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Transcript of Rio de Janeiro Março de 2014 - Federal University of Rio ...
COMPRESSIVE STRENGTH OF PULTRUDED GLASS-FIBER REINFORCED
POLYMER (GFRP) COLUMNS
Daniel Carlos Taissum Cardoso
Tese de Doutorado apresentada ao Programa de
Pós-graduação em Engenharia Civil, COPPE, da
Universidade Federal do Rio de Janeiro, como
parte dos requisitos necessários à obtenção do
título de Doutor em Engenharia Civil.
Orientadores: Eduardo de Miranda Batista
Kent Alexander Harries
Rio de Janeiro
Março de 2014
COMPRESSIVE STRENGTH OF PULTRUDED GLASS-FIBER
REINFORCED POLYMER (GFRP) COLUMNS
Daniel Carlos Taissum Cardoso
TESE SUBMETIDA AO CORPO DOCENTE DO INSTITUTO ALBERTO LUIZ
COIMBRA DE PÓS-GRADUAÇÃO E PESQUISA DE ENGENHARIA (COPPE) DA
UNIVERSIDADE FEDERAL DO RIO DE JANEIRO COMO PARTE DOS
REQUISITOS NECESSÁRIOS PARA A OBTENÇÃO DO GRAU DE DOUTOR EM
CIÊNCIAS EM ENGENHARIA CIVIL.
Examinada por:
________________________________________________
Prof. Eduardo de Miranda Batista, D.Sc.
________________________________________________
Prof. Kent Alexander Harries, Ph.D.
________________________________________________
Profa. Michèle Schubert Pfeil, D.Sc.
________________________________________________
Prof. Paulo Batista Gonçalves, D.Sc.
________________________________________________
Prof. Pedro Colmar Gonçalves da Silva Vellasco, Ph.D.
RIO DE JANEIRO, RJ - BRASIL
MARÇO DE 2014
iii
Cardoso, Daniel Carlos Taissum
Compressive Strength of Pultruded Glass-Fiber
Reinforced Polymer (GFRP) Columns / Daniel Carlos
Taissum Cardoso. – Rio de Janeiro: UFRJ/COPPE, 2014.
XIII, 181 p.: il.; 29,7 cm.
Orientadores: Eduardo de Miranda Batista,
Kent Alexander Harries
Tese (doutorado) – UFRJ/ COPPE/ Programa de
Engenharia Civil, 2014.
Referências Bibliográficas: p. 133-143.
1. Glass-fiber reinforced polymer. 2. Columns. 3.
Compressive strength. I. Batista, Eduardo de Miranda et
al. II. Universidade Federal do Rio de Janeiro, COPPE,
Programa de Engenharia Civil. III. Título.
iv
Acknowledgments
To CAPES, for the financial support.
To Bedford Plastics, for supporting this work either supplying material or sharing
information.
To the staff members of COPPE-UFRJ and University of Pittsburgh, for all the help
with my sandwich program and for being so patient with all my desperate last-minute
questions and documentation issues.
To the faculty members of COPPE-UFRJ and University of Pittsburgh, for the incentive
and support.
To Dr. Batista and Dr. Harries, for believing in this project since the beginning, for
making it possible and for bringing up the sun when days were cloudy. Thank you for
all advices, teachings, opportunities and endless support. Eternally grateful.
To the incredible friends I made in Pittsburgh. Thank you for the reception, for making
me feel at home during my stay, for the amazing support and for sharing with me your
best thoughts in the hardest moment of my life.
To my family and friends in Brazil. Thank you for being part of my life, for sharing and
letting me share smiles and tears, sad and happy moments. For the good thoughts and
vibrations. For all the love.
To Jordane and Bia. We shared our days together, we smiled and cried together. We
were together, after all… and we did it! I will always love you.
To my daughter Leticia. Thank you for being part of me and for teaching me so many
things. I would do everything again and again for you. I will always be with you and, in
the end, everything little thing is going to be all right. Among the things I learned: being
a father is having a reason to say “I love you” every day. I love you!
To God, for all.
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Resumo da Tese apresentada à COPPE/UFRJ como parte dos requisitos necessários
para a obtenção do grau de Doutor em Ciências (D.Sc.)
RESISTÊNCIA À COMPRESSÃO DE COLUNAS PULTRUDADAS DE POLÍMERO
REFORÇADO COM FIBRA DE VIDRO (PRFV)
Daniel Carlos Taissum Cardoso
Março/2014
Orientadores: Eduardo de Miranda Batista
Kent Alexander Harries
Programa: Engenharia Civil
Neste trabalho, desempenho e resistência de colunas pultrudadas de polímero
reforçado com fibra de vidro (PRFV) sujeitas a compressão concêntrica de curta duração
são estudados. Para realizar essa tarefa, uma extensa revisão bibliográfica dos
fundamentos é realizada, abordando as teorias de flambagem global e local para material
ortotrópico, modos de colapso em colunas perfeitas e teorias sobre o comportamento de
colunas 'reais'. As lacunas no conhecimento e na compreensão do comportamento desses
membros são identificadas e, para preenchê-las, métodos racionais e abrangentes
resultando em diretrizes de projeto simples e eficazes são desenvolvidos e apresentados.
Para avaliar os métodos propostos, os resultados são comparados com aqueles obtidos
experimentalmente e/ou a partir de análises numéricas via Método das Faixas Finitas
(MFF). Os programas experimentais foram dirigidos à resistência à compressão de
colunas de tubo quadrado de PRFV com diferentes esbeltezes de coluna e parede; e à
flambagem local de colunas curtas de seções tipo-I com diferentes relações entre larguras
de mesa e alma e altura da seção e espessura da parede. Os programas incluiram medições
da geometria da seção transversal, caracterização dos materiais, determinação
experimental de cargas críticas, imperfeições e resistência à compressão e observações a
respeito dos modos de colapso e comportamento pós-flambagem.
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Abstract of Thesis presented to COPPE/UFRJ as a partial fulfillment of the
requirements for the degree of Doctor of Science (D.Sc.)
COMPRESSIVE STRENGTH OF PULTRUDED GLASS-FIBER REINFORCED
POLYMER (GFRP) COLUMNS
Daniel Carlos Taissum Cardoso
March/2014
Advisors: Eduardo de Miranda Batista
Kent Alexander Harries
Department: Civil Engineering
In this work, performance and strength of pultruded glass-fiber reinforced
polymer (GFRP) columns subject to short term concentric compression are studied. To
accomplish this task, an extensive review of existing background theory is conducted,
including global and local buckling theories for orthotropic material, perfect column
failure modes and theories behind ‘real’ column behavior. Gaps in the knowledge and
understanding of the behavior of these members are identified and, to fill them in,
rational and comprehensive methods resulting in simple and effective design guidelines
are developed and presented. To evaluate the proposed methods, results are compared to
those obtained experimentally and/or from numerical analyzes via Finite Strip Method
(FSM). The experimental programs addressed the compressive strength of GFRP square
tube columns having different column and wall slenderness; and the local buckling of I-
sections stub columns having different flange-width-to-section-depth and depth-to-
thickness ratios. The programs included cross-section geometry measurements, material
characterization, experimental determination of critical loads, imperfections and
compressive strengths and observations on the failure modes and post-buckling
behavior.
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Index
1. Introduction 1
1.1. Overview 1
1.2. Motivation of Present Study 3
1.3. Objectives 5
1.4. Organization of the Thesis 6
2. Pultruded Glass Fiber Reinforced Polymer (GFRP) 8
2.1. Overview 8
2.2. Experimental Determination of Elastic Properties 10
2.2.1. Review of ASTM Standard Tests 10
2.2.2. Non-Standard Tests 15
2.2.3. Summary 18
2.3. Theoretical Prediction of Elastic Properties 19
2.3.1. Review of Existing Methods 19
2.3.2. Approximate Formulae for Elastic Properties 22
3. GFRP Perfect Columns and Plates 25
3.1. Overview 25
3.2. Crushing 25
3.2.1. Elastic Microbuckling 26
3.2.2. Plastic Microbuckling 26
3.2.3. Fiber Crushing 27
3.2.4. Closed-Form Methods 28
3.3. Global Buckling 29
3.3.1. Flexural Buckling 30
3.3.2. Torsional and Flexural-Torsional Buckling 32
3.4. Local Buckling 32
3.4.1. Literature Review 33
3.4.2. Assumptions 36
3.4.3. Formulation by Energy Method 38
3.4.4. Approximate Deflection Functions 39
3.4.5. Local Buckling Critical Stress 41
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3.4.6. Results and Comparison with Finite Strip Method 46
4. Real GFRP Columns and Plates 53
4.1. Overview 53
4.2. Factors Affecting Actual Behavior 54
4.2.1. Geometric Imperfections 55
4.2.2. Material Behavior and Imperfections 56
4.2.3. Post-Buckling Behavior of Plates 57
4.2.4. Other Parameters 58
4.3. Strength Curve 58
4.3.1. Literature Review 58
4.3.2. Plate Strength 59
4.3.3. Column Strength 65
4.3.4. Summary 70
5. Experimental Program 1: Compressive Strength of GFRP Square Tubes 71
5.1. Literature Review 71
5.2. Experimental Program 71
5.3. Cross-Section Geometry Measurement 72
5.4. Material Characterization 72
5.4.1. Longitudinal Modulus of Elasticity (EL,c) and Compressive Strength (FL,c) 73
5.4.2. In-Plane Shear Modulus (GLT) 75
5.4.3. Longitudinal Flexural Modulus of Elasticity (EL,f) 76
5.4.4. Transverse Flexural Modulus of Elasticity (ET,f) 78
5.5. Stub Column Tests 80
5.6. Column Compression Tests 81
5.7. Experimental Results 86
5.7.1. 25.4x3.2 Tubes 92
5.7.2. 50.8x3.2 Tubes 94
5.7.3. 76.2x6.4 Tubes 96
5.7.4. 88.9x6.4 Tubes 98
5.7.5. 102x6.4 Tubes 101
5.8. Stub Column Test Results 103
5.9. Experimentally Determined Imperfection Factors 104
5.9.1. Columns 104
5.9.2. Plates 105
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6. Experimental Program 2: Local Buckling of I-Sections 107
6.1. Literature Review 107
6.2. Experimental Program 109
6.3. Material Characterization 110
6.3.1. Longitudinal compression (EL,c and FL,c) 111
6.3.2. Transverse tension (ET,t) 112
6.3.3. 45 degree off-axis tension (GLT) 114
6.3.4. Longitudinal plate bending (EL,f) 116
6.4. Stub Column Tests 118
6.5. Stub Test Results 119
6.6. Validation of Approach to Capacity Prediction Proposed in Chapter 3 124
7. Conclusions 126
8. References 133
9. Appendix A – Column Tests Reports 144
x
List of Symbols
Lowercase Roman Letters
a coefficient
b plate width; coefficient
c coefficient
bf flange width
bw web width
d section depth
e eccentricity
f function
k plate buckling coefficient; spring constant
kcr critical plate buckling coefficient
ℓ half-wave length
ℓcr critical half-wave length
m number of half-waves in the longitudinal direction
n number of half-waves in the transverse direction; numbers of constituent plates
ns shear form factor
r radius of gyration
r0 polar radius of gyration
t thickness
tf flange thickness
tw web thickness
v column lateral deflection
w plate out-of-plane deflection
x0 distance between shear center and centroid of a section
x in-plane longitudinal axis
y in-plane transverse axis
z out-of-plane transverse axis
Uppercase Roman Letters
A area; coefficient
xi
Af flange area
B coefficient
Cw torsional warping constant
D11 plate flexural stiffness component
D22 plate flexural stiffness component
D12 plate flexural stiffness component
D66 plate flexural stiffness component
Ecsm modulus of elasticity of a continuous strand mat layer
EL,t tensile modulus of elasticity in the longitudinal direction
EL,c compressive modulus of elasticity in the longitudinal direction
EL,f flexural modulus of elasticity in the longitudinal direction
ET,t tensile modulus of elasticity in the longitudinal direction
ET,c compressive modulus of elasticity in the longitudinal direction
ET,f flexural modulus of elasticity in the longitudinal direction
Fe Euler buckling critical stress
Fcrg,F flexural buckling critical stress
Fcrg,FT flexural-torsional buckling critical stress
Fcrg,T torsional buckling critical stress
Fcrℓ local buckling critical stress
FL,t tensile strength in the longitudinal direction
FL,c compressive strength in the longitudinal direction
FL,f flexural strength in the longitudinal direction
FPP perfect plate strength
FT,t tensile strength in the longitudinal direction
FT,c compressive strength in the longitudinal direction
FT,f flexural strength in the longitudinal direction
Fv shear strength
Gcsm shear modulus of a continuous strand mat layer
GLT shear modulus
I moment of inertia
J Saint-Venant torsion constant
L colum/plate length
Kt buckling coefficient for twisting
Ke buckling coefficient for flexure
xii
M bending moment
N axial force; number of layers
Ncrℓ critical axial force per unit of width
Nrov number of roving layers
P load
Q elastic property
R radius
S surface area; section elastic modulus
T work done by a compressive force
U strain energy
Vf fiber content, in volume
Lowercase Greek Letters
α rotation
αp,L plate imperfection factor in the longitudinal direction
αp,T plate imperfection factor in the transverse direction
αc column imperfection factor
γLT shear strain in the longitudinal direction
γY yield shear strain
δ deflection
δ0 out-of-straigthness amplitude
εx axial strain
εy transverse strain
ζ coefficient
η ratio of flange-to-web widths; coefficient
θ angle of rotation
λc column relative slenderness
λp plate relative slenderness
νcsm Poisson’s ration of a continuous strand mat layer
νLT major Poisson’s ratio
νTL minor Poisson’s ratio
ξ coefficient
ρc column reduction factor
ρp plate reduction factor
xiii
σ stress
τLT shear stress in the longitudinal direction
φ fiber misalignment
χc column relative strength
χp,L plate relative strength in the longitudinal direction
χp,T plate relative strength in the transverse direction
Uppercase Greek Letters
∆0 out-of-flatness amplitude
Σ summation
1
1. Introduction
1.1. Overview
Composite materials are those constituted by two or more physically and/or
chemically distinct materials combined in order to achieve particular desirable material
or mechanical characteristics. In the case of two-material composites most often one
component is ‘reinforcement’, e.g. fibers or particles, which is embedded in the other
component called the ‘matrix’, e.g. polymer, metal or ceramic. Glass-fiber reinforced
polymer (GFRP), the subject of this work, is a composite material combining glass
fibers in a polymeric matrix.
Developed in the 1940’s for military applications during World War II, GFRP
has several advantages over traditional construction materials, including its light weight
and superior corrosion resistance, leading respectively to ease of fabrication,
transportation and installation, and reduced life-cycle maintenance costs. Other
advantages are the material’s high strength-to-weight ratio, low thermal conductivity,
electromagnetic transparency, low environmental impact and the ability to tailor the
geometry and therefore properties of the resulting GFRP components.
Nevertheless, the higher manufacturing cost was, for a long time, an obstacle to
making these materials competitive in anything other than niche markets such as the
marine, aerospace and sport equipment industries. This scenario has gradually changed
with the invention of the pultrusion process in 1951. Pultrusion is a highly automated
continuous manufacturing process that consists of pulling resin-impregnated reinforcing
material such as roving, mat or cloth, through a heated steel die, where the resin is cured
at elevated temperatures (Figure 1.1). Large scale production of pultruded GFRP
sections has contributed to reduce manufacturing costs, making these products
competitive and attractive to the construction industry. The process permits virtually
limitless versatility in fiber arrangement (referred to as ‘fiber architecture’) within a
section and section geometry. Nonetheless, typically commercialized cross sections
intended for structural applications are similar to standard hot-rolled steel sections, as
shown in Figure 1.2.
Figure 1.1
Figure 1.2 – Typical pultruded GFRP cross
The worldwide use of pultruded GFRP in primary load
systems has increased since
pedestrian bridges and bridge decks
of GFRP as the primary structural system for
reported cases (e.g.: KELLER, 2001)
realtively low – approximately
members is preferred rather than for flexure, where the deflection limit state
govern structural behavior and lead to inefficient use of the material. For this reason,
pultruded GFRP has been mostly used in trussed systems
applications are shown in Figure 1.3.
2
Figure 1.1 – Pultrusion process (CREATIVE, 2004).
Typical pultruded GFRP cross-sections (STRONGWELL, 2014a)
The worldwide use of pultruded GFRP in primary load-bearing structural
systems has increased since the 1990’s, with significant applications for
pedestrian bridges and bridge decks (BAKISet al., 2002, JACOB, 2006
s the primary structural system for buildings is very limited, with only a few
(e.g.: KELLER, 2001). Since the modulus of elasticity
approximately 1/6 to 1/12 that of steel – its use for tension/compression
s is preferred rather than for flexure, where the deflection limit state
govern structural behavior and lead to inefficient use of the material. For this reason,
has been mostly used in trussed systems. Some examples of
in Figure 1.3.
(STRONGWELL, 2014a).
bearing structural
for cooling towers,
, JACOB, 2006). So far, the use
, with only a few
. Since the modulus of elasticity of GFRP is
tension/compression
s is preferred rather than for flexure, where the deflection limit state will often
govern structural behavior and lead to inefficient use of the material. For this reason,
. Some examples of
3
(a) stair tower (BEDFORD, 2014) (b) pedestrian bridge at Blue Ridge Parkway, US (ET TECHTONICS, 2014)
(c) deck for Grasshopper Bridge, Zealand, Denmark (FIBERLINE, 2014)
Figure 1.3 – Some applications of pultruded GFRP.
To encourage the use of GFRP and allow engineers to safely design structures
made of this material, significant efforts are underway worldwide to develop codes and
standards for the design of GFRP structural members. These include the ASCE
Structural Plastics Design Manual (GRAY, 1984), the Eurocomp Design Code
Handbook (CLARKE, 1996), the Italian Guide for the Design and Construction of
Structures made of FRP Pultruded Elements (CNR, 2008) and the forthcoming ASCE
Standard for Load and Resistance Factor Design (LRFD) of Pultruded Fiber
Reinforced Polymer (FRP) Structures (ASCE FCAPS).
1.2. Motivation of Present Study
Pultruded GFRP has a compressive strength similar to mild structural steel,
typically ranging from 200 to 500 MPa. On the other hand, the longitudinal modulus of
elasticity is low, ranging from 17 to 35 GPa (the modulus of steel is 200 GPa). The
combination of the very low modulus-to-strength ratios with the use of slender sections
resisting compression leads to important stability (buckling) limit states that may reduce
the load-carrying capacity of members subject to compression or flexure. This issue is
4
aggravated by the considerably lower-still transverse and shear moduli, resulting from
the mostly unidirectional reinforcement of GFRP.
In the last 20 years, various authors have addressed the performance and
strength of pultruded GFRP members subject to short term concentric compression and
a significant number of works addressing global buckling, local buckling, buckling
interaction and compressive strength of GFRP columns can be found. However, there
remains a number of important gaps in the knowledge and understanding of the
behavior of these members. These are identified and served as the motivation for this
work, as explained in the following paragraphs.
First of all, most of the previous works have focused on so-called ‘WF-sections’
(doubly-symmetric wide flange I-shaped sections generally with a flange width equal to
their section depth). Little attention has been devoted to other section shapes despite
their potential advantages as compression members. A square tube, for instance, has i) a
weak-axis radius of gyration much greater than a comparable I-shaped section, leading
to improved global buckling strength for a given column length; ii) both flanges and
webs are supported along both their edges along the column length, resulting in
enhanced local buckling critical load; iii) the internal void can be used for utilities or
passive fire suppression systems (CORREIA et al., 2010); and iv) improved aesthetics
over open shapes. Despite the advantages, only a few works investigating the
compressive performance of square tube columns, typically focusing on long column
behavior, are available. Therefore, one motivation of this work arises from the necessity
of extending this investigation to square tube compression members having different
lengths and width-to-wall thickness ratios, resulting in a range of combinations of
global and sectional slenderness.
Since GFRP is an orthotropic material, closed-form equations to determine
critical loads are very complicated and must accommodate a range of material
properties and section geometries. To simplify such equations, existing design standards
and manuals assume simply-supported edge conditions at wall intersections. This
assumption leads to very conservative results. Recently proposed equations (ASCE
FCAPS) augment the admittedly conservative capacity calculated assuming such
simply-supported conditions with an empirically derived ‘fudge factor’; although this
factor is believed to be too simplistic and is based on limited data. On the other hand,
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some expressions have been developed which demonstrate good agreement with
numerical and experimental results; although these usually require enormous calculation
effort and must typically be applied on a case-by-case basis rather than generalized in a
design methodology. There is little consensus among researchers as to the best design
approach for local buckling of GFRP sections, which leads to the second motivation for
this work: the search of an appropriate design approach that must be both simple and
accurate.
Specifically considering the compressive strength of GFRP columns, some
explicit equations can be found in literature. However, these are typically empirical or
have coefficients empirically determined primarily from observations of tested WF-
sections. The lack of a rational and comprehensive approach valid for different sections
and ranges of material properties constitutes the third motivation for this work.
1.3. Objectives
The main objective of the thesis is to investigate the performance and strength of
pultruded GFRP members subject to short term concentric compression in an ambient
laboratory environment and present simple and effective design guidelines. To
accomplish this task, an extensive review of existing background theory is conducted,
including global and local buckling theories for orthotropic material, perfect column
failure modes and theories behind ‘real’ column behavior. To fill in the gaps, such as
the lack of simple and accurate expressions for local buckling of GFRP typical sections
and of a comprehensive equation for compressive strength that accounts for different
material properties and combinations of sectional and global slenderness, rational
methods are developed and presented. The Finite Strip Method (FSM) was used to
evaluate the proposed expressions for local buckling.
To augment available data on the performance and strength of GFRP columns
and to assess the proposed methods, two large experimental programs addressing
columns under concentric compression were carried out: i) square tubes made of
polyester matrix having different lengths and width-to-wall-thickness ratios; and ii) I-
sections stub columns made of vinyl ester or polyester matrices having different flange-
width-to-section-depth and depth-to-thickness ratios. The programs included cross-
section geometry measurements, material characterization, experimental determination
6
of critical loads, imperfections and compressive strengths and observations on the
failure modes and post-buckling behavior.
All tests were conducted at the Watkins-Haggart Structural Engineering
Laboratory (WHSEL) at the University of Pittsburgh, Pittsburgh, Pennsylvania, United
States, during the years of 2012 to 2013.
A secondary objective of this work consists of investigating the existing
methods to experimentally determine the material properties of pultruded GFRP,
focusing on the most commonly used ASTM Standards. Since most pultruded GFRP
sections are comprised of narrow and very thin plates, there is a dimensional obstacle to
the applicability and utility of such standards. This work describes the material
characterization of pultruded GFRP sections, presenting standard and non-standard test
methods and making initial design recommendations to estimate elastic properties
through the use of simple closed-form expressions.
1.4. Organization of the Thesis
The following organization was adopted in this thesis:
a. In Chapter 2, the characteristics and typical properties of pultruded GFRP
sections are presented. Current standardized and alternative approaches to
experimentally determine elastic properties are described and, finally,
closed-form equations are proposed as guidance for initial design.
b. In Chapter 3, the behavior of a ‘perfect’ column is addressed. Failure modes
by crushing, global buckling and local buckling for orthotropic materials are
described and expressions are presented. Closed-form equations to determine
the local buckling critical stress of typical pultruded GFRP sections – angles,
I-shaped, channels and rectangular tubes – comprised of orthotropic thin
walls subject to concentric compression are developed, presented and
compared to FSM results.
c. In Chapter 4, the compressive strength of a ‘real’ column is addressed. The
factors affecting the behavior are described, beam-column and imperfect
plate theories are presented and a design equation is proposed.
d. In Chapter 5, an experimental program investigating the behavior of square
tubes having different lengths and width-to-wall-thickness ratios is
7
described. Cross-section geometry, material properties, critical loads,
compressive strengths and failure modes are reported and discussion focuses
on observed post-buckling behavior and interaction between crushing, local
and global buckling. Column and wall imperfections are also estimated.
e. In Chapter 6, an experimental program investigating the local buckling of I-
sections is described. Stubs made of vinyl ester and polyester matrices
having different flange-width-to-section-depth ratios were tested and cross-
section geometry, material properties and critical loads are reported.
f. In Chapter 7, the conclusions are presented and suggestions for future
researches are made.
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2. Pultruded Glass Fiber Reinforced
Polymer (GFRP)
2.1. Overview
Pultrusion permits virtually limitless versatility in section geometry and fiber
content and architecture. However, typically in structural engineering applications,
angles, I-sections, channels and tubes made of isophthalic polyester and E-glass (E
indicates low electrical conductivity glass) reinforcement are used. Other resins such as
vinyl ester, fire retardant polyester and epoxy can be used to improve chemical
resistance, fire retardancy and mechanical properties, respectively; S-glass (high
strength glass) reinforcement can be adopted to enhance strength and stiffness.
