not unusual since at the corners, not only the upper and lower surface

have to be taken into account during the quenching, but also the edges. This can create

local differences (,another influential factor can be the flow direction during the tempering

process, but since this could not be traced back, no remarks were given).

Edges, corners and surface [13]

As can be seen in Figure 4.8 and 4.9, the edges of the panes are grounded. At the

corners, this is not the way as expected. It seems like the pane, which rolls through the

grinding machine, moves a bit at the start and towards the end. Because of that the

process does not produce a perfect symmetrical corner. The short length of the panes

can explain this. Probably the panes are not entirely supported and tilt a bit towards the

end and the beginning. Next to that, some pane specifications are etched in the corner.

These things would be taken into account if a pane breaks, since it could locally reduce

the strength.


Figure 4.8: Corner detail.17

Figure 4.9: Etched details in the glass.17

17

Strain gauges

To be capable of comparing the numerical calculations with experimental tests, strain

gauges have been applied on the glass panes. The positioning of the strain gauges has

been specified on the base of the analysis executed in the last chapter.

One square pane has been taken over from the test performed by Arend [1], which was

measured as TVG SQ 1. The positioning of the strain gauges on the pane can be seen

in Figure 4.10. A second square pane has been necessary, since during the four point

supported test, the first pane was broken. For that, the pane TVG SQ 2 was also used.

To save some expenses, only six strain gauges have been installed on the same

positions as the first pane, but a few were passed upon. For the rectangular pane, the

pane marked as TVG RE 1 was used.

Figure 4.10: Strain gauge positions on TVG SQ 1 and TVG RE 1.

17 Nehring, G., Beitrag zur Bemessung von kaltverformten Strukturen aus Dünnglas, ongoing PhD research (internal), Universität der Bundeswehr München, 2015.


C.2 ] Test setup properties

Two simple, but effective, test setups were designed and built. The focus was on the

supporting ways and the capability of testing both the square and the rectangular panes

that were available for testing. The first bearing method is the four point support, which

means that the four corners are fixed, as it has been described by Van Laar [18] and

others. Still, this seemed relevant, because the bending behaviour of rectangular thin

glass has not yet been described for a four point support and it will enable to make a

better comparison between the different bearing types. The second method is a two edge

linear support. It is a bearing type that can be seen more often in real practices and has

not yet been described. This will limit the movement of two edges and should by doing

so postpone the buckling effect.

As it was already said, for the creation of the desired double curved anticlastic shape

with these bearing types multiple methods were considered. However, only one corner

was lifted. This allowed to make an easier setup and provided more control over the

corner displacement. Next to that, the numerical modeling showed that the differences

between the various models were negligible.

Four point support

The basis for the setup was the design of Arend [1]. The metal frame, the point fixings

and the movable corner were all part of his design. It was adjusted to be capable to fit a

rectangular pane, too. To reduce effort, the length of the metal framework (1 m x 1 m)

was not adjusted. This means that the rectangular pane slightly popped out of the point

fixings and therefore not the corners but a part of the edge close to it was supported as

it can be seen on the middle drawing in Figure 4.11.


Figure 4.11: Drawings of the four point support test setup.

Further on, in the drawing the original setup can be seen at the left and at the right the

elevation, which shows the displaceable corner C. A detailed picture of that is displayed

in Figure 4.13. Here, the threaded bar, which can be freely rotated to lift the corner, is

visible as is the point fixing of Corner C. For the point fixings, a facade element of Pauli &

Sohn was used, which has an articulated raised head (, product information can be found

in Attachment C). This allows rotation in all directions, which was also possible in the

numerical model.

Furthermore, the points where the displacements are going to be measured (Midpoint

and Middle AD) are also marked on the drawings. There, the (HBM) inductive standard

displacement transducers will be attached, which will be explained later.

Figure 4.12: Four point support test setup.

Figure 4.13: Detail image of the adjustable corner C.


Two edge linear support

The basic concept for this setup is similar to that of the four point support. It makes it

possible to gradually move up one corner. The most difficult part was again being

capable of using one setup for both the square and the rectangular panes.

For the setup two timber beams were mounted with metal brackets on a large wooden

plate. One was fixed rotationally that it can rotate around corner D if lifted in corner C, as

can be seen in the drawings in Figure 4.14. For the second one, small holes were

prepared in the base plate so the brackets could be fixed with small bolts and unscrewed

again that the edge AB can be moved for the different pane widths.

As it can be seen in Figure 4.16, the base plate is only locally supported to leave space

for the displacement transducers. They were mounted in the middle of the pane and the

middle of the edge AD as marked in Figure 4.14. For that multiple holes where prepared

in the pane that they can be used for the square and the rectangular panes. Since the

metal four point supported setup was mounted on the wooden base plate to use these

transducers, the plate bended slightly under the weight. This was not tolerable since the

panes needed to be loaded afterwards. This would make the base plate bend even more.

No measurements were done in this (bended) situation and a wooden frame was added

to the construction to stiffen the base plate.

Figure 4.14: Drawing of the two edge linear support test setup.


Aluminum press-fit clamping bars were used, to keep the glass in place. The rubber

down part was pasted to the timber beams and the upper part can be screwed on the

beam to fixate the glass panes as required (see Figure 4.15). After that, the whole edge

CD, was able to be moved by the threaded bar in corner C. While the beam is more than

stiff enough, there was no worry about the edge not staying straight.

Figure 4.15: Detailed elevation drawing of corner D of the two edge linear support.

Figure 4.16: Side view of the two edge linear support test setup, before addition of the frame.

Figure 4.17: Overview of the setup.

Figure 4.18: Detailed view of edge BC in lifted position.


C.3 ] Test setup adjustments

After some preliminary tests, the setups showed that they were doing what they were

designed for, but nonetheless, some improvements were made. Here, the adjustments

and its influences are mentioned for both the setups.

