GLASS BEAM DESIGN FOR ARCHITECTS:
BRIEF INTRODUCTION TO THE MOST CRITICAL FACTORS OF GLASS BEAMS
AND EASY COMPUTER TOOL
by
Lei Fu
A Thesis Presented to the
FACULTY OF THE USC SCHOOL OF ARCHITECTURE
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
MASTER OF BUILDING SCIENCE
August 2010
Copyright 2010 Lei Fu
ii
ACKNOWLEDGEMENTS
I would like to express my gratitude to everybody who helped me to complete this work
and 2 years graduate study in USC. Without their help and support, I would never have
been able to finish this work. I would like to express my deepest thanks to Professor G.
Goetz Schierle, the committee chair of my thesis, for his guidance and encouragement
during the whole process. Also thanks to my committee members, Professor Douglas
Noble, Professor Anders Carlson and Professor Gail Borden, who helped me to find such
a good topic, solve structural problems and work on the right direction. Thanks to Mic
Patterson and Jeffrey Vaglio for their help on glass studies.
Thanks to Professor Marc Schiler, director of MBS program, who brought me to the U.S.,
and all faculties in the MBS program. Thanks to the School of Architecture for awarding
me the GRS scholarship. Thanks to all the teammates of MBS program to accompany me
these two years.
Thanks to my parents and families members. I love you.
iii
TABLE OF CONTENTS
Acknowledgements ………………………………………………………………………ii
List of Tables…………………………………………………………………………..…vi
List of Figures……………………………………………………………………….…ix
List of Functions………………………………………………………………………....xii
Abstract…………………………………………………………………………………xiii
Chapter 1: Introduction to Glass Beams…………………………………………………..1
1.1 Glass Beams…………………………………………………………..…………...1
1.1.1 Glass Beam Application in Architecture………………………………..1
1.1.2 Existing Buildings with Glass Beams……………………………..……5
1.1.3 Advantage of Glass Beams…………………………………….………10
1.2 History of Glass and Glass Beams…………………………………..…………...11
1.2.1 Glass in the Prehistoric Era…………………………………….………12
1.2.2 Glass Used for Architectural Purposes…………………….…………..12
1.2.3 Glass in Modern Architecture and the Pursuit of Transparency……….14
1.2.4 Structural Glass-Glass Fins and Glass Columns……………….………16
1.2.5 Glass Beams……………………………………………………………17
1.2.6 Further Development of Structural Glass Research……………………24
1.3 Current Research and Target of This Thesis………………………..……………25
Chapter 2: Properties of Glass and Glass Beam Manufacturing…………………………27
2.1 Glass as Structural Material…………………………………..………………….27
2.2 Glass Manufacturing……………………………………………………...……...29
2.2.1 Glass Manufacturing…………………………………………….29
2.2.2 Mechanical Properties of Glass………………………………..……..36
2.3 Glass Beam Composition………………………………………………..……….43
2.3.1 Lamination……………………………………………….……………43
2.3.2 Limit of Single Span Coming from Manufacturing Modulus………....44
2.3.3 Sealing……………………………………………..……………..…….46
2.3.4 Reinforcement………………………..…………….…………………..47
Chapter 3: Strength of Structural Glass Beams………………………………………….48
3.1 Griffith Flaw………………………………………………………………..……49
3.1.1 Higher Compression Strength but Lower Tensile Strength……..……..50
3.1.2 Higher Theoretical Strength but Lower Usable Strength……..……….51
3.1.3 Great Variety…………………………………………………….……51
iv
3.1.4 Strength of Glass under Long Duration Load………………………..53
3.1.5 Strengthened Glass…………………………………………………54
3.2 Strength of Annealed Monolithic Glass Panels…………………………...……..57
3.2.1 Standards and Industry Data……………………………………57
3.2.2 Strength of Glass……………………………………………………..66
3.3 Strength of Monolithic Glass Beams…………………………………………….67
3.3.1 Glass Supporting in Plane Load…………………………………...…...67
3.3.2 Glass under Long Duration Load………………………….…………72
3.4 Strength of Laminated Glass Beams…………………………………………….78
Chapter 4: Glass Beam Calculation……………………………………………………..79
4.1 Design Methods……………………………………………………………….…79
4.1.1 ASD……………………………………………………………….…..80
4.1.2 LRFD………………………………………………………………….80
4.2 LRFD Bending Calculation……………………………………………….….….82
4.2.1 Load and Load Factor………………………………………………...82
4.2.2 Section Modulus………………………………………………………84
4.2.3 Potential Breakage of Glass and Modification on Section Modulus….86
4.2.4 Resistance Factor………………………………………………...……88
4.3 Deflection Criteria………………………………………………………….……90
4.4 Buckling…………………………………………………….……………………93
4.4.1 Beam Buckling……………………………………………….………..96
4.4.2 Local Buckling………………………………………………………..102
Chapter 5: Excel Tool and the Largest Glass Beam……………………………………104
5.1 Excel Tool…………………………………………………………………........104
5.1.1 Design Area………………………………………………………….105
5.1.2 Material & Structure Area…………………………………………...107
5.1.3 Calculation Area……………………………………………………..108
5.2 The Biggest Glass Beams without Inter-brace………………………………….109
5.2.1 The Biggest Glass Beams for Roof Design……………………...…..109
5.2.2 Several Conclusions………………………………………………….111
5.3 Effect of Interlay Material on Beam Buckling……………………………..…..112
5.4 Rules of Thumb: Heat-treatment, Depth, Layers and Thickness……………….115
Chapter 6: Verification of the Excel Tool………………………………………………117
6.1 Buildings with Simple Glass Beams……………………………………………117
6.2 Comparison……………………………………………………………………..118
6.2.1 Workshop in Musée de Louvre……………………………………....118
6.2.2 Glass pavilion of Broadfield House Glass Museum…………………120
6.2.3 Glass Bridge in Rotterdam……………………….…………………...122
6.2.4 Glass Conservatory, “Teesdale”, Surrey……………………………..123
6.3 Conclusion…………………………………………………………………..….125
v
Chapter 7: Future Work………………………………………………………..……….126
7.1 Strength of Glass under Long-duration Load………………………………..…126
7.2 Lamination……………………………………………………………………...127
7.3 Buckling………………………………………………………………………...127
7.4 Connection Design……………………………………………………………...128
7.5 Environmental Impact and Protection………………………………………….128
Bibliography……………………………………………………………………………130
Appendices
Appendix A……………………………………………………………………………..133
Appendix B……………………………………………………………………………..135
vi
LIST OF TABLES
Table 1.1: Buildings with Glass Beams…………………………………………………...6
Table 2.1: Properties of Glass and Several Other Structural Materials………………….28
Table 2.2: Minimum Glass Thicknesses from ASTM 1300 (03)……………………..…34
Table 3.1: Strength Properties of Annealed Glass from Various Codes of Practice…….58
Table 3.2: Strength Properties of Thermal Treated Glass from Various Codes of
Practice………………………………………………………………………..59
Table 3.3: Strength Properties of Annealed Glass from Float Glass Industry………..….60
Table 3.4: Strength Properties of Thermal Treated Glass from Float Glass Industry...…61
Table 3.5: Standards for Glass Design and Practice in the US…………………………..62
Table 3.6: Allowable Design Stress for Various Probabilities of Breakage……………..65
Table 3.7: Test Result for Single Beams………………………………………………...69
Table 3.8: Test Result for Single Beams in TU Delft……………………………………71
Table 3.9: Tested Ration of Different Sets of Strength……………………...…………..72
Table 3.10: Glass Type Factor (GTF) for a Single Lite of Monolithic or Laminated
Glass from ASTM-1300 (03)…………………………..…………………... 73
Table 3.11: Load Duration Factor, Note-Calculated to 8/1000 Probability
of Breakage from ASTM E 1300 (03)……………………………………....74
Table 3.12: Modified Table of Load Duration Factor, Note-Calculated to 8/1000
Probability of Breakage from ASTM E-1300 (03)……… …………………75
Table 3.13: Load Duration Factor of Tempered Glass…………………………………..76
Table 3.14: Modified Load Duration Factor of Tempered Glass………………………..76
Table 3.15: Load Duration Factor of Heat-Strengthened Glass…………………………77
vii
Table 3.16: Summary of Glass Strength…………………………………………………78
Table 4.1: Allowable Design Stress for Various Probabilities of Breakage…………….86
Table 4.2: Resistance Factors of Common Structural Materials………………………...88
Table 4.3: Glass Coefficient of Deviation (%)…...…………………………………...…89
Table 4.4: Functions for Critical Elastic Buckling Moment Calculation………………..96
Table 4.5: Coefficient for Slenderness Factors of Bisymmetrical Beams
with no Intermediate Buckling Restraints………………………………….....99
Table 5.1: 20 Foot Beams for Roof Design…………………………………………….110
Table 5.2: 20 Foot Beams for Roof Design with Sliding when Buckling…………...…113
Table 6.1: Beam for Workshop in Musée de Louvre by the Excel Tool……………….119
Table 6.2: Beam for Glass Pavilion of Broadfield House Glass Museum
by the Excel Tool……………………………………………………….….121
Table 6.3: Beam for Glass Bridge in Rotterdam by the Excel Tool……………………123
Table 6.4: Beam for Glass Conservatory, ‘Teesdale’ by the Excel Tool………………124
Table A.1: Draft Used in this Thesis for Design Decisions & Criteria………………...133
Table B.1: 8 Foot Glass Beams…………………………………………………………136
Table B.2: 9 Foot Glass Beams…………………………………………………………137
Table B.3: 10 Foot Glass Beams…………………………………………………..……138
Table B.4: 11 Foot Glass Beams…………………………………………………..……139
Table B.5: 12 Foot Glass Beams…………………………………………………..……140
Table B.6: 13 Foot Glass Beams…………………………………………………..……141
Table B.7: 14 Foot Glass Beams…………………………………………………..……142
Table B.8: 15 Foot Glass Beams…………………………………………………..……143
viii
Table B.9: 16 Foot Glass Beams…………………………………………………..……144
Table B.10: 17 Foot Glass Beams…………………………………………………..…..145
Table B.11: 18 Foot Glass Beams…………………………………………………..…..146
Table B.12: 19 Foot Glass Beams…………………………………………………..…..147
Table B.13: 20 Foot Glass Beams…………………………………………………..…..148
ix
LIST OF FIGURES
Figure 1.1: Glass Beams Application in Architecture……………………………….……2
Figure 1.2: Check List of Glass Beam Design……………………………………………4
Figure 1.3: Case Study House #22 by Pierre Koenig, 1960, West Hollywood………….14
Figure 1.4: Library of Loyola University Chicago with Cable Net Glass Facade
by Solomon Cordwell Buenz, 2007, Chicago……………………………….15
Figure 1.5: Atrium of the Local Autority Office by J. Brunet and E. Saunier,
1994, St-Germain-en-Laye near Paris……………………………...………..17
Figure 1.6: Workshop of Musée de Louvre by J. Brunet and E. Saunier, 1993, Paris…..18
Figure 1.7: Entrance Pavilion of Broadfield House Glass Museum by Brent
Richards of Design Antenna, 1993, Kingswinford…………….........………21
Figure 1.8: Apple Retailer on Fifth Avenue by Rohlin, Cywinski and
Jackson, 2006, New York……………………………………………….…..23
Figure 1.9: Yurakucho Subway Station by Rafael Vinoly Architects, 1996, Tokyo….…24
Figure 2.1: Overview of Manufacturing and Processing Stages of Flat Glass……….….31
Figure 2.2: Comparison of Stress-Strain Graphics of Glass, Steel and Wood…….…….37
Figure 2.3: Comparison of Force-Displacement Graphics of Glass Beams……………..39
Figure 2.4: Simplified View of Molecular Structure of Glass……………………….…..40
Figure 2.5: Crack Pattern of Heat-strengthened Glass and Tempered Glass………….…43
Figure 2.6: Glass Roof for Wolfson Medical Building, University of
Glasgow, 2002, by Reiach and Hall Architects…………………………......45
Figure 2.7: Exploded View of Lays of Glass Fins Prior to Lamination……………....…46
Figure 3.1: Surface Flow and Scratch of Glass…………………………………………..49
Figure 3.2: Tension and Compression Stress in Glass…………………………………...50
x
Figure 3.3: Cracking Starts from Tensile Part of Glass Beams, Tested
in Faulty of Architecture, TU Delft……………………………………........51
Figure 3.4: Stress Cross Sectional Diagram of Heat-strengthened Glass
and Tempered Glass………………………………………………………....55
Figure 3.5: Compression/Tension Zone in Tempered Glass and Bending
Stress Decrease in the Bottom Surface……………………………………...56
Figure 3.6: Stress Cross-Sectional Diagram of Chemically Strengthened Glass………..57
Figure 3.7: Unfactored Load Chart for 6 mm (1.4 in) Glass with Four
Sides Simply Supported from ASTM E 1300 (03)……………………...…..66
Figure 3.8: Glass Sheets under out of Plane Load and in Plane Load……………...……68
Figure 3.9: Test Set-up with Glass Specimens Lying……………………………………70
Figure 4.1: Simple Beam under Uniformly Distributed Load…………………………...83
Figure 4.2: Section of Glass Beam and Stress Distribution…………………...…………84
Figure 4.3: Beam Deflection Area Method Visualized………………………………….91
Figure 4.4: Buckling of Cantilever Beams………………………………………………94
Figure 4.5: Local Buckling of Beams……………………………………………………95
Figure 4.6: Inter-brace of Beams……………………………………………………...…97
Figure 4.7: Plan View of Buckled Beams…………………………………………….101
Figure 4.8: yh and Applied Load…………………………………………………….102
Figure 5.1: Excel Tool for Glass Beam Design & Calculation…………………………105
Figure 5.2: Design Area………………………………………………………………...105
Figure 5.3: Material & Structure area…………………………………………………..107
Figure 5.4: Calculation Area……………………………………………………………108
Figure 5.5: Charts of 20 Foot Beams for Roof design with Sliding When Buckling…..114
xi
Figure 6.1: Workshop in the Musée de Louvre ………………………………………..119
Figure 6.2: Glass Pavilion of Broadfield House Glass Museum……………………….120
Figure 6.3: Glass Bridge in Rotterdam…………………………………………………122
Figure 6.4: Glass Conservatory, ‘Teesdale’, Surrey……………………………………124
Figure 7.1: Comparative Ranking of Resistance to Attach by Six
Common Environments……………………………………….…………...129
xii
LIST OF FUNCTIONS
Function 3.1: Strength of Glass under Long Duration Load…………………………….53
Function 4.1: Allowable Stress Design…………………………………………………..80
Function 4.2: Load and Resistance Factor Design……………………………………….81
Function 4.3: Section Modulus Calculation for Glass Beams…………………………...85
Function 4.4: Modified Section Modulus Calculation for Glass Beams………………...87
Function 4.5: Deflection Requirement to Avoid Vibration Damage…………………….90
Function 4.6: Deflection Criteria for Beam Design……………………………………...91
Function 4.7: Deflection Criteria for Glass Design……………………………………...92
Function 4.8: Moment of Interior………………………………………………………..92
Function 4.9: Beam Buckling Criteria…...........................................................................96
Function 4.10: Buckling Calculation for Beams without Intermediate Constraints……..97
Function 4.11: Local Buckling…………………………………………………………103
xiii
ABSTRACT
This thesis is written to discuss how to choose the right types of glass and dimensions for
glass beam design and construction. There is a tendency of structural use of glass recently
to achieve maximum transparency on buildings, and glass beam is one of the most
popular elements. However, there is only limited information on this new technique and
some of the information has not been published to the public, which make it difficult for
architects to design and build buildings with glass beams. The primary target of this
thesis is to provide an introduction about glass beams to explain how they work, and
create tables for size selection. Strength of structural glass is discussed and four primary
criteria three structure design, bending, deflection and buckling, are examined.