However, these alternate raw materials are more expensive and may increase the cost of
the pultruded product. Physical and mechanical properties of the typical constituents are
presented in Table 2.1.
Table 2.1 – Mechanical properties of typical constituents (CREATIVE, 2004).
Physical / Mechanical Properties
Resins Fibers
Polyester Vinyl ester Epoxy E-glass S-glass
Density (g/cm3) 1.1 1.1 1.1 2.6 2.5 Tensile strength (MPa) 77 81 76 3400 4585 Flexural strength (MPa) 123 138 115 - - Elongation to break (%) 4.5 5.0 6.3 4.8 5.4 Tensile modulus (GPa) 3.0 3.7 3.2 72 87
According to BAKIS et al. (2002), the volumetric fiber reinforcement content
(fiber volume ratio, or Vf) of a GFRP pultruded section usually ranges from 0.35 to 0.50
and is comprised of alternate roving and continuous strand mat (CSM) layers. In
practice, it has been observed that the fiber volume ratio may range from 0.30 to 0.60
and the roving volume corresponds to 50 to 70% of the total fiber volume. Furthermore,
the number of roving layers typically ranges from one to four in commonly pultruded
shapes. Resin-rich surface veils made of polyester or A-glass (high resistance to alkali)
9
are also used on outer surfaces to improve corrosion resistance. To provide protection
against sunlight, ultraviolet (UV) inhibitor are typically used. Additionally, it is
important to mention that different parts of a cross-section, such as the flanges and web
of an I-sections, may have different fiber content and fiber architectures (McCARTHY
and BANK, 2010). The typical layers of a pultruded section (pultrusion diagram) are
presented in Figure 2.1.
Figure 2.1 – Typical layers of a pultruded section (CREATIVE, 2004).
Commercially-available non-solid sections are available with different ranges of
wall widths (b) and wall-width-to-thickness ratios (b:t) (BEDFORD, 2010,
CREATIVE, 2004, STRONGWELL, 2014b). I-sections, for instance, may have flange
widths ranging from 38 to 305 mm and thickness ranging 4.8 to 19.1 mm. Other
sections such as tubes may have wall thickness as thin as 3.2 mm. In most of
commercially available sections, the thickness of webs and flanges in the same section
are equal.
Mechanical properties also vary as the formulation adopted by each
manufacturer is different. Typical mechanical properties reported by three different US
manufacturers are given in Table 2.2. It is important to recall that GFRP is a brittle
material, having an essentially linear elastic stress-strain behavior in the longitudinal
direction. For the transverse direction and shear, non-linear behavior is expected. GFRP
also undergoes important long-term deformation under sustained load (BANK and
MOSALLAM, 1992, CHOI and YUAN, 2003). This phenomenon is not addressed in
this work. To determine the mechanical properties, two different approaches can be
used: measuring them experimentally or predicting them theoretically (JONES, 1999).
In the following sections, these two approaches are presented in detail. The theoretical
10
determination of the compressive strength, important for the column problem, is
presented with more detail in Chapter 3.
Table 2.2 – Typical reported mechanical properties of GFRP pultruded sections.
Mechanical Properties BEDFORD (2010)
CREATIVE (2004)
STRONGWELL (2014a)
Long
itudi
nal Tensile strength (MPa) FL,t 207 228 207
Compressive strength (MPa) FL,c 207 228 207 Flexural strength (MPa) FL,f 207 228 207 Tensile modulus (GPa) EL,t 17.2 17.2 17.2
Compressive modulus (GPa) EL,c 17.2 20.7 17.2 Flexural modulus (GPa) EL,f 12.4 11.0 11.0
Tra
nsve
rse
Tensile strength (MPa) FT,t 48.0 51.7 48.3 Compressive strength (MPa) FT,c 103 114 103
Flexural strength (MPa) FT,f 68.9 75.8 68.9 Tensile modulus (GPa) ET,t 5.50 5.52 5.52
Compressive modulus (GPa) ET,c 6.90 6.90 5.52 Flexural modulus (GPa) ET,f 5.50 5.52 5.52
Shear strength (MPa) Fv 31.0 31.0 31.0 Shear modulus (GPa) GLT 3.10 2.90 2.93 Major Poisson’s ratio νLT - 0.35 0.33 Minor Poisson’s ratio νTL - 0.15 -
*Fiber contents for all manufacturers range from 0.3 to 0.6, in volume.
2.2. Experimental Determination of Elastic Properties
2.2.1. Review of ASTM Standard Tests
As mentioned previously, one of the ways to determine the material properties
of a GFRP pultruded material is experimentally. In fact, experimental material
characterization is required for a correlation to be drawn between theory and
experiment, whereas the theoretical determination of properties is adequate for design.
Usually, standard tests promulgated by the American Society for Testing and Materials
(ASTM) are adopted by researchers, although in some cases these tests cannot be
applied either due to dimensional limitations of the required test specimens as compared
to the pultruded sections or the lack of required apparatus for testing. In such cases,
properties are often estimated based on ‘experience’ or a theoretical approach. This
section focuses on the description of the ASTM standards often used to characterize
pultruded GFRP members, exploring their applicability and limitations. Throughout the
section, prismatic coupon designations refer to thickness x width x total length, in units
of mm. For most tests, the coupon thickness corresponds to the pultruded plate/wall
thickness, t. Experimental results reported were obtained from specimens extracted from
11
sections having different geometries, fiber architecture and formulation (E-
glass/polyester and E-glass/vinyl ester).
2.2.1.1. Tensile Moduli (EL,t and ET,t) and Poisson’s Ratios (νLT and νTL)
Tensile modulus of elasticity is usually determined according to ASTM D638
(2010) or ASTM D3039 (2008). Figure 2.2 illustrates the specimens recommended by
both ASTM D638 and D3039. The latter is more typical among researchers because it
the sample preparation is simpler, allowing the use of straight-sided untabbed rather
than ‘dog-boned’ or tabbed specimens (ADAMS et al., 2003). Typical coupon width
ranges from 12.7 to 25.4 mm and length from 300 to 350 mm. Typically, a width of
25.4 mm is preferred to avoid variability due to non-uniform fiber distribution. A
longitudinal strain gage or uniaxial extensometer is used to determine the strains in
order to calculate modulus of elasticity and a transverse strain gage can be used if
Poisson’s ratio is desired. The modulus of elasticity can be obtained as the slope of the
axial tensile stress (σx) versus strain (εx) graph and the Poisson’s ratio can be obtained as
the slope of the transverse (εy) versus axial strain (εx) graph.
(a) ‘Dog-boned’ specimen required by
ASTM D638 (b) Straight-sided tabbed specimen (tabs
are optional) used for ASTM D3039
Figure 2.2 – ASTM Standard coupons for tensile tests (ADAMS et al., 2003).
Researchers reporting ASTM D3039 longitudinal tensile tests include SONTI
and BARBERO (1996), ZUREICK and SCOTT (1997), KANG (2002) and GOSLING
and SARIBIYIK (2003). Tests were conducted on 3.2 to 12.7 mm thick coupons
extracted from polyester and vinyl ester-based GFRP with widths ranging from 15 to
50.8 mm and lengths from 203 to 457 mm. Longitudinal tensile moduli ranging from
17.4 to 28.3 GPa were reported in addition to average major Poisson’s ratios of 0.28.
Transverse tensile tests based on ASTM recommendations are rarely performed due to
the specimen dimensions required often being greater than those that may be cut from a
pultruded section.
12
2.2.1.2. Compressive Moduli (EL,c and ET,c)
Compressive modulus of elasticity is usually determined according to end
loading (ASTM D695, 2010) or shear-loading (ASTM D3410, 2008) test methods. Both
methods require short gage lengths to prevent buckling, which are usually calculated
beforehand in order to ensure that critical loads are much greater than squash loads
(ZUREICK and SCOTT, 1997). The use of strain gages on both faces is also an option
to account for flexural-induced strains. Typically, straight-sided coupons having a width
of 25.4 mm are adopted. Another method, combining end and shear loading, is provided
by ASTM D6641 (2009), the advantage of which is the shared load transfer, reducing
the risk of end crushing and slippage. The modulus of elasticity can be obtained as the
slope of the axial compressive stress (σx) versus strain (εx) graph. Test apparatus
recommended by ASTM D695, D3410 and D6641 are presented in Figure 2.3, where
the complexity of these apparatuses can be seen.
(a) ASTM D695 end
loading (b) ASTM D3410 shear
loading (c) ASTM D6641 hybrid
loading
Figure 2.3 – ASTM Standard compressive tests (ADAMS et al., 2003).
Compressive tests on longitudinal coupons were performed by ZUREICK and
SCOTT (1997), ZUREICK and STEFFEN (2000) and KANG (2002). 6.4 to 12.7 thick
coupons with widths ranging from 25.4 to 38.1 mm and gage lengths from 41 to 88.9
mm were tested. Longitudinal compressive moduli ranging from 16.8 to 30.7 GPa were
observed. Transverse compression tests based on ASTM recommendations are rarely
performed.
13
2.2.1.3. Flexural Moduli (EL,f and ET,f)
Flexural modulus of elasticity can be determined according to ASTM D790 (2010) or
D6272 (2010); three and four-point flexure tests, respectively. Typically, a 16:1 span-
to-thickness ratio is recommended to mitigate the influence of shear. Strain gages can be
applied at the mid-span to determine the bending strains or, in a simpler form, the
modulus of elasticity can be calculated based on mid-span deflection (δ) and applied
load (P). The methods recommended by ASTM D790 and D6272 are illustrated in
Figure 2.4. For a three-point flexure test in the longitudinal direction:
3
3
, 4 bt
PLE fL δ
= (2.1)
where L is the test span and b and t are the coupon width and thickness, respectively.
Thus, the modulus can be obtained as the slope of the graph P versus 4δb(t/L)3. For a
four-point test, this relationship becomes:
3
3
, 54
23
bt
PLE fL δ
= (2.2)
Three-point flexural tests on longitudinal coupons were performed by KANG
(2002) on 6.4x25.4x127 coupons over a 101.6 mm span and in the present study in
support of the testing reported in Chapter 5 and 6 on 3.2x40x254 and 6.4x45-57x254
coupons over a 203 mm span. Longitudinal flexural moduli ranging from 11.5 to 25
GPa were reported.
(a) ASTM D790 three-point flexure test
(ADMET, 2014) (b) ASTM D6272 four-point flexure test
(INSTRON, 2014a)
Figure 2.4 – ASTM Standard flexural tests.
14
2.2.1.4. Shear Modulus (GLT)
The in-plane shear modulus of a composite material can be determined by
various methods including i) Iosipescu (ASTM D5379, 2012); ii) ±45° tensile (ASTM
D3518, 2013); and iii) two and three-rail (ASTM D4255, 2007) tests. Usually, the
Iosipescu method is adopted by researchers because narrow coupons can be used,
specimen preparation is simple and the results are accurate; however a special fixture is
required (Fig. 2.5a). In this method, a ±45° rosette strain gage is used at the notch and
the shear modulus is obtained as the slope of the graph of average shear stress across the
notched section (load divided by area) (τ) versus the shear strain (γ), calculated as the
sum of the absolute values of the ±45° strain gage readings. The ±45° tensile test is a
good alternative when the Iosipescu fixture is not available although it requires a larger
specimen length (typically 230 mm) to mitigate undesirable deformation due to end
constraints. In this test, a 0°/90° biaxial rosette strain gage or a pair of longitudinal and
transverse strain gages are applied and the shear modulus is obtained as one half the
slope of the graph of the axial stress in the direction parallel to the applied force (σx/2 =
τLT) versus and the difference between measured longitudinal and transverse strains (εx –
εy) as shown in Fig. 2.5b. Finally, the two and three-rail methods require special fixtures
as well as specimens with larger dimensions and are usually not applied to pultruded
sections. According to a decision analysis-based evaluation of nine in-plane shear
methods (LEE and MUNRO, 1986), Iosipescu and ±45° tensile were ranked as
preferrable methods (1st places with the same score) while two and three-rail tests were
ranked in 4th and 6th places.
The Iosipescu method was adopted by BANK (1990), SONTI and BARBERO
(1996), ZUREICK and SCOTT (1997), ZUREICK and STEFFEN (2000) and KANG
(2002). 6.4 to 12.7 mm thick longitudinal coupons having widths ranging from 19 to
38.1 mm and lengths from 76.2 to 203 mm were tested from which shear moduli
ranging from 2.4 to 5.7 GPa were obtained.
15
(a) ASTM D5379 Iosipescu test: scheme (ADAMS et al., 2003) and picture
(INSTRON, 2014b)
(b) ASTM D3518 ±45° tensile test (ADAMS et al., 2003)
Figure 2.5 – ASTM Standard shear tests.
2.2.2. Non-Standard Tests
2.2.2.1. Transverse Tensile Modulus (ET,t) and Minor Poisson’s Ratio (νTL)
Due to the small transverse dimensions of typical pultruded sections, ASTM
standard coupons cannot be used in several situations. SONTI and BARBERO (1996),
GOSLING and SARIBIYIK (2003) and tests conducted in support of this study (see
Chapter 6) tested non-standard transverse coupons with short lengths – 9.5x25.4x88.9,
3.1x10x50 and 6.4x12.7x88.9, respectively – and obtained transverse tensile moduli
ranging from 9.6 to 12.2 GPa. The average minor Poisson’s ratio reported is 0.17.
GOSLING and SARIBIYIK (2003) also conducted a numerical investigation using the
finite element method to validate their results and discussed how geometric and material
parameters such as the ratio of orthotropy, coupon length, gage length, thickness and
clamping length may affect the results. The non-standard coupon recommended by
GOSLING and SARIBIYIK (2003) is shown in Figure 2.6a.
16
2.2.2.2. Longitudinal (EL,c) and Transverse (ET,c) Compressive Moduli
Since ASTM standards require special apparatus not always available, some
non-standard tests are also found in the literature. CORREIA et al. (2011) performed
compressive tests on 9.8x12.7x39 longitudinal coupons by end-loading without using an
anti-buckling apparatus (Fig. 2.6b). Similar tests were used in support of Chapter 6 of
this work using 6.4x12.7x50.8 longitudinal coupons. Resulting longitundinal
compressive modulus of elasticity ranged from 25.6 to 28.4 GPa. The same test was
performed by CORREIA et al. (2011) in the transverse direction and an average
modulus of 7.4 GPa was obtained. Longitudinal compressive modulus was also
obtained in the present work (Chapter 5) using full-sections tests on 25.4x3.2 and
102x6.4 mm square tubes (Fig. 2.6b) resulting in moduli ranging from 22.1 to 31.1 GPa.
To mitigate end effects, TEIXEIRA (2007) conducted full-section tests on square tubes
with different types of reinforcement – e.g. aluminum end plates, resin rings and glued
GFRP plates – and obtained moduli ranging from 25.6 GPa to 32.7 GPa. Full section
tests intrinsically consider material imperfections throughout the cross section,
including any residual stresses and non-uniform fiber distribution.
2.2.2.3. Transverse Flexural Moduli (ET,f)
Since many standard procedures cannot be applied in the transverse direction
due to dimensional limitations, non-standardized tests are adopted. For the square tubes
reported in Chapter 5 of this work, channel-shaped coupons were fabricated by cutting
off one of the tube walls. An eccentric static load was applied to the top flange while the
bottom flange was clamped to a flat surface (Fig. 2.6d). This results in a constant
moment in the channel web (similar to a so-called ‘omega’ strain gage). Strain gages
were applied to both faces of the web to measure the flexural strains. The transverse
flexural moduli obtained ranged from 10.5 to 13.5 GPa. Similar set-ups can be adopted
for other cross-sections.
2.2.2.4. Shear modulus (GLT)
±45° tensile tests were carried out in support of the study reported in Chapter 6
on non-standard 6.4x12.7x119 coupons having gage lengths of 81 mm (Fig. 2.6e).
Although the gage length is relatively short, the influence of end-clamping was
evaluated based on an approach proposed by ADAMS et al. (2003) and it was
17
concluded that the effect was marginal for the range of material properties studied. Two
strain gages (longitudinal and transverse) were applied at the specimen faces and the
shear modulus obtained ranged from 4.1 to 4.7 GPa.
Two other non-standard tests are also common: Timoshenko beam (BANK,
1989, ROBERTS and AL-UBAIDI, 2002) and torsion (SONTI and BARBERO, 1996,
TURVEY, 1998) tests. The Timoshenko beam does not require any special apparatus
and is easy to perform, although the shear modulus obtained is often lower than
expected (MOTTRAM, 2004a). Indeed, this is the case for Timoshenko beam tests
performed in support of Chapter 5 of the present work. The torsion test, on the other
hand, has proven to be a very good method, although a test rig able to apply torsional
loading is required. SONTI and BARBERO (1996) and TURVEY (1998) adopted
9.5x25.4x178 and 9.5x50.8x200-400 specimens, respectively, and obtained shear
moduli ranging from 2.8 to 4.5 GPa.
a) non-standard tensile coupon (GOSLING and SARIBIYIK, 2003)
b) end-loading compression with no buckling apparatus (CORREIA et al., 2011)
c) full-section compression
d) transverse flexure e) ±45° tension
Figure 2.6 – Non-standard methods.
18
2.2.3. Summary
A summary of the mean experimental results obtained by various authors in
addition to the tests conducted in support of the present study is presented in Table 2.3.
The fiber volume ratio (Vf) and number of roving layers (Nrov) are also shown. When
these data were not available, they were estimated based on observation of similar
pultruded sections. It is important to mention that some of the results presented refer to
works where the methods adopted to determine the material properties were not
described in detail (e.g., HAJ-ALI et al., 2001, TURVEY and ZHANG, 2006).
Table 2.3 – Experimental material properties reported in literature and from present study.
Pro
pert
ies
BA
NK
(19
90) SONTI and
BARBERO (1996)
ZUREICK and SCOTT
(1997)
TU
RV
EY
(19
98)
HA
J-A
LIet
al.
(200
1)
KA
NG
(20
02)
GO
SLI
NG
and
S
AR
IBIY
IK (
2003
)
TURVEY and ZHANG
(2006)
CO
RR
EIA
et a
l. (2
011)
Fla
nge
Web
WF
Tub
es
Fla
nge
Web
EL,t - 20.2 18.1 19.4 27.0 - 18.2 23.8 27.3 20.7 22.2 32.8 EL,c - - - 19.1 28.1 - 19.3 23.8 - 21.8 22.4 26.4 EL,f - - - - - - - 21.8 - - - 26.9 ET,t - 11.4 10.9 - - - 10.1 - 10.1 - - - ET,c - - - - - - 12.7 - - - - 7.4 ET,f - - - - - - - - - - - - GLT 2.6 3.6 4.2 4.3 4.7 3.3 4.5 2.7 - 3.7 3.8 - νLT - 0.28 0.29 - - - 0.28 - 0.29 - - - νTL - 0.19 0.18 - - - - - 0.15 - - - FL,c - - - 320 357 - - 379 388 267 294 376 Vf 0.30a 0.30a 0.30a 0.30 0.45a 0.38 0.34 0.35 0.45 0.30a 0.30a 0.55
Nrov - - - - - - 3 - - - 3a Present Study Notes
Pro
pert
ies Chapter 5 Chapter 6
a Estimated values, based on similar sections with reported data.
b Shear moduli obtained from Timoshenko-Beam test. Not plotted in Figure 2.8. c Moduli in GPa; strength in MPa.
Tub
es
25.4
x3.2
Tub
es
50.8
x3.2
Tub
es
102x
6.4
Fla
nge
VE
Web
VE
Fla
nge
PE
Web
PE
EL,t - - - - - - EL,c 22.1 - 31.1 25.6 - 25.7 - EL,f - 11.5 25 21.9 - 19.5 - ET,t - - - - 12.2 - 9.4 ET,c - - - - - - - ET,f - 10.5 13.5 - - - - GLT - 2.2b 2.7b - 4.7 - 4.1 νLT - - - - - - - νTL - - - - - - - FL,c 224 - 333 429 - 277 - Vf 0.36 0.39 0.49 0.51 0.56 0.43 0.49
Nrov 1 1 3 2 2 2 2
19
2.3. Theoretical Prediction of Elastic Properties
2.3.1. Review of Existing Methods
As mentioned previously, pultruded sections are comprised of different layers, each
with its own constituents and properties. Therefore, the problem can be divided into two
different levels: lamina, representing each layer, and laminate, representing the set of
laminae bonded together.
To determine the elastic properties of a composite lamina taking in consideration the
proportion of each constituent material, the procedures of micromechanics must be
adopted, i.e., interaction between fibers and matrix behavior must be studied. To do so,
some assumptions are made (JONES, 1999): i) the lamina is initially stress-free, linearly
elastic and macroscopically homogeneous and orthotropic; ii) the fibers are
homogeneous, linearly elastic, isotropic, regularly spaced and perfectly aligned and
bonded; and iii) the matrix is homogeneous, linearly elastic, isotropic and void-free.
According to JONES (1999), approaches to determine the elastic properties of a
lamina can be divided in two classes: mechanics of materials and elasticity. The former,
also called the rule-of-mixtures, is based on simple assumptions of load and
deformation-sharing between constituents, treating the constituents as springs connected
in series or parallel, and leads to simple expressions. This approach is not accurate for
determining transverse and shear moduli (JONES, 1999). The elasticity approach, on
the other hand, is based on the variational energy principles of classical elasticity theory
and generally leads either to upper and lower bounds or to complex problems that
require advanced or numerical analyzes in order to obtain the solutions; this is not
practical for an engineering design environment.
Based on the Hill’s ‘self-consistent’ elasticity method (HILL, 1965) where the
composite is modeled as a single fiber encased in a cylinder of matrix, with both
embedded in an unbounded homogeneous medium indistinguishable from the
composite, HALPIN (1969) derived a generalized formula after rearranging and making
simplifications to results obtained by HERMANS (1967). A review of the method is
presented by HALPIN and KARDOS (1976). The resulting expression, called the
Halpin-Tsai equation, is in good agreement with experimental data (TSAI, 1964) and
has been recommended by the Eurocomp Handbook (CLARKE, 1996) to determine the
20
elastic properties of unidirectionally reinforced composites. To determine a certain
elastic property, generalized as Q, the equation is presented in the following form:
)]( [
)]( [
mffmf
mffmfm
QQVQQ
QQVQQQQ
−−+−++
=ξ
ξξ (2.3)
where Qm and Qf are the matrix and fiber properties related to Q, respectively; Vf is the
fiber volume ratio; and ξ is the property coefficient associated with the property
represented by Q given in Table 2.4.
Table 2.4 – Coefficients for Halpin-Tsai equation (CLARKE, 1996).
Elastic Property, Q
EL νLT ET GLT
(Vf<0.65) (Vf ≥0.65) (Vf<0.65) (Vf ≥0.65) ξ ∞ ∞ 2.0 2.0 + 40 Vf
10 1.0 1.0 + 40 Vf10
Another approach based on the theory of elasticity is called the periodic
microstructure method. This is based on the Fourier series technique and assumes the
homogenization eigenstrain to be piecewise constant (LUCIANO and BARBERO,
1994). The method was compared to Halpin-Tsai equation (DAVALOS et al., 1996)
and similar properties were obtained with differences of less than 10%, except for major
Poisson’s ratio, where periodic microstructure led to results about 30% greater for
unidirectional layers.
Halpin-Tsai equation can be successfully used to determine the properties of a
roving layer, but it cannot be applied to the CSM layer, where the fibers are randomly
oriented and distributed. By assuming isotropic behavior for such lamina, NIELSEN
and CHEN (1968) proposed an integration over all possible fiber orientations. An
approximate solution for this integration was presented by HULL (1981) and relates the
random lamina properties to those of an unidirectionally reinforced lamina, as follows:
**
8
5
8
3TLcsm EEE += (2.4)
**
4
1
8
1TLcsm EEG += (2.5)
12
−=csm
csmcsm G
Eν (2.6)
21
in which EL* and ET
* are the longitudinal and transverse moduli assuming that all fibers
are oriented in the longitudinal direction, both calculated according Equation 2.3.
Hull’s equations were experimentally investigated by NAUGHTON et al.
(1985), who performed tests on chopped strand mat and woven roving composites. In
general, excellent agreement was achieved, except for the fact that experimental
transverse modulus was 18% lower than the longitudinal modulus, showing that the
material is not truly isotropic. However, the formulae proposed by HULL (1981) seem
quite reasonable for design applications.