Four point support

Two small adjustments had to be made before the real tests were performed. Both are

concerning the fixating of the pane. Since the panes are not filling up the whole area of

the point fixings, a rubber with the same thickness had to be placed on the free area that

they can be clamped correctly (see Figure 4.19 and 4.20).

The other point of attention was the metal screw-head in the middle of the support. It had

to be covered with a small piece of PVC tube, as it can be seen in Figure 4.20. The glass

corners could not accidentally get pushed against it when tightening the other clamps,

as happened with the square pane TVG SQ 1. This is normally included in the packaging

of the point fixing, but since these clamps were remainders of earlier tests, not all the

parts were there. And gently tightening the clamps was not precautious enough.

Figure 4.19: Point fixing detail, square.

Figure 4.20: Point fixing detail, rectangle.


The most important improvement was the edge clamping. For greater displacements, it

was noticed that the standard press-fit clamping bar created an unwanted curve in the

upper edge of the pane (see Figure 4.21). This creates secondary and undesired tensile

stresses in the glass that should be avoided. To do so, a hard rubber tube with a diameter

of 5 mm was fixed on to the beam and pressed into the fitting (for the normal rubber) of


the clamp so that the glass edges have a minimal amount of restrictions during the

bending (see Figure 4.22). The adjustment was inspired by the conclusions of Callewaert

[5].

On the other side of the clamp a wooden batten has been used to compensate the height

of the rubber tube and clamp the glass evenly, as it is also visible in Figure 4.22.

Figure 4.21: Detail flat clamping edge.

Figure 4.22: Detail round clamping edge.

This was not the only adjustment though. Originally, the displacement measuring of the

midpoint and the edge AD was planned to be done with inductive standard displacement

transducers, as it can be seen in Figure 4.23. But since the internal springs were slightly

pushing the panes up and thereby creating an extra supporting point, they were replaced

by displacement transducers with a magnetic head and a plunger. Those had such a

minimal weight (39.2 g) that there was no need to be scared of unwanted influences (see

Figure 4.24 and Attachment D).

Figure 4.23: Inductive displacement transducer.

Figure 4.24: Transducer with magnetic head.


A final small change was made: the corner lifting principle with the threaded bar, which

can be screwed up gradually had to be changed because it allowed too much movement

of the edge CD when loading the panes. For that, little wooden blocks were cut for every

5 mm displacement. Afterwards the beam was fixed using a clamp that the edge would

not be capable of moving anymore.

D ] Square experiments

Here, the experiments and the associated data is presented. First of all, the experiments

are divided by pane shape, starting with the square panes. Secondly, by bearing type,

being the four point support and the two edge linear support and thirdly, by the direction

in which the pane is put, horizontal or vertical. The idea behind the different directions

was to get a feeling about the influence of the own weight, but also to have the possibility

to load the panes and watch its response. This to look if the form activation takes place

and if the structure becomes stiffer after the corner displacements.

Next to that, also a comparison is made with the ANSYS model to find out if both are in

accordance with each other.

D.1 ] Four point support

Horizontal

As previously said, the first test was the four point supported square pane (TVG SQ 2).

After having leveled the supports that corner C was at the same height, the pane was

mounted upon the setup, taking into account the corner detailing and connecting the

displacement transducers (see Figure 4.25). Important was always to be sure in which

way to put the pane that the strain gauges were measuring the required values and to

make sure that they were all well connected to the computer.

Afterwards, a ritual was started, which has been performed on every single test. First of

all the values were set to zero (also the displacement transducers), although the pane

was already bent under its own weight and that downward movement would have been

interesting to capture. This was done while the displacement transducers were often

moved (for measurements of the rectangular panes) and sometimes the whole structure


was flipped into a vertical position and back. This made it impossible to have it set

perfectly to be capable of measuring the movement from the origin.

Secondly, the camera was put in position to capture the movement. Thirdly, when

horizontally tested, the pane was loaded with a known weight (a 3925 g steel sphere) in

the middle and in the middle of the edge AD (Midpoint and Edge AD in the graphs)

(see Figure 4.27 and 4.28). Then corner C was lifted up 5 mm, the values of the strain

gauges and the displacement transducers were noted and everything was repeated until

the corner was moved 105 mm (see Figure 4.26). After that, the test had to be stopped.

This was not because the glass pane would break otherwise, but since it was noticed

that the rubbers that hold the pane, were slightly getting torn out of the bearing together

with the pane.

Figure 4.25: Horizontal square four point support test,

0 mm corner displacement.

Figure 4.27: Horizontal square four point support test, 0 mm corner displacement, loaded midpoint.




0 mm corner displacement, loaded edge AD.

After all was done, the selected data, which results from the tests, was put into a graph

and compared to the FE-analysis, as it can be seen in Figure 4.29. All the graphs

throughout this section are built up in the same way and present the same measuring

points to be easily compared. First of all, the experimental data are represented by a dot

for each measurement. To keep the graph readable, not everything is presented, but the

most important parts are. First, the midpoint displacement (Midpoint) and the

displacement of the middle of edge AD (Edge AD) are presented for all the corner


displacements. Second, the differences of the displacements with that ones in the loaded

situation are added (Δz midpoint and Δz Edge AD). Third, the maximum normal stress is

compared to the corner displacement on a second Y-axis. This is not always in the same

location, but it is shown in the drawing at the top left of each graph (DMS).

All dots were connected with trend lines to have a better view on the course of the data

(full line for displacements, dotted line for the normal stresses). They were compared to

the midpoint displacement and the maximum normal stress, which was calculated with

the FE-model. These lines were then also full for displacements and dotted for normal

stresses, but instead of being a trend line of measurement points, they were just lines

since they were calculated more continuously.

To be capable of comparing the FE-data more easily with the experimental data, it was

choosen to shift the initial displacements and stresses to the origin for the horizontal test,

as it has been done at the start of the experiments.