1
CHAPTER 1
INTRODUCTION TO GLASS BEAMS
1.1. Glass Beams
1.1.1. Glass Beam Application in Architecture
Although Glass is one of the most important and widely used construction materials, its
application has been restricted to being a planar load resistant enclosure material for
thousands of years, and now is experiencing an innovative transition to being a primary
load carrying structural material. Point load carrying glass columns, linear load carrying
glass beams and walls as well as glass compression bars have already been designed and
built. As a result of the combination of force and full translucency which has never been
seen before, structural glass has quickly become popular since its recent inception, and
glass beams, one of the most important elements of structural glass, has been widely
explored and used, especially in Western Europe, such as Germany, the Netherlands and
the UK.
This new structural element is very popular in courtyard extension projects and entrance
pavilion projects of various building types, as it is almost perfect for architects to create
“invisible buildings” where people could live or play in a controlled climate and at the
same time enjoy natural views and sunshine. Glass beam design is very flexible. Glass
2
can be connected to almost all kinds of architecture elements, such as steel beams,
masonry walls, concrete columns, glass columns and fins. Theoretically, it could be
strong enough to support decks made of different kinds of materials. However, because
of its transparency and other issues, most of the time, glass beams are used to support
clear glass panes, and sometimes they are used to support patterned or frosted glass panes.
As far as we know, no one has designed glass beams supporting opaque decks such as
concrete or wood floors because there are lots of cheaper methods to support opaque
decks and people cannot create transparent structures with opaque decks.
Figure 1.1: Glass Beams Application in Architecture
GLASS BEAMS
Retailer
Museum
Courtyard Covering
Pavilion
Subway
In-between Bridge
Single House
Office
Structural Glass:
Glass Fins
Glass Columns
Glass Walls
Glass Arches
Compression
bars
Decks:
Empty
Clear Glass
Patterned Glass
Frosted Glass
Other Materials
Attaching to:
Glass
elements
Architectural
elements of
other materials
Roofs
Bridges
Floors
Composition:
Glass Panes
Interlay
Sealing
Reinforceme
t
Extension
Preservation
Exhibition
Entrance
Solar House
3
Glass selection is a huge challenge to designers not only because of transparency and
aesthetics issues, but also because glass has some delicate structural properties and thus
every inappropriate decision architects make could lead to structural failure. As a result,
there are more issues need to be taken care of, and sometimes those issues are interrelated
with each other. For example, it is necessary to use safety glass to build glass beams, and
wired glass is one of the most widely used safety glass types. Is it a good choice to use
wired glass for glass beams? The answer is probably “no,” as wired glass cannot be heat
strengthened which means that the strength will be hugely reduced. Color is another
interesting issue that has nothing to do with structure but could lead to structural failure
of glass beams under certain circumstances. If designers add too much color which leads
to a huge amount of solar radiation absorption raising the temperature of beams up to 60
or 80 0C (140-1760F), PVB (polyvinylbutyral) will be melted and the laminated glass will
be delaminated. The following is a list of the factors that could lead to structural failure
and are important in the architectural design process. A more detailed discussion will be
continued in other chapters.
4
Figure 1.2: Check List of Glass Beam Design
Because of the brittleness and elasticity of glass and suspicion about the mechanical
properties of glass, glass beams are currently mainly used only to support glass roofs,
which have limited design load requirements. Some architects also have designed glass
pedestrian bridges connecting different buildings, and glass floors supported by glass
beams have only been used in a few family residence projects. And presently, in order to
achieve high transparency, glass beam has been only used to support clear glass decks.
GLASS
BEAMS
Check list of
structure and
thermal property
of glass beam:
Strength
Stiffness
Buckling
Stress
concentration on
connection
Thermal
expansion
Thermal shocking
resistance
Resistance factor
Residual strength
Long term effect
Possibility of
failure
Depth & Thickness
Layers
Safety glass
Frosted glass
Patterned glass
Color
Coating
Thermal Strengthen
Chemical Strengthen
Bolt
Clamping
Bolt +
Clamping
Adhesive
Interlay
Material:
PVB
SGP
CIP
EVA
Manufacturer and
fabricator capacity
Maintenance
Critical Temperature
Moisture
UV light
Building Type
Structure system
Span
Deck Material
5
Typically, because of the restrictions of glass manufacturing, single span glass beams are
limited to 6 meters (19-20 feet). Composite glass beams with reinforcement spanning
more than 15 meter (49 feet) have been created and tested, and the potential and capacity
of this kind of structural elements are still being explored.
1.1.2. Existing Buildings with Glass Beams
In order to help readers get a better understanding of glass beams, a selective list of
buildings with glass beams will be presented in this section. These samples almost cover
all building types with glass beams and different types of glass beams to give readers a
general idea of glass beam application. These buildings are organized chronologically to
show the development of glass beams and tendency of this movement.
6
Table 1.1: Buildings with Glass Beams
Glass Bridge of Kraaijvanger Urbis
Architectural Practice, 1994, Rotterdam,
Netherlands, (Schittich et al. 1999, p.280)
Architects: Dirk Jan Postel, Kraaijvanger.
Urbis. Rotterdam
Engineer: Rob Nijsse, ABT Velp,
Dimension: 3*(10mm*300mm*3.2m)
Span: 3.2m
Glass Pavilion of Broadfield House Glass
Museum, 1994, Kingswinford, UK
(Richards 2006, p.69)
Architect: Design Antenna
Engineer: Dewhurst Macfarlane
Dimension: 3(10mm*300mm*5.3m)
Full Span: 5.3m
Workshops in the Musée du Louvre,
1993, Paris, France (Compagno 2002,
p.24)
Architect: J. Brunet and E. Saunier
Engineer/Consultant:
Dimension: 4(15mm*600mm*4m)
Full Span: 4m
7
Table 1.1: Continued
Arab Urban Development Institute
Reading Room, 1998, Riyadh, Saudi
Arabia, (Wurm 2007, p.153)
Architects: Nabil Fanous Architects
Engineer: Dewhurst Macfarlane and
Partners
Dimension: 2*(15mm*UN*2.67m)
Full Span: 8m
Yurakucho Subway Station Cantilever
Canopy, 1996, Tokyo, Japan, (Rafael
Vinoly Architects)
Architects: Rafael Vinoly Architects
Engineer/Consultant: Dewhurst
Macfarlane and Partners
Dimension:Mutiple*
Full Span: 15 feet
Demountable Glass Pavilion, 1995-1996,
RWTH Aachen, Germany (Wurm 2007,
p.108)
Construction: Department of Building
Construction, RWTH AachenConcept and
Design: U. Knaach and W. Fuhrer
Dimension:
Full Span:
8
Table 1.1: Continued
Glass Bridge of Schwabisch Hall, 2005,
Schwabisch Hall, Germany, (Wurm 2007,
p.174)
Architects: Kraft+Kraft Architekten
Engineer: Ludwig und Weiler GMBH
Dimension: 4*(12mm*UN*3m)
Full Span: 6.2m
International Chamber of Commerce
(IHK), 2003, Munich, Germany, (Wurm
2007, p.144)
Architects: Betsch Architekten
Specialist Contractor: Andreas Oswald
GMBH, Munich
Dimension: 2*(12mm*0.9m*4.5m)
(10mm+19mm+10mm)*0.9m*4.5m
Full Span: 14m
Wolfson Medical Building, 2002,
University of Glasgow, UK, (Wurm 2007,
p.168)
Architects: Reiach and Hall Architects
Engineer/Consultant: Arup, London
Dimension: 2*(19mm*1.3m*3.9m)
Maximal Span: 15.5m
9
Table 1.1: Continued
Refectory at the TU Dresden, 2006,
Dresden, Germany, (Wurm 2007, p.176)
Architects: Maedebach, Redeleit &
Partner Architects
Engineer: Leonhardt, Andra& Partner
Consultant/Test: Prof. Bernhard Weller,
Thomas Schadow
Dimension: 4*(12mm*350mm*1.45m)
Maximal Span: 5.75m
Apple Retailer on Fifth Avenue, 2006,
New York, USA (Seele)
Architect: BCJ Architects
Engineer/ Consultant: Eckersly
O’Callahan/ Seele/ Dewhurst Macfarlane
and Partners)
Dimension: 5(1/2”*1’*10’10”)
Full Span: 32’
Great Western Dock, 2005, Bristol, UK,
(Wurm 2007, p.171)
Architects: Alec French Architects
Engineer: Arup, London
Special Contractor: Space Decks LTD
Dimension: 3*(10mm*0.9m*4.5m)
Full Span: 14m
10
1.1.3. Advantage of Glass Beams
Besides the unique aesthetic values of glass beams, they also have several other
advantages. Since the beginning of human civilization, people have been dreaming of
living in a controllable shelter that could not only protect themselves from the
environment and living conditions but also provide them with the benefit of sunshine and
natural ventilation. Glazing, which is transparent and operable, is one common way to
achieve that goal, and maximum glass application and transparency are two of the
important themes of recent architectural history. Glass envelopes have been widely used
on modern buildings, and transparent glass structures are the latest trend in this progress.
Glass beams show significantly high benefits for certain types of projects, such as
existing building extensions and historic building preservation, where envelopments with
minimal visual interruption are needed. “Invisible” extension projects are the most
popular glass beam application in buildings so far. This kind of structure could support
additional function requirements without breaking the original fabric relationship of
existing buildings.
Although elasticity and brittleness of glass lead to great concerns about structural glass,
glass has several advantages as a structural material. Glass has high compressive strength,
11
approximately two times that of steel, and some theoretical tensile strength. The strength/
density ratio, especially compression strength and theoretical tensile strength, of glass is
higher than most other structural materials, including steel. In addition, glass has a
relatively high elastic modulus, which is one third of steel but still more than two times
that of concrete.
Glass beams are also very environmentally friendly because of their transparency and
recyclability. Generally speaking, the more transparency buildings have, the more they
utilize natural energy. Although people have already created highly transparent envelopes,
opaque structural elements still block a huge amount of natural light and view. With the
help of structural glass, people can make maximum use of sun-shine and enjoy the views
outside. Environmentally friendly design must consider the lifecycle view of the product,
and production of new materials produces pollutants: waste heat, CO2, dust and so on. As
cullet (waste glass that is crushed to be melted to form new glass) is an essential
ingredient in the manufacture of float glass, it is possible that glass application could
produce zero waste and needs minimal energy over its life cycle.
1.2. History of Glass and Glass Beams
This section includes a brief history of glass and structural glass to help readers
understand why people like to build structure with glass, a brittle material that is not
12
favored in existing design theory. Several existing cases will also be discussed in this
section and organized primarily chronologically and then grouped into categories based
on structural rules to demonstrate the development of structural glass elements and glass
beams.
1.2.1. Glass in the Prehistoric Era
Naturally occurring glass, especially obsidian, has been used by Stone Age societies all
over the world, and the history of manmade glass dates back to the middle of the third
millennium BC in coastal north Syria, Mesopotamia or Old Kingdom Egypt based on
recent archaeological evidence. In the Late Bronze Age in Egypt and West Asia, people
discovered various methods to get glass. Most of the glass products in that period were
colored glass ingots, vessels and common beads. Although they were produced with
different methods, much of those early productions relied on grinding techniques
borrowed from stone working, which means that the glass was ground and carved in a
cold state. During the Hellenistic Period, new techniques of glass production were
introduced making it possible to manufacture larger and colorless glass. (Wikipedia 2010)
1.2.2. Glass Used for Architectural Purposes
Using glass for architectural design is the innovation of ancient Roman architects. Two
revolutions happened in the first century BC: invention of glass blowing method, which
13
makes it possible to make glass panes big enough for architecture, and the introduction of
manganese oxide, which makes it possible to produce clear glass. The Romans were so
excited about this progress that they began to use these relatively small pieces of glass
with poor optical qualities (compared with modern glass panes) in the most important
buildings in Rome and the most luxurious villas of Herculaneum and Pompeii.
(Wikipedia 2010)
Glassblowing is a technique that involves inflating the molten glass into a bubble, or
parison, with the aid of the blowpipe, or blow tube (Wikipedia 2010). This technology
provides the ability to make glass products of different shapes and create large glass
panes. In order to make flat glass sheets appropriate for architectural use, the process of
heating, blowing and cooling should be done multiple times. However, there is still a
limit to the flatness of this technique. After the process of glassblowing, glass plates are
cut to fit a window, and the edges of those disks are usually thicker. When installed in the
window frame, those glass panes would be carefully placed thicker side down to provide
stability and to prevent water accumulation. As a result, window panes were always
thicker at the bottom and thinner on the top, which suggested glass is a kind of liquid
material.