To determine the average properties of the composite, a laminate analysis
through classical lamination theory must be carried out (JONES, 1999), for which the
laminate is assumed to be thin and no debonding between layers occurs. The predicted
properties obtained from this methodology, along with the laminae properties, were
compared to experimental properties of pultruded material and excellent agreement was
achieved (DAVALOS et al., 1996, SONTI and BARBERO, 1996). Laminate behavior
is schematically represented in Figure 2.7, where it can be seen that the laminae behave
as parallel springs. For orthotropic composites, longitudinal and transverse extensional
moduli, shear modulus and major Poisson’s ratio can be respectively determined as:
∑=
==N
iiiLcLtL tE
tEE
1,,,
1 (2.7)
∑=
==N
iiiTcTtT tE
tEE
1,,,
1 (2.8)
∑=
=N
iiiLTLT tG
tG
1,
1 (2.9)
∑=
=N
iiiLTLT t
t 1,
1 νν (2.10)
in which i is an index representing each of N lamina having thickness ti and elastic
properties EL,i, ET,i, GLT,i and νLT,i; and tis the laminate thickness.
The minor Poisson’s ratio can be obtained from the following relationship
(JONES, 1999):
22
LT
tL
tTTL E
Eνν
,
,= (2.11)
To obtain the flexural properties, classical lamination theory must be used along
with the transformed section method. Assuming that the fibers are located in the middle
of each lamina, the laminate longitudinal and transverse flexural moduli can be given
as:
( )∑=
+−+=
N
iiiiL
iifmiffmfL ztE
tVEVEE
tE
1
2,
33
,3
,3, 12
12 (2.12)
( )∑=
+−+=
N
iiiiT
iifmiffmfT ztE
tVEVEE
tE
1
2,
33
,3
,3, 12
12 (2.13)
where Vf,i is the fiber volume ratio of lamina i; and zi is the distance from the center of
lamina i to the neutral axis of the laminate as shown in Figure 2.7.
Figure 2.7 – Schematic behavior of a laminate subject to in-plane and flexural loads.
2.3.2. Approximate Formulae for Elastic Properties
As can be seen from Equations 2.7 to 2.13, thickness and fiber content of each
lamina must be known. DAVALOS et al. (1996) suggested that the specification
(surface veil, CSM or roving) and thickness of each lamina must be provided by the
material producer. However, these data are usually not promptly available and the
method itself requires several calculation steps, which are not practical to use in design.
An additional issue with accuracy is apparent when one considers that during
pultrusion, in situ fiber architecture varies from the theoretical design. In particular,
rovings are often seen to ‘drift’ through the thickness of the laminate affecting both
dimensions t and z.
23
In this section, approximate closed-form expressions for elastic properties of
pultruded GFRP are developed by the author and presented. These expressions are
based on regression of average experimental data available in literature and are easily
applied to the preliminary design of pultruded GFRP sections. The source data, given in
Table 2.3, includes properties of different pultruded shapes, fiber contents and sections
made of both polyester and vinyl ester resins.
In Figure 2.8, experimental longitudinal, transverse and shear moduli are plotted
against the fiber content. Straight lines obtained from regression are also shown. While
longitudinal and shear moduli increase with fiber volume, as expected, transverse
modulus falls marginally. This is believed to reflect a reduced proportion of randomly
oriented fiber material at greater longitudinal fiber contents. The regression expressions
are:
ftLcL VEE 2.397.8,, +== (GPa) (2.14)
ftTcT VEE 7.38.11,, −== (GPa) (2.15)
fLT VG 2.12.3 += (GPa) (2.16)
In Figure 2.9, the ratio of flexural-to-extensional longitudinal modulus is plotted
against the number of roving layers in the laminate. As expected, the ratio is very low
when only one roving layer is present since the fibers are likely close to the neutral axis
and therefore have little contribution to the flexural properties. The ratio increases with
the number of rovings and is asymptotic to 1.0 as should also be expected. Although
only a few points are available, a representative expression is suggested:
( ) tLrovfL ENE ,, /66.011.1 −= (2.17)
None of the works considered determined experimentally both transverse
extensional and flexural moduli but a theoretical analysis based on the method
described by DAVALOS et al. (1996) shows that both are approximately equal, which
leads to:
tTfT EE ,, = (2.18)
24
Only a few works dedicated effort to determine experimentally Poisson’s ratios
and a regression analysis cannot be carried out. The average major and minor Poisson’s
ratio obtained experimentally were 0.28 and 0.17, respectively, and it is quite reasonable
to adopt these numbers for design purposes.
Figure 2.8 – Experimental elastic moduli vs. fiber content (source data in Table 2.3).
Figure 2.9 – Experimental ratio of longitudinal flexural-to-extensional moduli vs.
number of roving layers (source data in Table 2.3).
GLT = 3.2+1.2Vf0
5
10
15
20
25
30
35
0.30 0.35 0.40 0.45 0.50 0.55 0.60
Ela
stic
Mo
duli
(GP
a)
Fiber Content, Vf
Tensile
Compressive
Shear
0.00
0.20
0.40
0.60
0.80
1.00
1 2 3 4
Flex
ura
l/Ext
ensi
ona
l Mo
duli
Number of Roving Layers, Nrov
EL,f /EL,t = (1.11 – 0.66/Nrov)
25
3. GFRP Perfect Columns and Plates
3.1. Overview
Perfect or ideal columns (PC) are defined as initially perfectly straight members
subject to a concentric compression force (TIMOSHENKO and GERE, 1961). Such
ideal members are useful to study the instability of compression members associated
with the bifurcation of equilibrium, i.e., to determine the condition (load) for which the
member loses the ability to resist increasing loads in the original configuration and
begins to exhibit large deflections that change their original shape (buckling). The
compression load for which bifurcation occurs is called the ‘critical load’ although this
does not coincide with the failure load of a real imperfect column (ZIEMIAN, 2010) as
will be described in detail in Chapter 4. Similarly, perfect or ideal plates (PP) can be
adopted for the study of plates subject to compression; these are initially perfectly flat
and subject to concentric in-plane compression load.
In the case of a perfect column with a cross-section comprised of perfect plates
(PC-PP), bifurcations associated with the overall column lateral deflection (global
buckling) and the out-of-plane deflection of the individual plates (local buckling) can be
mathematically described. These modes of behavior are naturally uncoupled although,
in the presence of imperfections, may interact and cause significant reduction of load-
carrying capacity (BATISTA, 2004). However, this phenomenon does not occur for PC-
PP and, in this chapter, each buckling mode is addressed separately, assuming perfectly
elastic material. A third failure mode occurs if the column material compressive
strength is exceeded (crushing) prior to global or local buckling. Each behavior,
crushing, global and local buckling are discussed in relation to pultruded GFRP
members in the following sections.
3.2. Crushing
The term ‘crushing’ generally refers to all collapse modes that may occur at the
constituent material level. When a long fiber composite is loaded in compression, the
most important associated failure modes are (FLECK, 1997):
26
3.2.1. Elastic Microbuckling
Initially investigated by ROSEN (1965), this failure mode is associated with the
fiber buckling, where each fiber behaves as a compressed member within continuous
lateral restraint provided by the matrix. This failure mode can be subdivided in two
classes:
a. Transverse (or extensional) mode: adjacent fibers bend in opposite directions
and the matrix is subject to transverse stress, as shown in Figure 3.1a;
b. Shear mode: adjacent fibers bend in the same direction and the matrix is
subjected to shear, as shown in Figure 3.1b. For unidirectional composites
with fiber content greater than 0.30, the shear mode usually governs the
strength (FLECK, 1997). The critical compressive stress associated with this
mode is given as:
GEd
GF cL ≅
+=22
1, 3 λπ (3.1)
in which G is the unidirectional composite shear modulus; E is unidirectional
composite longitudinal modulus; d is the fiber diameter; and, λ is the buckling
wave length.
The lowest value for the strength is obtained assuming that the wave length is
much larger than the fiber diameter, which results in the second term of the equation
being very small and the strength being approximately equal to the shear modulus of the
material. Since the Eq. 3.1 was obtained from bifurcation analysis, it does not account
for fiber waviness, which may reduce the capacity. Experimental investigation
conducted by JELF and FLECK (1992) showed that Rosen’s theory is accurate for
linear elastic matrix material.
3.2.2. Plastic Microbuckling
Assuming a rigid-perfectly plastic material, fibers with an initial misalignment
but no kink band, ARGON (1972) derived a simple expression for the critical
compressive stress, depending on the shear yield strain (γY) and fiber misalignment
angle (φ). Argon’s work was extended by BUDIANSKY (1983), who considered an
elastic-perfectly plastic matrix and obtained a similar expression as follows:
27
Y
cL
GF
γφ /12, += (3.2)
In Equation 3.2, FLECK (1997) recommends φ = 2° and γY = 1%. The failure
mode is illustrated in Figure 3.1b. Experimental evidence from several sources has
shown that this mode often governs the compressive strength for polymer matrix
composites (FLECK, 1997).
3.2.3. Fiber Crushing
This failure mode, illustrated in Figure 3.1c, occurs when the matrix stiffness
and strength is sufficient to prevent microbuckling and the fibers fail when the uniaxial
strain equals their intrinsic crushing strain. Tests conducted by PIGGOT and HARRIS
(1980) showed that, for glass fiber, the fiber crushing stress is FLc,3 = 1.7 GPa.
(a) Elastic microbuckling (JONES, 1999) (b) Plastic microbuckling (JELF and
FLECK, 1992)
(c) Fiber crushing (JELF and FLECK, 1992)
Figure 3.1 – Compressive failure modes of composite materials.
28
3.2.4. Closed-Form Methods
A closed-form equation for crushing stress was derived by BARBERO (1998)
which assumed a statistical distribution of fiber misalignment within the cross section
and a continuum damage model where the loads carried by fibers that failed are
redistributed until a certain limit. The real functions for the problem were approximated
by polynomial functions and the following equation was proposed:
b
cL aGF
+= 1,
χ (3.3)
in which χ is a dimensionless parameter that depends on shear properties and fiber
misalignment. For glass-polyester composites with fiber content of 0.40, this number is
found to be χ = 5.15; and a = 0.21 and b = -0.69 are constants obtained from data fitting.
Later, an average value of χ = 5.83 was obtained by BARBERO et al. (1999a),
after determining experimentally the shear properties of pultruded rods having various
formulations and measuring their distribution of fiber misalignment using an optical
technique described by YURGARTIS (1987). The rods were tested under compression
and the maximum and average differences observed between predicted (Equation 3.3)
and experimental results were 46% and 27%, respectively, with Equation 3.3 providing
conservative estimations for all situations. In the same work, the authors showed that
the compressive strength of a typical CSM lamina is much lower than those usually
obtained for unidirectional composites (rovings) and, therefore, suggested that the
strength of CSM layers in a pultruded section should be neglected and roving layers
assumed to carry all the load. Finally, compressive tests were performed on full I-
sections and square tubes; results were compared to the proposed method, with a
maximum difference of about 20% observed.
Although the strength of an unidirectional composite can be readily calculated
from the expressions presented previously, the applicability to pultruded GFRP sections
requires that specification of the material used for roving layers and their thickness are
known. However, such data are often not readily available and are rarely presented in
data sheets, tables and manuals provided by manufacturers. Thus, an expression,
suitable for initial design, obtained from regression of available experimental mean
results obtained by various authors (including tests performed in support of Chapters 5
29
and 6 of the present work) is proposed. The experimentally obtained compressive
strengths are shown in Table 2.3 and are plotted against the fiber volume ratio in Figure
3.2. The proposed expression is:
fcL VF 352191, += (MPa) (3.4)
Figure 3.2 – Experimental compressive strengths vs. fiber content.
3.3. Global Buckling
Global buckling of column members is a generic term used by engineers and
usually includes buckling modes where the half-wave length is of the same order of
magnitude as the length of the compression member and for which the section remains
undistorted. This includes the following buckling modes: flexural, torsional and
flexural-torsional. In the available literature, major effort has been directed toward
global buckling of doubly symmetric sections, where the flexural mode governs the
behavior. To date, few works have been dedicated to investigate the compressive
buckling behavior of monosymmetric or asymmetric cross-sections (HEWSON, 1978,
ZUREICK AND STEFFEN, 2000). In the following sections, expressions for these
buckling modes are presented.
0
100
200
300
400
500
0.30 0.35 0.40 0.45 0.50 0.55 0.60
Co
mp
ress
ive
Stre
ngth
(MP
a)
Fiber Content, Vf
30
3.3.1. Flexural Buckling
Considering a column of length L and radius of gyration with respect to the
weak axis r subject to a compressive force, the classical equation for critical buckling
stress (Euler stress), Fe, is given as (ZIEMIAN, 2010):
( )2
2
/ rLK
EF
e
Le
π= (3.5)
in which EL is the cross-sectional flexural modulus of elasticity; and Ke is the ‘effective
length factor’ dependent on end conditions, presented in Table 3.2.
Table 3.2 – Buckling coefficients for columns, Ke.
fix-fix fix-pin fix-fix (free to translate)
pin-pin fix-free pin- fix (free to translate)
Ke 0.5 0.7 1.0 1.0 2.0 2.0
Equation 3.5 has been recommended by standards to determine global buckling
critical stress of pultruded GFRP columns (GRAY, 1984, CLARKE, 1996, CNR, 2008).
Experiments with slender columns have shown that this expression leads to reasonable
approximation of the critical stress (BARBERO and TOMBLIN, 1993, ZUREICK and
SCOTT, 1997). However, since GFRP has a very low in-plane shear modulus, an
expression reported by TIMOSHENKO and GERE (1961) which accounts for the effect
of shear is recommended (ZUREICK and SCOTT, 1997, LANE and MOTTRAM,
2002), as follows:
+=
LTeseFcrg GFn
FF/1
1, (3.6)
where ns is the shear form factor, shown in Table 3.3; GLT is the in-plane shear modulus;
and Fe is the Euler critical stress, given by Eq. 3.5.
31
Table 3.3 – Shear form factors for typical cross-sections, ns.
I-section (strong axis) I-section (weak axis) Square tubes ns A/Af 1.2 A/Af 2.0
* A is the cross-section area; and Af is the flange area.
Whereas Equation 3.6 was derived by assuming the shear force as a function of
the cross-section shape, a more accurate equation obtained by considering the
deformation of an element cut out from the column (Fig. 3.3a) was also derived by
TIMOSHENKO and GERE (1961):
LTs
LTesFcrg Gn
GFnF
/2
1/41,
−+= (3.7)
This expression will be adopted in this work for correlation with experimental
critical loads obtained from Southwell plots for a wide range of column lengths
(Chapters 5 and 6).
(a) Deformation of an element cut out from a column;
b) Comparison between Equations 3.6 and 3.7.
Figure 3.3 – Shear effect on columns (adapted from TIMOSHENKO and GERE, 1961).
A comparison of Equations 3.6 and 3.7 is presented in Figure 3.3b. As can be
seen, the difference between the equations increases with Euler stress and as shear
modulus decreases. In other words, the greater the EL/GLT and the shorter the L/r, the
greater the difference between the two equations. As pointed out by TIMOSHENKO
and GERE (1961), Equation 3.6 leads to conservative critical stresses, whereas
Equation 3.7 is more accurate. A comparison between the results predicted by these
equations and experiments is presented in Chapter 5.
0.00
0.25
0.50
0.75
1.00
0.00 0.25 0.50 0.75 1.00
Fcr
g,F/F
e
nsFe/GLT
32
3.3.2. Torsional and Flexural-Torsional Buckling
Torsional buckling is likely to occur for thin-walled open cross-sections with
low torsional rigidity and with shear center coinciding with the centroid, e.g. a
cruciform section. The torsional buckling critical stress is given as (TIMOSHENKO
and GERE, 1961):
( )
+= JG
LK
CE
ArF LT
tt
wLTcrg 2
2
20
,
1 π (3.8)
where Cw is the torsional warping constant of cross section; J is the Saint-Venant
torsion constant of cross section; r0 is the polar radius of gyration of cross section with
respect to the shear center; and KtLt is the effective length for twisting: Kt = 0.5 when
ends are restrained against warping and Kt = 1.0 when ends are free to warp.
Flexural-torsional buckling is a naturally coupled mode likely to occur for thin-
walled open cross-sections with low torsional rigidity where shear center does not
coincide with the centroid, e.g. angles and channels. Disregarding transverse shear
effects, the flexural-torsional buckling critical stress for a monosymmetric section is
given as (TIMOSHENKO and GERE,1961):
( )
+
−−−
−+
=2
,,
200,,
200
,,,
])/(1[411
])/(1[2 Tcrgxe
TcrgxeTcrgxeFTcrg
FF
rxFF
rx
FFF (3.9)
in which: Fe,x is the Euler buckling (Eq. 3.5) in the direction perpendicular to the axis of
symmetry; Fcrg,T is the torsional buckling (Eq. 3.8); and x0 is the distance between the
shear center and the centroid.
3.4. Local Buckling
Plate bending and buckling equations are usually presented as functions of the
following stiffness parameters based on elastic properties; these are adopted in the
present work:
( )TLLT
fL tED
νν−=
112
3,
11 (3.10)
33
( )TLLT
fT tED
νν−=
112
3,
22 (3.11)
2212 DD LTν= (3.12)
12
3
66
tGD LT= (3.13)
where t is the plate thickness; EL,f and ET,f are the longitudinal and transverse plate
flexural moduli, respectively; GLT is shear modulus; νLT is the major in-plane Poisson’s
ratio; and νTL = νLTET,f/EL,f is the minor in-plane Poisson’s ratio.
3.4.1. Literature Review
To analytically determine the local buckling load of columns subjected to
concentric compression, three different approaches may be used: i) full section buckling
analysis with appropriate continuity conditions at plate intersections; ii) buckling
analysis of individual plates rotationally restrained by adjacent plates; and iii) buckling
analysis of individual plates without interaction between plate elements. The following
paragraphs describe works related to each of these approaches.
One of the first attempts to determine the local buckling critical load of full-
section plate assemblies was based on the principles of moment distribution
(LUNDQUIST et al., 1943), but the complexity of the plate stiffness expressions was an
obstacle to obtaining explicit solutions. Buckling coefficients for steel sections for a
range of geometries were obtained by KROLL et al. (1943) and can be also found in the
Japanese Handbook of Structural Stability (CRCJ, 1971). However, in general, little
attention has been devoted to this kind of analysis and there is no known literature
focusing on full-section analysis to obtain closed-form equations for sections having
orthotropic material properties (such as GFRP).
On the other hand, accurate predictions of buckling modes of thin-walled
sections and their associated critical loads have been obtained (AMOUSHAHI and
MOJTABA, 2009, SILVESTRE and CAMOTIM, 2003) using approaches based on the
Finite Strip Method (FSM) (LI and SCHAFER, 2010) or Generalized Beam Theory
(GBT) (BEBIANO et al., 2008). A very interesting approach was adopted by BATISTA
34
(2010), who proposed a set of closed-form equations for typical cold-formed steel
sections from regression analysis of FSM and GBT results. Such equations allow direct
and accurate computation of the effect of local buckling and have been incorporated for
practical structural design (ABNT 14762, 2010). No similar work for orthotropic
materials is known.
An alternative approach to full-section buckling analysis was introduced by
BLEICH (1952), who proposed considering plates as individual elements rotationally
restrained by their adjacent plates. For uniaxially compressed infinitely long plates
made of isotropic linear elastic material, the equation for local buckling critical stress is
presented in the following form:
( ) [ ]qpb
tEFcr 2
112
2
2
2
+
−=
νπ
l (3.14)
where E is the modulus of elasticity;ν is the Poisson’s ratio; b is the plate width;t is the
plate thickness; and p and q account for restraint conditions along the long edges. These
latter parameters are functions of the coefficient of elastic restraint and can be obtained
from a transcendental equation resulting from the exact stability condition. Bleich
referred to the final term in brackets as the buckling coefficient, k.
Recently, this approach was adopted by other authors to develop comprehensive
closed-form equations for the local buckling of sections made of orthotropic material.
Early attempts to solve the problems of rotationally restrained long plates were made by
BANK and YIN (1996) and QIAO et al. (2001), but these involved numerical analysis
and explicit expressions were not achieved. The first set of closed-form equations for
this problem was obtained by KOLLAR (2002, 2003). The form of these equations is
similar to Eq. 3.14, but includes terms related to orthotropic properties, as can be seen in
Table 3.4. KOLLAR (2003) compared the resulting equations with numerical and
experimental results available in literature and concluded that, for I-sections, the
equations overestimate the web buckling by up to 6.5% and underestimate the flange
buckling by no more than 5.5%, which is quite reasonable for a design approach. The
accuracy and sensitivity of Kollar’s equations were investigated by McCARTHY and
BANK (2010), who concluded that they correlate better with experimental results than
equations derived based on simply-supported flange-web junctions. QIAO and SHAN
35
(2005) also developed explicit expressions for GFRP sections and compared the results
with those obtained from Finite Element Method (FEM) analyzes, concluding that their
equations overestimate the critical load by up to 4.7%. Although the equations
presented by KOLLAR (2003) and QIAO and SHAN (2005) are very accurate, they
require independent calculations for each plate of the cross section (e.g. web and flanges
of an I-section) and the calculation of section-specific parameters such as the elastic
restraint coefficient, substantially increase calculation effort. Kollar’s equations were
included in an appendix of the Italian standard (CNR, 2008) to specifically address the
local buckling of I-sections.
Table 3.4 – Critical stresses for uniaxially compressed orthotropic long plates with
various edge conditions (KOLLAR, 2003).
Edge conditions Critical Stress, Fcrℓ
( )( )
++++11
66122
11
222
112 2
62.02139.412D
DD
D
D
tb
D ξξπ
+
11
66
11
222
112
12'
3
D
D
D
D
tb
D
ζπ
for 1.17ζ’ (D11/D22)0.5> 1
+
−
11
66
11
22
22
112
112
12'12.47D
D
D
D
D
D
tb
D ζπ for
1.17ζ’ (D11/D22)0.5 ≤ 1
( )( )[ ] ( )
+−+−−+−
ζνηνηπ
12.41
1711611.15
11
222
112 K
KD
D
tb
D
for K ≤ 1
( )( )[ ]νηνηπ −−+− 1611.1511
222
112
KD
D
tb
D for K> 1
where: k is the spring constant and GIt is the torsional stiffness of restraining stiffener; ν = D12/(2D66+D12); K = (2D66+D12)/(D11D22)
0.5; ζ’ = D22b/(GIt); ξ = 1/[1+10D22/(kb)] for plates rotationally restrained by springs; ξ = 1/[1+0.61ζ’1.2] for plates rotationally restrained by stiffeners; η = 1/[1+(7.22-3.55ν)D22/(kb)]0.5.
The simplest approach often used to estimate the local buckling critical load of
orthotropic sections is obtained by assuming simply-supported conditions at the plate
intersections junction; this approach has been adopted by most available standards
36
(GRAY, 1984, CLARKE, 1996). Solutions for long plates with both longitudinal edges
simply-supported (SS) and one edge free and the other simply-supported (FS) were
obtained by LEKHNITSKII(1968) and are given, respectively, as:
( )[ ]661222112
2
, 222 DDDDtb
F SScr ++= πl (3.15)
tb
DF FScr 2
66,
12=
l (3.16)
or, in terms of the elastic properties:
( ) ( )
−++
−= TLLT
L
LT
L
TLT
L
T
TLLT
fLSScr E
G
E
E
E
E
b
tEF ννν
ννπ
1422112
2,
2
,l (3.17)
( ) ( )
−
−=
= TLLTL
LT
TLLT
LLTFScr E
G
b
tE
b
tGF νν
πννπ
148
1124
2
222
,l (3.18)
As can be seen, Equations 3.17 and 3.18 can be written in a form similar to
Equation 3.14, with the final terms in brackets representing the buckling coefficients, k.
In the following sections, the development of closed-form equations for local
buckling computation of typical GFRP pultruded shapes, taking into consideration full
section buckling, is addressed.
3.4.2. Assumptions
To develop equations for the local buckling critical load of typical GFRP
sections, the following assumptions are made:
a. classic plate theory hypotheses apply, which include out-of-plane deflections
much greater than the plate thickness and negligible transverse shear effects;
b. plate intersections are free to rotate, but not to translate, i.e., the junction
lines remain straight in the buckled configuration;
c. the member is considered to be infinitely long, i.e., multiple half-wave
critical lengths can be accommodated within the column and the influence of
end conditions is negligible. This assumption leads to an expression that
37
reflects a lower bound critical stress, independent of the actual column
length, which is a typical and appropriate design hypothesis;
d. deflected-shapes are approximated by double-sinusoidal and polynomial-
sinusoidal functions addressing the end conditions and the compatibility of
rotation between constituent plates. The assumed buckled shapes for typical
GFRP sections are presented in Figure 3.4.