_______________________________________________________________________________________________________________________________________________________________________________________________________________

Figure 4.29: Data from the square four point supported horizontal testing of TVG SQ 2 and the FE-model.

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Max. Normal Stress FEM


Based on the data shown in Figure 4.29, it can be concluded that the pane responds

stiffer to the specific loads, for an increasing corner displacement. This is visualized by

the downward going Δz midpoint and Δz Edge AD. It would be a mistake to read it in that

way though. Instead, since the bearings were free in all rotational directions, too many

external factors influenced the loaded pane and as it can be seen in the Figures 4.25 and

4.26. When the corner C is displaced over a certain distance, the point fixing, which has

only a limited rotational freedom, can rotate less and less since it moves together with

the angle of the diagonal AC. The stiffening effect can therefore completely be assigned

to the prevention of the rotation at corner C (,this also makes that the pane is not capable

of stiffening up during the cold bending).

It can be seen that the results for the midpoint displacement and the maximum normal

stress are approaching the FE-model very well. The values for the experimental midpoint

displacement are slightly lower than the numerical values are. This can be explained with

the positioning of the displacement transducer, which might not be perfectly centered.

The differences from the maximum normal stress can be explained in the same way: the

strain gauges might not have been on exactly the same position as the numerical

maximum and therefore show slightly deviating values.

Further on, a buckling point cannot be detected. Since the pane was already curved

before the start of the corner lifting (due to its own weight), a HP-shape has never been

formed during the run through. It stayed the whole time in a “double cylindrical" deformed

shape, as if it had already buckled (see Figure 4.26).

Vertical

The same setup was now put vertically. This allows to investigate the effects of the cold

bending under conditions where the own weight is of less importance. The pane starts

from a perfectly straight position, which was not the case for the horizontal tests.

As it can be seen in the graph in Figure 4.30, the pane has a clear buckling point. After

reaching a peak midpoint displacement, all the values skip and the subsequent course

is not in the same trend as before. Because of that, the trend lines in the graph have been

cut in three parts to display a more fluent course.


For this case, the FE-model is not approaching the experimental data, but the

development is similar. It reaches a maximum value where it buckles and afterwards it

has a (rather) linear course. Two reasons for the deviation from the FE-calculation could

be that firstly, the point fixings have a specific diameter (of 60 mm), which is not

implemented in the numerical model and secondly, that the pane is very unstable after it

buckled. Therefore, it is difficult to have straightforward data. With very little effort the

panel can be pushed back and forth, so it seems to be buckled in the other direction as

it can be seen in Figures 4.31 and 4.32. When 'popped' back and forth the value at the

beginning is not the same as at the end.

_______________________________________________________________________________________________________________________________________________________________________________________________________________

Figure 4.30: Data from the square four point supported vertical testing of TVG SQ 2 and the FE-model.

When the pane buckles, the movement of the edge can be noticed very well. Until the

buckling point, it moves slightly up, therefore the edge AD is not straight as would be

when creating a perfect HP. When buckled, it seemed to have taken over the movement

of the midpoint displacement, which stays rather constant, while the middle of the edge

AD keeps descending.

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A question to be asked now, while comparing the experiment with the FE-data, is why

the numerical model displays a smooth transition at the buckling point and the

experiments do not. The data from the experiment shows that the pane buckles equally

abrupt as the models with greater thicknesses presented in chapter 3. For thicknesses

greater than 4 mm there also seemed to be a clear difference between the peak value

and the converged final value.

Figure 4.31: Buckled square pane, four point

supported, backward position, 200% enlarged width.

Figure 4.32: Buckled square pane, four point

supported, forward position, 200% enlarged width.

D.2 ] Two edge linear support

Horizontal

After testing the square panels with a four point support, the two edge linear support was

the next. Here, with the added adjustments, it was possible to load the pane (in the

horizontal setup) with the steel sphere (of 3925 g). Now one can have a look at how the

structure responded to it.

The data of the executed tests is presented in Figure 4.33. It can be concluded that a

square thin glass pane has a decreasing deflection (under a known load), for an

increasing corner displacement. This is valid both for the midpoint displacement as for

the displacement of the edge AD and can it run up to 40% for a 105 mm corner

displacement. It can also be noticed that the maximum stresses in the bearings decrease

when loading the pane (presented as Δ Max. Normal Stress in the graph), which is another

important advantage for cold bending.


__________________________________________________________________________________________________________________________________________________________________________________________________________

Figure 4.33: Data from the square two edge linear supported horizontal testing of TVG SQ 1 and the FE-model.

Comparing the graph with the one of the four point support it can be noticed that the

maximum normal stresses are higher, but for that the midpoint displacement is higher,

too. It can also clearly be seen that the overlap between the experimental data and the

numerical data is really good. Only the midpoint displacement is a bit lower, but it can be

explained again by the positioning of the displacement transducer. It might have been a

bit aside from the middle and therefore a small deviation can be seen here.

During the bending, a real HP-shape is never formed though. The pane also does not

show a peak value for the midpoint displacement, nor does it buckle. It just forms a

cylindrical-like shape, similar to the four point supported setup as it can be seen in Figure

4.34 and 4.35.

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∆ Max. Normal Stress Max. Normal Stress FEM


Figure 4.34: Horizontal square two edge supported

test, 0 mm corner displacement.

Figure 4.35: Horizontal square two edge supported test, 105 mm corner displacement.

Vertical

A similar phenomenon happens as in the case of the four point supported setup. After

having reached a peak midpoint displacement, the pane buckles and becomes unstable.

The corner displacement at which it happened was higher though (45 mm compared to

35 mm). It is an interesting fact to know, since a greater curved shape can be created

with the linear supported pane as with the four point supported, before the instability

appears (see Figure 4.36).

_______________________________________________________________________________________________________________________________________________________________________________________________________________

Figure 4.36: Data from the square two edge linear supported vertical testing of TVG SQ 1 and the FE-model.