14
1.2.3. Glass in Modern Architecture and the Pursuit of Transparency
Although some architects prefer solid materials, glass was a great contribution to modern
architecture. The history of modern architecture could be read as a history of architecture
transforming from solid to transparent, as a pursuit of freedom and openness. Examples
of this pursuit of transparency are abundant: from the Crystal Palace in the 1850s,
Ludwig Mies Van Der Rohe to the Case Study houses in California and international
style high rise buildings built all over the world.
Figure 1.3: Case Study House #22 by Pierre Koenig, 1960, West Hollywood
(Wikipedia 2009)
15
Engineers have already established a fairly well developed construction system with glass
panes and mullions, with pretty good transparency. However, architects and engineers are
still not satisfied with it, as no matter how much transparency, they still use opaque
mullions. In order to get better transparency, glass structures with cable and bolt
connection are used, such as cable truss structures and cable net structures. But the same
issue of transparency still exists. How can designers create a kind of structure with 100%
transparency?
Figure 1.4: Library of Loyola University Chicago with Cable Net Glass Facade by
Solomon Cordwell Buenz, 2007, Chicago (Wikipedia 2009)
16
1.2.4. Structural Glass - Glass Fins and Glass Columns
Although glass as a building material was used for more than two thousand years, the
structural properties of this material did not become a matter of serious research until the
late 1980s. Glass, which was mainly used for facade and decoration, began to be
considered as a kind of structural material to explore the possibility to design buildings
with maximum transparency, and different glass structural elements began to appear.
Glass fins are the oldest and most widely used glass structures, which were first designed
in the 1950s, and mainly designed to resist lateral wind load. Foster and Partners
designed and built the linear-shaped Sainsbury Center for Visual Art in Norwich
(England) between 1974 and 1978 using 60 cm (1.97 feet) wide, 25 mm (1 inch) thick
fins of toughened glass for the 30×7.5 m (98.43×24.61 feet) glass wall. The success in
use of glass fins to resist wind load prompted the idea of glass columns and beams.
Benthem Crouwel Architects completed the house in Almere ( the Netherlands) in 1984
using story-high glazing in the living room consisting of 12mm(0.47 inches) thick
toughened glass designed to resist wind load with 15mm(0.59 inches) thick fins of
toughened glass. The glass fins also serve as a bearing point for the lightweight roof.
Ten years later architects J. Brunet and E. Saunier built the glass roof for the atrium of
the Local Authority Office in St-Germain-en-Laye near Paris, which demonstrated the
17
spectacular use of glass columns. 22×22 cm (8.44×8.66 inches) glass columns designed
with three layers of toughened glass to support the load of a 24×24 m (78.74×78.74 feet)
glass roof. They are approved for a loading of six tonnes (13228 pounds), but have been
calculated to support 50 tonnes (110230 pounds). At end of the 1980s, glass beams, as
well as all glass structures, began to be used widely. (Basic information from Intelligente
Glasfassaden: Material, Anwendung, Gestaltung by Andrea Compagno)
Figure 1.5: Atrium of the Local Autority Office by J. Brunet and E. Saunier, 1994,
St-Germain-en-Laye near Paris
1.2.5. Glass Beams
Compared to glass fins and glass columns, glass beams evolved late. In general,
they are used as glass roofs or floors and have to carry live load as well as medium- and
long-term dead loads.
18
1.2.5.1. Early Glass Beams
Just one year before the project of the Local Authority Office in St-Germain-en-Laye, J.
Brunet and E. Saunier created a glass roof with glass beams for the Workshop of the
Musée de Louvre in Paris. The glass construction covers a three-story light well that is
16m (52.49 feet) long and 4m (13.12 feet) wide with laminated skylight panes composed
of four layers of 15mm (0.59 inches) thick toughened glass. The panes are supported by
60cm (1.97 feet) high laminated glass beams of four 15 mm thick strips of toughened
glass. Those beams were estimated to support five tonnes. Surprisingly, exhaustive tests,
taken later to study the behavior of the material, revealed that the glass beams could
actually resist of 12.2 to 14 tonnes.
Figure 1.6: Workshop of Musée de Louvre by J. Brunet and E. Saunier, 1993, Paris
(Compagno 2002, p. 24)
19
This building shows the idea of connecting different buildings parts with glass roofs or
glass floors supported by glass beams. The glass structure is fixed on-to solid nearby
elements to provide enough stiffness, while providing perfect transparency to provide
views of its surrounding. The architectural office of Kraijvanger and Urbis, Rotterdam
(Netherlands) built a glass bridge 3.2 m (10.50 feet) long to provide a first floor link
between the offices of two adjacent buildings. Architects Ottavio di Blasi Associati also
proposed a glass bridge of the basilica of Aquileia (Italy) to protect the valuable 4th
century mosaic floor.
1.2.5.2. All Glass Structure
The same year as the project of the Workshop of Musée de Louvre, one year before Local
Authority Office in St-Germain-en-Laye near Paris by J. Brunet and E. Saunier, a more
exciting and inspired building was built in Kingswinford near Dudley, England. The
Entrance Pavilion of Broadfield House Glass Museum designed by Brent Richards of
Design Antenna with Dewhurst Macfarlane (structural engineer) is probably the first all-glass
structure in the world supported by glass columns and glass beams, and was the
largest all-glass structure in the world for more than ten years. The structure is 11m long
by 5.3m wide (36.01×17.39 feet), with 3.5m (11.48) high glass columns supporting the
roof.
20
Kingswinford used to be famous for its glass industry and became the center of glass
manufacture in England since the sixteenth century competing with Venetians. This
entrance pavilion was designed to show off the achievement of this field. It is built
against the rear wall and projecting side wall of Broadfield House, and its gable end is of
rendered blockwork. The structure is a relative simple one-way beam to column system
with 300mm high (0.98 feet) beams 1.1m (3.61 feet) apart. The roof is designed to
support 0.75 KN/m2 (15.75 psf) snow loading, making it strong enough for a man to work
on it for cleaning. Both beams and columns are made of three sheets of 10mm (0.39
inches) glass laminated together, making them 32mm (1.26 inches) thick (with two inter-layers
of 1mm PVB). At the rear the beams rest on the shoes fixed to the wall; at the front
they were connected to the columns by cutting and splicing the laminated sheets to form
mortis-and-tenon joints which were bonded onsite with resin laminate. No metal
connections are used on this building.
21
Figure 1.7: Entrance Pavilion of Broadfield House Glass Museum by Brent
Richards of Design Antenna, 1993, Kingswinford (Firman Glass)
1.2.5.3. Two Way Structural System
Twelve years later, the span record of the Entrance Pavilion of Broadfield House Glass
Museum was broken by the Apple Cube on Fifth Avenue, New York, which is 10 m by
10 m by 10 m (32 feet by 32 feet by 32 feet). This cube, following the idea of Pei’s
Louvre Pyramid of Musée de Louvre, as well as the all glass stair and elevator, is the
entrance for the Apple retailer underneath Grand Army Plaza, which shows the great
ambition of Apple and the magic of glass application. This building is so successful that,
according to a research of photos on Flickr, it is one of the top 20 popular buildings in the
world with which people would like to take pictures.
22
The roof structure is based on a lamellar principle whereby each 10 feet and 10 inches
long beam section across the two-way grid is supported via a pin connection to another
10 feet and 10 inches long glass beam section spanning the other direction. This kind of
structure eliminates the need for moment connections through the glass and creates
longer spans with shorter single beams. The roof beams are laminated from 5 pieces of
1/2” heat strengthened glass with both ends laminated to a thin stainless steel shoe insert
that allows the post connection of a fin plate. The two way glass beam system was first
used in the Glass Reading Room of the Arab Urban Development Institute and became
popular in later designs. In 2006, Maedebach, Redeleit & Partner designed and built a
huge glass roof supported by two way glass beams covering an interior courtyard of
24×30m (78.74×98.43 feet).
23
Figure 1.8: Apple Retailer on Fifth Avenue by Bohlin, Cywinski and Jackson, 2006,
New York (Apple Inc. 2010)
1.2.5.4. Interlocking Fins
The Yurakucho Subway Station in Tokyo, designed by Rafael Vinoly Architects with
Dewhurst Macfarlane and Partners and finally finished in 1996, shows a new approach of
using glass in beam design. The canopy, 10.6m (34.78 feet) long, 4.6m (15.09 feet) wide
and 4.8m (15.75 feet) high at the apex, is supported by three parallel cantilevers
composed of several triangular shaped laminated blades. Those triangular blades are
bolted to interlock to each other, with one bolt in the center and two at both ends to
prevent rotation and torsion. 40mm thick acrylic panels are used there for earthquake
safety. Interlocking fins were also used in later designs, such as the glass roof for the
24
International Chamber of Commerce (IHK) in Munich, 2003, designed by Betsch
Architects.
Figure 1.9: Yurakucho Subway Station by Rafael Vinoly Architects, 1996, Tokyo
(Rafael Vinoly Architect)
1.2.6. Further Development of Structural Glass Research
Further experiments and practice continued, aiming to make optimum use of the high
compression strength of glass. Glass arches and glass domes, designed on this idea, were
presented at Glasstec ’98 in Dűsseldorf. “Glasbogen 2” (glass arch 2) consisted of
fourteen 1.64*4m (5.38*13.12 feet) laminated glass panes tested under various loads to
the destruction point after the exhibition. The glass dome, built by Seele of Gersthofen
25
(Germany), had a diameter of 12.3m (40.35 feet) with 2.5m (8.20 feet) rise with
triangular panes of laminated glass. Two years earlier, a tensegrity glass structure was
presented at Glasstec ’96. For the Central Hall of the new Lehrter Bahnhof station, Berlin,
glass was used as compression elements of the cable truss, and it was demonstrated to be
able to carry loads at a stress of 60 ksi by both FEM calculation and experiment. This
stress level is higher than the yield stress of stell used as columns in building today. All
of those projects show new approaches of using glass as a structural material. However,
they will not be covered in this thesis, which is focused on glass beams, and thus the in-plane
bending resistance of horizontal structural elements.
1.3. Current Research and Target of This Thesis
Numerous studies are underway about glass beams by research institutes, glass designers
and manufacturers mainly in Western Europe, especially in Germany, the Netherlands
and the UK. TU Delft and RWTH Aachen have sponsored glass research centers working
on structural glass, and publish papers every year. Valuable data and studies can also be
found in UK, Switchland and some other areas. Dewhurst Macfarlane & Partners, Seele
and Arup are the most experienced and well known engineers and glass consultants in
this area. They have been involved in several of the most challenging projects and create
innovative details. Most of their research is presented and published in the GPD
conference (Glass Performance Days), and available online, www.glassfiles.com, or in
the GPD publications.
26
In general, there are two branches of glass beam study: first, glass property and laminated
glass application in architecture; and secondly, glass beam reinforcement. Although glass
has been used for thousands of years and thoroughly studied, it has not been used as
linear load carrying material. The new application requires us to reconsider this material
from a new perspective, and to be restudied again. As there is still a lot of uncertainty
about glass and it is the base of the whole study, this thesis will be focused on glass beam
theory and design without any reinforcement.
27
CHAPTER 2
PROPERTIES OF GLASS AND GLASS BEAM MANUFACTURING
Properties of structural materials are determinant factors for structure form and size
selection, while mechanical properties, such as density, strength, stiffness and Poisson’s
ratio, are the most important properties for structural design. Those properties determine
load carrying capacities and the deformation of the structure. Other properties such as
chemical resistance, thermal resistance, moisture resistance and so on also limit
application of glass. Those factors will be discussed in this chapter. Glass and glass beam
manufacturing processes also have a strong influence on design and will also be
discussed in this chapter.
2.1. Glass as Structural Material
Glass is a material that has been used by humans in everyday life for thousands of years;
however, it has not been seen as a structural material until recent decades. People are
afraid of structural use of glass mainly for two reasons: brittleness and transparency. The
brittleness of glass makes glass break before yielding, which means that nothing could
foreshadow the breaking of glass beams and thus people have no warning to get away
from potentially dangerous structures. Transparency makes glass beams look light and
not strong enough to sustain load and also cause acrophobia. However, if we take a look
28
at the major structural properties of glass, we will find out glass can be a good structural
material.
Steel S235 Softwood S10
Concrete
C20/25
Glass Soda-
Lime glass
Refractive Index _ _ _ 1.5
Density (lb/ft3) 500 38 140 159
E-Modulus (ksi) 30457 1600 4200 10100
Tensile Strength (ksi)
34.8 (yield
strength) 2.03 0.32 6.5
Elongation at break (%) 25 0.7 _ 0.0006-0.17
Compressive Strength (ksi) 34.1 0.58-3.77 2.9 approx. 72.5
Limiting Tensile Stress
(ksi) 31.61 1.3 0.15 1.74/2.61
Safety Factor 1.1 1.3 1.8 2.5
Breaking Length (m) 2800 1500 45 480/720
Thermal Conductivity
(W/m×k) 75 0.5/0.2 1.6 1
Thermal Shock Resistance _ _ _ 40
Coefficient of Thermal
Expansion (10-6*1/k) 12 5 or 35 10 9
Table 2.1: Properties of Glass and Several Other Structural Materials (Original
Chart from Wurm 2007, p. 36)
Good engineers are always looking forward to create stronger structures with lighter
materials. Strength and Strength/Density ratio are two of the most important factors for us
to evaluate structural materials. Although tensile strength of glass is only about 20% of
steel, it is still many times stronger than concrete and wood, and the compressive strength
of glass is even greater than that of steel. The compressive strength/density ratio of glass
29
is nearly seven times that of steel, which is quite impressive. Although tensile
strength/density ratio is not as good, it is still nearly 60% of that of steel.
2.2. Glass Manufacturing
Glass manufacturing should be discussed first for two reasons. First, manufacturing
processes and abilities limit the accessibility and economical efficiency of the material
and structure designers choose. Also, not just any kind of glass is suitable for glass beam
design, and it is meaningless to talk about “properties of glass” in general. Qualified
types of glass should be verified first, and then discussed regarding their manufacturing,
composition and properties.