(a) Angle (b) Channel
(c) I-section (d) Rectangular Tube
Figure 3.4 – Sections studied, local axis definition and buckling modes assumed (in
each view, only in-plane wall modes are shown).
e. thickness, t, and material properties are assumed to be uniform throughout
the cross-section. This assumption is usually valid for GFRP pultruded
sections, although it is acknowledged that in some cases cross-sections are
intentionally manufactured to be non-homogeneous (McCARTHY and
BANK, 2010). For instance, SONTI and BARBERO (1996) reported ratios
of flange-to-web longitudinal and shear moduli of an I-section of 1.12 and
0.84, respectively. Nevertheless, for design purposes, the lesser or average
properties can be adopted and limitations on the differences of constituent
plates properties can be established for the application of the equations.
Similar equations for specified non-homogeneous sections can be obtained
38
using the method described although the homogeneity assumption is made in
this study in order to obtain a simple expression.
3.4.3. Formulation by Energy Method
The Rayleigh Quotient energy method (BAZANT and CEDOLIN, 1991) is used
to determine local buckling critical loads. In this method, the strain energy in bending,
U, and the work produced by compressive load, T, are calculated for an assumed
approximate deflected shape, w, and the critical load per unit of width, Ncrℓ, can be
obtained from the condition of neutral equilibrium (U = T). Given the expressions for U
and T for orthotropic plates (LEISSA, 1985), the critical load for a plate assembly
subject to unidirectional compression in the x-direction is given as:
∑∫∫
∑∫∫
=
=
∂∂
∂∂∂
+∂∂
∂∂
+
∂∂
+
∂∂
=n
i S i
i
n
i S ii
ii,
i
i
i
ii,
i
ii,
i
ii,
cr
dxdyx
w
dxdyyx
wD
y
w
x
wD
y
wD
x
wD
N
i
i
1
2
1
22
662
2
2
2
12
2
2
2
22
2
2
2
11 42
l (3.19)
in which i is an index referring to each of n constituent plates; S is the plate surface area;
x and y are the local longitudinal and transverse in-plane axes, respectively, as defined
in Figure 3.4 for typical cross section shapes; and D11, D22, D12 and D66 are plate
stiffness parameters defined in Eqs. 3.10 to 3.13.
Equation 3.19 can also be written in terms of the elastic properties, as follows:
( )( )
∑∫∫
∑∫∫
=
=
∂∂
∂∂∂
−
+∂∂
∂∂+
∂∂+
∂∂
−
=n
i S i
i
n
i S
ii
i
i,L
i,LTi,TLi,LT
i
i
i
i
i,L
i,Ti,LT
i
i
i,L
i,T
i
i
i,TLi,LT
ii,L
cr
dxdyx
w
dxdy
yx
w
E
G
y
w
x
w
E
E
y
w
E
E
x
w
tE
N
i
i
1
2
122
2
2
2
22
2
22
2
2
3
14
2
112νν
ν
νν
l (3.20)
The closer the selected deflected-shape function approximates reality, the more
accurate the result. However, this may lead to complex expressions and it is the
engineer’s role to select a function that leads to both a simple and accurate result. In this
work, double-sinusoidal and polynomial-sinusoidal functions, both addressing the end
39
conditions and the compatibility of rotation between plate elements were adopted. In
general, for each constituent plate, the functions take the form:
( )l/sin)(),( xyfyxw π= (3.21)
in which f(y) is chosen to be either a polynomial or a sinusoidal function; and ℓ is the
half-wave buckling length (Fig. 3.4).
3.4.4. Approximate Deflection Functions
The functions f(y) were selected based on observation of the governing buckling
modes of each cross section studied as shown in Figure 3.4. The following sections
present the functions adopted for webs and flanges of each section, identified with
subscripts w and f, respectively. For consistency of discussion, the section orientation is
considered (see Figure 3.4) such that the ratio bf/bw ≤ 1.0 for angles and rectangular
tubes.
3.4.4.1. Angles
Angles are comprised of two plates each having one edge free and the other
rotationally restrained. In this case, for a fictitious small rotation α applied at the plate
intersection, each plate is assumed to rotate freely with respect to the intersection, which
itself remains straight as shown in the section view in Figure 3.4a. Therefore, the
assumed functions are:
y)y(f)y(f fw α== (3.22)
3.4.4.2. I-shapes and Channels
I-shapes and channel sections are comprised of one web and two flanges, the
former rotationally restrained along both longitudinal edges and the latter rotationally
restrained at the plate intersection with the web only. Applying fictitious small and
opposite rotations α at the plate intersections, the flanges are assumed to rotate freely
with respect to the intersection, which remains straight (Figures 3.4b and 3.4c). The web
bends symmetrically. Assuming that the bending shape can be described by a sinusoidal
shape, the functions adopted for web and flanges, respectively, are:
40
( ) ( )www bybyf /sin/)( ππα= (3.23)
yyf f α=)( (3.24)
3.4.4.3. Rectangular Tubes
Rectangular tubes are comprised of four plates – two flanges and two webs –
each rotationally restrained along both longitudinal edges. Once again, considering
fictitious small rotations α applied at all plate interfaces, the plates bend symmetrically
in cross sectional (Figure 3.4d). Assuming that the bending shape can be described by a
sinusoidal shape, the deflected shape functions for webs and flanges, respectively, are:
( ) ( )www bybyf /sin/)( ππα= (3.25)
( ) ( )fff bybyf /sin/)( ππα= (3.26)
However, the case of a rectangular box is not as simple as the previous cases. In
reality, the flange stiffness is much higher than that of a plate with one of the edges free.
This results in restraint of the web and changes the anticipated curvature. This effect is
more pronounced for lower values of bf/bw. For bf/bw = 0, the deflected shape of the web
approximates that of a single plate with both longitudinal edges clamped. For bf/bw =1
(square tube), the deflected shape of the web approximates that of a plate with simply-
supported edges. Therefore, alternative functions f(y) need to be selected. These are
found by considering the tube section as a system of bars with joints free to rotate, but
held rigidly in space. Unit distributed loads are applied to the bars, as shown in Figure
3.5 and the deflected shape functions determined as follows:
−+
++
−=
wwwww b
y
b
y
b
y
b
yyf 1
11)(
3
ηηη
(3.27)
−+
+
+
−=
fffff b
y
b
y
b
y
b
yyf 1
11
11
1)(
3
4
η
ηηη (3.28)
where η = bf/bw.
41
Figure 3.5 – Cross section of rectangular tube treated as a system of bars subject to unit
distributed loads.
3.4.5. Local Buckling Critical Stress
Using the approximate deflected shapes described in the previous section and
assuming uniform plate thickness and material properties, it is possible to obtain the
strain energy in bending, U, and the work produced by compressive load, T, for each
cross section. Then, from Eq. 3.20, the critical load per unit of width, Ncrℓ, for each
cross section can be obtained. The critical stress, Fcrℓ, is obtained by dividing Ncrℓ by the
plate thickness t and can be generally expressed as:
( )2
,2
112
−=
wTLLT
fLcr b
tEkF
ννπ
l (3.29)
in which k is the shape-specific buckling coefficient. This equation has the same form as
BLEICH’s (1952). Expressions for k are presented in the following sections.
3.4.5.1. Angles
For angles:
( )( ) 11
6632
2
1
112
D
Dbk w
ηπη
+++
=l
(3.30)
Or, in terms of the elastic properties:
42
( )( ) ( )
fL
LTTLLT
w
E
Gbk
,32
2
11
112 ννηπη −
+++
=l
(3.31)
In Figure 3.6, the buckling coefficient k given by Eq. 3.30 (for the case of η =
1.0) is plotted against the ratio ℓ/bw for an isotropic case and for an orthotropic condition
with EL,f/ET,f = 2.0 and EL,f/GLT = 6.5 (relatively typical of pultruded GFRP). It can be
seen that each curve asymptotically trends to a constant as ℓ/bw increases. In fact, if the
half-have length ℓ is much greater than the web width bw, the first term of equation 3.31
becomes negligible. In this case, the critical buckling coefficient, kcr, that leads to the
minimum critical stress is:
( )( )( )
fL
LTTLLTcr E
Gk
,32
11
112 ννηπη −
++= (3.32)
Figure 3.6 - Buckling coefficient, k, vs. ℓ/bw for angles.
3.4.5.2. I-shapes
For I-sections:
( ) ( )[ ]
( )3/1
4142/32
11
66122
22
2
ηπη
+++++
=D
DDbLDbk ww
l (3.33)
Or, in terms of the elastic properties:
0.00
0.50
1.00
1.50
2.00
2.50
3.00
0.00 1.00 2.00 3.00 4.00 5.00
Bu
cklin
g co
effic
ien
t k
ℓ/bw
Isotropic
Orthotropic EL,f/ET,f = 2.0; EL,f/GLT = 6.5
νLT = 0.32η = 1.0
43
( ) ( )( )
( )3/1
14142/
32
,,
,2
,
,
2
ηπ
ννην
+
−+++
+
= fL
LTTLLT
fL
fTLTw
fL
fT
wE
G
E
Eb
E
E
bk
l
l (3.34)
In Figure 3.7, the buckling coefficient k given by Eq. 3.34 (for the case of η =
1.0) is plotted against the ratio ℓ/bw for an isotropic case and for an orthotropic condition
with EL,f/ET,f = 2.0 and EL,f/GLT = 6.5. As can be seen, there is a critical half-wave
length, ℓcr, that leads to the minimum value of k, kcr. Minimizing Eq. 3.34 with respect
to ℓ, ℓcr is obtained as:
( ) 4/132
4/1
,
, 3/1 ηπ+
=
fL
fTwcr E
Ebl (3.35)
Figure 3.7 - Buckling coefficient, k, vs. ℓ/bw for I-sections.
Substituting ℓ = ℓcr in Eq. 3.34 and manipulating, the critical buckling
coefficient, kcr, is obtained as follows:
( )( )
( )3/1
14142
3/1
232
,,
,
,
,
32 ηπ
ννην
ηπ +
−+++
+= fL
LTTLLT
fL
fTLT
fL
fTcr
E
G
E
E
E
Ek (3.36)
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
5.00
0.00 0.50 1.00 1.50 2.00 2.50
Bu
cklin
g c
oef
ficie
nt k
ℓ/bw
νLT = 0.32η = 1.0
(ℓcr/bw ; kcr)
(ℓcr/bw ; kcr)
44
3.4.5.3. Channels
The expression obtained for k for channel sections is similar to that obtained for
I-shaped sections (Eq. 3.34), with the only difference being that the denominator of the
second term is 1+4π2η
3/3, instead of 1+π2η
3/3:
( ) ( )( )
( )3/41
14142/
32
,,
,2
,
,
2
ηπ
ννην
+
−+++
+
= fL
LTTLLT
fL
fTLTw
fL
fT
wE
G
E
Eb
E
E
bk
l
l (3.37)
In Figure 3.8, the buckling coefficient k given by Eq. 3.37 (for the case of η =
1.0) is plotted against the ratio ℓ/bw for an isotropic case and for an orthotropic condition
with EL,f/ET,f = 2.0 and EL,f/GLT = 6.5. As observed for I-sections, there is a critical
buckling coefficient, kcr, associated with a critical half-wave length, ℓcr. Minimizing k
with respect to ℓ in Eq. 3.37:
( ) 4/132
4/1
,
, 3/41 ηπ+
=
fL
fTwcr E
Ebl (3.38)
and,
( )( )
( )3/41
14142
3/41
232
,,
,
,
,
32 ηπ
ννην
ηπ +
−+++
+= fL
LTTLLT
fL
fTLT
fL
fTcr
E
G
E
E
E
Ek (3.39)
Figure 3.8 - Buckling coefficient, k, vs. ℓ/bw for channels.
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
5.00
0.00 0.50 1.00 1.50 2.00 2.50 3.00
Buc
klin
g c
oef
ficie
nt k
ℓ/bw
Isotropic
Orthotropic EL,f/ET,f = 2.0 EL,f/GLT = 6.5
νLT = 0.32η = 1.0
(ℓcr/bw ; kcr)
(ℓcr/bw ; kcr)
45
3.4.5.4. Rectangular Tubes
For tubes, two cases were considered: i) k obtained by adopting f(y) as
sinusoidal deflected shapes (Eqs. 3.25 and 3.26); and ii) adopting f(y) as polynomial
deflected shapes (Eqs. 3.27 and 3.28). For the first case, the buckling coefficient is:
( ) [ ]
( )ηηηη
+−+++
=23
11
66122
22
242/
D
DDbDbk ww l
l (3.40)
or:
( ) ( )
( )ηηη
νννη
+−
−++
+
=23
,,
,2
,
,
2142/
fL
LTTLLT
fL
fTLTw
fL
fT
wE
G
E
Eb
E
E
bk
l
l (3.41)
Figure 3.9 shows a plot of the buckling coefficient k against ℓ/bw for the case of
η = 1.0 and, once again, a minimum value for k can be observed. It is interesting to note
that, for the isotropic condition, the critical length occurs when ℓ/bw = 1.0 and the
buckling coefficient is equal to 4.0, the well-known coefficient for simply-supported
plates (ZIEMIAN, 2010). Minimizing Eq. 3.41 with respect to ℓ:
( ) 4/123
4/1
,
, ηηη +−
=
fL
fTwcr E
Ebl (3.42)
Figure 3.9 - Buckling coefficient, k, vs. ℓ/bw for rectangular tubes (Equation 3.41).
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
0.00 0.50 1.00 1.50 2.00 2.50 3.00
Buc
klin
g co
effic
ient
k
ℓ/bw
νLT = 0.32η = 1.0
(ℓcr/bw ; kcr)
(ℓcr/bw ; kcr)
46
Substituting ℓ = ℓcr in Eq. 3.41, the critical buckling coefficient, kcr, is obtained
as:
( )
( )
( )ηηη
νννη
ηηη +−
−+
++−
=23
,,
,
,
,
23
1422 fL
LTTLLT
fL
fTLT
fL
fTcr
E
G
E
E
E
Ek (3.43)
Adopting the polynomial approximations for f(y) given by Eqs. 3.27 and 3.28,
the buckling coefficient assumes a form similar to Eq. 3.41, although the terms
involving η are significantly more complex:
( ) [ ]
11
66122
,
,2 42/
D
DDBbE
EA
bk
wfL
fT
w
+++
=l
l (3.44)
in which parameters A and B depend on the box geometry as follows:
1102112634263122110
50430245040302450412345678910
356
4 +++−+−+−++++++=
ηηηηηηηηηηηηηη
πA (3.45)
1102112634263122110
12962108442084210961212345678910
2345678
2 +++−+−+−++++++++++=
ηηηηηηηηηηηηηηηηηη
πB (3.46)
Following the same procedure adopted to obtain Eqs. 3.42 and 3.43, the critical
length and buckling coefficient are obtained as:
4/14/1
,
, 1
=
AE
Eb
fL
fTwcrl (3.47)
( )
−++=
fL
LTTLLT
fL
fTLT
fL
fTcr E
G
E
EB
E
EAk
,,
,
,
, 1422 ννν (3.48)
3.4.6. Results and Comparison with Finite Strip Method
Figures 3.11 to 3.14 present plots of the critical buckling coefficients, kcr,
described in the previous sections, versus the flange-to-web width ratios (bf/bw) for
angles (Eq. 3.32), I-shapes (Eq. 3.36), channels (Eq. 3.39) and rectangular-tubes (Eqs.
3.43 and 3.48) sections, respectively. The isotropic condition is considered in addition
47
to three different cases of orthotropy: i) EL,f/ET,f = 1.0 and EL,f/GLT = 3.5; ii) EL,f/ET,f =
1.5 and EL,f/GLT = 5.0; and iii) EL,f/ET,f = 2.0 and EL,f/GLT = 6.5, representing a range of
pultruded GFRP properties. The curves are compared to those obtained using the finite
strip method, implemented using CUFSM 4.04 (LI and SCHAFER, 2010). By default,
deflected shapes similar to those represented by Equation 3.21 are used in CUFSM,
with third-order polynomial functions adopted for f(y). Preliminary analysis showed that
discretizing each constituent plate into two strips was adequate to obtain accurate
results, i.e., results converged to those obtained with more refined configurations. Thus,
angles, I-sections, channels and tubes were modeled with 4, 10, 6 and 8 strips,
respectively. Figure 3.10 presents the interface of the program with the 8-strip
discretization adopted for a rectangular tube section. Additionally, the critical buckling
coefficients for the simply-supported condition at plate intersections recommended by
current standards and guidelines (GRAY, 1984, CLARKE, 1996) for the third
orthotropic case, EL,f/ET,f = 2.0 and EL,f/GLT = 6.5, are also plotted. These were obtained
according to Equations 3.17 and 3.18. The following sections provide discussion of
each section studied.
Figure 3.10 – Interface of program CUFSM 4.04 and discretization adopted for a
rectangular tube section.
48
3.4.6.1. Angles
As can be seen on Figure 3.11, excellent agreement is achieved between the
proposed expression for angles (Eq. 3.32) and the finite strip method, regardless of the
ratio bf/bw. For the range of typically commercially-available pultruded angles (0.33
<bf/bw< 1.0; see Table 2.2), the maximum difference observed was 1%. It is important
to note that, for bf/bw = 0, the critical buckling coefficient approximates the case of a
plate with one edge simply-supported and the other free, i.e., kcr = (12/π2)(1–
νLTνTL)GLT/EL,f. For the isotropic condition with ν=νLT=νTL = 0.30 and GLT =
EL,f/[2(1+ν)], kcr = 0.426, which is approximately equal to the exact coefficient used for
steel (k = 0.425) (ZIEMIAN, 2010). The basic assumption of a simply-supported
condition at the plate intersection (Eq. 3.18) is shown to be very conservative, with a
difference of up to 25% from the proposed approximate solution and those obtained
using the FSM.
Figure 3.11 – Critical buckling coefficient, kcr, vs. η = bf/bw for angle sections.
3.4.6.2. I-Shapes
Good agreement is achieved between the proposed expression for I-shapes (Eq.
3.36) and the finite strip method, as can be seen in Figure 3.12. For the range of
typically commercially-available pultruded I-shapes (0.45 <bf/bw< 1.05; see Table 2.2),
Eq. 3.36 leads to results up to 6% higher than those obtained from the finite strip
method. The critical buckling coefficient approximates that of a plate with both edges
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
Crit
ica
l Bu
cklin
g co
effic
ien
t kcr
η = bf/bw
Eq. 3.32Eq. 3.18FSM
Typical angles
νLT = 0.32
Orthotropic 3
49
simply-supported for bf/bw = 0. For the isotropic condition with bf/bw = 0, Eq. 3.36 leads
to kcr = 4.0, the well-known coefficient for long simply-supported plates (ZIEMIAN,
2010). The basic assumption of simply-supported conditions at plate intersections (Eqs.
3.17 and 3.18) leads to results up to 49% lower than those obtained using the FSM for
the range of typical section geometries.
Figure 3.12 – Critical buckling coefficient, kcr, vs. η = bf/bw for I-sections.
3.4.6.3. Channels
Similar to I-sections,good agreement is achieved for channels, as can be seen in
Figure 3.13. For the range of typically commercially-available pultruded channels (0.15
<bf/bw< 0.53; see Table 2.2), the critical buckling coefficients obtained from Eq. 3.39
are up to 10% higher than those obtained using the finite strip method. This difference
tends to increase for higher values of bf/bw. Once more, it is noted that the critical
buckling coefficient approximates that of a plate with both edges simply-supported for
bf/bw = 0, with kcr = 4.0 for the isotropic condition. The assumption of simply-supported
conditions at plate intersections (Eqs. 3.17 and 3.18) leads to results as low as 55% of
those obtained using the FSM for the range of typical section geometries.
0.00
1.00
2.00
3.00
4.00
5.00
6.00
0.00 0.20 0.40 0.60 0.80 1.00 1.20
Crit
ica
l Bu
cklin
g c
oef
ficie
nt k
cr
η = bf/bw
Eq. 3.36Eqs. 3.17/3.18FSM
Typical I-sections
νLT = 0.32
50
Figure 3.13 – Critical buckling coefficient, kcr, vs. η = bf/bw for channel sections.
3.4.6.4. Rectangular Tubes
As shown in Figure 3.14, excellent agreement was achieved using the
polynomial-sinusoidal function as an approximate deflected shape (Eq. 3.48), but poor
agreement was achieved using the double-sinusoidal function (Eq. 3.43). For the range
of typically commercially-available pultruded rectangular tubes (0.25 <bf/bw< 1.0; see
Table 2.2), the critical buckling coefficients obtained using Eq. 3.43 are up to 20%
higher than those obtained from the finite strip method, while those obtained from Eq.
3.48 are only 3% higher. Nonetheless, for square tubes (bf/bw = 1), both equations and
FSM calculations converge to the same solutions. Once again, the critical buckling
coefficient approximates that of a plate with both edges simply-supported for bf/bw = 1,
with k = 4.0 for the isotropic condition. Using Equation 3.48 for bf/bw = 0 and an
isotropic material, the approximations represent the condition of a plate with both edges
clamped and kcr = 6.98, which is the approximately equal to the exact coefficient used
for steel (k = 6.97) (ZIEMIAN, 2010). The assumption of simply-supported conditions
at plate intersections (Eqs. 3.17 and 3.18) leads to results up to 32% lower than those
obtained using the FSM for the range of typical section geometries.
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
5.00
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
Crit
ica
l Bu
cklin
g c
oef
ficie
nt k
cr
η = bf/bw
Eq. 3.39FSMEqs. 3.17/3.18
Typical channel sectionsνLT = 0.32
51
Figure 3.14 – Critical buckling coefficient, kcr, vs. η = bf/bw for rectangular tube
sections.
Equations 3.45 and 3.46, while accurate, are cumbersome. To allow for practical
engineering use, these expressions can be simplified to account only for the range of
typically available box sections. Parameters A and B can be approximated by simpler
functions of η obtained from regression over the range of interest, as follows:
η4.24.3 −=A for 0.25 ≤ η ≤ 1.0 (3.49)
213.132.181.0 ηη −+=B for 0.25 ≤ η ≤ 1.0 (3.50)
Substituting Eqs. 3.49 and 3.50 into Eq. 3.44, the critical local buckling
coefficient for rectangular tubes with 0.25 ≤ η ≤ 1.0 becomes:
( ) ( )
−+−++−=
L
LTTLLT
L
TLT
L
Tcr E
G
E
E...
E
E..k νννηηη 14213132181042432 2 (3.51)
Using Equation 3.51 results in critical buckling coefficients that are only 3%
greater than the comparable FSM solutions for the range of typical sections.
3.4.6.5. Summary
A summary of the resulting equations, their range of applicability in terms of
bf/bw and the observed relative differences of the various values of kcr from those
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
Crit
ica
l Bu
cklin
g c
oef
ficie
nt k
cr
η = bf/bw
Eq. 3.43Eq. 3.48Eq. 3.17FSM
Typical rectangular tube sections
νLT = 0.32
Orthotropic 3
52
obtained using finite strip method (FSM) is presented in Table 3.5. The observed
differences between the proposed equations and those promulgated in current standards
and guidelines are also shown. This summary assumes that plate thickness, t, and
isotropy, ET/EL, are the same for all plates comprising the cross section.
Table 3.5 – Summary of results.
Section kcr Applicable bf/bwa
Greatest relative difference from FSM solution in range of
applicability Proposed
Eqs. Eqs. 3.15 and 3.16
Angles Eq. 3.30 0.33 < η=bf/bw < 1.0 1% 25% I-shapes Eq. 3.36 0.45 < η=bf/bw < 1.05 6% 49% Channels Eq. 3.39 0.15 < η=bf/bw < 0.53 10% 55%
Rectangular Boxes
Eq. 3.43 0.25 < η=bf/bw < 1.0
20% 32% Eq. 3.48 (or
Eq. 3.51) 3%
a Applicable bf/bw refers to the geometry of typical commercially-available sections.
53
4. Real GFRP Columns and Plates
4.1. Overview
Real columns (RC) are those affected by geometric imperfections and material
imperfections. In such cases, the applied compressive force produces second-order
bending moments that result in increased lateral deflections and therefore reduced
critical capacities compared to those described for ‘perfect’ columns in the previous
chapter. To analyze this problem and obtain the non-linear load-lateral deflection
response, beam-column theory (TIMOSHENKO and GERE, 1961) has to be used. A
similar behavior is observed for real plates (RP).
For a solid cross-section column with initial out-of-straightness and linear elastic
behavior, the section is subjected to combined compression-flexure stresses resulting
from the action of compressive force and second order bending moments; failure is
expected when the compressive stress reaches the material strength. If the column has a
thin-walled section comprised of perfect plates (RC-PP condition), failure occurs when
the compressive stress in a certain plate equals its local buckling critical stress. Finally,
real columns comprised of real plates (RC-RP) have their compressive capacity affected
by second-order effects on both the column and constituent plates; in this case failure
occurs when the compressive stress at the most compressed face of a certain plate
equals the material strength.