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When comparing the FEA to the experiments, it can be noticed that the peak midpoint

displacement is really nearing the numerical data, although the critical corner

displacement is a bit higher. Afterwards, the jump back as calculated in the FEA is not

happening in the experiment. This can probably be explained by the overall lack of

stability of the pane. It can be seen that the curve is going up again after the pane

buckled, which was not the fact for the four point supported pane.

Looking at the stresses then, it can be noticed that the skip made in the experimental

data is not visible in the FEA. Furthermore, the maximum normal stresses are overall a

bit lower for the experiment. It can be noticed though that both paths take a similar

course.

Looking at the pane itself, the same can be stated as was done for the four point

supported pane. As it can be seen in the reflections in Figure 4.37 and 4.38, the pane

does not have a preferred state after being buckled. It can be pushed back and forth and

stay at the back or front, as if it was a natural state. This is something which has never

been described for thicker panes and raises some questions about their behaviour. It

seems rather unlikely that the lack of stability is typical for thin glass since it only gets into

this state after buckling. An explanation can be that the stresses rise up to a level where

the pane breaks during the ‘popping’ movement. This has never been noticed.

Figure 4.37: Vertical square two edge supported test,

45 mm corner displacement, "popped" backward.

Figure 4.38: Vertical square two edge supported test,

45 mm corner displacement, "popped" forward.


E ] Rectangle experiments

In this section, the tests with the rectangular panes are described. Measurements are

always done horizontally and vertically like for the square panes. On this way, a

comparison can be made concerning the influence of the own weight. The horizontal

tests are again loaded with specific masses to have an idea in which amount the form

really gets activated and if the structure gets stiffer or not.

E.1 ] Four point support

As already mentioned before, the four point supported setup has been modified that

rectangular panes also fit in. But instead of clamping the corner, a part of the edges is

fixed. Because of that it was not possible to measure the displacement of the edge AD,

but the displacement at the middle between the point fixings was measured as it could

be seen in Figure 4.11.

Horizontal

It is already known what happened with the horizontal test of the four point supported

square pane. Hence, it can be noticed that the same phenomena is visible here. As it

can be seen in Figure 4.39, the deflection for the loaded situations seems to decrease

for increasing corner displacements, but again the same can be stated as before. The

rotational freedom of the point fixing in corner C gets more and more limited for greater

corner displacements. Again, this does not allow to state anything about whether or not

the pane stiffens for increasing displacements.

The corner displacement for this test was restricted to 75 mm due to this limited freedom.

As can be seen in Figure 4.40, the point fixing at corner B had reached a limit of the

rotational movement, which did not allow a greater corner displacement without the pane

slipping in the bearing. This is of course an effect, which was not welcomed to be

included.


_______________________________________________________________________________________________________________________________________________________________________________________________________________

Figure 4.39: Data from the rectangle four point supported horizontal testing of TVG RE 1 and the FE-model.

Here again, no peak value is reached. The pane does not show a buckling point and

cannot be formed into a HP. Due to its own weight it is not capable of overcoming the

initial bending. A shape is formed, which resembles to a single curvature (see Figure

4.40).

Further on, similar trends between the FE-calculations and the test data can be seen,

although the point fixings are not exactly in the corner. The differences that can be seen

in the graph, are expected to be caused by that. It accentuates that the courses of the

numerical analysis and the experiments are very similar, which affirms both.

Figure 4.40: Rectangle four point supported horizontal test, 75 mm corner displacement.

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Vertical

The same setup was now switched to the vertical position that the own weight would

have less of an impact on the pane. It is to notice that the experimental data matches the

FE-data very well (see Figure 4.41). The positioning and size of the point fixings can

explain the small differences that are prominent.

_______________________________________________________________________________________________________________________________________________________________________________________________________________

Figure 4.41: Data from the rectangle four point supported vertical testing of TVG RE 1 and the FE-model.

Again, it has also to be said that after the panel has reached its peak midpoint

displacement, it is very unstable. Some differences with the quadratic setup have been

noticed though. First of all, it can be observed that for the rectangular shape, the corner

displacement can be higher before the pane buckles. Next to that, it was also not

possible to stabilize the shape in the other direction as was possible for the square

panes. An explanation for that can be that a real buckling point with an intense skip as

that of the square panes was not noticed during the experiments, although the reflections

clearly showed a distorted image after the maximum midpoint displacement (see Figure

4.42 and 4.43).

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The overall movement of the pane appeared to be smoother during the tests. It can also

be seen in the graph. Because of that it needs to be considered that the large amount of

rotational freedom in the bearing points had something to do with the lack of stability

after the buckling, instead of being a property of the buckled pane.

Figure 4.42: Vertical rectangle four point supported

test, 45 mm corner displacement.

Figure 4.43: Vertical rectangle four point supported test, 55 mm corner displacement.

E.2 ] Two edge linear support

Horizontal

For the setup the load at the midpoint was increased by 5790 g, so that it counts up to a

total of 9715 g (see Figure 4.44). This was done while the distance between the bearings

was so small that the deflection of the initial load was too small to have a clear view on

its influence.

Figure 4.44: Rectangular two edge linear support horizontal test, 75 mm corner displacement, loaded midpoint.


On the opposite of the square pane the central stiffening did not significantly appear (see

Figure 4.44). The displacement is rather constant for increasing corner displacements.

This probably happens because the distance between the supports is already so small

that the bending does not have a big enough influence on this scale.

On the other hand, if looking at the deflection of the edge AD for the (3925 g) load, it can

be recognized that the pane reacts stiffer for an increasing corner displacement. This is

actually more interesting than at the center, since the deflections here are even bigger

than at the midpoint, although the load has less than half the weight.

_______________________________________________________________________________________________________________________________________________________________________________________________________________

Figure 4.45: Data from the rectangle two edge linear supported vertical testing of TVG RE 1 and the FE-model.