2.2.1. Glass Manufacturing
The manufacturing process of glass includes primary processes, mechanical processes,
thermal treatment, lamination and coating. The primary process produces basic jumbo
glass sheets, usually of annealed glass. When they are needed, those sheets will be cut
into required size, drilled and edges ground. Thermal treatments will be applied after the
mechanical process. Any mechanical processes occurring after thermal treatment will
cause strength damage of heat-strengthened glass. Lamination provides a method to bond
different pieces of glass together, and shows many benefits from the bond and interlay
30
material. The coating process covers the surface of glass with additional materials and
layers to improve certain properties of glass.
Most of any further treatments after the primary process have a negative effect on the
strength of glass except for thermal treatment. Cutting, drilling and grinding cause
mechanical surface damage, and coating as well as lamination causes chemical surface
damage. There are two ways to improve the strength properties of glass, thermal
treatment and chemical strengthening.
31
Figure 2.1: Overview of Manufacturing and Processing Stages of Flat Glass
(Wurm 2007, p. 34)
32
2.2.1.1. Composition of Glass
The most widely used glass today is soda-lime glass, which accounts for about 90% of
glass in the market. It is prepared by melting a mixture of silicon dioxide (silica), sodium
carbonate (soda), limestone, dolomite, aluminum oxide and small quantities of fining
agents at temperatures locally up to 1675 0C (3047 0F). For all kinds of glass, the primary
ingredient is silica (SiO2), which melt at 2300 0C (4200 0F) at a viscosity of 10 Pa.s (100
P). To lower the melting point, sodium carbonate (Na2Co3) is added. And because soda
makes the glass water soluble, lime (CaO) and aluminum oxide (Al2O3) are added.
Generally, soda-lime glass contains 73% SiO2, 14% Na2O and 9% CaO by weight. To
improve properties of glass, other recipes may be used which leads to different types of
glass.
2.2.1.2. Primary Manufacturing Processes: Float Glass, Rolled Glass, Drawn Glass
Millions of tons of flat glass sheets are produced every year in the world and 90% of
them are float glass. The glass floating method was invented by Alastair Pilkington in the
1950s. In honor of the inventor, the float glass process is also called the Pilkington
Process. During the floating process, glass is melted at approximately 1100 0C (2012 0F)
and then poured continuously from the furnace to an approximately 50 meters long
(164.04 ft) shallow bath of molten tin. The glass floats on the tin and spreads out until it
33
solidifies at approximately 600 0C. Thickness is controlled by the speed at which the
solidifying glass sheet is drawn out of the bath. Usually, monolithic glass sheets are
between 2mm (0.08 in) to 19mm (0.75 in) thick. After cooling, float glass is cut into
jumbo sheet of 3.21×6m (10.53×19.69 ft). Some glass factories produce super-sized glass
sheets as long as 12m (39.38 ft) for special purposes. There are four leading glass
manufacturers in the world – Nippon Sheet Glass (which took over Pilkington in 2005),
Asahi, Saint-Gobain and Guardian- providing about two-third of global production of
high quality float glass (Approximately 25 million tons). (Wurm 2007, p. 46)
Other methods are also used to produce flat glass sheets, such as glass rolling method and
drawing method. The rolling method is mainly used to produce glass sheets with
beautiful patterns and colors. The drawing method is mainly used to produce special flat
glass to specific needs. However, because of the high dimensional accuracy, geometrical
precision and optical quality, float glass is considered to be the only glass suitable to be
used in buildings if further processes are needed.
34
Table 2.2: Minimum Glass Thicknesses from ASTM E 1300 (03) (ASTM 2004, p.
626)
2.2.1.3. Thermal Treatment: Annealed Glass, Heat-strengthened Glass and
Tempered (Toughened) Glass
After completing all mechanical work, thermal treatment is applied to annealed glass
providing greater resistance to mechanical and thermal loads. Heat treatment could be
carried out on all types of basic glass products except wired glass. In this process,
annealed glass is heated up-to about 650 0C (1202 0F) and cooled with air blown over
both sides quickly. The thermal treatment process has strong influence on the mechanical
properties of glass. Generally speaking, heat-strengthened glass is two to four times
stronger than annealed glass, while tempered glass is four to five times stronger than
annealed glass. Thermal shock resistance is also significantly improved by heat-treatment.
35
Because of the huge amount of energy stored in the glass, tempered glass has different
fracture behavior from annealed glass and is thought to be safty glass. However, heat
treatment has no affect on the stiffness of glass, which is defined by Young’s modulus
(E-modulus), and may be detrimental to optical properties of glass. Tempered glass and
heat-strengthened glass cannot be further cut or grinded. Tempered glass also has
particular phenomenon called spontaneous fracture because of the high pre-compression
on the surface, which leads to great concern for its application as beams.
2.2.1.4. Lamination
Laminated glass is made up of several layers of monolithic glass sheets and interlayer
materials. Almost all kinds of glass panes (and other materials besides glass) could be
used as single sheets. The materials used as the interlayer material are polyvinylbutyral
(PVB), cast-in-place resin (CIP), ethylene vinylacetate (EVA), and Sentry Glass Plus
(SGP) by Dupont. PVB is the most widely used one, which is used in more than 95
percent of all laminated safety glass. The thickness of the interlayer is a multiple of the
film thicknesses of 0.38 and 0.76 mm. Laminated glass is considered to be the ideal glass
for structural use because the lamination process substantially improves the load-bearing
behavior, robustness, and especially post fracture integrity. Because of the interlayer
material, laminated glass can hold its broken fragments, and have the ability to carry load
after breakage. This ability is very crucial for structural design.
36
2.2.2. Mechanical Properties of Glass
Mechanical properties of a material include but are not limited to elasticity, plasticity,
strength, brittleness, ductility, hardness and toughness. Mechanical properties are critical
factors in structural materials selection.
2.2.2.1. Elasticity, Brittleness and Lamination
Elasticity describes the physical property of a material to return to its original shape after
stress is removed, while plasticity describes permanent deformation. In modern structural
design theories, plasticity is advantageous for structural materials, as plastifying material
can still hold certain amounts of load under extreme load. Steel, an elastic/ plastic
material, shows this tendency, while glass is an ideal elastic/brittle material. The behavior
of glass is similar to concrete in tension or wood in compression or tension. The absence
of yield stress, or the phenomena of yield strength being greater than the ultimate strength,
implies that glass will break without yielding. The absence of plastic deformation and
yield stress makes glass highly susceptible to local over-stressing when used as a
structural material. In order to make it easy to understand and use the same terminology
as used in other material design, we would like suggest yield strength and ultimate
strength are the same thing in glass design, which means the actual stress leading to glass
breakage.
37
Figure 2.2: Comparison of Stress-Strain Graphics of Glass, Steel and Wood (Wurm
2007, p. 38)
In order to deal with the elasticity and brittleness of glass, safety glass has been adoped
for most uses and engineers developed several technologies following different ideas.
Wired glass and laminated glass are two of the most widely accepted safety glass. Wired
glass is produced by feeding wire mesh into the liquid glass to make sure when glass
breaks, pieces will not drop down and could still be held together by wire mesh to have
some residual strength. Laminated glass is produced by laminating several glass sheets
with interlaying material. When laminated glass breaks, pieces from different sheets will
overlap each other to make sure the whole structure will not fall down and keep residual
strength. Both of those two methods could improve mechanical properties of glass but
laminated glass is preferred and all architectural cases with glass on top of people or
38
supporting load are required to be laminated glass by code. The biggest problem of wired
glass for structural glass design is that it is produced through a rolling process and rolled
glass cannot be further tempered or strengthened. As a result, all glass beams have to be
built with laminated glass, and all the existing cases are built with laminated glass too.
The strategy of lamination provides glass with a kind of plasticity and ductility. Because
stress-strain graphics can only show mechanical property of material, it is not that helpful
for us to understand how the lamination process works, and we would like to use force-displacement
graphics instead. The force-displacement curve of steel and monolithic
beams is the same as the stress-strain curve, while laminated glass beams show some new
features. The drawing below is an imaginary force-displacement curve of a four-layer
glass beam, which is strong enough to allow breakage of two layers, and we assume glass
sheets break layer by layer before the whole structure collapses.
39
Figure 2.3: Comparison of Force-Displacement Graphics of Glass Beams
Because actual strength of glass sheets falls into a certain range around the average
strength of glass, some sheets will break first under stress below that level and some will
break under stress above that level. Whenever one lay break, the total force will stay the
same while displacement will increase due to increased stress in the remaining sheets.
Because of the lamination, sometimes glass sheets will not break with stress above its
strength because other glass sheets will hold it. After the collapse of structure, pieces in
different layers overlap each other to provide residual strength. Although several tests
have been performed to study the residual strength of laminated glass, residual strength of
laminated glass beams is still little studied and unknown.
40
2.2.2.2. Strength
The strength of a material is the ability to withstand stress without failure. Material
strength is defined as tensile strength, compressive strength, bending strength and shear
strength. Tensile strength resists tension, compressive strength resists compression,
bending strength resists bending (combined tension and compression) and shear strength
resists shear (a sliding action).
Figure 2.4: Simplified View of the Molecular Structure of Glass (Wurm 2007, p. 36)
As mentioned before, the main chemical composition of glass is silicon oxide, which is
super strong with a theoretical tensile strength of 8 KN/mm2 (1160 ksi) (some argue this
value could be as high as 15-20 KN/mm2), which is about 70 times the yield strength of
steel. However, this value, or even a close value, has never been observed in reality or
testing. Glass fibers incorporated in a matrix of resin have been tested in 1976 by Gordon
to have a usable strength greater than 2 KN/mm2 (290 ksi). And the widely accepted
tensile strength of annealed glass is only 25-30 N/mm2 (2.9-4.35 ksi), while thermal
41
treatment could make it several times stronger. However, strength and design strength of
glass is such a complex and important issue that it needs a lot of discussion later.
2.2.2.3. Stiffness
Stiffness defines the ability of a material to resist deformation under load, and is
measured by the elastic modulus (E-modulus), which is also called Young’s modulus (Y-modulus)
in honor of English scientist Young, who defined it in 1807. A higher E-modulus
causes smaller deformation under stress. Certain stiffness is needed for
structural design, and the widely accepted E-modulus of glass is 70 KN/mm2 (10.150 ksi),
which is about 1/3 of the E-modulus of steel. Thermal treatment has no affect on the E-modulus
of glass.
2.2.2.4. Thermal Strain and Thermal Shock Resistance
Unrestrained objects will expand and contract when temperature increases and decreases.
The coefficient of thermal expansion defines the tendency of material to expand or
contract with changing temperature. Glass has a big coefficient of thermal expansion,
which is about seven times greater than that of steel and concrete. This implies glass
structures will either have large thermal stresses if restrained or larger thermal extension
if unrestrained.
42
Thermal shock describes the phenomenon of cracking caused by rapid temperature
change, which is more crucial for brittle materials such as glass and ceramic because of
their low toughness, low thermal conductivity and high thermal coefficient. Thermal
treatment could significantly improve thermal shock resistance of glass. Typically, it is
40 Kelvin for annealed float glass, but 150 Kelvin for tempered glass and 300 Kelvin for
tempered borosilicate glass.
2.2.2.5. Crack Pattern
The crack pattern is one of the most useful and important mechanical property of glass
for structural glass design for several reasons. Glass generates different patterns under
different levels of thermal treatment. It is important for the safety of the glass and ability
of it to resistant load after it is broken. Certain crack patterns can also imply impurity of
the glass and weakness of glass under different kinds of loads that need special treatment.
43
Figure 2.5: Crack Pattern of Heat-strengthened Glass and Tempered Glass (Wurm
2007, p. 55)
2.3. Glass Beam Composition
There will be a brief introduction of how to make a glass beam and problems in this area.
Glass beams are not simply made up of monolithic glass sheets hung on top of columns,
but of laminated glass sheets with several further processes. Every process will change
properties of glass and influence the design method.
2.3.1. Lamination
Research on existing glass beam structures and code requirements indicate that all glass
beams are built by laminated glass. Unlike laminated wood boards, which are usually
44
laminated horizontally, glass beams are always laminated vertically for two reasons. First,
vertically laminated glass provides alternative load paths for the beam, which means that
even if one piece of the laminated glass is broken, the other sheets could still carry loads.
Secondly, even if all layers of glass sheets are broken, the broken glass pieces are still
held together by the interlay materials with some residual load-bearing capacity. The
strength of laminated glass beams is mainly determined by the strength of monolithic
glass panes, while the strength of interlayers and crack pattern will have strong effects on
the residual load-bearing capacity.
2.3.2. Limit of Single Span Coming from Manufacturing Modulus
As mentioned before, 90% of architectural flat glass is float glass, while the size of
jumbo sheets before cutting is usually 6 meters (19.69 feet) long and there will be a huge
price premium for manufacturing, transporting and resembling oversized glass sheets.
Oversized glass sheets can only be achieved from limited manufacturers with a limited
selection of glass recipes, mainly low-iron glass. As a result, it is a great challenge to
build glass beams, as well as other glass elements, longer than 6 meters. Therefore, the
single span of one piece of a glass beam is always no longer than 6 meters.
There are several ways to build glass beams longer than 6 meters. Reiach and Hall
Architects in Edinburgh designed and built a glass roof supported by glass beams
45
spanning 15.5 meters (50.85 feet). As no single glass sheets could be that long, they
connected single beams of 3.9 meters long by double shear steel splice plate connection
located near the top and bottom. In the project of Yurakucho Subway Station in Tokyo,
designed by Rafael Vinoly Architects with Dewhurst Macfarlane and Partners, the
designers invented the idea of interlocking glass fins to create a cantilever canopy
spanning 10.6m (34.78 feet) long. While in the design of Apple Retailer on Fifth Avenue
in New York, glass columns of 10 m (32 feet) high are needed. Manufacturers spliced
five layers of 10 feet long monolithic glass fins with a central overlap of 3.3 m (11 feet).
Figure 2.6: Glass Roof for Wolfson Medical Building, University of Glasgow, 2002,
by Reiach and Hall Architects ((Wurm 2007, p. 168)
46
With the ongoing development of the building industry, people want glass sheets bigger
and bigger. In the Apple Cube project, engineers, working with manufactures and
fabricators, invented a creative way to build glass columns made 10 m (32 feet) long
laminated glass. They laminated different sheets interlocking with each other, and used
an autoclave, which is usually used for aircraft wings, to work on the lamination process.