As mentioned in Chapter 3, RC-RP constitutes a coupled buckling problem and
may lead to significant reduction of load-carrying capacity (BATISTA, 2004). The
degree of interaction between these buckling modes is dependent on the initial
imperfections, both of the entire column and the constituent plates, and can be
accurately obtained from a non-linear analysis considering initial imperfections and
post-buckling behavior.
In this chapter, an approximate method to determine a lower bound compressive
strength equation for the RC-RP problem based on available second-order theories for
54
columns and plates is proposed. Plate and column slenderness – useful parameters for
the subsequent discussion – are respectively defined as:
lcr
cLp F
F ,=λ (4.1)
crg
crcL
crg
PPc F
FF
F
F },min{ , l==λ (4.2)
in which FL,c is the material crushing strength, defined in section 3.2; Fcrg is the global
buckling critical stress, defined in section 3.3; Fcrℓ is the local buckling critical stress,
defined in section 3.4; and FPP is the perfect plate compressive strength, defined as the
lesser of FL,c and Fcrℓ.
Columns can be classified based on their plate and column slenderness defined
by Equations 4.1 and 4.2, respectively, as presented at Table 4.1. Ranges of
slendernesses are presented as guidance and will be used in subsequent discussion.
Table 4.1 also indicates the expected failure modes for each slenderness combination.
Table 4.1. – Classification of columns and plates and expected failure modes.
λc λp
Short Intermediate Long (λc ≤ 0.7) (0.7 <λc< 1.3) (λc ≥ 1.3)
Compact Crushing
Interaction between crushing and global
buckling Global buckling
(λp ≤ 0.7)
Intermediate Interaction between crushing and local
buckling
Interaction between crushing, local and global bucklings
Global buckling (0.7 <λp< 1.3)
Slender Local buckling
Interaction between local and global
bucklings Global buckling
(λp ≥ 1.3)
4.2. Factors Affecting Actual Behavior
The behavior of real columns is affected by geometric imperfections and
material particularities. The most important factors are described in the following
subsections:
55
4.2.1. Geometric Imperfections
Geometric imperfections are related to manufacture and assembly tolerances. In
columns, two types of geometric imperfections are usually considered:
a. Initial out-of-straightness or ‘crookedness’ (Figure 4.1a) usually depends on
the manufacturing process (and may also result from poor storage practices)
and is related to member length. For pin-ended columns, it is typically
assumed that the initial crookedness takes the shape of a half sine wave with
amplitude δ0 at midheight (Figure 4.1a). An applied compression load,
therefore induces second order bending moments along the length of the
column. Estimated values are given by manufacturers as a fraction of the
column length, typically δ0<L/500.
(a) out-of-straightness in columns (b) out-of-flatness in plates
Figure 4.1 – Some geometric imperfections in columns and plates related to
manufacture.
b. Initial eccentricity can be associated with several factors such as the
deviation of the vertical axis of the column due to dimensional and assembly
tolerances (bolt holes, imperfect contact, etc.). Unlike initial crookedness
which is largely dependent on column length, initial eccentricity is more a
function of section size. As occurs in the case of out-of-straightness, the
column experiences a second order moment associated with the eccentricity,
e0, of the applied load.
Geometric imperfections of plates can also be divided in two groups:
c. Intrinsic imperfection of plates comprising the cross section affect the
constitutive relationships assumed for these plates (GODOY, 1996). The
56
most important parameter is the variation of plate thickness. In design, this is
addressed by considering an average thickness and manufacturer tolerances.
The misalignment of fiber layers is another example of an intrinsic
imperfection (see Section 2.3), in this case leading to variation of the
flexural modulus of elasticity.
d. Geometric imperfection of plates (Figure 4.1b) is analogous to column out-
of-straightness; the most important factor being plate out-of-flatness.
Measuring the initial deflected surface is difficult and impractical for design
purposes. Typically, a double sine wave surface with the same shape as the
buckling mode is assumed. The out-of-flatness amplitude is usually provided
by the manufacturer as a fraction of the plate width, typically ∆0<b/125.
4.2.2. Material Behavior and Imperfections
The most important material behavior and imperfections are:
a. Material stress-strain behavior of GFRP differs from that of steel columns
(from which most of our understanding of real columns come). Significantly,
a plastic hinge will not form when the compressed GFRP column deflects
laterally and no redistribution of internal stress occurs;
b. Highly orthotropic behavior of pultruded GFRP makes the material weaker
in the transverse direction affecting even ‘elastic’ redistribution of internal
stress. Additionally, tensile, compressive and flexural strengths may be
different in the longitudinal direction and, therefore, the mechanisms of
collapse are not always the same. For I-sections, for example, tearing failure
at the flange-to-web junction is often a governing failure mode (TURVEY
and ZHANG, 2006);
c. Residual stresses result from the pultrusion process which involves elevated
temperature curing and subsequent cooling of the profile. Two types of
residual stress may be present:
i. Micro-mechanical stress results from the mismatch of thermal properties
between fiber and matrix: the matrix experiences residual tensile stresses
while fibers are subjected to residual compression. The internal forces
resulting from this effect balance each other. This effect results in micro-
defects, although it is assumed that the effects of these are captured by the
57
experimentally determined material strength. A complete review of this
subject is provided by PARLEVLIET et al. (2007);
ii. Macro-mechanical stress results from the effects of uneven cooling of
the cross section. This effect depends on section shape, thickness,
temperature and thermal and elastic properties. No research was found
in the literature regarding this effect in pultruded FRP sections. As an
attempt to obtain an order of magnitude for this effect, one can compare
it with the hot-rolling process for carbon steel. In the pultrusion process,
the curing temperature is about 200°C, which is about 20 to 30% the
temperature involved in hot-rolling steel. Additionally, the modulus of
elasticity of GFRP is about 10% that of steel, and the coefficient of
thermal expansion is nearly the same. Assuming that residual stresses
are proportional to temperature and modulus of elasticity, it can be
estimated that residual stresses due to uneven cooling in a GFRP
pultruded profile will be about 2 to 3% of those exhibited in carbon
steel. Thus it is expected that macro-mechanical residual thermal
stresses have little impact on real column performance. In the absence
of stress redistribution (brittle material), this effect is intrinsically
captured if the material compressive strength is obtained from a full-
section compressive test.
4.2.3. Post-Buckling Behavior of Plates
In general, for isotropic materials, after the plate buckling load is reached,
moderate to large in-plane transverse deflections accompany the out-of-plane buckling
deflections. Membrane stretching of the middle surface of the plate accompanies the
transverse deflections and the plate can continue to carry additonal load while remaining
stable. For composite plates, however, the post-buckling reserve strength is less
pronounced than for metallic plates (LEISSA, 1985). Post-buckling of GFRP I-sections
was observed by TOMBLIN and BARBERO (1994) and TURVEY and ZHANG
(2006), who tested stub columns and obtained ultimate strengths greater than the local
buckling critical stresses. However, the TOMBLIN and BARBERO (1994) point out
that “…in most cases, permanent damage of the section occurred via internal cracks,
delamination, etc., as was evident when the column was reloaded”. A theoretical review
58
of post-buckling behavior of orthotropic plates is presented by BLOOM and COFFIN
(2000).
4.2.4. Other Parameters
Other parameters affecting GFRP real column behavior and strength include the
presence of holes and notches, moisture absorption and temperature variation. These
parameters may be controlled through design and will not be discussed further here.
4.3. Strength Curve
4.3.1. Literature Review
The first attempt to develop a compressive strength curve for GFRP columns
was made by BARBERO and TOMBLIN(1994), who observed that the failure loads of
I-shaped intermediate columns were as much as 25% lower than theoretical critical
loads, demonstrating that coupled buckling significantly affects capacity. The authors
proposed an empirical equation for the ultimate strength of the column, Fu, based on
ZAHN’s (1992) universal equation for wood design and determined the appropriate
interaction constant, c = 0.84, to fit this equation to experimental results. This
coefficient was later revised to c = 0.65 by BARBERO and DE VIVO (1999) in order to
include additional experimental data. BARBERO (2000) suggested use of the finite
element method to investigate imperfection sensitivity of intermediate columns and to
obtain a refined interaction constant. The equation, which is adopted by the Italian code
(CNR, 2008) with c = 0.65, is given as:
2
222 1
2
11
2
11
c
cc
cr
u
cc
/
c
/
F
F
λλλ
−
+−
+=
l
(4.3)
Alternative compressive strength equations were proposed by PUENTE et al.
(2006) and SEAGATITH and SRIBOONLUE(1999). The former’s proposal was based
on DUTHEIL’s (1961) expression with adjusted coefficients to fit experimental data
available in the literature and obtained from their own tests on long circular and
rectangular tubes:
59
−+= 01
1122
.,minF
F
ccr
u
λφφγl
(4.4)
in which ϕ = 0.5[1 + 0.12(λc2 – 0.25) + λc
2] and γ = 1.2.
Importantly, both Equations 4.3 and 4.4 assume slender plates; i.e.: Fcrℓ<<Fc,L.
SEANGATITH and SRIBOONLUE (1999) proposed adopting the Euler equation for
long columns and a linear relationship experimentally adjusted for intermediate and
short columns:
−≤=r
L
)r/L(
EF c,L
u 6
5225
2
2π (MPa) (4.5)
where EL,c is the compressive modulus of elasticity in the longitudinal direction; L is the
column length; and r is the radius of gyration.
As can be seen in Equations 4.3 to 4.5, current strength curve equations
proposed in the literature adopt empirical coefficients to fit experimental results. These
expressions do not explicitly consider the influence of plate slenderness.
4.3.2. Plate Strength
It is known that a compressed perfect plate (PP) may fail either by crushing of
material or by local buckling. For a real plate (RP), imperfections need to be considered
as they may significantly reduce the ultimate load capacity. In an attempt to obtain an
analytical approach for determining the plate strength curve, the case of a uniaxial
compressed plate with all edges simply-supported is considered (Figure 4.2) and the
following assumptions are made:
a. classic plate theory hypotheses apply, which include out-of-plane deflections
much greater than the thickness and negligible transverse shear effects;
b. stretching in the middle plane is neglected and no post-buckling strength is
experienced;
c. the material behavior is assumed to be linear elastic;
d. mechanical properties and thickness are assumed to be uniform throughout
the plate;
60
e. residual stresses are negligible;
f. the initially deflected surface, w0, is described by a double sinusoidal
function:
( ) ( )b/ysinL/xmsinw ππ00 ∆= (4.6)
where, as shown in Figure 4.2, L and b are the plate length and width,
respectively; m is the number of half-waves in the longitudinal direction (x);
∆0 is the out-of-flatness amplitude; and x and y are the in-plane axes.
Figure 4.2 – Uniaxial compressed simply-supported plate.
For long plates (L>>b), m ≈ L/ℓcr, where ℓcr is the critical half-wave length.
Therefore, the out-of-flatness can be rewritten as:
( ) ( )byxw cr /sin/sin00 ππ l∆= (4.7)
g. Failure criterion is governed by maximum stress. Assuming that the plate
strength under combined axial-bending in the longitudinal direction, FL,cf, is
described by a linear interaction between pure compressive and flexural
strengths:
cxcL
cLfLfLcfL F
FFFF ,
,
,,,, σ
−−= (4.8)
61
Where FL,f and FL,c are the pure flexural and compressive strengths,
respectively; and σx,c= Nx/t is the compressive stress induced by the applied
compressive load per unit of width acting in the x direction, Nx;
For this problem, the governing differential equation in terms of the additional
out-of-plane deflection w is (adapted from BLOOM and COFFIN, 2000):
( ) ( )02
2
4
4
2222
4
66124
4
11 22 wwx
Ny
wD
yx
wDD
x
wD x +
∂∂=
∂∂+
∂∂∂++
∂∂ (4.9)
where D11, D12, D22 and D66 are the plate stiffness parameters defined by Equations 3.10
to 3.13.
From classical plate theory (CPT) (TIMOSHENKO and GERE, 1961), it can be
demonstrated that the additional out-of-plane deflection,w, is proportional to the
initially deflected surface:
cxcr
cx
Fww
,
,0 σ
σ−
=l
(4.10)
The bending moment per unit of width in the longitudinal direction, Mx, is given
as:
cxcr
cx
cr
x FD
bD
bw
dy
wD
dx
wDM
,
,122
2
112
2
02
2
122
2
11 σσπ−
+=
∂+∂−=ll
(4.11)
For a plate with all edges simply-supported:
( ) 4/11122 / DDbcr =l (4.12)
and,
( )661222112
2
422 DDDDtb
Fcr ++= πl (4.13)
In which case, Equation 4.11 can be rewritten as:
62
lcrcx
cxx FDDDD
DDDDtwM
/1422
/
,
,
66122211
122211110 σ
σ−
+++
= (4.14)
The maximum bending moment occurs at x = ℓcr/2 and y = b/2, where w0 = ∆0
and the maximum compressive stress due to combined axial and bending moment in the
section, σx,cf, is given as:
lcrc,x
c,xc,x
max,xc,xcf,x F/DDDD
DD/DD
t
b
bt
M
σσ
σσσ−
+++
∆+=+=
14226
6
66122211
1222111102
(4.15)
Considering that compressive stress σx,c is increased to a limit a Fu,L until failure
occurs in the longitudinal direction (σx,cf = FL,cf) and substituting Equation 4.8 into 4.15:
lcrL,u
L,uL,uL,u
c,L
c,Lf,L
f,L F/F
F
DDDD
DD/DD
t
b
bFF
F
FFF
−
+++
∆+=
−−
14226
66122211
122211110 (4.16)
Manipulating Equation 4.16:
lcrLu
Lp
Lu
cL
FFF
F
/11
,
,
,
,
−+=
α (4.17)
in which:
+++
∆=
66122211
12221111
,
,0,
422
/6
DDDD
DDDD
F
F
t
b
b fL
cLLpα (4.18)
Equation 4.17 can be rearranged and written as a quadratic equation in terms of
non-dimensional parameters:
( ) 011 ,2
,2
,2 =+++− LppLpLpp χλαχλ (4.19)
where λp is the relative plate slenderness defined by Equation 4.1 and χp,L= Fu,L/FL,c is
the normalized plate strength in the longitudinal direction.
Equation 4.19 can be solved in terms of the normalized plate strength χp,L,
resulting in:
63
( )
2
222,
2,
, 2
411
p
ppLppLp
Lp λλλαλα
χ−++−++
= (4.20)
Although Equation 4.20 has been developed for a plate under uniaxial
compression with all edges simply-supported, it has the same form as the typical
column strength curves developed for steel, such as those defined by Perry-Robertson
(ZIEMIAN, 2010) and DUTHEIL (1961), with the imperfection factor in the
longitudinal direction αp,L defined as a function of geometry, initial imperfection, plate
stiffness and strength parameters (Eq. 4.18). For other boundary conditions, different
expressions for αp,L are obtained, although these always depend on the same parameters.
For αp,L = 0, the problem simplifies to the perfect plate condition, with χp,L = 1.0 for λp ≤
1.0 and χp,L = 1/λp2 for λp> 1.0.
Similar investigation can be done for a failure in the transverse direction. In this
case, the failure is associated with a state of pure transverse bending, resulting from the
second order effects caused by the compressive load. Using similar procedures, it can be
shown that:
2,
,
1
pTp
Tp λαχ
+= (4.21)
where χp,T = Fu,T/FL,c is the normalized plate strength in the transverse direction and αp,T
is the transverse direction plate imperfection factor which assumes a form similar to Eq.
4.18, but as a function of the transverse flexural strength, FT,f:
+++
∆=
66122211
22221112
,
,0,
422
/6
DDDD
DDDD
F
F
t
b
b fT
cLTpα (4.22)
It can be noted from Equations 4.18 and 4.22 that the factors αp,L and αp,T are
functions of b/t, which is related to the plate slenderness λp. Therefore, these equations
can be manipulated and rewritten in terms of the elastic properties and plate slenderness,
as follows:
( )
( )( ) p
TLLTLTfTLTfTfLcL
fTLTfTfLfL
fL
cLLp
GEEEF
EEEE
F
F
bλ
ννν
να
−++
+∆=
1422
/6
,,,,
2
,,,,
,
,0, (4.23)
64
( )
( )( ) p
TLLTLTfTLTfTfLcL
fTfTfLfTLT
fT
cLTp
GEEEF
EEEE
F
F
bλ
νννν
α−++
+∆=
1422
/6
,,,,
2
,,,,
,
,0, (4.24)
Failure can occur in longitudinal or transverse directions and the ultimate
compressive stress is the lesser of Fu,L and Fu,T. Figure 4.3 shows a plot of the
normalized plate strength for both directions plotted against the plate slenderness for
typical pultruded GFRP material properties and different degrees of plate imperfections.
From Figure 4.3, it can be seen that the adoption of manufacturer’s out-of-flatness
amplitude (∆0 = b/125) leads to a reduction of capacity up to 40% compared to the
perfect plate condition. It can also be noted that failure in transverse direction is not
critical. Actually, longitudinal and transverse capacities are relatively close for slender
plates and, in this case, failure may be initiated in either mode or a combination of both.
Combination modes were reported by TURVEY and ZHANG (2006) and observed in
the present study (Chapter 6) for I-sections with slender flanges, where longitudinal and
transverse cracks were observed.
Figure 4.3 – Normalized plate strength vs. plate slenderness.
The plate capacity reduction factor ρp, representing the residual capacity with
respect to the ideal condition, shown in Figure 4.4 plotted against the plate slenderness
for the same representative case as shown in Figure 4.3, can be defined as the ratio of
RP-to-PP normalized strengths:
0.00
0.20
0.40
0.60
0.80
1.00
1.20
0.00 0.50 1.00 1.50 2.00 2.50 3.00
No
rma
lized
Pla
te S
treng
th
Plate Slenderness, λp
∆0/b= 0 (perfect plate)
EL,f = 20 GPaET,f = 10 GPaGLT = 4 GPaνLT = 0.28FL,c = 240 MPaFL,c/FL,f = 0.65FL,c/FT,f = 1.60
χp,L (Eq. 4.20)
χp,T(Eq. 4.21)
65
0
,
p
Lpp χ
χρ = (4.25)
where χp0 is the PP normalized strength; that is: χp0 = 1.0 for λp ≤ 1.0 and χp0 = 1/λp2 for
λp> 1.0.
Figure 4.4 – Plate reduction factor (Eq. 4.25) vs. plate slenderness.
4.3.3. Column Strength
As mentioned previously, a PC-PP may fail either by crushing (very short
columns), local buckling (short columns) or global buckling (long columns). For a RC-
RP, column and plate imperfections must be considered. Accurately predicting the
actual behavior of such members is not an easy task and can be done only by carrying
out a sophisticated non-linear analysis. Nonetheless, an attempt to obtain an
approximate strength curve for flexural buckling of a pin-supported column is
developed here. For this case, the following assumptions are made:
a. beam column theory hypotheses apply, which include negligible transverse
shear effects;
b. the column bending stiffness remains constant until failure;
c. the initial deflected shape (out-of-straightness), v0, is described by a
sinusoidal function:
0.00
0.20
0.40
0.60
0.80
1.00
1.20
0.00 0.50 1.00 1.50 2.00 2.50 3.00
Pla
te R
edu
ctio
n Fa
cto
r
Plate Slenderness, λp
∆0/b= 0 (perfect plate)
EL,f = 20 GPaET,f = 10 GPaGLT = 4 GPaνLT = 0.28FL,c = 240 MPaFL,c/FL,f = 0.65FL,c/FT,f = 1.60
66
( )Lxv /sin00 πδ= (4.26)
where x is the longitudinal axis of a column of length L and δ0 is the
amplitude of the initial out-of-straightness.
d. Failure occurs when the compressive stress at a certain plate comprising the
cross-section reaches its ultimate compressive stress, Fu,L = χp,LFL,c = ρpFPP,
in which FPP is the PP strength, given as the lesser of Fc,L and Fcrℓ.
The solution for the final ordinates of the lateral deflection curve v(x) of a pin-
supported beam-column under these conditions is well known and can be approximated
as (TIMOSHENKO and GERE, 1961):
+
−=
L
x
F
e
Fxv
crg
c
crgc
πσπ
δσ
sin4
/1
1)( 0
0 (4.27)
where σc = P/A is the compressive stress induced by the applied compressive load P
acting in the longitudinal direction; A is the cross-sectional area of the column; e0 is the
eccentricity of the compression load, assumed equal at both ends; and Fcrg is the global
buckling critical stress. For the flexural buckling case considered, Eq. 3.6 is suggested,
although the formulation in this section does not consider shear effects.
From Equation 4.27, the maximum deflection at the mid-height (x = L/2), is:
+
−=
crg
c
crgc F
e
Fv
σπ
δσ
00max
4
/1
1 (4.28)
The maximum corresponding bending moment at the mid-height is:
+
−+
−= 00max
41
/1δσσ
πσσ
crg
c
crg
c
crgc
c
FFe
F
AM (4.29)
In Equation 4.29, the term in parenthesis that modifies e0 ranges from 1.0 to
approximately 4/π, as σc varies from 0 to Fcrg. Conservatively, setting this term equal to
4/π results in:
67
+−
= 00max
4
/1δ
πσσ
eF
AM
crgc
c (4.30)
Resulting in the maximum compressive stress at an outer fiber of the column:
( )
+−
+= 00max,
4
/1δ
πσσσσ e
S
A
Fcrgc
ccc (4.31)
where S is the elastic section modulus.
Compressive stress σc is increased to a limit of Fu when failure occurs (σc,max =
ρpFPP):
( )
+−
+= 00
4
/1δ
πρ e
S
A
FF
FFF
crgu
uuPPp (4.32)
Equation 4.32 can be rearranged and written as a quadratic equation in terms of
non-dimensional parameters:
( ) 011 222 =+++− cpcccc χρλαχλ (4.33)
in which λc is the relative column slenderness defined by Equation 4.2; χc= Fu/FPP is the
normalized column strength in the longitudinal direction; and αc is the imperfection
factor given as:
S
Aec
+= 00
4 δπ
α (4.34)
Equation 4.33 can be solved in terms of the normalized column strength χc,
resulting in:
( )
2
2222
2
411
p
pppcpcc
c λλρλαρλα
χ−++−++
= (4.35)
The equation developed has, once again, the same form as the typical column
strength curves, but with an imperfection factor αc based on geometric imperfections
and cross sectional properties (Eq. 4.34). The additional parameter, ρp, accounts for
plate slenderness, λp. Although the expression has been developed for a column with
68
negligible shear deformation and with a section comprised by perfect compact plates, it
provides a reasonable representation of real behavior, which can be far from that of a
perfect column. It is important to note that, just as for plates, Equation 4.35 simplifies to
the PC-PP condition when αc = 0 and ρp = 1.
The column capacity reduction factor ρc, representing the residual capacity with
respect to the ideal condition (PC-PP), can be defined as the ratio of (RC-RP)-to-(PC-
PP) normalized strengths:
0c
cc χ
χρ = (4.36)
where χc0 is the PC-PP normalized strength; that is: χc0 = 1.0 for λc ≤ 1.0 and χc0 = 1/λc2
for λc> 1.0.
Figure 4.5 shows examples of representative column strength curves taking into
account the interaction between plate and column behavior. Figure 4.6 shows the
reduction factors plotted against the relative column slenderness. It can be seen that
maximum reduction of capacity is occurs for λc = 1. Additionally, it can be noted that,
for a relative slenderness λc = 0, the normalized strength is less than 1.0, reflecting the
initial eccentricity and its resulting first order bending moment. Since GFRP is a brittle
material, no redistribution of stresses can be assumed and the capacity may be reduced
even for very short members.
Figure 4.5 – Column strength curves (Eq. 4.35)vs. relative column slenderness.
0.00
0.20
0.40
0.60
0.80
1.00
1.20
0.00 0.50 1.00 1.50 2.00 2.50 3.00
No
rma
lized
Co
lum
n S
treng
th χ c
Relative Column Slenderness λc
PC-PP
ρp = 1.0 (RC-PP)
αc = 0.15
ρp = 0.70ρp = 0.85
69
Figure 4.6 – Column reduction factors (Eq. 4.36) vs. relative column slenderness.
A contour plot of ρc as a function of λc and λp is presented in Figure 4.7. As can
be seen, the maximum reduction of capacity is observed for columns having λc = λp =
1.0
Figure 4.7 – Contour plot of the column reduction factor ρc(Eq. 4.36) as a function of λc
and λp.