As can be seen in the graph, the experimental data matches the numerical data very well

(see Figure 4.45). Only the midpoint displacement seems to be somewhat lower, but

again this can emerge with the positioning of the displacement transducer. Further on, a

buckling point cannot be observed. The midpoint seems to be rising in a constant pace

for the tested corner displacements.

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Max. Normal Stress FEM


Vertical

It can be seen that for a rectangular pane, the buckling point is not that easily reached

as for the square pane. In addition, it can be noticed that the supporting type can have

a major influence on the buckling behaviour of the glass pane. As it can be seen in Figure

4.46, the midpoint displacement for the two edge supported rectangle pane in vertical

position goes straight up as it is typical for the cold bending before buckling.

Further on, it can be conceived that the graph approaches the FE-calculation very

closely. The maximum normal stress might be a bit higher although the midpoint

displacement is a bit lower. Overall they approach each other very well, which validates

again the numerical model and the test setup.

_______________________________________________________________________________________________________________________________________________________________________________________________________________

Figure 4.46: Data from the rectangle two edge linear supported vertical testing of TVG RE 1 and the FE-model.

Another point, which could be noticed during the experiment and which is less visible in

the graph, is that the free edges slightly revealed a S-shaped curve from a corner

displacement of 65 mm on (see Figure 4.48). Although very minimal, from that corner

displacement on the edge displacement slightly decreases in the graph.

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Figure 4.47, 4.48: Overview of the rectangular two edge linear support test setup, 105 mm corner displacement.

F ] Laminated glass experiments

In a later phase, laminated thin glass panes with the same dimensions were made

available for testing by the company LiSEC (as were the monolithic ones). Although it

was not prepared numerically, and due to the complexity of the model it was not

attempted anymore in the limited time frame. The opportunity was taken though to do

some preliminary testing with the same setups that were already made and extensively

tested. The basis of comparison will also be the tests performed on the monolithic panes

and the numerical investigation about pane thicknesses.

In this section, the panes and their properties will be defined first. Afterwards, the

performed tests are described, as has been done before for the monolithic panes. In this

part, the tests have been conducted without strain gauges and the graphs are therefore

built up slightly different.

In each graph, both the tests in the horizontal (HO) and vertical (VE) direction are

mentioned, so that a comparison can easily be made. In the horizontal position, the

panes have been loaded with the 3925 g steel sphere (and the 5790 g extra weight for

the midpoint of the rectangular panes) to investigate the deflections and the stiffening of

the shape.

For the measurements of the loaded situation, a new ritual had to be added to the run

through. Since the PVB foil shows a time dependent creep and relaxation effect (the


displacement values change over time when loading the pane), the measurements were

only saved after the values for the specific corner displacement had been constant for

10 seconds. This enabled to possibility to create reliable and comparable data.

Due to the limited amount of test specimens the rectangular tests were limited to a corner

displacement of 75 mm to make sure that breakage was not possible. For the square

pane, the corner displacement was restricted to 100 mm.

F.1 ] Pane properties

For the measurements of the panes, the same process was executed as before. First of

all the average width and length was defined by measuring the edges and the middle of

the pane. Then, the average thickness of the pane was determined by measuring the

thickness of the four corners and the thickness of the PVB was noted as it can be seen

in Table 4.5.

Table 4.5: Laminated pane measurements.

Laminated LAvg. [mm] WAvg [mm] tAvg [mm] PVB thickness [mm] VSG SQ 1 1000 1000 4.92 0.76 VSG RE 1 1100 359 5.73 1.52 VSG RE 2 1100 360 4.93 0.76

After that, a laser-optical measurement was performed on both the panes of the

laminated glass. This was done at the same locations as for the monolithic panes as it

can be seen in Figure 4.49. Here again, for each point three measurements were made

in both directions to have a reliable value. For the representation of the data in Table 4.6,

the average values (of the three points for the square panes and two points for the

rectangular panes) were chosen instead of an evaluation as before, because only a

limited amount of panes was investigated.

Table 4.6: Laminated pane laser-optical measurements.

Laminated UAvg. [MPa] MAvg [MPa] UAvg [MPa]

VSG SQ 1 X Y

-71.1 -73.2

34.6 33.8

-70.4 -72.6

VSG RE 1 X Y

-69.3 -71.1

33.5 31.9

-70.1 -71.2

VSG RE 2 X Y

-69.1 -70.8

34.4 32.5

-69.5 -70.1

Figure 4.49: SCALP measurement points.


F.2 ] Square experiments

As in the last section, the experiments for the laminated glass panes are divided in the

same order. First, the square experiments starting with the four point supported ones,

and finishing at the end with the rectangular two edge linear supported experiments.

Four point support

Here, it is obvious to notice the effect of the own weight on the displacements, both for

the midpoint and the edge AD (see Figure 4.50). For the vertical setup (VE) the midpoint

goes up linearly until a corner displacement of 80 mm. After that, the midpoint seems to

be moving quicker from its origin. It can be perceived as a buckling effect, although the

pane did not have the lack of stability that was noticed for the monolithic pane. Another

proof for the buckling can be found in the displacement of the edge AD. There, the trend

is going slightly downwards until the 80 mm corner displacement and after that it moves

up again.

The horizontal setup shows a different path. Here the midpoint moves similar to the

monolithic pane, taking a slope, which increases less and less for increasing corner

displacements. Also, the edge AD takes quite a strong downward curve compared to the

vertical setup.

_______________________________________________________________________________________________________________________________________________________________________________________________________________

Figure 4.50: Data from the square four point supported horizontal and vertical testing of VSG SQ 1.

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Midpoint HO Edge AD HO Midpoint VE Edge AD VE ∆z Midpoint HO ∆z Edge AD HO


For this setup, it is to notice again that the deviations seem to decrease for increasing

corner displacements. For a first time for the four point supported setup, these values

seem to be reliable. Since the pane is now laminated, it responds stiffer than the

monolithic thin glass pane. Therefore the corners rotate less for the specific loadings.