Figure 2.7: Exploded View of Lays of Glass Fins Prior to Lamination, (O’Callaghan,
2007, p. 3)
2.3.3. Sealing
Although glass is a kind of inert inorganic material which is not easily broken down
under normal condition, the interlayer materials, such as PVB, CIP, EVA and SGP are
organic materials which can easily get mechanical or chemical damage. In order to
protect interlayers of glass beams, especially when beams are exposed to outside
environments or moisture, edge sealing is needed. It is also a good idea to connect the
47
other side of beams in contact with roof or floor with silicon because in this way, the roof
or floor panels could work as braces to prevent the beams from buckling and out of plane
bending damage. During the process of lamination, the edge of some types of laminated
glass has already been sealed with transparent double-sided adhesive tapes. Silicone is
also a good choice for edge sealing. However, improper sealing has a negative effect on
the optical properties of glass beams.
2.3.4. Reinforcement
In order to deal with the brittleness of glass, minimize the potential for damage on tensile
edge of the glass beam and improve the residual load-bearing capacity, researchers are
trying to apply additional reinforcement to glass beams. Generally speaking, there are
two methods: reinforcement on the top or bottom of glass beams and reinforcement in the
interlayer materials. Following the idea of commonly used single material or composite I-beams,
steel, wood and glass flanges on both the top and bottom of glass beams have
been developed and tested. Following the idea of reinforced concrete, which uses steel
reinforcing elements on the tension side only of beams to resist tension stress, metal
fibers or sheets and glass fibers are used on the bottom of glass beams. Fibers could also
be used in the process of lamination as a part of the interlayer materials to improve the
capacity of broken glass beams to resist load.
48
CHAPTER 3
STRENGTH OF STRUCTURAL GLASS BEAMS
Strength is always the first and most important factor for structural material selection and
structural design. As mentioned in Chapter 2, glass has high theoretical strength, however,
yield strength, especially tensile strength of structural glass panels, is lower than the
theoretical strength. The strength of glass is not mainly governed by the chemical
components of glass but by the size of surface flaws, and glass strength can easily be
influenced by manufacture, fabrication, installation, maintenance and environment.
Because there is plenty of data on strength of glass panels but only limited tests on glass
beams, designers are more certain about strength of glass panels than that of glass beams
and those tests on glass beams are not sufficiently thorough to be able to draw
conclusions. As a result, in order to decide on an allowable strength of glass beams, it is
currently better to determine the strength of individual glass panels first and use that
strength to find the overall glass beams strength. Two tests will be referenced in this
chapter because annealed glass sheets are a type of glass made through the float process,
and other types of glass further treatments to annealed glass sheets. It is a generally
acceptable method to determine the strength of annealed glass first and get the strength of
other types of glass from this strength with modifications. This method will also be used
in this thesis.
49
3.1. Griffith Flaw
In order to explain brittleness and elasticity of glass as well as the relatively low
allowable stress, especially tensile stress, of glass, A.A Griffith proposed his theory in the
1920’s, which is widely accepted and called Griffith Flaw Theory. Based on his
observation on bulk glass, he suggested that the surface of bulk glass contains tiny flaws,
which cause stress concentrations, and the strength of glass is a function of glass flaw
size. His theory explains the mystery of glass strength and several unique phenomena that
are shown in glass beam tests. Furthermore, those flaws have already been observed in
real life.
Figure 3.1: Surface Flow and Scratch of Glass (Button & Brain 1993, p. 213 &
214)
50
3.1.1. Higher Compression Strength but Lower Tensile Strength
Glass has high compression strength, about 7.2 ksi for annealed glass panels, but poor
tensile strength, about 2.8 ksi. Glass, especially glass beam, typically breaks from its
tensile rather than compressive strength. It is easy to understand and explain this
phenomenon with Griffith Flaw Theory. When one wants to break a piece of paper into
two pieces, it is much easier to try to tear it than to stretch it. If one wants to stretch it, it
is easier to stretch paper with a split, even a very tiny split, than paper without a split, and
the paper always breaks from the split first. However, if one wants to crush a brick, tiny
cracks or flaws will not greatly reduce the strength, and the brick does not necessarily
break from cracks first. As a result, compression strength has never been tested on glass
panels, and strength of glass always implies tensile strength or bending strength.
Figure 3.2: Tension and Compression Stress in Glass (Wurm 2007, p. 37)
51
Figure 3.3: Cracking Starts from Tensile Part of Glass Beams, Tested in Faulty
of Architecture, TU Delft (Annealed Glass Left, Tempered Glass Right) (Bos et al.
2005, p. 3)
3.1.2. Higher Theoretical Strength but Lower Useable Strength
Accepted strength of annealed glass panels is no more than 1% of the theoretical strength
of glass. Although most parts of glass could resist high stress, glass will break from
Griffith flaws first because the fabric of glass structure is damaged there and bending
stress is higher than average. However, researchers cannot test stress on those flaws and
can only get average stress in the glass when it breaks, which is generally accepted as the
yield or ultimate strength of glass.
3.1.3. Great Variety
Strength of float glass shows a great variety; however, the strength will not drop below
certain values. Most of the time strength of glass will fall in the range of 25-100 MPa
(3.6-14.5 ksi), which demonstrates the size of Griffith flaws will also fall in a certain
52
range. Generally speaking, strength of glass sheets shows a kind of relationship with
surface/volume ratio. And normally smaller and thicker glass sheets are stronger than
bigger and thinner sheets. This could also be explained by Griffith Flaw theory.
Griffith suggested surface flaws have the following features: 1. the size of these flaws
shows great variability, however, they will not be bigger than a certain value without pre-damage;
2. distribution of those flaws appears to be random. Because flaws of different
size distribute randomly, the larger size the glass panel is, the more likely there will be
bigger Griffith flaws on the surface, and the lower the strength could be. However, this is
only based on statistical analysis. The strength of a 100 ft2 glass panel is not necessarily
lower than the strength of a 1 ft2 glass panel. However, the possibility of a 100 ft2 glass
panel to have low strength is 100 times that of a 1 ft2 glass panel. This phenomenon is
also proven by tests and accepted by most standards. In the current version of ASTM
(American Society of Testing and Materials) E-1300, design stress of glass is not simply
provided by one table but by pages of graphics with dozens of charts relating to the size
of glass. Since glass beams need to be built by large sheets, designers have to be
conservative to choose the right strength.
53
3.1.4. Strength of Glass under Long Duration Load
Another property of the strength of glass is that it will drop with time. Thus, allowable
stress for glass under long-duration load is smaller than that under short-duration load. As
is proven in both theory prediction and real life, strength of glass does drop. Every time a
glass panel is loaded, there will be tension area on the surface, and the tension tends to
stretch existing flaws bigger or create new flaws. Every time there is a process on the
surface of glass, there is also the possibility to create new flaws or enlarger existing flaws.
As a result, glass will lose its strength, and the relationship between load duration and
strength is usually described as:
Function 3.1: Strength of Glass under Long Duration Load (Peter 1999, p. 31)
Based on various test results, most researchers agree that the number n lies somewhere
between 12 and 20 for annealed glass. Pilkington suggests a value of 16, while Schott
Glasswerke uses 20.
The strength reduction will not go on without limitation until strength of glass becomes
zero. Manufactures generally agree that strength of glass will drop to a certain value
above zero and then stop dropping. Both Pilkington and Schott Glaswerke suggest this
54
figure to be 7 N/mm2 (1.02 ksi) and theoretically it will take glass panels 3170 years to
lose their strength from 25N/mm2 to 7 N/mm2 (Peter, 1999, p. 31). However, because
strength of glass could also be influenced by other factors, some experts believe this
process needs only 50 years in real life for glass without special protection.
3.1.5. Strengthened Glass
Griffith Flaw theory could also help designers to understand and find ways to improve
the strength of glass. Heat strengthened glass and chemically strengthened glass are two
of the most widely used types of strengthened glass on the market. Both heat
strengthening and chemical strengthening methods pre-compress the surface of glass
panels to achieve higher strength, but heat strengthened glass is particularly preferred as
strengthened safety architectural glass because thermal treatment not only improves the
strength of glass but also changes the fracture pattern of glass. When heat strengthened or
tempered glass breaks, energy stored in the glass will dissipate and cracks will spread all
over the sheet. As a result, those glass sheets will break into small blunt pieces without
sharp edges.
3.1.5.1. Heat-treatment
Heat-treated glass is a kind of further fabricated glass. During that process, glass is heated
to a uniform temperature and then cooled quickly. Because cooling and strengthening
55
happen first on the surfaces of glass and then on the core, the surface will be pre-compressed
and the core will be pre-tensioned with a parabolic distribution of stress.
Typically, the pre-compressed zone is 20% of the total thickness of the glass on both
sides. Because of the pre-compressed surface, when glass panels are loaded, the surface
will retain compression or only have a little bit of tension, which will not enlarge Griffith
flaws. Because of pre-compression, strength of glass under long-duration load will also
improve. Based on the toughness of glass, there are full tempered (toughened) safety
glass and partially tempered heat-strengthened glass.
Figure 3.4: Stress Cross Sectional Diagram of Heat-Strengthened Glass and
Tempered Glass (Wurm 2007, p. 55)
56
Figure 3.5: Compression/Tension Zone in Tempered Glass and Bending Stress
Decrease in the Bottom Surface (Wurm 2007, p. 55)
3.1.5.2. Chemically Strengthened Glass
Glass with high sodium could be further strengthened in a hot potassium chloride bath.
Sodium ion exchange and densification of the molecular structure create high
compressive stresses on the surface. Chemically strengthened glass could be cut to
limited extent, and chemical strengthening process could be applied to nearly all kinds of
glass sheets. However, because of the small depth of penetration of the liquid, chemically
strengthened glass is still highly susceptible to surface defects and fracture patterns may
not be improved.
57
Figure 3.6: Stress Cross-Sectional Diagram of Chemically Strengthened Glass
(Wurm 2007, p. 54)
3.2. Strength of Annealed Monolithic Glass Panels
Both theory and tests have shown that it is hard to define glass strength in general, even
annealed monolithic glass panels. Because this thesis is written for architects, not
engineers or glass experts, it is important to simplify the problem and provide some easy
rules of thumb.
3.2.1. Standards and Industry Data
Standards are always the first thing to check for design. However, all the published
strength is strength of glass panels tested under out of plane load, and most of the time, it
is strength under short-duration load, which does not define design load. The reason for
this is that historically glass has been designed to withstand short-duration loading from
wind applied to its surface, not its edge as in the case of a glass beam.
58
3.2.1.1. Standards in Europe and North America
Based on research of Peter Robert Crompton in 1999, a list of standards and industry data
is created to provide people with a general idea of strength of glass. BS stands for British
Standards, which are used in UK. DIN stands for Deutsches Institut für Normung, which
is the German Institute for Standardization. CGSB stands for Canadian General Standards
Board and provides general guidelines for structural use of glass. ASTM standards are
standards used in the US. LF stands for load factor.
BS 6262 BS 5516 BS 7449 DIN 1249 CGSB 12.20 ASTM C1048 ASTM E1300
1982 1991 1991 1988 1989 1990 1994
Bending Strength
-Remote from cut edge
-Near (o r on) cut edge
Allowable Stress
-Remote from cut edge 2.90-4.35 4.35 (2.18-3.63)/1.5 2.18-3.63
-Near (o r on) cut edge 2.90-4.35 2.9/1.5
Variation With Area Not assessed None apparent None apparent Yes Yes
Variation With Thickness Not assessed Yes None apparent None apparent None apparent
Allowable Stress under
Long-duration Load
-Remote from cut edge 1.09-1.63 0.87-1.02 (0.87-1.45)/LF 1.31-2.18
-Near (o r on) cut edge 0.87-1.03 1.16/LF
Table 3.1: Strength Properties of Annealed Glass from Various Codes of Practice
(ksi) (original from Crompton, 1999, Table 1a)
59
BS 6262 BS 5516 BS 7449 DIN 1249 CGSB 12.20 ASTM C1048 ASTM E1300
1982 1991 1991 1988 1989 1990 1994
Bending Strength
-Remote from cut edge 17.40 10.00
-Near (or on) cut edge 17.40 9.72
Allowable Stress
-Remote from cut edge 7.98 7.25 (8.70-17.40)/1.5 8.70-14.50
-Near (or on) cut edge 7.25 11.60/1.5
Variation With Area Minimal Yes Yes Yes
Variation With Thickness Minimal Yes None apparent None apparent
Allowable Stress under
Long-duration Load
-Remote from cut edge 1.09-1.63 1.16-8.27 (6.96-11.60)/LF 7.83-13.05
-Near (or on) cut edge 1.16-8.28 9.28/LF
Table 3.2: Strength Properties of Thermal Treated Glass from Various Codes of
Practice (ksi) (original from Crompton, 1999, Table 1b)
Strength properties of glass are described in different ways, and the variety of these codes
proves the statements on the first section of this chapter. All these standards only provide
design stress of glass, which is used in the ASD method of design (Allowable Stress
Design), instead of considering the strength of glass. Probably this is because it is so
difficult to define the strength, however, there is a permissible strength used in the glass
industry, under which the size and distribution of flaws can be controlled. Most of these
codes provide clues about the relationship of glass size and strength. Strength of glass
under long-duration load is smaller than that under short-duration load. And the strength
of glass shows great variety. On the issue of allowable stress of annealed glass remote
from cut edges, the codes do not agree with each other. As a result, although most of the
time codes are the first and most important design information, these codes and standards
investigated need to be improved and more information is needed for glass beam design.
60
3.2.1.2. Industry Data
Glass manufacturers test and publish strength of their products. This section is also based
on research done by Peter Robert Crompton in 1999. He provided a list of major glass
manufacturers, and strength of their products.