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
0.00 0.50 1.00 1.50 2.00 2.50 3.00
Co
lum
n R
educ
tion
Fact
or ρ c
Relative Column Slenderness λc
ρp = 1.0 (RC-PP)
αc = 0.15
ρp = 0.70ρp = 0.85
0.00
0.50
1.00
1.50
2.00
0.00 0.50 1.00 1.50 2.00 2.50 3.00
Rel
ativ
e P
late
Sle
nder
ness
λ p
Relative Column Slenderness λc
αp = 0.28 λpαc = 0.15
0.50
70
4.3.4. Summary
In this chapter, a compressive strength equation with explicit coefficients for
plate and column imperfection was presented (Eq. 4.35). Different from existing
methods (BARBERO and TOMBLIN, 1994, PUENTE et al., 2006, SEANGATITH and
SRIBOONLUE, 1999), the proposed expression accounts for plate and column
slendernesses, λp and λc. In Eq. 4.35, λp is implicitly considered in the plate reduction
factor ρp, defined in Eq. 4.25.
71
5. Experimental Program 1: Compressive
Strength of GFRP Square Tubes
5.1. Literature Review
The majority of previous studies on the compressive strength of GFRP columns
have addressed I-sections comprised of slender flange plates, from which the following
conclusions have been obtained: (i) short column compressive strength is governed by
local buckling (YOON, 1993, TOMBLIN and BARBERO, 1994, TURVEY and
ZHANG, 2006); (ii) long columns fail by global buckling (BARBERO and TOMBLIN,
1993, ZUREICK and SCOTT, 1997); and (iii) intermediate columns fail by interaction
between local and global buckling (BARBERO and TOMBLIN, 1994, BARBERO et
al., 1999b, LANE and MOTTRAM, 2002). Important investigations on the strength of
square tubes, where the walls are either compact or intermediate, were conducted by
ZUREICK and SCOTT (1997), SEANGATITH and SRIBOONLUE (1999) and
HASHEM and YUAN (2001). These authors concluded that short columns may fail
either by crushing or local buckling while long columns fail by global buckling.
Although crushing vs. global buckling interaction was observed by SEANGATITH and
SRIBOONLUE (1999) for intermediate columns, coupled buckling was not reported by
any of these authors. Additionally, none of the works previously cited establish a
relationship between wall slenderness and failure modes, which seems to be very
important to the comprehension of the overall column behavior. The definitions of
short, intermediate and long columns and slender, intermediate and compact plates are
those defined previously in Table 4.1.
5.2. Experimental Program
In this chapter, an experimental campaign addressing the compressive strength
of square tubes is described. In this work, square tube columns having different lengths
and sections, resulting in a range of combinations of global and sectional slenderness,
were tested. Five square tube section geometries were considered: 25.4x3.2, 50.8x3.2,
72
76.2x6.4, 88.9x6.4 and 102x6.4, in which the section designation refers to the nominal
outside square dimension and wall thickness in units of mm. All pultruded GFRP
sections were made with a non-fire retardant polyester resin and were provided by the
same manufacturer.The experimental program was carried out in four stages: cross-
section geometry measurement, material characterization, stub tests and column
concentric compression tests, as described in the following sections.
5.3. Cross-Section Geometry Measurement
Since the section dimensions given by the manufacturer are nominal, the real
values must be obtained. The external dimensions and walls thicknesses of each section
were measured with a digital caliper and cataloged. Average dimensions (± one
standard deviation) of the tubes of each size are presented in Table 5.1. As can be seen,
no significant variation was observed for the section widths although some marked
variation was observed for wall thicknesses. The most significant variation between
measured and nominal thickness was observed for the 50.8x3.2 (measured thickness
was 90% of nominal) and the highest coefficient of variation was observed for the
25.4x3.2 sections (9.1%). It is important to note that significant thickness variations
were observed within individual cross sections. For instance, a single 25.4x3.2 section
presented wall thicknesses ranging from 3.70 mm to 2.74 mm. In this work, apparent
imperfections along length were considered (section 5.9) and, therefore, out-of-
straightness along length was not measured.
Table 5.1 – Actual and average dimensions of square tubes tested.
Section measured calculated
Width (mm) Thickness
(mm) Area (mm²)
Moment of Inertia (mm4)
25.4x3.2 25.5 ± 0.2 3.19 ± 0.29 285 24,123 50.8x3.2 50.7 ± 0.2 2.88 ± 0.10 551 210,835 76.2x6.4 75.9 ± 0.2 6.23 ± 0.28 1736 1,415,909 88.9x6.4 88.8 ± 0.3 6.20 ± 0.19 2048 2,341,876 102x6.4 100.9 ± 0.1 6.33 ± 0.21 2395 3,587,989
5.4. Material Characterization
In order to obtain a good correlation between theory and experiments, the
material properties were experimentally obtained. Average values (showing one
standard deviation) are given in Table 5.2. These values were obtained using tests
73
described in the following sections. The properties were obtained from a limited number
of tests on samples having both 3.2 and 6.4 mm wall thicknesses. Fiber content was
estimated by weighing 50 mm long specimens of each section and assuming fiber and
resin densities according to table 2.1. Since different sections were found to have
different fiber contents (Table 5.2), the properties for the sections not tested were
extrapolated by multiplying the ratio of the experimental to predicted properties for the
sections tested (Table 4) by the predicted properties for the desired section, determined
according to DAVALOS et al. (1996) for roving and CSM volumes being 70 and 30%
of the fiber content, respectively.
5.4.1. Longitudinal Modulus of Elasticity (EL,c) and Compressive Strength (FL,c)
Square tube specimens having lengths of 75 mm and 125 mm were extracted
from tubes with sections of 25.4x3.2 mm and 76.2x6.4 mm, respectively. The sample
lengths were arbitrarily chosen to have 2 to 3 times the tube width. Three specimens
were extracted from both tube sizes. To ensure uniform contact with the test plattens, a
diamond blade saw was used to cut the samples and the ends were finished parallel
using a bench-sanding machine. The full-tube coupons were instrumented with four
strain gages (one per face) and tested in concentric compression, as shown in Figure 5.1.
The tests were conducted under displacement control at a loading rate of 0.15 mm/min
until failure. All data were recorded automatically.
Figure 5.1 – Compression test of 125 mm long 76.2x6.4 mm specimen to determine
longitudinal modulus of elasticity and material strength.
74
The measured values of compressive strength, FL,c, are given in Table 5.2. A
large coefficient of variation was observed for the 25.4x3.2 tubes (19.2%). This is
expected since ultimate compressive behavior is dependent on many manufacturing
factors such as the presence of small defects, quality of fiber wet-out and fiber
alignment, as well as testing factors such as ensuring uniform contact. The 76.2x6.4
section presented a lower coefficient of variation (7.2%), indicating that small defects
are less critical to the performance of the larger section. Brittle failure was observed for
all tubes tested. The measured values of longitudinal compressive modulus, EL,c,
obtained as the slope of the compressive stress-strain curves, are also given in Table 5.2.
The average ultimate compressive strain, εuL,c, was 0.010. Stress-strain curves are
presented in Figure 5.2.
Table 5.2– Material properties (± one standard deviation) obtained from tests and
adopted for sections not tested.
25.4x3.2 50.8x3.2 76.2x6.4 88.9x6.4 102x6.4 Vf 0.36 0.39 0.49 0.47 0.44
EL,c (MPa) 22,100 ± 500 1 23,700 31,100 ± 1200 1 29,900 28,200 EL,f (MPa) 10,800 11,500 ± 300 1 25,000 ± 3000 1 24,000 22,700 ET,f (MPa) 9900 10,500 ± 600 1 13,500 ± 1400 1 12,800 12,000 GLT (MPa) 2100 2200 ± 100 1 2700 ± 100 1 2600 2400 FL,c (MPa) 220 ± 40 1 240 330 ± 30 1 320 300
(1) Obtained from tests; all other values were estimated from extrapolation, as described previously.
Figure 5.2 – Stress-strain curves for full section compression tests.
0
50
100
150
200
250
300
350
400
0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014
Axi
al S
tress
(MP
a)
Axial Strain
Sample 1 (EL,c = 21,506 MPa)Sample 2 (EL,c = 22,151 MPa)Sample 3 (EL,c = 22,736 MPa)Sample 4 (EL,c = 30,915 MPa)Sample 5 (EL,c = 32,647 MPa)Sample 6 (EL,c = 29,737 MPa)
25.4x3.2
50.8x6.4
EL,cEL,c
EL,cEL,c
EL,c
EL,c
75
5.4.2. In-Plane Shear Modulus (GLT)
This property was determined according to the method described by BANK
(1989) and is based on Timoshenko beam theory. For a three-point bending test (Figure
5.3), the response may be determined as:
LT
s
c,L G
n
r
L
EPL
Aw +
=2
12
14 (5.1)
where w is the mid-span deflection; P is the point load at the middle of the span; L is the
tested span; A is the cross-sectional area; ns is the shear form factor; and r is the radius
of gyration.
(a) Schematic representation of the test.
(b) View of the test.
Figure 5.3 – Timoshenko beam test adopted for determination of in-plane shear
modulus.
76
Tests were performed on two samples of each of 50.8x3.2 and 76.2x6.4 tubes
over spans having values of (L/r)2 varying from 100 to 500. The deflections were
measured with a dial gage (0.025 mm precision) at applied loads of 2670 N and 5340 N,
respectively, for the 3.2 and 6.4 mm thick tubes. These values were selected in order to
avoid localized crushing. Additionally, since the shear stress-strain relationship is not
linear (ZUREICK and SCOTT, 1997), large shear strains must be avoided in order to
ensure that the shear modulus obtained experimentally represents the initial (elastic)
value. To obtain the real Timoshenko beam deflection, the mid-span deflection used in
the calculation was the net deflection, accounting for material settlement over the
supports. Support deflection was measured with a digital caliper, taking a distance of 3
times the wall thickness above the support as reference as shown in Figure 5.3. For each
specimen, (4Aw/PL) was plotted against (L/r)2, a straight line fit was made and the shear
modulus obtained from the inverse of the intersection coefficient (BANK, 1989). The
measured values of shear modulus, GLT, are given in Table 5.2 whereas
(4Aw/PL)vs.(L/r)2 curves are presented in Figure 5.4.
Figure 5.4 – (4Aw/PL) versus (L/r)2 curves obtained from Timoshenko beam tests.
5.4.3. Longitudinal Flexural Modulus of Elasticity (EL,f)
Three-point bending tests were performed on two 254 mm long longitudinal
strip samples extracted from each of the 50.8x3.2 and 76.2x6.4 sections. Strip widths
(b) of 40 mm and 57 mm were used for the 3.2 mm and 6.4 mm thick (t) samples,
0
500
1000
1500
2000
2500
3000
3500
0 100 200 300 400 500 600 700
4Aw
/(P
L) (
mm
²/N
) x1
0-6
(L/r)2
Sample 1 (GLT = 2015 MPa)Sample 2 (GLT = 2210 MPa)Sample 3 (GLT = 2870 MPa)Sample 4 (GLT = 2625 MPa)
25.4x3.2
50.8x6.4
GLtGLt
GLtGLt
77
respectively and a 203 mm test span, L, was adopted for both cases. Free weights were
used and the mid-span deflections, δ, were measured with a dial gage (0.025 mm
precision). The test set-up is presented in Figure 5.5. P was plotted against (4δbt3)/L3
and a straight line fit was made for each tested specimen, as shown in Figure 5.6, the
longitudinal flexural modulus being its slope.
The measured values of longitudinal modulus of elasticity for plate bending, EL,f,
are given in Table 5.2. A greater variation in EL,f is observed for the thicker samples
(11.9% for the 6.4 mm coupons versus 2.7% for the 3.2 mm coupons). This is believed
to be associated with deviations in the position of the inner and outer roving layers for
6.4 mm thick sections. Since the positions of these rovings play an important role in the
bending stiffness, a small misalignment can substantially affect the apparent stiffness. In
the thinner 3.2 mm sections, there is only one layer of roving located at the middle of
the section and, therefore, a small misalignment does not affect the stiffness as
significantly.
Figure 5.5 – Three-point bending test adopted for determination of longitudinal
flexural modulus.
78
a) 25.4 x 3.2
b) 50.8 x 3.2
Figure 5.6 – P versus (4δbt3)/L3 curves obtained from a strip three-point bending
tests.
5.4.4. Transverse Flexural Modulus of Elasticity (ET,f)
The transverse flexural modulus of elasticity was obtained from a non-standard
transverse bending test. Tests were performed on two 102 mm long channel-shaped
samples extracted from each of 50.8x3.2 and 76.2x6.4 tube specimens. The channel
shape was made by cutting one wall off the tube. The bottom flange was clamped
0
5
10
15
20
25
30
35
40
45
50
0.000 0.001 0.002 0.003 0.004 0.005
Loa
d P
(N)
4δbt3/L3
Sample 1 (EL,f = 11,830 MPa)
Sample 2 (EL,f = 11,195 MPa)25.4x3.2
EL,f
EL,f
0
50
100
150
200
250
300
0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014
Loa
d P
(N)
4δbt3/L3
Sample 3 (EL,f = 28,015 MPa)
Sample 4 (EL,f = 22,052 MPa)50.8x6.4
EL,f
EL,f
79
rigidly to a flat surface and eccentric loads (Pe) were applied at the top flange as shown
in Figure 5.7. Free weights were used and strains were measured in the constant
moment region which results in the vertical web using strain gages applied to both faces
of the web to measure the flexural strain. ET,f was obtained as the slope of the straight
line fit of Pe/Sp vs. (ε1–ε2)/2 (Figure 5.8) where ε1 and ε2are the recorded strains on the
outer and inner faces of the web wall and Sp is the elastic section modulus of the web
wall section tested (i.e.: 102t2/6; where 102 mm is the length of the specimen). The
measured values of transverse modulus of elasticity for plate bending, ET.f, are given in
Table 5.2. The major Poisson’s ratio, νLT, was assumed to be equal to 0.32 for all
sections.
(a) Schematic representation of the test.
(b) View of the test.
Figure 5.7 – Transverse plate bending test set-up on 76.2 x 6.4 mm tube.
80
a) 25.4 x 3.2
b) 50.8 x 3.2
Figure 5.8 – Pe/Sp versus (ε1–ε2)/2 (Flexural stress vs. strain) obtained from
transverse modulus tests.
5.5. Stub Column Tests
Stub column tests were performed in order to gather information about local
buckling and post-buckling behavior of GFRP square tube columns, as well as to
estimate typical initial wall imperfections. Three 254 mm long 50.8x3.2 tubes were
tested under concentric compression in a 600 kN-capacity servo-hydraulic universal
0
2
4
6
8
10
12
0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012
Fle
xura
l Str
ess σ
= P
e/S
p(N
)
Flexural Strain ε = (ε1 - ε2)/2
Sample 1 (ET,f = 11,084 MPa)
Sample 2 (ET,f = 9916 MPa)25.4x3.2
ET,f
ET,f
0
1
2
3
4
5
6
7
0.0000 0.0001 0.0002 0.0003 0.0004 0.0005
Fle
xura
l Stre
ss σ
= P
e/S
p(N
)
Flexural Strain ε = (ε1 - ε2)/2
Sample 3 (ET,f = 14,904 MPa)
Sample 4 (ET,f = 12,185 MPa)25.4x3.2
ET,f
ET,f
81
testing machine. The length was chosen to be five times the width in order to be
sufficiently long to minimize any influence of the boundary conditions. The ends of
each stub were machined parallel. Each stub was instrumented with two electrical
resistance strain gages carefully positioned on the inside and outside of the wall at the
center of one of the walls. Tweezers, wooden rods and a rigid metallic ruler were used
to manipulate and glue the strain gage on the inner wall and no sign of debonding was
observed. The position of the gages was chosen such that they coincided with the
theoretical peak of the buckled shape. Tests were conducted under displacement control
at a rate of 0.10 mm/min. Figure 5.9 shows a view from one of the tests in which local
buckling is clearly evident on both visible faces. Stub column test results are described
in Sections 5.8 and 5.9.
Figure 5.9 –Stub column tests on 50.8 x 3.2 mm tube.
5.6. Column Compression Tests
The primary activity of the present study is a series of column buckling tests.
Square tubes with the following nominal sections were tested: 25.4x3.2 mm, 50.8x3.2
mm, 76.2x6.4 mm, 88.9x6.4 mm and 102x6.4 mm. In all, 74 tests were conducted, with
82
column lengths determined in order to cover a range of slenderness values intended to
result in long, intermediate and short columns (see Table 4.1). Table 5.3 presents the
distribution of tests according to the combination of plate and column slenderness.
Figure 5.10 presents a summary of recent experimental works on GFRP tubes and I-
sections plotted according to the combination of column (λc; Eq. 4.2) and plate (λp; Eq.
4.1) slenderness. When material compressive strength was not reported by the author, it
was assumed based on the manufacturer of the section. The present study (solid circles
in Figure 5.10) will significantly augment the available data over all column lengths and
considers sections primarily having intermediate plate slenderness – an important region
with little available data.
Table 5.3 – Distribution of compression tests according to combination of slenderness.
λc λp
Short (λc ≤ 0.7)
Intermediate (0.7 < λc < 1.3)
Long (λc ≥ 1.3)
Compact (λp ≤ 0.7)
0 12 7
Intermediate (0.7 < λp < 1.3)
11 18 12
Slender (λp ≥ 1.3)
5 7 2
Figure 5.10 – Summary of recent experimental works plotted according to plate and
columns slenderness (specimens are hollow tubes except as indicated).
0.0
0.5
1.0
1.5
2.0
2.5
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Rel
ativ
e P
late
Sle
nder
ness
λ p
Relative Column Slenderness λc
I-shaped sectionsZureick and Scott (1997)Seangatith and Sriboonlue (1999)Hashem and Yuan (2001)Seangatith (2004)Present study
short columns
interm.columns long columns
com
pact
pl
ates
inte
rm.
plat
essl
ende
r pl
ates
83
A diamond blade saw was used to cut the tubes to the specified lengths and the
ends were machined parallel with a belt sander as required. The columns were tested in
concentric compression on a 900kN-capacity servo-hydraulic controlled universal test
machine. Rollers were located at both columns ends in order to reproduce a one-
dimensional pinned-pinned condition, i.e., an effective length coefficient Ke close to 1.0.
The tests were conducted under displacement control at an axial loading rate of
approximately 750 µε/min (based on initial specimen length). The tests were ended
when either crushing was observed or the lateral deflection at midheight, δ, exceeded
L/50. In the latter case, the columns were unloaded at the same rate and retested in their
other principal direction over a different length, if no damage was observed. The
columns were instrumented with 3 draw-wire transducers (DWT) to measure lateral
deflections. DWTs were positioned at the section corner and mid-wall, at midheight, in
order to capture the effects of both local and global buckling. A third DWT was
positioned arbitrarily for control in some cases, e.g. to obtain comparative Southwell
plots or capture local buckling in another region. For 25.4x3.2 and 76.2x6.4 sections,
only one DWT was used at the mid-wall at midheight. The wires were connected to the
column using a glued nut; thus no holes were drilled. All data were recorded
automatically. Figure 5.11 shows the test layout and Figure 5.12 shows the roller
fixture.
84
(a) Layout of compression test, as designed. (b) View of test layout, as built.
(c) Position of DWTs.
Figure 5.11 – Column compression test.
85
(a) Schematic representation. (b) View of extremity, as built.
Figure 5.12– Roller detail.
The geometry of the rollers affects the column loading and end conditions. Due
to the presence of the roller, when the column deflects laterally, the line of action of the
compressive force P shifts, which causes a small restoring eccentricity, as shown in
Figure 5.13. This condition is similar to the compression of bars with rounded ends and
is accounted for by means of an appropriately corrected effective length coefficient, Ke,
obtained from the following (TIMOSHENKO and GERE, 1961):
R
L
Ktan
K ee 222−=
ππ (5.2)
in which L is the column length, including the end bearing plates; and R is radius of the
roller.
86
Figure 5.13– Effect of rounded supports over column behavior.
Table 5.4 presents the resulting effective length coefficients, Ke, for increasing
values of L/2R. As L/2R approaches 0, Ke tends to 0.5 (perfect fixed end conditions)
and, for large values of L/2R, Ke tends to 1.0 (perfect pinned end conditions). Thus the
effective K-factor is reduced for shorter columns in the same test set-up. This effect is
accounted for in the test results presented in the following sections.
Table 5.4 – Effective length coefficients Ke based on the ratio L/2R.
L/2R 0 1 2 4 6 8 Ke 0.50 0.561 0.639 0.769 0.838 0.877
L/2R 10 20 30 40 50 1000 Ke 0.901 0.950 0.967 0.975 0.980 0.999
5.7. Experimental Results
Table 5.5 provides the theoretical material compressive strengths, FL,c, and the
theoretical local buckling critical stresses, Fcrℓ, for each section tested. The five cross
sections represent compact, intermediate and slender sections based on the classification
previously described. Since sections having b/t < 13 were tested, transverse shear
effects may be important (BARBERO, 2011). In this case, the local buckling critical
stress was determined according to KIM et al. (2009):
87
( ) ( )
++
++=
65
8485
65
84 2
2
2
2
2
2 tGG
D
b
tGG
D
btb
DF TZLZTZLZcr
πππl
(5.3)
in which b and t are the plate width and thickness, respectively; D = D11 + 2(D12 + 2D66)
+ D22 is the plate flexural stiffness assuming that the half-wave length is approximately
equal to the plate width; GLZ and GTZ are the longitudinal and transverse out-of-plane
shear moduli, assumed to be equal to the in-plane shear modulus, GLT. For
longitudinally reinforced orthotropic plates, the half-wave length is a little longer than
the plate width, which results in a different expression for D. However, for the range of
material properties considered in this study, the difference resulting from the previous
definition is less than 3%.
Table 5.5 – Classification of cross sections tested.
Sections FL,c (MPa) Fcrℓ (MPa) λp Classification Table 5.2 Eq. 5.3 Eq. 4.1 -
25.4x3.2 220 547 0.64 Compact 50.8x3.2 240 118 1.43 Slender 76.2x6.4 330 365 0.95 Intermediate 88.9x6.4 320 253 1.12 Intermediate 101.6x6.4 300 191 1.26 Intermediate
Complete column test reports are presented in Appendix A. Table 5.6 provides a
summary of the 74 tests conducted. Experimental critical (Fcr) and ultimate stresses (Fu)
are presented, as well as the theoretical prediction for a perfect column (Fcrg,F),
determined according to Eq. 3.7. For the theoretical prediction of Fcrg,F, the effective
lengths were considered as the specimen length plus the end plates thicknesses,
multiplied by the effective length factor, Ke, calculated to take into account the restoring
eccentricity caused by the roller geometry, as described by Eq. 5.2. The Southwell
Method (SOUTHWELL, 1932), in which the critical stress is determined as the slope of
the δ versus δ/σc curve (the so-called Southwell plot where δ is the lateral deflection at
midheight and σc is the applied compressive stress) was used to obtain the experimental
critical loads. Examples of Southwell plots are provided in the following sections. Low-
load non-linearities of Southwell plots (SPENCER and WALKER, 1975) were
observed, especially for shorter columns. In such cases, the initial points were rejected
and only the linear portion of the plot was considered for straight line fitting. For short
columns, Southwell plots were also used to verify local buckling, as proposed by
TOMBLIN and BARBERO (1994). Table 5.6 presents the failure modes observed for
88
each test and references to subsequent photos showing these. The following
abbreviations are used to describe the failure modes:
CE: crushing at column end;
CM: crushing at column midheight;
GB-CE: global buckling followed by crushing at column end;
GB-CM: global buckling followed by crushing at column midheight;
GB-L/50: global buckling with large lateral deflection (δ>L/50);
LB-CE: local buckling followed by crushing at column end;
LB-CM: local buckling followed by crushing at column midheight;
CB-CE: coupled buckling followed by crushing at column end;
CB-CM: coupled buckling followed by crushing at column midheight.
When the tests were stopped based on midheight deflections exceeding δ>L/50,
the true ultimate stress Fu was not achieved. This value may be greater than the final
stress achieved, FL/50, but less than the critical stress Fcr. These cases are indicated in
Table 5.6 (GB-L/50) and the value of FL/50 is reported in these cases. This value is
necessarily a lower bound for Fu.
Table 5.6– Summary of compression tests.