Because of that, it can be stated that the shape stiffening enacts as well. It counts only

until a corner displacement of 75 mm though. From this point on, the rotational freedom

of the corner C is prevented because the point fixing reached its limit as can be seen in

Figure 4.51.

Looking at the shapes of the horizontal and vertical setup, it can easily be stated that the

vertical setup comes closer to a HP-shape and the horizontal setup has more of a

cylinder shape throughout the entire run through. Again, this is presumably caused by

the original deflection under its own weight at the starting point.

Figure 4.51: Square four point supported horizontal test, 100 mm corner displacement


For this setup almost the same can be stated as for the four point supported pane. Two

points have to be mentioned though. Firstly, as it can be seen in Figure 4.52, again the

deviations decrease for higher corner displacements. This proves the form activation,

although the shape is not a perfect hyperbolical paraboloid.

Secondly, in the case of the vertical setup, the midpoint displacement proceeds linearly

for increasing corner displacements, even for displacements greater than 80 mm. This

proves that the buckling point is postponed because of the boundary conditions. Also,

the movement of the edge is very limited. This indicates that the shape is a very close fit


to the original intention, although the edge presents a slight S-curve as can be seen in

Figure 4.54.

_______________________________________________________________________________________________________________________________________________________________________________________________________________

Figure 4.52: Data from the square two edge linear supported horizontal and vertical testing of VSG SQ 01.

Figure 4.53: Square two edge supported,


Figure 4.54: Square two edge supported, S-curved edge detail, 250% stretched.

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F.3 ] Rectangle experiments

The experiments with rectangular laminated thin glass panes are described in this

section. Next to doing the normal set of test, also a small comparison is made between

two interlayer thicknesses. Since the glass is already that thin, the thickness of the

interlayer can also be an influential parameter.

Four point support

Looking at Figure 4.55 and 4.56, two important items can be noted. First of all, in both

situations (horizontal and vertical position) and for both interlayer thicknesses, the pane

has slightly smaller deflections for the specific loads for increasing corner displacements.

Here again, it can be stated that the pane stiffens out during the process, since the

rotational freedom of the point fixings is maintained.

_______________________________________________________________________________________________________________________________________________________________________________________________________________

Figure 4.55: Data from the rectangle four point supported horizontal and vertical testing of VSG RE 1.

Furthermore, it can be noticed that for the thinner PVB interlayer, there is less difference

between the midpoint displacement of the horizontal and the vertical setup as it is for the

thicker interlayer. A difference between both panes was also noticed during the testing.

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Dis

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The thicker interlayer needed a lot more time to settle after being loaded. Therefore, it

can be concluded that the thinner interlayer reacts stiffer.

_______________________________________________________________________________________________________________________________________________________________________________________________________________

Figure 4.56: Data from the rectangle four point supported horizontal and vertical testing of VSG RE 2.

In both cases a shape closely approaching to a HP was formed, as it can be seen in

Figures 4.57 and 4.58.

Figure 4.57: Rectangle four point supported,

75 mm corner displacement, VSG RE 1.

Figure 4.58: Rectangle four point supported,


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In comparison to the tests on the square panes, the influence of the horizontal and

vertical tests is almost not visible anymore. This because the pane is supported over its

two long edges with an in between distance of only 350 mm. The laminated panes do

not bend anymore under their own weight with such a support.

For both panes it can be noticed that for the (increased) load in the middle, the pane is

not stiffening. For the (normal) load at the edge though, which causes greater deflection

in the beginning, the deflection decreases for increasing corner displacements. This

proves again that there is a positive effect on the stiffness due to the cold bending.

_______________________________________________________________________________________________________________________________________________________________________________________________________________

Figure 4.59: Data from the rectangle two edge linear supported horizontal vertical testing of VSG RE 1.

Looking at the created shapes itself, they appear to be real hyperbolical paraboloids, as

can be seen in Figure 4.61 – 4.63.

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_______________________________________________________________________________________________________________________________________________________________________________________________________________

Figure 4.60: Data from the rectangle two edge linear supported horizontal vertical testing of VSG RE 2.

Figure 4.61: Rectange two edge linear supported,


Figure 4.62: Rectange two edge linear supported,


Figure 4.63: Rectange two edge linear supported, 75 mm corner displacement, VSG RE.

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G ] Summary

To wrap up the chapter, some final conclusions about the influential factors of the cold

bending of thin glass are made. Many graphs were already presented, but excluding the

laminated glass panes, they only presented information about one pane shape, which

was supported in a specific way (four point support or two edge linear support). Here, as

it can be seen in Figure 4.64, a combination of data from the vertical tests of the

monolithic panes is provided.

For the representation of the data, the basic outlining remains the same. The used lines

are trend lines, so that the courses can be distinguished and not only the measurement

points. Further on, the full lines represent the displacements and the dotted lines the

maximal normal stresses, measured by strain gauges.

_______________________________________________________________________________________________________________________________________________________________________________________________________________

Figure 4.64: Data from the vertical square and rectangle experiments of TVG SQ 1, TVG SQ 2 and TVG RE 1. 4P = four point support; 2L = two edge linear support.

Looking at the graph, the same conclusions can be made as was done for the numerical

study of the boundary conditions (in chapter three). Linearly supporting the edges

instead of only point fixing the corners allows a greater corner displacement before

0

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Midpoint 4P SQ Midpoint 2L SQ Midpoint 4P RE

Midpoint 2L RE Max. Normal Stress 4P SQ Max. Normal Stress 2L SQ

Max. Normal Stress 4P RE Max. Normal Stress 2L RE


buckling. Next to that, the influence of the width-to-length ratio can be seen. The

rectangular panes allow greater corner displacements than the quadratic and it can be

noticed that the maximum normal stresses are lower for the four point supported panes

as for the two edge linear supported.