Pilkington Saint Gobatn Schott PPG Hankuk Nippon
Bending Strength
-Remote from cut edge 7.25-13.05 6.00 5.37-7.11 7.11
-Near (o r on) cut edge 7.25-13.05 6.00 5.08 7.11
Allowable Stress
-Remote from cut edge 4.06-5.95
-Near (o r on) cut edge 2.61-4.06
Variation With Area None suggested Yes None suggested None suggested None suggested
Variation With Thickness Yes None suggested None suggested None suggested None suggested
Allowable Stress under
Long-duration Load
-Remote from cut edge 1.09 2.18-4.06 1.74
-Near (o r on) cut edge 1.09 2.18-4.07 1.31
Table 3.3: Strength Properties of Annealed Glass from Float Glass Industry (ksi)
(original one from Crompton, 1999, Table 2a)
61
Pilkington Saint Gobatn Schott PPG Hankuk Nippon
Bending Strength
-Remote from cut edge 21.32
-Near (o r on) cut edge 21.32
Allowable Stress
-Remote from cut edge 8.56
-Near (o r on) cut edge 8.56
Variation With Area None suggested None suggested
Variation With Thickness Yes None suggested
Allowable Stress under
Long-duration Load
-Remote from cut edge 5.66 7.11
-Near (o r on) cut edge 5.66 4.93
Table 3.4: Strength Properties of Thermal Treated Glass from Float Glass Industry
(ksi) (original from Crompton, 1999, Table 2b)
Manufacturers prefer to use strength instead of allowable stress, because it represents the
quality of their products. Generally, the strength from manufacturer data is twice the
allowable stress from the codes, and most manufacturers agree that strength of glass
panels is somewhat between 5.08 to 7.25 psi (35-50 N/mm2). Because designers are more
interested in the optical, thermal and acoustic properties of glass, strength data is not
widely available.
3.2.1.3. ASTM Standards
ASTM standards are the standards used in the US. ASTM E 1300: Standard Practice for
Determining Load Resistance of Glass in Building defines strength properties of glass in
62
practice. There are also other standards that concern strength properties such as ASTM
C1036: Standard Specification for Flat Glass, ASTM C 1048: Standard Specification for
Heat-Treated Flat Glass-Kind HS, Kind FT Coated and Uncoated Glass and so on. A list
of standards for different kinds of glass design and construction practice in the US is
listed below based on research of Peter Robert Crompton 1999, page 137. Most of those
standards are about building envelope construction.
American Society
for Testing and
Materials
Standard test methods for strength of glass by flexure
(determination of modulus of rupture). ASTM C 158
Standard terminology of glass and glass products. ASTM C
162
Standard test method for annealing point and strain point of
glass by fiber elongation. ASTM C 336
Standard test method for annealing point and strain point of
glass by beam bending. ASTM C 598
Standard specification for flat glass. ASTM C 1036
Standard specification for heat-treated glass. ASTM C 1048
Standard test method for determining tensile adhesion
properties of structural sealants. ASTM C 1135
Table 3.5: Standards for Glass Design and Practice in the US (original one from
Crompton, 1999, p. 137)
63
Table 3.5: Continued
Standard specification for laminated architectural flat glass.
ASTM C 1172
Standard test method for shear strength of adhesive bonds
between rigid substrates by the block-shear method. ASTM D
4501
Standard test method for rate of air leakage through exterior
windows, curtain walls and doors. ASTM E 283
Standard test method for structural performance of exterior
windows, curtain walls and doors by uniform static air
pressure difference. ASTM E 330
Standard test method for water penetration of exterior
windows, curtain walls and doors by uniform static air
pressure difference. ASTM E 331
Standard test method for structural performance of glass in
exterior windows, curtain walls and doors under the influence
of uniform static loads by destructive methods. ASTM E 997
Standard test method for structural performance of glass in
windows, curtain walls and doors under the influence of
uniform static loads by non-destructive methods ASTM E 998
64
Table 3.5: Continued
Standard practice for determining the minimum thickness and
type of glass required to resist a specified load. ASTM E 1300
Standard test method for bond integrity of transparent
laminates. ASTM F 521
Standard guide for selection of test methods for interlayer
materials for aerospace transparent enclosures. ASTM F 942
American Welding
Society
Recommended practice for stud welding
American
Architectural
Manufacturers’
Association
Field checks of metal curtain walls for water leaking. AAMA
501.2
Glass design for sloped glazing
Structural properties of glass
ASTM E 1300 provides two ways to define allowable stress of annealed glass panels. As
is shown in Table 3.6, in versions before 2002, allowable stress under short term load was
listed in a table. Load duration is 60 seconds, and the relationship between strength of
glass panels for various thicknesses and sizes is not apparent. The strength of glass is also
decided by the allowable possibility of breakage, and 8 out of 1000 is the most widely
65
used one in the glass industry and design practice. This table also suggests, under 3
second duration load, allowable stress of heat-strengthened glass is always twice that of
annealed glass, and allowable stress of tempered glass is always four times that of
annealed glass, which is also widely accepted in practice. Although allowable stress is
different from strength, they are likely to follow the same rule.
Table 3.6: Allowable Design Stress for Various Probabilities of Breakage (GANA
2008, p. 59)
In later ASTM versions load duration becomes 3 seconds, and graphics showing effect of
thickness and size were introduced. Those charts suggest strength of glass panels with
four sided simple supports is between 0.5 ksi and 10 ksi, and glass panels with high
slenderness are generally stronger.
66
Figure 3.7: Unfactored Load Chart for 6 mm (1.4 in) Glass with Four Sides Simply
Supported from ASTM E 1300 (03) (ASTM 2004, p. 674)
3.2.2. Strength of Glass
Most codes and industry data suggest strength of annealed monolithic glass panels is
greater than 6 ksi and the minimal value is 5.37 ksi. However, those values are mean
value of tests or average strength of samples. Because designers are going to use glass as
structure elements, it is better to be more conservative. In the thesis Assessment of Design
Procedures for Structural Glass Beams by Peter Robert Crompton, strength of 25 N/mm2
(3.63 ksi) is suggested for several reasons. It is the minimal tested strength from many
tests in different institutes; it is also predicted by statistical analysis from his tests; based
67
on Griffith Flaw theory, glass strength below 3.63 ksi means there are visible flaws on
the surface and those pieces of glass are not qualified as undamaged architectural glass.
(Peter 1999, p. 47-47) As a result, 3.63 ksi (25 N/mm2) will be used as strength of
annealed monolithic glass panels.
3.3. Strength of Monolithic Glass Beams
Although a rationale for determining the strength of monolithic glass panels has been
decided, this strength cannot be used directly for glass beam design because glass beams
are loaded in a different way to glass panels and show different features. Generally
speaking, there are two major differences: first, glass beams undergo in plane bending
instead of out of plane bending; secondly, glass beams support long duration loads. Those
two issues will be discussed in this section.
3.3.1. Glass Supporting in Plane Load
In order to decide strength of glass beams and help designers to do better structural
glass design, several research institutes did tests and published data on both monolithic
annealed glass beams and glass panels. These tests are good resources to understand the
different characteristics of glass beams and glass sheets under in plane load and out of
plane load. It is more interesting to know the relationship between strength of glass
panels and glass beams rather than simply the strength of glass beams. In other words, the
68
primary task of this section is to define a ratio of “strength of glass beams versus strength
of glass panels”.
Figure 3.8: Glass Sheets under out of Plane Load and in Plane Load
3.3.1.1. Tests by Fair and Williams in 1999
In 1996, Fair and Williams did a series of three point bend test on glass beams in batches
of 16 at the University of Oxford and Peter Robert Crompton published the data in 1999
in his paper Assessment of Design Procedures for Structural Glass Beams. “The beams
were simply supported on smooth curved steel supports 500 mm (19.69 in) apart. A
central point load was applied by a steel dowel with a soft piece of rubber hosing between
69
it and the glass. The glass samples were 600 mm (23.62 in) long ×100 mm (3.94 in) wide
×10 mm thick (0.39 in).” The result is published in Table 3.7 below. (For more
information about the test, see Assessment of Deign Procedures for Structural Glass
Beams, Crompton, page 43-45).
Failure Stres
Statistics
In Plane
Bending
Out of Plane
Bending
In Plane
Bending
Out of Plane
Bending
Mean 5.69 6.86 19.83 24.92
Std Deviation 0.49(0.65) 1.12 1.44 1.91
Std Deviation(%
of average) 8.6(11.4) 16.3 7.3 7.6
Range 4.84‐6.66
(4.84‐8.12)
5.40‐9.02 17.00‐22.58 20.78‐26.97
Annealed Glass Toughened Glass
Table 3.7: Test Result for Single Beams (ksi) (original one from Peter, 1999,
Table 2, p. 45, Coefficient of variability (Std Deviation/Mean ×100%) is added by
author)
3.3.1.2. Tests in Delft University of Technology
TU Delft has a unique group working on structural glass, and numbers of tests were done
there. Bos, Louter and Veer organized and did a serious of tests on glass beams and
published their data in their paper The Strength of Architectural Glass. “Glass beams of
size 1000 mm long and 100 mm wide were cut from a single glass plate with a thickness
of 10 mm. These were professionally cut on professional cutting machines and finished
by grinding and polishing. One third of the specimens were pre-stressed using full
70
thermal tempering, one third of the specimens were pre-stressed using heat strengthening.
All specimens were wrapped in PET foil for safety. For the heat strengthened and fully
tempered glass multiple layers PET foil were necessary. For annealed float glass a single
layer of foil was sufficient. The beams were tested in 4 point bending on a Zwick Z 100
universal testing machine with the specimen lying or standing. To avoid buckling of the
standing specimens, the specimen was supported on the sides at 5 points along the length.
1mm thick Teflon sheet was used as an intermediary between the metal supports and the
glass to avoid inducing high contact stresses. ” (Veer et al., 2008, page 2).
Figure 3.9: Test Set-up with Glass Specimens Lying (Veer et al. 2008, p. 2)
The paper includes a table with Weibull analysis results as reproduced in Table 3.8 below.
71
Lying Standing Lying Standing Lying Standing
Average Failure
Strength (ksi) 6.09 3.87 15.08 10.01 22.83 13.73
Minumum Tested
Value (ksi) 3.74 3.07 8.53 7.96 13.94 10.53
Maximum Tested
Value (ksi) 8.50 5.67 24.22 13.89 29.75 17.71
Calculated Weibull
Strength (ksi) 3.48 2.90 7.83 7.54 12.76 11.17
Coefficient of
Deviation (%) 21.80% 18.30% 27.70% 15.30% 18.90% 12.40%
Annealed Heat Strengthened Tempered
Table 3.8: Test Result for Single Beams (ksi) in TU Delft
3.3.1.3. Discussion and Conclusion
All tests suggest that strength of glass beams is lower than that of glass panels. Several
reasons could cause this reduction. First, the cutting and grinding process will create
more and bigger flaws on the edge, and reduce the strength of glass beams. And although
those tests were done very thoughtfully, upright glass beams are not as stable as laying
flat glass panels. The ratio of “Strength of glass under in plane load/Strength of glass
under out of plane load” is listed in Table 3.9 below. Because the average strength ratio
of the second set of tests shows significant difference from the other two results, this set
of data will be omitted in the discussion below. Additionally the Weibull strength should
be the most useful one for glass strength analysis because it is a professional method for
glass strength analysis.
72
Both tests suggest strength of annealed glass beams is a bit more than 80% of that of
glass panels. The first set of tests suggest tempered glass will reduce strength more than
annealed glass will, while the second set suggest that heat-treated glass will reduce
strength less than annealed glass will. It is hard to explain this disagreement. Maybe it is
because they used different methods to calculate the strength, or used different methods
to do the tests, or they bought glass samples from different manufacturers. In this thesis, a
reduction to 80% of in plane strength is suggested, which means the strength of annealed
monolithic glass beams should be only 80% of the strength of the same type of glass
panels, and the strength for annealed monolithic glass beam calculations should be
80%×3.63=2.90 ksi (20 N/mm2). It should be noted that the 80% strength chosen is
slightly less than both test results reported to limit overestimating glass beam strength.
Annealed
Heat-
Strengthened Tempered
Strength of
glass under in
plane
load/Strength of
glass under out
of plane load
Test one
(mean) 82.90% 79.57%
Test two
(average) 63.55% 66.38% 60.14%
Test two
(Weibull) 83.33% 96.30% 87.53%
Table 3.9: Tested Ration of Different Sets of Strength
3.3.2. Glass under Long Duration Load
As mentioned before, the strength of glass will drop significantly under long duration
load. Although designers are not sure about how much reduction they get exactly, most
experts suggest a value between 3/8 to 1/3 for load period longer than one year, and uses
73
1/3 in practice. ASTM E 1300 (03) will be studied in this discussion. Most research is
based on the study of annealed glass, and more research is needed on heat-treated glass.
3.3.2.1. Annealed Glass under Long Duration Load
ASTM E 1300 (03) provides several ways to decide strength of glass under long-duration
load, and the most widely used one is Table 1 in ASTM E 1300 (03), Glass Type Factor
(GTF) for a Single Lite of Monolithic or Laminated Glass. However, this GTF is for
load-duration of one month which does not apply to typical duration of glass beams.
Table 3.10: Glass Type Factor (GTF) for a Single Lite of Monolithic or Laminated
Glass from ASTM E-1300 (03) (ASTM 2004, p. 626)
In addition to GTF, another useful table is Table X6.1 in ASTM E 1300 (03), Load
Duration Factor, which is designed for annealed glass with 8/1000 probability of
breakage. ASTM suggests annealed glass retains only 31% of its original strength under
load longer than one year. As a result, strength of monolithic annealed glass beams is
3.63×80%×31%=0.90 ksi (6.2 N/mm2).
74
Based on Table X 6.1 of ASTM 1300 E 1300 (03), Function 3.1 “σn×tf = constant” will
be studied.
Table 3.11: Load Duration Factor, Note-Calculated to 8/1000 Probability of
Breakage from ASTM E 1300 (03) (ASTM 2004, p. 681)
After modification of Table 3.11 using Function 3.1, Table 3.12 is provided below. The
glass type factor is used in this table instead of strength, σ, and the unit of duration is one
minute. ASTM suggests n = 16.