Theoretical Experimental L(mm) Ke Fcrg,F (MPa) λc Fcr (MPa) Fu (MPa) Failure Mode
(1) Eq. 5.2 Eq. 3.7 Eq. 4.2 Southwell
plot - -
25.4x3.2 tubes 218 0.889 365 0.78 - 220 CE (Fig. 5.18b) 225 0.889 347 0.80 345 203 GB-CE 228 0.889 339 0.81 354 216 GB-CM
251 a 0.902 284 0.89 275 170 GB-CM (Fig. 5.17a) 251 b 0.902 284 0.89 304 197 GB-CE 270 0.908 248 0.95 240 189 GB-CM 277 0.908 238 0.97 257 196 GB-CE 278 0.908 236 0.97 240 181 GB-CM 290 0.914 218 1.01 233 176 GB-CM 298 0.914 208 1.04 224 179 GB-L/50 330 0.921 172 1.14 189 181 GB-L/50 350 0.927 153 1.21 145 136 GB-L/50 400 0.936 118 1.38 109 96 GB-L/50 441 0.943 98 1.51 90 84 GB-L/50 470 0.946 86 1.61 82 64 GB-L/50 489 0.949 80 1.68 74 61 GB-L/50 520 0.951 71 1.78 67 56 GB-L/50
89
Theoretical Experimental L(mm) Ke Fcrg,F (MPa) λc Fcr (MPa) Fu (MPa) Failure Mode
(1) Eq. 5.2 Eq. 3.7 Eq. 4.2 Southwell plot
- -
589 0.956 55 2.01 52 43 GB-L/50 679 0.962 42 2.32 46 40 GB-L/50 (Fig. 5.17b)
50.8x3.2 tubes 304 0.914 706 0.41 - 119 LB-CM 306 0.914 700 0.41 - 119 LB-CM 342 0.927 582 0.45 - 107 LB-CM (Fig. 5.20a) 344 0.927 577 0.45 - 109 LB-CM 380 0.933 492 0.49 - 109 LB-CM 596 0.956 228 0.72 - 109 LB-CM (Fig. 5.21) 597 0.956 228 0.72 - 106 LB-CM 818 0.970 127 0.96 131 92 CB-CM (Fig. 5.20b) 874 0.972 113 1.02 129 81 CB-CM 876 0.972 112 1.03 124 105 CB-CM 901 0.972 106 1.05 119 96 CB-CM 960 0.974 94 1.12 90 78 GB-L/50 1206 0.980 61 1.39 58 58 GB-L/50 (Fig. 5.20c) 1749 0.981 30 2.00 27 25 GB-L/50
76.2x6.4 tubes 390 0.839 1226 0.52 - 235 CE 548 0.877 710 0.68 - 259 CE 644 0.882 551 0.78 - 199 CE (Fig. 5.24a) 688 0.887 493 0.82 - 218 CE 768 0.892 409 0.90 382 259 GB-CE (Fig. 5.24b) 813 0.892 373 0.94 368 250 GB-CE (Fig. 5.23a) 914 0.889 309 1.04 275 185 GB-CE 994 0.902 261 1.13 293 239 GB-CE 1089 0.908 220 1.23 210 162 GB-L/50 1340 0.921 148 1.50 139 113 GB-L/50 1354 0.927 144 1.52 147 133 GB-L/50 1414 0.927 133 1.58 133 109 GB-L/50 1513 0.933 116 1.70 111 96 GB-L/50 1778 0.943 84 1.99 83 68 GB-L/50 (Fig. 5.23b)
88.9x6.4 tubes 405 0.839 1405 0.42 - 168 LB-CM 430 0.839 1298 0.44 - 208 LB-CM (Fig. 5.27a) 582 0.882 795 0.56 - 210 CB-CM 711 0.887 585 0.66 - 212 CB-CM 876 0.889 420 0.78 368 193 CB-CM 1011 0.902 325 0.88 197 177 LB-CM 1025 0.902 317 0.89 284 185 CB-CM 1117 0.908 271 0.97 273 181 CB-CM
1225 0.914 229 1.05 208 166 CB-CM (Fig. 5.27b/5.28)
1413 0.927 174 1.21 163 156 GB-L/50 1544 0.933 146 1.32 137 140 GB-L/50
90
Theoretical Experimental L(mm) Ke Fcrg,F (MPa) λc Fcr (MPa) Fu (MPa) Failure Mode
(1) Eq. 5.2 Eq. 3.7 Eq. 4.2 Southwell plot
- -
1876 0.946 100 1.59 93 90 GB-L/50 2070 0.951 82 1.76 76 79 GB-L/50 (Fig. 5.27c) 2179 0.953 74 1.85 70 71 GB-L/50
101.6x6.4 tubes 378 0.839 1710 0.33 - 162 LB-CM 428 0.839 1459 0.36 - 175 LB-CM 582 0.882 903 0.46 - 164 LB-CM (Fig. 5.30a) 833 0.892 525 0.60 - 167 LB-CM 875 0.889 489 0.62 - 163 LB-CM 1236 0.914 267 0.85 - 161 LB-CM
1394 0.927 212 0.95 181 150 CB-CM (Fig. 5.30b/5.31)
1544 0.933 175 1.04 140 140 CB-CM 1635 0.936 157 1.10 149 120 CB-CM 1802 0.943 130 1.21 112 94 GB-L/50 1994 0.949 107 1.34 93 81 GB-L/50 2040 0.949 102 1.37 83 83 GB-L/50 2247 0.953 85 1.50 74 68 GB-L/50 (Fig. 5.30c)
1 The column lengths include the steel bearing plates: 50 mm is added for 25.4 and 50.8x3.2 tubes; 76
mm is added for the remaining tube sections.
A summary of all experimental results are presented graphically in Figures 5.14
and 5.15, which show the relative critical and ultimate stresses (normalized by the
perfect plate strength FPP, i.e., the minimum of FL,c and Fcrℓ given in Table 5.5),
respectively, plotted against column slenderness λc (given by Equation 4.2). The error
bars in Figure 5.15 represent the ‘distance’ between FL/50 and Fcr for the columns that
exhibited large lateral deflections (true failure stress lies between these values). Eq.
4.35, proposed in the present study, is also plotted in Figure 5.15 for specific
parameters, along with other equations proposed in literature. It can be seen that the
proposed Eq. 4.35 captures the experimental data quite well, while the equations
proposed in the literature are unconservative in many cases.
91
Figure 5.14 – Experimental normalized critical stresses plotted against the
relative column slenderness λc.
Figure 5.15 – Experimental normalized ultimate stresses χc plotted against the
relative column slenderness λc.
A ‘map’ of failure modes and experimental reduction factors according to
column and plate slenderness is shown in Figure 5.16. As expected, intermediate
columns with intermediate plates experienced the greatest reduction of capacity with
respect to the perfect condition, i.e., the lowest reduction factor ρc. It can also be
concluded that local buckling was noticed for sections with λp > 1.0 and global buckling
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
0.75 1.25 1.75 2.25
No
rmal
ized
Crit
ical
Stre
ss F
cr/F
PP
Relative Column Slenderness λc
25.4x3.2 mm50.8x3.2 mm76.2x6.4 mm88.9x6.4 mm101.6x6.4 mm
1/λc²
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0 0.5 1.0 1.5 2.0 2.5
No
rma
lized
Stre
ng
th F
u/F
PP
Relative Column Slenderness λc
25.4x3.2 mm50.8x3.2 mm76.2x6.4 mm88.9x6.4 mm101.6x6.4 mmPerfect ColumnEq. 4.35 (α=0.34; ρp=0.92)Barbero and De Vivo (1999)Puente et al. (2006)
92
with large displacements for λc > 1.0. Short columns experienced either crushing or
local buckling and intermediate columns generally exhibited interaction between
individual failure modes. The following sections provide a discussion of the tests of
each section size.
Figure 5.16 – Map of failure modes and column reduction factors according to
column and plate slenderness.
5.7.1. 25.4x3.2 Tubes
25.4x3.2 mm columns with relative slenderness ranging from λc = 0.78 to 2.32
were tested. Column lateral deflection was observed for all, except the shortest column
(λc = 0.78), which failed by crushing at the column end. Columns with λc ≥ 1.04
exhibited large lateral deflections (δ > L/50) and columns with 0.80 < λc < 1.04 failed
by crushing either close to the mid height or at their ends prior to exhibiting large
deflections. Figure 5.17 shows views from tests on intermediate and long 25.4x3.2
0.0
0.5
1.0
1.5
2.0
0.0 0.5 1.0 1.5 2.0 2.5
Rel
ativ
e P
late
Sle
nde
rnes
s λ p
Relative Column Slenderness λc
Crushing at end
Global buckling with crushing at end
Global buckling with crushing at midheight
Global buckling with δ > L/50
Local buckling with crushing at midheight
Coupled buckling with crushing at midheight
short columnsintermediate
columns long columns
com
pact
pla
tes
inte
rm. p
late
ssl
ende
r pl
ates
― ρ = 0.5 to 0.6― ρ = 0.6 to 0.7― ρ = 0.7 to 0.8― ρ = 0.8 to 0.9― ρ = 0.9 to 1.0
0.55
0.60
0.6
0
0.8
0
0.7
0
αL,c = 0.014αc = 0.34
αp,L = 0.014 αc = 0.34
93
columns and Figure 5.18 shows crushing failure modes observed in intermediate
columns. Figure 5.19 shows the representative stress versus lateral deflection curves and
corresponding Southwell Plots for the two columns shown in Figure 5.17 having critical
stresses, determined as the slope of the Southwell plots, of 275 MPa and 46 MPa,
respectively.
(a) intermediate column (λc = 0.89). (b) long column (λc = 2.32).
Figure 5.17 – Views from tests of 25.4x3.2 columns.
(a) crushing at mid-height (λc = 0.89) b) crushing at column end (λc = 0.78)
Figure 5.18 –Failure modes observed for short and intermediate 25.4x3.2
columns.
94
a) Stress versus deflection curve (λc = 0.89). b) Southwell Plot (λc = 0.89).
c) Stress versus deflection curve (λc = 2.32) d) Southwell Plot (λc = 2.32)
Figure 5.19– Representative stress vs. lateral deflection and Southwell Plots for
25.4x3.2 columns.
5.7.2. 50.8x3.2 Tubes
50.8x3.2 columns with relative slenderness ranging from λc = 0.41 to 2.00 were
tested. Global buckling with large lateral deflections was observed for columns with
slenderness greater than λc = 1.12 (Figure 5.20c). Local buckling was clearly observed
for columns with slenderness less than λc = 0.72 (Figure 5.20a). Local buckling
initiation was quickly followed by development of out-of-plane deflections and a failure
mode governed by kinking at the column at mid height. The combination of the kinking
and out-of-plane deflections resulted in eventual rupture of the tube along the flange-
web junction as shown in Figure 5.21. Coupled buckling was observed for columns with
slenderness ranging from λc = 0.93 to 1.05 (‘ripples’ associated with local buckling of
the compression face of the specimen are clearly evident in Figure 5.20b). Both local
and global buckling modes were clearly observed and the local buckling led to rapid and
catastrophic failure.
Figure 5.22 shows representative load versus lateral deflection curves and
corresponding Southwell plots for short, intermediate and long 50.8x3.2 columns. The
0
20
40
60
80
100
120
140
160
180
0 1 2 3 4 5 6
Stre
ss σ
(MP
a)
Lateral Deflection δ (mm)
y = 275.3x - 3.3
0
1
2
3
4
5
6
0 0.01 0.02 0.03 0.04
δ(m
m)
δ/σ (mm/MPa)
0
5
10
15
20
25
30
35
40
45
0 5 10 15
Stre
ss σ
(MP
a)
Lateral Deflection δ (mm)
y = 46.3x - 2.3
0
2
4
6
8
10
12
14
16
0.0 0.1 0.2 0.3 0.4δ
(mm
)
δ/σ (mm/MPa)
Failure mode: GB-CE
Failure mode: GB-L/50
95
Southwell plots for the intermediate columns clearly show a bilinear behavior – the
hallmark of coupled global-local buckling. The global behavior is affected by the loss of
stiffness resulting from the local buckling. For the case (λc = 0.96) shown in Figure
5.22d, the ‘initial’ critical buckling stress is found to be 131 MPa. When local buckling
initiates, the slope of the δ-δ/σ curve falls to anticipate a critical buckling stress of 77
MPa. This reduction is attributed to the loss of member stiffness associated with local
buckling.
a) short (λc = 0.45) b) intermediate (λc = 0.96) (c) long (λc = 1.39)
Figure 5.20 – Views from tests of 50.8x3.2 columns (local buckling denoted with
arrows).
Figure 5.21 – Failure mode observed for short and intermediate 50.8x3.2 mm
columns (λc = 0.72).
96
a) Stress versus wall deflection curve (λc = 0.41). b) Southwell Plot (λc = 0.41).
c) Stress versus deflection curve (λc = 0.96). d) Southwell Plot (λc = 0.96).
e) Stress versus deflection curve (λc = 1.39). f) Southwell Plot (λc = 1.39).
Figure 5.22– Representative stress vs. lateral deflection and Southwell Plots for
50.8x3.2 columns.
5.7.3. 76.2x6.4 Tubes
76.2x6.4 columns with relative slenderness ranging from λc = 0.52 to 1.99 were
tested. Column lateral deflection was observed for columns with slenderness greater
than λ = 0.95. Columns with slenderness greater than λc = 1.23 exhibited large lateral
deflections (δ>L/50). All columns having λc ≤ 1.13 failed by crushing at their ends,
usually with significant delamination. Local buckling was not observed. Figure 5.23
shows views from tests of intermediate and long 76.2x6.4 columns and Figure 5.24
shows the failure modes observed. Figure 5.25 shows the representative load versus
0
20
40
60
80
100
120
140
0.0 0.5 1.0 1.5
Stre
ss σ
(MP
a)
Wall transverse deflection ∆ (mm)
y = 121.4x - 0.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0 0.005 0.01 0.015
∆(m
m)
∆/σ (mm/kN)
0102030405060708090
100
0 2 4 6 8 10 12
Stre
ss σ
(MP
a)
Lateral Deflection δ (mm)
y = 131.0x - 1.4
y = 77.1x + 0.6
0
2
4
6
8
10
12
14
0.00 0.05 0.10 0.15 0.20δ
(mm
)
δ/σ (mm/MPa)
0
10
20
30
40
50
60
70
0 5 10 15 20 25 30
Stre
ss σ
(MP
a)
Lateral deflection δ (mm)
y = 59.0x - 0.2
0
5
10
15
20
25
30
0 0.1 0.2 0.3 0.4 0.5
δ(m
m)
δ/σ (mm/MPa)
Failure mode: LB-CM
Failure mode: CB-CM
Failure mode: GB-L/50
X
CB
97
lateral deflection curves and corresponding Southwell plots for intermediate and long
76.2x6.4 columns.
(a) intermediate column (λc = 0.94). (b) long column (λc = 1.99).
Figure 5.23 – Views from tests on 76.2x6.4 columns.
(a) intermediate column (λc = 0.78). (b) long column (λc = 0.90).
Figure 5.24 – Failure modes observed for short and intermediate 76.2x6.4
columns.
98
a) Stress versus deflection curve (λc = 0.94). b) Southwell Plot (λc = 0.94).
c) Stress versus deflection curve (λc = 1.99) d) Southwell Plot (λc = 1.99)
Figure 5.25– Representative stress vs. lateral deflection and Southwell Plots for
76.2x6.4 columns.
5.7.4. 88.9x6.4 Tubes
88.9x6.4 columns with relative slenderness ranging from λc = 0.42 to 1.85 were
tested. Column lateral deflection was observed for columns with slenderness greater
than λc = 0.66. Large lateral deflections (δ> L/50) were observed for λc ≥ 1.21. Local
buckling in the shorter specimens was not easily observed with the naked eye, however
the Southwell plots indicated the presence of local buckling for columns with
slenderness less than λc = 0.56. Kinking, followed by rupture along the flange-web
junction was the predominant failure mode. Since local buckling was evident for the
shorter columns, coupled buckling was assumed for columns with slenderness ranging
from λc = 0.66 to 1.05. Once again, local buckling was not clearly observed visually for
the columns and, in this case, the Southwell plots did not show significant bilinearity.
Nonetheless, the failure mode was governed by kinking, just as for the short columns.
Additionally, a notable difference in lateral deflection along the centerline and edge of
the tube at mid-height was observed, as shown in Figure 5.26. The graph of this figure
results from subtraction of mid-wall and tube edge displacements. Due to the nature of
0
50
100
150
200
250
300
0 2 4 6 8 10 12
Stre
ss σ
(MP
a)
Lateral deflection δ (mm)
y = 367.8x - 4.7
0
2
4
6
8
10
12
0 0.01 0.02 0.03 0.04 0.05
δ(m
m)
δ/σ (mm/MPa)
0
10
20
30
40
50
60
70
80
0 10 20 30 40
Stre
ss σ
(MP
a)
Lateral deflection δ (mm)
y = 82.6x - 7.3
0
5
10
15
20
25
30
35
40
0 0.1 0.2 0.3 0.4 0.5 0.6δ
(mm
)
δ/σ (mm/MPa)
Failure mode: GB-CE
Failure mode: GB-L/50
99
the test set-up, this is unlikely indicative of twist and is therefore interpreted as having
captured the relative deflection of a local buckle.
Figure 5.27 shows views from tests on short, intermediate and long columns and
Figure 5.28 shows the typical failure mode observed. Figure 5.29 shows representative
stress versus lateral deflection curves and corresponding Southwell plots for some
88.9x6.4 columns. A reversal of deflections is observed in Figure 5.29a, for the wall
transverse deflections.
Figure 5.26– Stress versus wall transverse deflection for a 88.9x6.4 column that
exhibited coupled buckling (λc = 0.78).
a) short (λc = 0.44) b) intermediate (λc = 1.05) (c) long (λc = 1.76)
Figure 5.27 – Views from tests on 88.9x6.4 columns.
100
150
200
-0.6 -0.4 -0.2 0.0 0.2 0.4
Stre
ss σ
(MP
a)
Wall transverse Deflection ∆ (mm)
100
Figure 5.28 – Failure mode observed for short and intermediate 88.9x6.4
columns (λc = 1.05).
a) Stress versus wall deflection curve (λc = 0.44). b) Southwell Plot (λc = 0.44).
c) Stress versus deflection curve (λc = 0.78). d) Southwell Plot (λc = 0.78).
e) Stress versus deflection curve (λc = 1.76). f) Southwell Plot (λc = 1.76).
Figure 5.29– Representative stress vs lateral deflection and Southwell Plots for
88.9x6.4 columns.
0
50
100
150
200
250
0.0 0.1 0.2 0.3 0.4 0.5
Stre
ss σ
(Mp
a)
Wall transverse deflection ∆ (mm)
y = 211.1x - 0.0
0.0
0.1
0.2
0.3
0.4
0.5
0 0.0005 0.001 0.0015 0.002 0.0025
∆(m
m)
∆/σ (mm/MPa)
0
50
100
150
200
250
0 1 2 3 4 5
Stre
ss σ
(MP
a)
Lateral deflection δ (mm)
y = 367.6x - 3.6
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
0.000 0.005 0.010 0.015 0.020 0.025
δ(m
m)
δ/σ (mm/MPa)
0
10
20
30
40
50
60
70
80
90
0 10 20 30 40 50
Stre
ss σ
(MP
a)
Lateral deflection δ (mm)
y = 75.8x - 0.2
0
10
20
30
40
50
0 0.1 0.2 0.3 0.4 0.5 0.6
δ(m
m)
δ/σ (mm/MPa)
Failure mode: LB-CM
Failure mode: CB-CM
Failure mode: GB-L/50
101
5.7.5. 102x6.4 Tubes
102x6.4 columns with relative slenderness ranging from λc = 0.33 to 1.50 were
tested. In general, the behavior and failure modes were very similar to the ones observed
for 88.9x6.4 columns. The only notable difference is regarding the local buckling,
which could be observed with the naked eye for some short specimens. These were
characterized by half-waves with very small amplitude and length ranging from 1 to 2
times the tube width. Column lateral deflection was observed for columns with
slenderness greater than λc = 0.63 and large lateral deflections (δ >L/50) were achieved
for λc ≥ 1.21. The Southwell plots indicated the presence of local buckling for columns
with slenderness less than λc = 0.60.
Figure 5.30 shows views from tests on short, intermediate and long columns.
Local buckling can barely be seen at Figure 5.30a. Figure 5.31 shows the typical failure
mode observed. Figure 5.32 shows the representative load versus lateral deflection
curves and corresponding Southwell plots for 102x6.4 columns.
a) short (λc = 0.46) b) intermediate (λc = 0.95) c) long (λc = 1.50)
Figure 5.30 – Views from tests on 102x6.4 columns (local buckling denoted
with arrows).
102
Figure 5.31 – Typical failure mode observed for short and intermediate 102x6.4
columns (λc = 0.95).
a) Stress versus wall deflection curve (λc = 0.33). b) Southwell Plot (λc = 0.33).
c) Stress versus deflection curve (λc = 0.95). d) Southwell Plot (λc = 0.95).
e) Stress versus deflection curve (λc = 1.50). f) Southwell Plot (λc = 1.50).
Figure 5.32– Representative stress vs lateral deflection and Southwell Plots for
102x6.4 columns.
020406080
100120140160180200
0.0 0.5 1.0 1.5 2.0
Stre
ss σ
(MP
a)
Wall transverse deflection ∆ (mm)
y = 170.8x - 0.0
0.0
0.5
1.0
1.5
2.0
0 0.002 0.004 0.006 0.008 0.01
∆(m
m)
∆/σ (mm/MPa)
0
20
40
60
80
100
120
140
160
0 5 10 15 20
Stre
ss σ
(MP
a)
Lateral deflection δ (mm)
y = 181.5x - 3.3
0
5
10
15
20
0.00 0.05 0.10 0.15
δ(m
m)
δ/σ (mm/MPa)
0
10
20
30
40
50
60
70
80
0 10 20 30 40 50
Stre
ss σ
(MP
a)
Lateral deflection δ (mm)
y = 73.7x - 3.4
0
10
20
30
40
50
0.00 0.20 0.40 0.60 0.80
δ(m
m)
δ/σ (mm/MPa)
Failure mode: LB-CM
Failure mode: CB-CM
Failure mode: GB-L/50
103
5.8. Stub Column Test Results
Figures 5.33 and 5.34 present plots of the axial stress vs. local flexure-induced
strain and the corresponding Southwell plots (SOUTHWELL, 1932), respectively. The
flexure-induced strain, εf, was determined as (εA–εB)/2, where εA and εB are the recorded
strains on both faces of the plate. Low-load non-linearities of Southwell plots
(SPENCER and WALKER, 1975) were barely observed. Local buckling was visible for
all specimens, with number of observed half-waves ranging from 4 to 5. In Figure 5.33,
a reversal of deflection can be observed for stub #3, which can be explained by the
proximity of the strain gage to a inflection point. Local buckling critical stresses were
consistent with the theoretically determined values presented in Table 5.5.
Figure 5.33 – Compressive stress vs. flexure-induced strain of stubs.
Figure 5.34 – Southwell plots of stubs.
0
20
40
60
80
100
120
140
-4,000 -3,000 -2,000 -1,000 0 1,000 2,000 3,000 4,000 5,000 6,000
Stre
ss σ
(MP
a)
Flexure-induced strain εf (µε)
Stub #1 (Fcrℓ = 124.9 MPa)
Stub #2 (Fcrℓ = 125.2 MPa)
Stub #3 (Fcrℓ = 126.3 MPa)
y = 124.9x - 29.2
y = 125.2x + 24.2
y = 126.3x + 1.6
0
500
1,000
1,500
2,000
2,500
3,000
3,500
4,000
4,500
5,000
0 5 10 15 20 25 30 35 40
Fle
xure
-ind
uce
d s
train
ε f(µε)
εf /σ (µε/MPa)
Stub #1
Stub #2
Stub #3
104
In Figure 5.35, compressive stress is plotted against axial shortening. It can be
seen that buckling is followed by a plateau, which is indicative of little post buckling
reserve of strength.
Figure 5.35 – Compressive stress vs. axial shortening of stubs.
5.9. Experimentally Determined Imperfection Factors
The real geometric imperfection can be determined by measurement using
several techniques currently available (SINGER et al., 2002, FEATHERSTON et al.,
2008). However, this is time consuming and the results have little practical application.
Measuring the imperfection of plates comprising a GFRP square tube is even more
difficult, since wall thickness varies throughout the column and, therefore, there is no
correspondence between the outer surface measurements and the initially deflected
surface. To address this problem, apparent initial imperfections defined by sinusoidal
functions proportional to the buckled shapes are assumed as described in Chapter 4.
5.9.1. Columns
The apparent imperfection of a column can be determined by means of a
Southwell plot analysis (SOUTHWELL, 1932). By plotting lateral displacement, δ, vs.
lateral displacement / applied stress, δ/σ, for a pinned-pinned column with initial out-of-
straightness and eccentricity, it can be shown that the intercept of the straight line fit
corresponds to -(4e0/π + δ0) (TIMOSHENKO and GERE, 1961). The column
20
40
60
80
100
120
140
0.0 0.5 1.0 1.5 2.0
Stre
ss σ
(MP
a)
Axial Shortening ∆u (mm)
Stub #1
Stub #2
Stub #3
105
imperfection factor αc can then be obtained by multiplying (4e0/π + δ0) by A/S (Eq.
4.34). Table 5.7 summarizes maximum apparent imperfections and αc-factors according
to each section considered. The largest factors were obtained for the 25.4x3.2 tubes.