Further on, the horizontal two edge linear supported tests showed that a form-activation

was achieved and the deflections for known loads got smaller for increasing corner

displacements, although the shape was never a perfect HP. The same can be expected

for the vertical setups and as long as the buckling point is not reached, even though they

were not explicitly tested. From there on, the pane is not stable anymore and is not

capable of carrying loads.

Looking at the shape again, the tests with the laminated thin glass displayed the best

results. In the vertical position and linearly supported at two edges, a shape, which was

as close as possible to a HP, was formed for both the quadratic as for the rectangular

pane.

80

5.

Application of thin glass

A ] Overview

Having studied the behaviour of thin glass during cold bending, it also seemed relevant

to create a small design to show the possibilities of it outside of the laboratory. In this

design, the material and its shape should be the main subject, but of course, designing

something without a context is not so evident.

An indoor element that is not accessible, was decided upon, because the panes were

not loaded with real wind or snow loads during the tests. Since pre-tensioned thin glass

is still a new and unknown product, a promotional element was chosen as a subject.

B ] Concept

With the upcoming Bau 2015 (world’s leading trade fair for architecture, materials and

systems) [2], an element for an exhibition booth was put forward. It cannot be anything,

which is just presented though. It needs to be something functional. Since exhibitions are

typically very short term, the structure should be light and the assembly should be as

simple as possible.

With the performed tests in mind, the most suitable panes seem the quadratic 1 m x 1 m

laminated panes in a setup where the edges are supported. It is known though, that 1 m

x 2 m (w x l) laminated thin glasses are also produced, since two of them came in the

same delivery as the panes used before. Although no tests were performed on these,

since the setups did not allow tests on such large panes, an estimation of an allowable

deformation was made based on the previous experiments.

Knowing that the square (1 m x 1 m) laminated pane enabled a corner displacement of

100 mm without breaking, nor buckling. A good assumption could be that a 1 m x 2 m

Application of thin glass 81

rectangular pane would allow a corner displacement of 150 mm, which is less than

combining two square panes.

This assumption is based on the result from the numerical research made in chapter 3

and the experiments in chapter 4. First, it is known that a two edge linear supported

quadratic pane does not buckle for a corner displacement of 100 mm in the vertical

position. Second, it is noted that the width-to-length ratio has a positive influence on the

critical corner displacement. Third, the larger the pane gets the smaller the internal

stresses get for a constant ratio.

C ] Design

If taking all elements into account now, a possible application could be a see-through,

cold bent, glass dividing wall for an exposition booth. This allows the exposition of the

cold bent glass, but it can also function as a practical element on an exhibition by splitting

up the site in multiple parts, although simultaneously staying visibly connected.

For the design, a square ground floor of 10 m x 10 m is assumed that is divided in four

equal parts. This seems like an average exposition site and allows to keep the design

simple. A walkway with a width of 1 m is left free that an easy passage from one zone to

another is possible, as it can be seen in Figure 5.1.

Figure 5.1: Elevation and floorplan of the setup,

left: concept drawing; right: twisted design with cold bent glass.


Since the cold bent glass ought to be the main subject of the wall, the use of different

materials is reduced to two, being timber and glass. This also makes the structure, which

is built up from four identical timber frame walls around a central pole, light enough to be

carried by two people and put together without big machinery (see Figure 5.2). The thin

glass allows this, since a laminated pane of two times 1 m x 2 m x 2 mm (w x l x t) only

weighs about 20 kg.

Figure 5.2: Partial section.

The different walls would be made up from timber beams that are designed to fit together

with a carpentry connection so that they can easily be positioned correctly and are

afterwards screwed together to create sufficiently stiff connections (for the stability) (see

Figure 5.4). Small slots would have to be made into the beams to enable sliding in the

glass panes. This connection of glass and timber should not create any problems since

timber only has a Mohs’ hardness of 2 - 3 compared to 5.5 for glass.

Figure 5.3: Connection detail of the frame.

Figure 5.4: Connection frame and substructure.


To ensure the stability of the whole structure, a connection to the underground is

necessary. Often a temporal floor is put up on an exposition site, which could make this

possible. For that, tapped slots could be prepared in the lower beam with which the walls

could be connected to a base structure made out of a timber frame. In that frame thread

inserts would have to be prepared to bolt the wall to it and guarantee a secure

construction (see Figure 5.4). Afterwards, some OSB plates and a typical exposition

carpet could finish the flooring for the booth.

When everything comes together then, a possible perspective of the booth could be seen

in Figure 5.4.

Figure 5.4: Rendered perspective.

84

6.

Conclusions and recommendations

A ] Conclusions

In the underlying exploratory investigation, the most important factors for the cold

bending of thin glass (t < 3 mm) have been mapped in an attempt to create an affordable

and easy alternative for the production of doubly curved anticlastic shapes. For that,

numerical analyses and experimental tests have been performed on quadratic and

rectangular shaped thin glass panes. Creating the double curved shape was integrated

in the boundary conditions and achieved by lifting one corner. This was done for the

numerical as well as the experimental research.

Based on the numerical analyses it can be concluded that many factors influence the

critical corner displacement where the pane buckles. The key factors were the boundary

conditions and the width-to-length ratio. The two edge linear supported setup allowed a

greater corner displacement (before buckling) than the four point supported setup.

Additionally, a similar result can be seen for decreasing w/l-ratios: the smaller the ratio,

the higher the critical displacement.

Next to that, a relation between the glass thickness and the buckling point was found.

For the four point supported vertical setup this resulted in: ΔZBuckling = 16.0 · t, for

thicknesses smaller than or equal to 3 mm. It was noticed that for increasing thicknesses,

the internal tensile stresses also increased during the cold bending. A similar effect was

visible for decreasing pane sizes, with a constant width-to-length ratio. The smaller the

pane got, the higher the internal stresses rose, although the critical buckling

displacement remained constant.