75
Annealed Duration (min) Constant
1 0.05 0.05
0.93 1/6 0.052189
0.83 1 0.050728
0.72 10 0.052158
0.64 60 0.047537
0.55 720 0.050482
0.53 1440 0.055818
0.47 10080 0.057151
0.43 43200 0.059017
0.36 525600 0.041831
0.31 7095600 0.051615
Table 3.12: Modified Table of Load Duration Factor, Note-Calculated to 8/1000
Probability of Breakage from ASTM E-1300 (03)
Based on Function 3.1, the modified table shows a relative stable constant, which is
considered acceptable. The average constant in the table (considering durations up to and
including 1 year duration or 525600 minutes) is 0.52, and annealed glass needs 13.5 years
to lose 69% of its strength, the value of strength ASTM suggests for load duration over
one year.
3.3.2.2. Tempered Glass under Long Duration Load
There is no single table defining strength of tempered glass under load of different
periods. However based on Table 3.10, the strength of tempered glass will be ¾ of its
original strength, if it is loaded with load for 1 month. Strength of tempered glass is
shown in Table 3.13 similar to Table 3.12.
76
Tempered Duration
(min)
Constant
1 0.05 0.05
1/6
1
10
60
720
1440
10080
3/4 43200 0.05
525600
7095600
Table 3.13: Load Duration Factor of Tempered Glass
As a result a proper “n” for tempered glass is 47.5. A new table is created to define the
strength of tempered glass and we can find out tempered glass could hold 67% of its
original strength under long-duration load (13.5 years). Thus, the strength of monolithic
tempered glass beams is 3.63×4×80%×67% = 7.83 ksi (53.6 N/mm2).
Tempered Duration Constant
1 0.05 0.05
0.9749717 1/6 0.05
0.9388796 1 0.05
0.8944525 10 0.05
0.861341 60 0.05
0.8174393 720 0.05
0.8055974 1440 0.05
0.7732617 10080 0.05
3/4 43200 0.05
0.7115002 525600 0.05
0.6735635 7095600 0.05
Table 3.14: Modified Load Duration Factor of Tempered Glass
77
3.3.2.3. Heat-strengthened Glass under Long Duration Load
The same method (65% strength after 1 month of loading per Table 3.10) will used on
tempered glass will be used on heat-strengthened glass and we can get constant “n” =
31.7. As a result, heat-strengthened glass could hold 67% of its original strength under
long-duration load (13.5 year) and strength of monolithic heat-strengthened glass beams
is 3.63×2×80%×55%=3.19 ksi (22 N/mm2). Strength for different types of glass is
provided below, which shows the significant benefit from thermal treatment of glass
regarding long term load.
Heat-
Strengthened Duration Constant
1 0.05 0.05
0.962731989 1/6 0.05
0.909825373 1 0.05
0.846081702 10 0.05
0.799585565 60 0.05
0.739301134 720 0.05
0.723311147 1440 0.05
0.68024587 10080 0.05
0.65 43200 0.05
0.600476105 525600 0.05
0.553144392 7095600 0.05
Table 3.15: Load Duration Factor of Heat-Strengthened Glass
78
Annealed Heat-Strengthened Tempered
Glass Panels (3 second) 3.63 7.26 14.52
Glass Beams (3 second) 2.904 5.808 11.616
Glass Beams (beyond 1
year) 0.90024 3.194 7.829184
Table 3.16: Summary of Glass Strength
3.4. Strength of Laminated Glass Beams
Many studies have been done to find a way to define the strength of laminated glass.
Traditionally, glass is used as a building envelope material under short duration out of
plane load (wind, seismic loading); force and load transfer through interlayer materials
and are shared by different layers of monolithic glass sheets. However, under in-plane
bending, theoretically, laminated glass shows no significant difference from monolithic
glass. As a result, laminated glass sheets will be treated as monolithic glass with the same
thickness, if it does not buckle. Since interlay materials take only 3% of the whole beam
by volume, and are not designed as structural materials, they are not considered in
bending calculation.
79
CHAPTER 4
GLASS BEAM CALCULATION
Structures fail for various reasons. Some conditions are more likely to happen and some
less. Design criteria vary based on building location and importance factor. Typically,
U.S. building codes are based on designing buildings of all types to have a life span of
more than 50 years. In Los Angeles, buildings must be designed for seismic loads as well
which could be problematic for brittle materials like glass. Seismic load will not be
discussed in this thesis. Breakage under everyday load, such as dead load, live load and
wind load, will be explored. Because of the property of glass, some special issues will be
explored in this chapter.
4.1. Design Methods
Allowable Stress Design (ASD) and Load and Resistance Factor Design (LRFD) are the
two most widely used design methods used by designers and engineers. ASD is a
traditional method, while LRFD is relatively new. Both these two methods examine
strain and stress in the structure to assure the material is strong enough to resist the load
without breaking and stiff enough to minimize strain deformation.
80
4.1.1. ASD
In Allowable Stress Design, designers calculate actual stress in material to assure
it is below allowable stress limits, which are typically the yield strength divided by a
safety factor. ASD could be described as a function as:
Stress ≤ breaking strength / safety factor
Function 4.1: Allowable Stress Design
Stress in the structure is determined by structure types and load applied; strength is
decided by the material used and the Safety Factor (SF) is decided by the experience and
research of experts. Because there are unpredictable conditions a safety factor is used to
reduce the possibility of failure to an acceptable level, but not too high to get material
waste. However, as structural glass has been used for only a few decades, the experts
have less experience and so it is not easy to define a proper Safety Factor right now.
4.1.2. LRFD
Load and Resistance Factor Design is a new design method used for concrete
design first in the 1960s. Instead of comparing actual stress with factored strength, LRFD
81
uses factored load and nominal resistance (reduced yield strength), and LRFD for beam
design can be described as a function as:
(Factored Load / Section Modulus) ≤ (Yield Strength / Resistance Factor)
where Factored Load = Actual Load x Load Factor
Function 4.2: Load and Resistance Factor Design
Instead of single Safety Factor used to limit allowable stresses, LRFD uses two separate
factors, the Load Factor and the Resistance Factor. The load Factor is primarily defined
by the type of load and increases the load to be checked, and the Resistance Factor is
primarily defined by the material used to build the structure and the type of loading and
reduces nominal capacities based on structural behavior. This method offers safety
factors, based on probability, used to calculate and evaluate the structure. The Load
Factor also offers more rational safety to LRFD, amplifying uncertain live load by 1.6 but
predictable dead load only by 1.4. More and more engineers use LRFD to design and
teach in collage. Calculation of this thesis is based on LRFD. Because LRFD is more
exact than ASD, sizes of structure elements are usually smaller than by ASD but LRFD
provides less safety.
82
4.2. LRFD Bending Calculation
Glass beams are structural elements primarily loaded in bending, so they should satisfy
the function below:
(Factored Load / Section Modulus) ≤ (Yield Strength / Resistance Factor)
where Factored Load = Actual Load x Load Factor
Function 4.2: Load and Resistance Factor Design
Several items need to be decided: load and Load Factor, section modulus, yield strength
of structural glass and Resistance Factor.
4.2.1. Load and Load Factor
This thesis is written for glass beam design in the Southern California area to resist live
load and dead load, and the life span of the glass beams is assumed to be 50 years. The
beam is assumed to be a simple beam, which means that it is supported with a roller
support on one end, a pin support on the other end, and loaded with uniformly distributed
load.
83
Figure 4.1: Simple Beam under Uniformly Distributed Load
Although code required live loads are different for different types of buildings, in order to
simplify the problem, the most widely used values in the California Building Code are
going to be used in this thesis, which are 50 psf for office floors and 20 psf for roofs. The
typical load cases are 1.4 DL or 1.2 DL + 1.6 LL, so the factor changes on dead load
depending on whether it is combined with live load or not.
In order to create ideal transparency, glass beams are more likely to support glass panels
on top and are not likely to support equipment. As a result, dead load is mainly the glass
panels on top of the beam. Usually designers use laminated glass with each sheet thicker
than 3/8”, and use more layers on floor design because that provides more safety. Two
layers of laminated glass, each layer of ½” heat-strengthened glass would be enough to
support the roof load, and three layers of laminated glass with each layer of 1/2'” heat-strengthened
glass would be enough to support the floor load. Compared to the weight of
84
glass, the weight of the interlay material could be omitted. Weight of glass beams should
be calculated as dead load, and estimated to be one third of glass panels on top in this
thesis based on case study and calculation. Generally accepted density of glass is 163
lbs/ft3 (pcf), and distributed dead roof load is about 17 2/3 psf and distributed dead load
for floor design is assumed 26 1/2 psf. The load Factor for dead load is 1.4 as required by
code.
4.2.2. Section Modulus
Figure 4.2: Section of Glass Beam and Stress Distribution
Section Modulus is determined from the cross section and composition of the beam.
Glass beams could be built with glass sheets of different thickness, however, in order to
simplify the problem, only beams built with glass sheets of equal thickness will be
85
discussed in this thesis. And this thesis only assumes laminated rectangular glass beams
of glass sheets and interlayer material. The effect of the interlayer material is omitted in
this simplified calculation for several reasons.
First, compared with glass sheets, interlayer material is only a tiny part of the beam cross
section. PVB is the most popular interlayer material for laminated glass, and the
thickness of one PVB film is 0.38 millimeter (0.015”). As a result, if designers choose
PVB as the interlayer material, the resulting thickness of the interlayer is a multiple of
PVB film thickness of 0.015”, and interlayer with 2 or 3 layers of film is preferred, which
will be explained later. Taking a glass beam made up of 3 layers of 3/8” glass sheets for
example, the total thickness of beam is 3×3/8”+2×2×0.015”=1.125” and PVB takes only
5% of the beam by volume. Additionally, bending strength of PVB is less than glass.
Finally, more research is still needed about the effect of interlayer material on laminated
glass under in plane bending. As a result, the thickness and effect of the interlayer
material is omitted and laminated glass is simplified to be monolithic glass sheets with
the same thickness. These simplifications allow the section modulus of the rectangular
beam shown above to be calculated as:
Section Modulus S=nbd2/6 (n=3 in the case)
Function 4.3: Section Modulus Calculation for Glass Beams
86
4.2.3. Potential Breakage of Glass and Modification on Section Modulus
Before strength based design of glass, designers needed to spend time on modification to
the section to provide more safety to the beam because of the elasticity and brittleness of
glass. Elasticity and brittleness are the two main reasons that lead to conservative design
for the structural use of glass because there is a possibility that glass sheets will break
below assumed design strength, it is difficult to predict when this will happen, and it is
almost impossible to define the yield strength of glass without testing. In steel structures,
designers do not need to worry about the safety of the structure if stress is below yield
strength since ductile yielded would be evident before failure, however, this does not
work on glass. ASTM E 1300 proposes different design stress values corresponding to
the possibility of breakage (Table 4.1).
Table 4.1: Allowable Design Stress for Various Probabilities of Breakage (GANG
2008, p. 59)
87
In order to reduce the possibility of failure of glass beams, one could suggest that they
should be strong enough to allow one glass sheet to break and redistribute load to the
remaining sheets, which means the beam will not fail until two or more sheets break.
Although manufacturers are trying to provide products with better mechanical properties,
not all glass sheets will reach the required strength, and 8 out of 1000 probability of
breakage under required stress is allowed by most codes. If designers design a glass beam
that is just strong enough to sustain the design load, there is a 8 out of 1000 probability
that the beam will fail. In order to reduce this unacceptable possibility, additional layers
and redundancy are necessary. If a 3-layer beam is strong enough to sustain load with
only 2 layers, the possibility of beam failure is reduced to 0.8%×0.8%=64 out of one
million. And because it is almost impossible for those two layers to break at the same
time, without a disaster like an earthquake or terrorist attack, one has time to replace
broken beams with new beams. Of course, designers can design beams allowing two
sheets break, but that would be too expensive and an overconservative use of material.
With the modification to the number of layers to minimize breaking failure of glass
beams, the Section Modulus function should be:
Section Modulus S = (n-1)bd2/6
(n = layers of glass sheets of the beam.)
Function 4.4: Modified Section Modulus Calculation for Glass Beams
88
4.2.4. Resistance Factor
In LRFD, the resistance factor modifies the mechanical properties of the material.
Although theoretically precise structural materials will not fail if the stress is below yield
strength, not all fabricated elements of the material will meet this requirement due to
variations in the processing, handling, curing, mixing, and other aspects of the production
of structural materials. Also, because structural behavior is idealized in analysis designers
cannot accurately predict how the structure will behave to real boundary conditions,
tolerance errors, and imprecise section properties. This would also make it dangerous to
make full use of the strength. As a result, comparing factored stress with yield strength
multiplied with a resistance factor reduces the probability of failure. Standards and codes
recommend resistance factor for different materials and loading types.
Table 4.2: Resistance Factors of Common Structural Materials (Schierle 2008, p.
143)
89
Generally speaking, the quality of materials with a smaller range and coefficient of
deviation is better and could be designed with larger resistance factor. Based on the
results of two sets of tests referenced in Chapter 3, structural properties of glass are
relatively stable. Compared with timber, which is a widely used structural material,
strength of glass is easier to predict.
Annealed Heat-Strengthened Tempered
Lying Standing Lying Standing Lying Standing
Set
One 16.3 8.6 7.6 7.3
Set
Two 21.8 18.3 27.7 15.3 18.9 12.4
Table 4.3: Glass Coefficient of Deviation (%) (From tests in University of Oxford
and TU Delft University of Technology)
The first set of tests shows a smaller coefficient of deviation. Although the second set is
not that good, compared with timber, a widely used structural material, glass is still not as
bad as imagined. The typical timber coefficient of deviation is between 10% and 30%,
while the coefficient of deviation of glass is between 7% and 28%. From this point of
view, glass deserves a higher or similar resistance factor as timber, whose resistance
factor for bending is 0.85.
However, glass is a brittle and elastic material, and designers need to be conservative in
structural design with glass. The smallest value used on major structural materials is 0.8
90
for masonry bending design, which happens to be another brittle and elastic material like
glass. As a result, the author suggests a resistance factor for glass between 0.85 and 0.80.
Glass beams with no more than 5 layers have a resistance factor of 0.85 because they
have less surface area and less Griffith flaws, while glass beams with 5 layers or more
have a resistance factor of 0.80.
4.3. Deflection Criteria
Deflection does not necessarily lead to structural failure, however, sometimes large
deflection damages the function and if it is large enough to be visible, people will feel
unsafe. The generally accepted deflection requirements for beam design are span/360 for
live load only and span/240 for combined live load and dead load. Another thing to check
is vibration, and a simple rule of thumb in metric units is
F=16/ (d1/2) Hz
Function 4.5: Deflection Requirement to Avoid Vibration Damage
Assuming d is the midspan deflection of a beam under permanent load in mm and F is the
first natural frequency in Hz. Engineers design for F>5 Hz to avoid dynamic excitation
by foot traffic or by wind.
91
Figure 4.3: Beam Deflection Area Method Visualized
The generally accepted deflection equation used on traditional, timber and steel beams
design is:
Δ=5/384 WL3/(EI) (Except for E, this equation is ONLY defined by geometry)
Δ: Deflection
W: total load applied as uniform distributed load
L: Span of Beam
E: Elastic Modulus
I: Moment of Inertia
Function 4.6: Deflection Criteria for Beam Design
92
However, a more conservative function is proposed for glass design, especially for out of
plane bending, and referenced and verified by Jan Belis et. al (Monolithic Calculation
Model for the Out-of-plane Bending Laminated Glass Beams). In his paper, Belis also
suggested the interlayer material has almost no effect on the in plane bending of glass
beams and designers can treat laminated glass beams as monolithic glass beams of the
same thickness.
Δ=11/768 WL3/(EI)
Function 4.7: Deflection Criteria for Glass Design
Because bending deflection is more likely determined by the geometry instead of the
material, and there is only a small difference between those two functions, the classic
beam deflection function is used in this thesis. The moment of inertia equals:
I =nbd3/12
“n”: number of layers in the laminated beam
“b”: thickness of single layer
“d”: the depth of beam.
Function 4.8: Moment of Interior
93
4.4. Buckling
The extraordinary slenderness of glass beams leads to great concern regarding buckling
damage. The typical width/depth ratio of rectangular timber and concrete beams is 1/3 to
1/5, while most of the as-built glass beams have a width/depth ratio as high as 1/10 or
even higher. There are two main types of buckling, global buckling of beams and local
buckling of beams. “Buckling of beams” refers to the phenomenon that whole beam will
displace out of plane under in plane load by twisting, while “local buckling” refers to the
phenomenon that a part of the beam will deform and fold out of plane. Buckling only
happens under compression and bending. Even a beam that has almost ideal pinned
supports has compression on top and any fixity of the supports or imperfections in the
beam geometry or load direction can cause the beam to buckle if the loads are large
enough.
94
Figure 4.4: Buckling of Cantilever Beams (Trahair, 2009)
95
Figure 4.5: Local Buckling of Beams (Lamont 2001, Figure 4.6)
There is no specific function available to describe buckling of glass beams. However,
designers have been building glass fins and fin buckling has been studied before. As both
glass beams and glass fins take in plane load, the buckling behavior is the same and
buckling design method of fins will be used in beam design. In Structural Use of Glass in
Buildings published by the Institution of Structural Engineer in the UK, a glass fin
buckling calculation method of the Australian Standard AS 1288-1994 is provided, and
will be used in this thesis.
96
4.4.1. Beam Buckling
In buckling design and calculation, the design moment shall not exceed more than the
critical elastic buckling moment (MCR) divided by a factor of 1.7.
M ≤MCR/1.7
Function 4.9: Beam Buckling Criteria
Design moment (M) is the un-factored bending moment, which equals to wL2/8 for
simple beams under uniform distributed load. Functions used for MCR are different
depending on intermediate buckling restraints.
Beams with Intermediate
Buckling Restraints
MCR=(g1/Lay) [(EI y) (GJ) ]1/2
Beams without Intermediate
Buckling Restraints
MCR=(g2/Lay) [(EI y) (GJ) ]1/2[1-g3(yh/Lay) [(EI y)
/(GJ) ]1/2]
Continuously Restrained
Beams below Neutral Axis
MCR=[(π/Laɸ)2(EI y) [d2/4+y0
2]+(GJ) ]/(2y0+yh)
Table 4.4: Functions for Critical Elastic Buckling Moment Calculation
As stated before, only beams with no intermediate buckling restraints will be discussed in
this thesis. Appendix H in Structural Use of Glass in Buildings gives information about
buckling calculations of other beams. The formula for beam without intermediate
buckling restraints will be used in the following discussion.
97
Figure 4.6: Inter-brace of Beams
MCR=(g2/Lay) [(EI y) (GJ) ]1/2[1-g3(yh/Lay) [(EI y) /(GJ) ]1/2]
MCR: Critical elastic buckling moment
g2 & g3: Slenderness factors
Lay: Distance between effectively rigid buckling restraints
E: Elastic Modulus
G: Torsional elastic modulus
J: Torsional moment of inertia
Iy: Moment of inertia along the weak axis
yh: Height above the beam centroid of the point of load application
Function 4.10: Buckling Calculation for Beams without Intermediate Constraints
98
Several issues need to be explained further.
First, the value of the coefficients for slenderness factors g2 and g3 are determined by
connections and applied load. In order to prevent unnecessary movement which is likely
to cause damage of the glass, it is better to use fixed connections properly designed to
resist y-y axis (weak axis) rotation. As a result, g2 equals 6.1 and g3 equals 1.8.
99
Table 4.5: Coefficient for Slenderness Factors of Bisymmetrical Beams with no
Intermediate Buckling Restraints (Institution of Structural Engineer 1999, p. 152)
100
Lay refers to the distance between effectively rigid buckling restraints, and in the present
case, Lay equals the beam span.
E is the elastic modulus, which cannot be changed by thermal-treatment, and the
generally accepted E for glass is E = 10.4×106 psi. Please note that the formula has
coefficients that depend on the units being used and thus properties are being shown only
in these units.
Iy refers to moment of inertia along the weak axis. Because glass beams are made up of
laminated glass, when they buckle, they will displace horizontally and different layers
have a tendency to slide relative to each other. Iy has to be calculated differently
depending on whether the sliding happens or not. The possibility of sliding is defined by
strength, stiffness and thickness of the interlayer material as well as load duration. It is
hard to describe this phenomenon with functions or numbers at this time. It will not
happen without un-predictive load and could be observed. For simplified calculation, the
thesis assumes sliding will not happen and further discussion will be mad in Chapter 5.
101
Figure 4.7: Plan View of Buckled Beams
G is the torsional elastic modulus and is 28.3 GPa (4104.57 ksi) for glass fin calculations,
and will be used in glass beam calculation in this thesis.
J is the torsional moment of inertia and is J = bd3/3 (1-0.63b/d), where b and d are the
breadth and depth of the beams.
yh is the height above the centroid of the point of the load application. Is most cases, yh
refers to half the height of the beam, if the load is loaded right above the beam. However
sometimes, yh could be different if load is loaded as illustrated in the graphic below. In
this thesis, only the first and most widely used situation will be discussed and yh equals
half the height of the beam.
102
Figure 4.8: yh and Applied Load
4.4.2. Local Buckling
Local buckling will not cause deformation of the whole beam, and usually happens at the
free edge. Yoxon described a simplified check method in the following formula:
103
Mmax<Et3/6(1+v)
where
Mmax is the maximum unfactored bending moment in the beam
E is the elastic modulus
t is the thickness of beam
v is Poisson’s ratio.
Function 4.11: Local Buckling
Both experience of full-size tests and non-linear finite element analysis have proven that
Et3/6(1+v) determines the buckling limit. As mentioned in the discussion of beam
buckling, we shall assume different layers will not slide relative to each other and t is
equal to b×n in this thesis.
104
CHAPTER 5
EXCEL TOOL AND THE LARGEST GLASS BEAM
Based on the design approach and parameters chosen in Chapter 3 and Chapter 4, an
Excel tool is created for glass beam design to help architects find the required depth of
beams in 2 inch increments under user provided design criteria. This tool is introduced in
this chapter to explore the capacity and selection of glass beams.
5.1. Excel Tool
The Excel Tool is composed of three areas, “design area” on the left in brown, “material
& structure area” on the left in blue and “calculation area” in blue on the right. The
design area is for architects to input design requirements in the first part and get the
correct beam size they need in the second part. This area is most useful for architects. The
material & structure area shows us default setting of properties of glass, which is based
on the discussion of the previous two chapters, and criteria for structural design. The
“calculation area” is a supporting area to run the tool and shows the users how the tool
works. This is an open source tool based on Excel and every parameter could be
customized by users, however, it is suggested that most designers do not change anything
except the “Selection” row and “Customized Value” row in the design area.
105
Figure 5.1: Excel Tool for Glass Beam Design & Calculation
5.1.1. Design Area
Figure 5.2: Design Area
106
There are several things need to be decided by architects in order to run the tool, and
those requirements are listed in the first part of this area. The “basic information” row
lists allowable choices to select for design. The “customized value input/default value”
provides either default values for each section in blue or space to input customized value
within bold borders. Designers need to make a selection for every decision in the “select
your information row” utilizing drop down lists. Default values may be used if the user
has no specific preferences or is uncertain of decisions to make for specific options.
The second part of this area provides an overview of the beam sizes necessary to meet the
requirements. The first two rows are a report of calculations shown in black. There are
minimal depths to meet the requirements of bending stress, deflection criteria and
buckling criteria. Smaller numbers indicates the structure is less likely to fail because of
that criterion, and the tool will automatically select the biggest number as the depth of
beam to ensure that all the criteria checked are satisfied. The last two rows show an
overview of the beam.
107
5.1.2. Material & Structure Area
Figure 5.3: Material & Structure Area
This area is designed for engineers or advanced architects to process further study of their
beam and make modification. There are properties of glass and structural criteria in this
area. Glass properties were chosen based on the conclusions of Chapters 3 and 4, and the
spreadsheet uses the most common values here. The user may change those values if they
have customized glass products which are qualified as structural glass.
108
5.1.3. Calculation Area
Figure 5.4: Calculation Area
109
This area is a supplementary part of the tool and shows how the tool works. The tool
checks the depth of the beam in two inch increments in a range from 0 to 50 inches,
because beams with greater depth do not make any sense. If the tool shows designers
need to use beams with depths greater than 50 inches, it is likely that the span and loading
provided should not be supported by glass beams. The user should not make any change
to this part. A table showing how decisions change properties of glass and structure
criteria will be provided in Appendix A.
5.2. The Biggest Glass Beams without Inter-brace
Because of the industry capacity of glass manufacturing and laminated glass fabrication,
the largest span of glass beams would be 19-20 feet. People could build glass beams
bigger than this, but the cost would be huge and could lead to a number of problems. This
study limits the span to 20 feet and provides a selection process to decide on glass
products and required depths. Levels of heat treatment, thickness of glass sheet as well as
numbers of layers will be explored.
5.2.1. The Biggest Glass Beams for Roof Design
The calculation report of 20 foot beams with different levels of heat treatment thickness
of glass sheets and layers is shown in a list below. All beams with depth larger than 50
inches are shown in red because this calculation indicates glass beams are inappropriate
110
for the parameters chosen. The generally accepted depth/span ratio of horizontal
resistance structure ranges from 1/10 (trusses or suspension structures) to 1/20 (beams),
and depths out of this range should not be used either. Beams deeper than 24” (1/10 of
span) are shown in orange and beams shallower than 12” (1/20 of span) are shown in blue.
As a result, properly sized beams are shown in black.
Table 5.1: 20 Foot Beams for Roof Design (AG stands for annealed glass, HG stands
for heat-strengthened glass and TG stands for tempered glass)
111
5.2.2. Several Conclusions
Several conclusions are listed below.
1. Bending stress is still the governing criterion for glass beam design even though
other criteria were considered.
2. Heat-treatment on glass could greatly reduce the required size of beams.
3. Most of the annealed beams fall in the range of red or orange, which indicates that
the strength properties of annealed glass are so poor that annealed glass should
not be qualified as structural glass.
4. Tempered glass is so strong that some tempered beams even fall in the blue area.
Although strength properties of tempered glass are quite good, sometimes this
type of glass can suffer spontaneous breakage, especially for area supported by a
less developed glass industry. As a result, heat-strengthened glass is
recommended as a first choice, and then tempered glass.
5. The more layers designers use, the smaller the resulting depth they get for the
beam. They can also cut the overall area of cross section by selecting laminated
glass with more layers and thinner sheets. However, because of the capacity of the
glass fabrication industry, the more layers they have on laminated glass, the less
available those glass beams are. And also, by doing so, the beams get more
surface area and edge and introduce more Griffith Flaws, which leads to reduction
112
of the strength quality. As a result, three or four layers are recommended, which
are also the most widely used numbers in existing glass beams.
6. Simply increasing thickness of glass sheets will cut down the depth of beams but
increase the area of cross section and material needed to build the beam.
Similar tables will be offered in Appendix B.
5.3. Effect of Interlayer Material on Beam Buckling
As is mentioned in Chapter 4, as glass beams are constructed by laminated glass, when
the beams buckles out of plane, different sheets will slide relative to each other if the
interlayer material is not strong enough to hold them. Most previous discussion is based
on the assumption that those sheets will not slide, however, if the interlayer material fails
first and glass sheets could slide freely, buckling, instead of bending, could be the
governing factor. Similar tables like Table 5.1 are created in order to explore the
importance of buckling damage with sliding.
113
Table 5.2: 20 Foot Beams for Roof Design with Sliding when Buckling (AG stands
for annealed glass, HG stands for heat-strengthened glass and TG stands for
tempered glass)
114
Figure 5.5: Charts of 20 Foot Beams for Roof Design with Sliding When Buckling
115
As shown above, buckling will be the governing factor instead of bending stress, when
designers use thinner glass sheets with low levels of heat-treatment. And most of the time,
the transition between buckling governing and bending governing happens on thickness
of 1/2 inch or 3/8 inch, which are also the most widely used thicknesses and
recommended thickness.
The assumption that interlayer material will completely fail and different layers will slide
freely towards each other is one extra condition that is not likely to happen on well
fabricated laminated glass beams. And a balance between free sliding and no sliding is
needed. A lot of papers have been published about the effect of interlayer material on the
structural property of laminated glass. The interlayer effect on laminated glass beams is
quite new and requires huge amount of further experiment and study. As a result, at this
time, it is better to assume that the interlay material could be strong enough.