The experimental results were consistent with the former numerical analyses. For the

vertical tests the same conclusions could be made as for the numerical analysis. The

boundary conditions as well as the width-to-length ratio appeared again to be an

influential factor on the critical corner displacement.

Conclusions and recommendations 85

The primary deformation of the setups created a close fit to a perfect hyperbolical

paraboloid. This cannot be stated about the horizontal test. Here, the initial deflection

caused by the own weight prohibited the desired deformation and produced a similar

shape as was formed when the pane buckled in the vertical setup, the "double cylindrical"

deformed shape (although the lack of stability was not visible there). This can also be

seen in the numerical data. Furthermore, in the horizontal experiments it was noted that

a form-activation took place. This was proven by the decreasing deflections for a

constant load and increasing corner deflections. Additionally, it can run up to about 40%

less deflection for a corner displacement of 105 mm for the two edge linear supported

square pane.

To take everything into account, some final tests have been performed on laminated thin

glass, too. They were truly promising for the practical application since the buckling point

was only reached after larger corner displacements. Therefore, they can enable a reliable

usage in doubly curved architecture.

B ] Recommendations

Now, knowing more about the behaviour of thin glass during cold bending, two directions

could be challenging and interesting for a follow-up research. Bending laminated thin

glass has already shortly been touched in this thesis and yielded promising results for

the creation of double curved glass architecture. A further study could concentrate on

the internal stresses created by cold bending of laminated thin glass and the behaviour

of multiple interlayers during the bending process. The main focus would be the correct

modelling of the interlayers in a numerical environment.

Another direction, which could form a fascinating subject for a further investigation, would

be the boundary conditions. Clamping the panes with point fixings and linear press-fit

bars has proven to be very effective. But such devices are also always visible on both

sides of the glass. The current technics of adhesively bonded connections are very

promising and could allow a fully glazed doubly curved outlook of buildings and

structures. Interesting would be the stress diagram and distribution in the connections

during the shaping process. For that, a correct numerical modelling will again be the

center of attention.

86

Bibliography

[1] Arend, S., Untersuchung zum Tragverhalten von Schalen aus Dünnglas, Master thesis, Universität der Bundeswehr München, 2014.

[2] BAU 2015, http://www.bau-muenchen.com/en/Home, (accessed January 2015). [3] Beer, B., ‘Complex geometry façades – Introducing a new design concept for cold-

bent glass’, Glass Performance Days, 2013, pp. 516-522. [4] Breuninger, U., Stumpf, M., Fahlbusch, M., ‘Tragstruktur der Loggia in

Wasseralfingen’, Bautechnik, vol. 80, no. 6, June 2003, pp. 355-361. [5] Callewaert, D., ‘Refined test set-up for pure torsion of laminated glass’, Glass

Performance Days, 2007, pp. 118-121. [6] Eekhout, M., ‘The New, Cold Bent Glass Roof of the Victoria & Albert Museum,

London’, Glass Performance Days, 2009, pp. 408-412. [7] Galuppi, L., Massimiani, S., Royer-Carfagni, G., ‘Large deformations and snap-

through instability of cold-bent glass’, Challenging Glass 4 & COST Action TU0905 Final Conference,London, Taylor & Francis Group, 2014, pp. 681-690.

[8] Gebbeken, N., ‘Einführung in die Methode der finite Elemente – Stabtragwerke’,

Universität der Bundeswehr, 2000 cited in Arend, S., ‘Untersuchung zum Tragverhalten von Schalen aus Dünnglas’, Master thesis, Universität der Bundeswehr München, 2014.

[9] Gebhardt, C., Praxisbuch FEM mit ANSYS Workbench: Einführung in die lineare und

nichtlineare Mechanik, München, Carl Hanser Verlag, 2014. [10] Hanoar, A., Principles of structures, Oxford, Blackwell Science Ltd, 1998. [11] LiSEC, http://www.lisec.com/Innovation/Flexibles-D%C3%BCnnglas/LiSEC-hat-

das-Glasvorspannen-revolutioniert, 2013, (accessed November 2014). [12] Luible, A., Stabilität von Tragelementen aus Glas, PhD thesis, École Polytechnique

Fédérale de Lausanne, 2004. [13] Nehring, G., Beitrag zur Bemessung von kaltverformten Strukturen aus Dünnglas,

ongoing PhD research (internal), Universität der Bundeswehr München, 2015.

Bibliography 87

[14] Schittch, C., Staib, G., Balkow, D., Schuler, M., Sobek, W., Detail: Glass

Construction Manual, Basel, Birkhäuser Verlag AG, 2007. [15] Schott AG, http://www.schott.com/advanced_optics/german/syn/advanced_optic

s/products/optical-materials/thin-glass/, (accessed December 2014)

[16] Siebert, G., Maniatis, I., Tragende Bauteile aus Glas: Grundlagen, Konstruktion, Bemessung, Beispiele, Berlin, Wilhelm Ernst & Sohn, 2012.

[17] Staaks, Koud torderen van glaspanelen in blobs, Master thesis, Technische

Universiteit Eindhoven, 2003.

[18] Van Laar, H., Stabiliteit van in het werk vervormde glaspanelen voor dubbelgekromde gevels, Master thesis, Technische Universiteit Delft, 2004.

[19] Wurm, J., Glas als Tragwerk: Entwurf und Konstruktion selbsttragender Hüllen,

Basel, Birkhäuser Verlag AG, 2007.

88

Appendix

A ]Deformation methods ...................................................................................... 89

B ]Laser-optical measurements ........................................................................... 91

C ]Point fixing details ............................................................................................ 92

D ]Displacements transducer details ................................................................... 93

Appendix 89

A ] Deformation methods

Appendix 90

Appendix 91

B ] Laser-optical measurements

Appendix 92

C ] Point fixing details

Appendix 93

D ] Displacements transducer details

exploratory investigation on the cold bending of...

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