This result partially reflects the unavoidable human error inherent in positioning and
aligning the specimens being relatively similar for all specimens, resulting in the load
eccentricity being the greatest for the smallest specimen.
Table 5.7 – Maximum experimentally calculated column imperfections and αc-factors.
25.4x3.2 50.8x3.2 76.2x6.4 88.9x6.4 101.6x6.4 λc 0.89 1.12 1.99 0.97 1.34
(4e0/π + δ0) 3.30 3.15 7.33 3.86 5.40 A/S 0.150 0.066 0.047 0.039 0.034 αc 0.50 0.21 0.34 0.15 0.18
5.9.2. Plates
Imperfection of plates cannot be determined in a similar fashion, because the
position of the buckled shape peak is unknown a priori and therefore strain gages
cannot be located to capture the largest flexure-induced strain. Thus, an approximate
solution is proposed, assuming that, for stub columns with negligible overall lateral
deflection, the total axial shortening ∆u is equal to the sum of the shortening resulting
from axial force ∆ua and from plate bending ∆ub. Using classical plate theory to
determine ∆ub, it can be shown that:
∫ ∫
∂∂+=∆+∆=∆
L b
cLba dxdy
x
w
bE
Luuu
0 0
2
, 2
1σ (5.3)
in which L is the column length; and b is the tube width.
Choosing length L as a multiple of the width (L = mb, m as integer) and adopting
w = ∆0σ/(Fcrℓ–σ) sin(πx/b) sin(πy/b), as previously defined in Section 4.3.2, Eq. 5.3 can
be written as:
σ
πσ
σσ b
m
FE
Lu
cr 8
22
20
−∆+=∆
l
(5.4)
By plotting ∆u/σc vs. [σ/(Fcrℓ–σ)]2π
2m/(8bσ) for a stub test, the apparent initial
imperfection can be obtained as the square root of the slope of the straight line fit while
106
the intercept with the ∆u/σ axis corresponds to L/EL,c. Figure 5.36 presents this plot for
all three stubs tested, limited to σ/Fcrℓ < 0.95 in order to avoid capturing post-buckling
effects, which are not considered in the formulation. Since ∆u = 0 and [σ/(Fcrℓ–σ)]2 ~ 0
for σ = 20 MPa, the absolute values of σ and ∆u shown in Figure 5.33b were used.
Neglecting the sub 20 MPa behavior and making a straight line fit, the greatest value of
∆02 obtained was 0.00079 for stub #1, corresponding to an apparent imperfection
amplitude of 0.028 mm. The αp,L-factor can then be obtained from Eq. 4.18. For the
50.8x3.2 tube, t = 2.88 mm and material properties presented in Table 5.2 and taking
FL,c/FL,f =0.65, αp,L = 0.014.
Figure 5.36 – ∆u/σ vs. [σ/(Fcrℓ–σ)]2π2m/(8bσ) for stub columns tested.
y = 2.6E-04x + 9.7E-06
8.0E-06
9.0E-06
1.0E-05
0.0E+00 2.0E-04 4.0E-04
∆u
/σ(m
m3 /N
)
[σ/(Fcrℓ–σ)]2π2m/(8bσ) (mm/N)
Stub #1
Stub #2
Stub #3
107
6. Experimental Program 2: Local
Buckling of I-Sections
6.1. Literature Review
A review of the most relevant experimental campaigns that have addressed the
local buckling of pultruded GFRP sections was presented by MOTTRAM (2004b).
Significantly, the columns adopted by most of the cited works cannot be considered
stubs (YOON, 1993, MOTTRAM et al., 2003, LANE and MOTTRAM, 2002), as they
are not short enough to mitigate the influence of overall lateral deflections (global
buckling).Throughout this chapter, I-section designation refers to the nominal depth d,
flange width bf, and plate/flange thickness t (d x bf x t) in units of mm.
The first known work performed on I-section stub columns (TOMBLIN and
BARBERO, 1994) investigated the local buckling behavior of different sections
(102x102x6.4; 152x152x6.4; 152x152x9.5; 203x203x9.5) under compression.
Specimen lengths ranged from 2 to 4 times the predicted half wave-lengths.
Acknowledging the fact that the buckling shape is previously unknown, the stubs were
instrumented with dial gages in different positions in order to capture the deflection at
different points. The authors remarked that deflection growth under constant load
following buckling might exist affecting the reported buckling loads. The experimental
critical loads, determined using the Southwell method (SOUTHWELL, 1932), were
compared to those theoretically determined and a maximum difference of 24.2% was
reported. Since the elastic properties were estimated from a micromechanical approach
rather than determined experimentally, the correlation between tests and theory is not
conclusive. The critical loads for the shortest columns were up to 36% greater than
those of the longest.
TURVEY and ZHANG (2004, 2006) tested 102x102x6.4 I-section stub columns
having lengths ranging from 200 to 800 mm (approximately 1 to 4 times the half wave
length). The columns were instrumented with strain gages and displacement transducers
and the critical loads were obtained from Southwell plots. The results were compared to
108
those obtained using the Finite Element Method (FEM) based on experimentally
determined material properties and assuming constituent plates restrained against
rotation at their ends. A maximum difference of 10% was observed between
experimental results and FEM predictions. The critical load obtained for the shortest
column was 32% greater than that of the longest.
The differences between the critical loads for short and long columns reported
by TOMBLIN and BARBERO (1994) and TURVEY and ZHANG (2004, 2006) are
remarkable and can be explained by the fact that the plates comprising the sections are
not simply-supported at their ends because the line of action of the force shifts as the
plate deflects laterally. This results in the boundary conditions falling between simply-
supported and clamped conditions and the buckled shape to be similarly undetermined,
as shown in Figure 6.1. This problem is similar to the compression of bars with round
ends (TIMOSHENKO and GERE, 1961) described in relation to Figure 5.13.
Additionally, columns with lengths that are not integer multiples of their critical half-
wave lengths tend to have greater critical loads. However, the longer the column, the
less important these effects are. In the present work, a similar effect is expected.
Figure 6.1 – Effect of end conditions on local buckling.
109
6.2. Experimental Program
Pultruded GFRP I-shaped stubs with different flange width-to-section depth
ratios (bf/d) were tested. Four sets of nine specimens were extracted from 102x102x6.4
and 76x76x6.4 I-sections made with both polyester (PE) and vinyl ester (VE) matrices,
for a total of 36 stubs. Flange widths of the original wide flange I-sections were cut
down in order to obtain flange-width-to-section-depth ratios (bf/d) of 1.0 (flange not
cut), 0.75 and 0.5. Dimensions of each stub were measured at various locations on each
specimen with a digital caliper and cataloged. Average dimensions as well as the
calculated cross-sectional area, A, of each specimen are presented in Table 6.1. No
significant deviations were observed within a cross-section. All sections were provided
by the same manufacturer and flange and web plates have the same thickness and
similar fiber architecture. The experimental program was carried out in two stages:
material characterization and stub compression tests, as described in the following
sections.
Table 6.1 – Average dimensions of the stubs tested.
Stub L (mm)
bf(mm) tf (mm)
d (mm)
tw (mm)
bw (mm)
bf/bw A (mm2)
VE1-1 400 100 6.34 101 6.31 95.1 1.06 1830 VE1-2 400 76.4 6.31 101 6.27 95.1 0.80 1520 VE1-3 266 48.5 6.33 101 6.25 95.0 0.51 1170 VE2-1 400 100 6.34 102 6.32 95.2 1.06 1830 VE2-2 400 69.2 6.35 102 6.3 95.2 0.73 1440 VE2-3 266 48.3 6.35 101 6.24 95.1 0.51 1170 VE3-1 400 100 6.34 101 6.31 95.1 1.06 1830 VE3-2 400 74.7 6.36 101 6.31 95.0 0.79 1510 VE3-3 266 49.5 6.36 102 6.32 95.1 0.52 1190 VE4-1 305 74.8 6.33 76.3 6.4 70.0 1.07 1350 VE4-2 305 53.4 6.34 76.4 6.41 70.1 0.76 1090 VE4-3 152 35.4 6.35 76.3 6.43 70.0 0.51 859 VE5-1 305 74.8 6.32 76.3 6.43 70.0 1.07 1350 VE5-2 305 54.7 6.33 76.3 6.42 70.0 0.78 1100 VE5-3 152 33.7 6.33 76.2 6.42 69.9 0.48 835 VE6-1 305 74.9 6.36 76.2 6.46 69.8 1.07 1360 VE6-2 305 52.8 6.37 76.2 6.44 69.8 0.76 1080 VE6-3 152 36 6.35 76.1 6.45 69.8 0.52 866 PE1-1 400 100 6.28 101 6.17 95.0 1.06 1810 PE1-2 400 74.1 6.26 101 6.23 95.0 0.78 1480 PE1-3 266 47.9 6.25 101 6.18 95.2 0.50 1150 PE2-1 400 100 6.26 101 6.2 95.0 1.06 1810 PE2-2 400 73.5 6.26 101 6.18 94.9 0.77 1470
110
Stub L (mm)
bf(mm) tf (mm)
d (mm)
tw (mm)
bw (mm)
bf/bw A (mm2)
PE2-3 266 49.5 6.25 102 6.2 95.3 0.52 1170 PE3-1 400 100 6.27 102 6.21 95.2 1.05 1810 PE3-2 400 73.4 6.27 102 6.21 95.2 0.77 1470 PE3-3 266 50.1 6.24 101 6.18 95.2 0.53 1170 PE4-1 305 74.9 6.27 76.2 6.32 69.9 1.07 1340 PE4-2 305 54.8 6.26 76.2 6.33 69.9 0.78 1090 PE4-3 152 33.6 6.27 76.2 6.34 69.9 0.48 825 PE5-1 305 74.9 6.28 76.2 6.34 69.9 1.07 1340 PE5-2 305 53.4 6.30 76.2 6.34 69.9 0.76 1080 PE5-3 152 37.6 6.30 76.2 6.34 69.9 0.54 877 PE6-1 305 74.9 6.29 76.2 6.36 69.9 1.07 1350 PE6-2 305 53.7 6.30 76.2 6.34 69.9 0.77 1080 PE6-3 152 34.6 6.30 76.2 6.34 69.9 0.49 839
6.3. Material Characterization
In order to obtain a good correlation between theory and experiment, material
properties were obtained experimentally from a limited number of samples extracted
from the web or flange of VE and PE 102x102x6.4 sections as described in the
following paragraphs. All samples were tested to failure and no unloading test was
performed. Results (showing one standard deviation) are given in Table 6.2.
Extrapolating from the experimentally determined values and accounting for the
measured fiber volume ratios (Table 6.2), material properties for each section were
estimated as described in Chapter 5.4 for use in all subsequent calculations. Fiber
content was estimated by weighing 50 mm long specimens of each section and
assuming fiber and resin densities according to table 2.1. The major Poisson’s ratio, νLT,
was assumed to be equal to 0.32 for all sections.
111
Table 6.2 – Material properties (± one standard deviation) of stub test specimens.
Fiber Content, Vf(determined experimentally) VE 102x6.4 VE 76x6.4 PE 102x6.4 PE 76x6.4
Flange 0.51 0.49 0.43 0.42 Web 0.56 0.55 0.49 0.47
Section 0.54 0.52 0.46 0.45 Experimentally Determined Mechanical Properties
Fc (MPa) 429 ± 72 277 ± 30 EL,c(MPa) 25,600 ± 4600 25,700 ± 2400 EL(MPa) 21,900 ± 1300 19,500 ± 100
ET,t~ET (MPa) 12,200 ± 800 9430 ± 830 GLT (MPa) 4720 ± 120 4080 ± 20
Calculated Mechanical Properties (Representative for Section) Fc (MPa) 453 438 294 289 EL,c(MPa) 27,100 26,100 27,300 26,800 EL(MPa) 23,100 22,300 20,700 20,300
ET,t~ET (MPa) 11,800 11,200 8760 8560 GLT (MPa) 4550 4290 3810 3630
6.3.1. Longitudinal compression (EL,c and FL,c)
Longitudinal compressive modulus and strength, EL,c and Fc, were determined
from tests on sets of three 12.7 mm wide x 6.4 mm (t) x 50.8 mm tall coupons extracted
from the flanges of both VE and PE sections. The coupons were instrumented with one
electrical resistance strain gage and tested in concentric compression at a rate of 0.10
mm/min until failure. The test set-up and axial stress-strain curves are shown in Figures
6.2 and 6.3, respectively. The initial modulus (continuous line in graph) was determined
from regression through axial stress values ranging from 0 to 20% of the failure stress.
Figure 6.2 – Compression test adopted for determination of longitudinal
compressive modulus (specimen is 12.7 mm wide in this image).
112
a) Polyester
b) Vinyl ester
Figure 6.3 – Axial stress versus strain curves obtained from longitudinal compression
tests.
6.3.2. Transverse tension (ET,t)
Transverse tension modulus, ET,t, was determined from tensile tests on sets of
three untabbed 12.7 mm wide x 6.4 mm (t) x 88.9 mm long transverse coupons
extracted from webs of both VE and PE sections. Coupons were tested with a free
length of approximately 50.8 mm. The coupons were instrumented with one electrical
resistance strain gage and tested in tension at a rate of 0.10 mm/min until failure. Due to
the untabbed short grip lengths, failure at the ends was observed in some cases; this is
not a major concern since the aim of the test was to obtain the initial modulus of
0
50
100
150
200
250
300
350
0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014
Axi
al S
tress
(MP
a)
Axial Strain
Sample 1 (EL,c = 22,419 MPa)
Sample 2 (EL,c = 27,914 MPa)
Sample 3 (EL,c = 26,759 MPa)
PE
EL,c
EL,c
EL,c
0
100
200
300
400
500
600
0.000 0.005 0.010 0.015 0.020 0.025
Axi
al S
tress
(MP
a)
Axial Strain
Sample 4 (EL,c = 27,182 MPa)
Sample 5 (EL,c = 30,335 MPa)
Sample 3 (EL,c = 19,310 MPa)
VE
EL,c
EL,c
EL,c
113
elasticity. The test set-up and axial stress-strain curves are shown in Figures 6.4 and 6.5,
respectively. The initial modulus (continuous line in graph) was determined from
regression through axial stress values ranging from 0 to 20% of the failure stress.
Figure 6.4 – Tension test adopted for determination of transverse tensile
modulus (6.4 mm is thickness; strain gage is located on 12.7 mm breadth).
114
a) Polyester
b) Vinyl ester
Figure 6.5– Axial stress versus strain curves obtained from transverse tension tests.
6.3.3. 45 degree off-axis tension (GLT)
Shear modulus, GLT, was determined from 45 degree off-axis tensile tests on sets
of three untabbed 12.7 mm wide x 6.4 mm (t) x 119 mm long transverse coupons
extracted from webs of both VE and PE sections. These were tested with a gage length
of approximately 81.3 mm. The coupons were instrumented with two orthogonally-
oriented electrical resistance strain gages, one oriented parallel (εx) and one
perpendicular (εy) to the axis of loading (thus oriented ±45o with respect to the
pultrusion direction). Coupons were tested in tension at a rate of 0.10 mm/min until
failure. Failure at the ends due to the untabbed short grip lengths were observed, but
0
10
20
30
40
50
60
70
0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018
Axi
al S
tress
(MP
a)
Axial Strain
Sample 1 (ET,t = 8254 MPa)
Sample 2 (ET,t = 10,044 MPa)
Sample 3 (ET,t = 9973 MPa)
PE
ET,c
ET,c
ET,c
0
10
20
30
40
50
60
70
0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016
Axi
al S
tress
(MP
a)
Axial Strain
Sample 4 (ET,t = 13,992 MPa)
Sample 5 (ET,t = 10,991 MPa)
Sample 6 (ET,t =11,553 MPa)
VE
ET,c
ET,c
ET,c
115
once again, this is not a concern since only the initial shear modulus was of interest. The
test set-up and shear stress-strain curves are shown in Figures 6.6 and 6.7, respectively.
The initial modulus (continuous line in graph) was determined from regression through
shear stresses ranging from 0 to 20% of the failure stress. The shear modulus was
obtained as follows (ROSEN, 1972):
yx
x
LT
LTLT
/G
εεσ
γτ
−== 2
(6.1)
in which τLT and γLT are the shear stress and strain in the longitudinal direction (i.e.:
parallel to the pultrusion direction); σx is the applied tensile stress; and εx and εy are the
measured strains in the directions parallel and perpendicular to the applied tension
force, respectively. The influence of the coupon end conditions was evaluated according
to ADAMS et al. (2002) and it was concluded that, for the free length adopted and
range of material properties studied, coupon end effects can be neglected.
Figure 6.6 – 45 degree tension test adopted for determination of in-plane shear
modulus (6.4 mm is thickness; strain gage is located on 12.7 mm breadth).
116
a) Polyester
b) Vinyl ester
Figure 6.7– Shear stress versus strain curves obtained from 45 degree tension tests.
6.3.4. Longitudinal plate bending (EL,f)
The longitudinal plate bending modulus, EL,f, was determined from three-point
bending tests of 44.5 wide x 6.4 mm (t) deep x 254 mm long longitudinally-oriented
coupons tested over a 203 mm simple span, L (Figure 6.8). Sets of three coupons were
extracted from flanges of both VE and PE sections. Mid-span deflections, δ, were
measured with a dial gage (0.025 mm precision) and loads were obtained from the
universal testing machine. Loads, P, were recorded every 1.27 mm of deflection and EL,f
was obtained as the slope of the straight line fit for P vs. 4δbt3/L3, where b is the
specimen width. Experimental results are shown in Figure 6.9.
0
5
10
15
20
25
30
35
40
45
50
0.000 0.005 0.010 0.015 0.020 0.025
Sh
ear S
tress
τ LT
(MP
a)
Shear Strain γLT
Sample 1 (GLT = 4058 MPa)
Sample 2 (GLT = 4076 MPa)
Sample 3 (GLT =4096 MPa)
PE
GLT
GLT
GLT
0
5
10
15
20
25
30
35
40
45
0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018
Sh
ear S
tress
τ LT
(MP
a)
Shear Strain γLT
Sample 4 (GLT = 4559 MPa)
Sample 4 (GLT = 4822 MPa)
Sample 6 (GLT =4790 MPa)
VE
GLT
GLT
GLT
117
Figure 6.8 – 3-point bending test adopted for determination of longitudinal flexural modulus.
a) Polyester
b) Vinyl ester
Figure 6.9– P versus 4δbt3/L3 curves obtained from three point bending tests.
0
200
400
600
800
1000
1200
0.000 0.010 0.020 0.030 0.040 0.050 0.060
Loa
d P
(N)
4δbt3/L3 (mm2)
Sample 1 (EL,f = 19,455 MPa)
Sample 2 (EL,f = 19,337 MPa)
Sample 3 (EL,f = 19,626 MPa)
PE
EL,f
EL,f
EL,f
0
200
400
600
800
1000
1200
1400
0.000 0.010 0.020 0.030 0.040 0.050 0.060
Loa
d P
(N)
4δbt3/L3 (mm2)
Sample 4 (EL,f = 22,208 MPa)
Sample 5 (EL,f = 22,703 MPa)
Sample 6 (EL,f = 20,842 MPa)
VE
EL,f
EL,f
EL,f
118
6.4. Stub Column Tests
Thirty six stub column tests were performed in order to obtain the local buckling
critical stresses of GFRP I-shaped sections. Since the exact buckling shapes were not
known a priori (TOMBLIN and BARBERO, 1994), the stub lengths, L, (Table 6.1)
were arbitrarily chosen in order to accommodate 2 or 3 half-wave lengths, but not
necessarily integer multiples of the critical lengths (Eq. 3.35). Preliminary calculations
showed that the lengths adopted resulted in relative slenderness (VANEVENHOVEN et
al., 2010) lower than 0.5 and therefore, negligible interaction between local and global
buckling was expected.
Stubs with bf/d = 1.0 and 0.75 were instrumented with two electrical resistance
strain gages positioned on both faces of the flange edge at a height coinciding with the
peak of the idealized buckled shape. For stubs with bf/d = 0.5, the strain gages were
located on the web, since the flanges are very narrow. The stubs were tested under
concentric compression in a 600 kN-capacity servo-hydraulic universal testing machine
under displacement control at a rate of 0.10 mm/min. Figure 6.9 shows views from
different test set-ups.
a) bf/d=0.99; L=400 mm (VE3-1)
b) bf/d=0.73; L=400 mm (PE3-2)
c) bf/d=0.48; L=266 mm (VE2-3)
Figure 6.9 - Representative stub tests with different set-ups.
119
6.5. Stub Test Results
Figures 6.10 and 6.11 present plots of the stress vs. local flexure-induced strain
for stub columns extracted from 102x102x6.4 and 76x76x6.4 sections, respectively. The
flexure-induced strain, εf, is determined as (εA–εB)/2, where εA and εB are the recorded
strains on either face of the plate. The local buckling critical stresses, Fcr, shown in
Table 6.3, were obtained experimentally as the slope of Southwell Plots
(SOUTHWELL, 1932) of flexure-induced strain vs. flexure-induced strain/stress, i.e.,
same procedure adopted for square tube stub columns (section 5.8). Some of the
Southwell plots were compared to Lundquist plots (LUNDQUIST, 1938) –
recommended for plate buckling – and negligible differences were observed. Buckling
coefficients, k, are also reported. These were obtained by dividing the experimental
critical stresses by π2EL,f/[12(1-νLT νTL)](t/bw)2, in which t is the average thickness. In
agreement with the graphs shown in Figures 6.10 and 6.11, more pronounced transverse
deflections were typically observed for stubs extracted from 102x102x6.4 with bf/d =
1.0 (see Fig. 6.9a). Deflections were much less pronounced for stubs extracted from
76x76x6.4 with bf/d = 0.5 (see Fig. 6.9c) in which the values of local buckling critical
stress and material compressive strength are closer, leading to failure before large
deflections developed. Strain reversal was noted in some cases (Figs 6.10 and 6.11) as a
result of the proximity of the strain gages to an inflection point of the buckled shape.
Table 6.3- Summary of stub test results.
Stub η Fcr
(MPa) k Fu
(MPa) Stub η Fcr (MPa) k
Fu (MPa)
VE1-1 1.06 170 1.92 171 PE1-1 1.06 147 2.02 151 VE1-2 0.80 219 2.49 226 PE1-2 0.78 200 2.74 201 VE1-3 0.51 294 3.34 301 PE1-3 0.50 274 3.79 247 VE2-1 1.06 166 1.87 167 PE2-1 1.06 147 2.01 153 VE2-2 0.73 243 2.74 249 PE2-2 0.77 210 2.89 214 VE2-3 0.51 289 3.29 291 PE2-3 0.52 274 3.80 281 VE3-1 1.06 172 1.94 174 PE3-1 1.05 149 2.04 155 VE3-2 0.79 234 2.62 229 PE3-2 0.77 194 2.67 198 VE3-3 0.52 313 3.52 312 PE3-3 0.53 270 3.75 277 VE4-1 1.07 286 1.80 280 PE4-1 1.07 236 1.73 221 VE4-2 0.76 358 2.25 365 PE4-2 0.78 286 2.09 255 VE4-3 0.51 475 2.94 415 PE4-3 0.48 408 2.97 319 VE5-1 1.07 279 1.75 278 PE5-1 1.07 227 1.65 217 VE5-2 0.78 410 2.56 396 PE5-2 0.76 481 3.48 277 VE5-3 0.48 385 2.39 420 PE5-3 0.54 402 2.91 348 VE6-1 1.07 246 1.52 248 PE6-1 1.07 217 1.58 214 VE6-2 0.76 364 2.25 345 PE6-2 0.77 275 1.99 291 VE6-3 0.52 393 2.42 401 PE6-3 0.49 445 3.22 335
120
a) VE sections
b) PE sections
Figure 6.10 - Compressive stress vs. flexure-induced strain of stubs extracted from VE
and PE 102x102x6.4 sections.
0
50
100
150
200
250
300
350
-2,000 0 2,000 4,000 6,000 8,000 10,000 12,000 14,000
Axi
al S
tress
σ(M
Pa
)
Flexure-induced strain εf (µε)
VE2-1 (Fcr = 166 MPa)
VE3-1 (Fcr = 172 MPa)
VE1-3 (Fcr = 294 MPa)
VE2-2 (Fcr = 243 MPa)VE3-2 (Fcr = 234 MPa)
VE3-3 (Fcr = 313 MPa)
VE2-3 (Fcr = 289 MPa)
VE1-2 (Fcr = 219 MPa)
VE1-1 (Fcr = 170 MPa)
0
50
100
150
200
250
300
350
-2,000 0 2,000 4,000 6,000 8,000 10,000 12,000
Axi
al S
tress
σ(M
Pa
)
Flexure-induced strain εf (µε)
PE2-1 (Fcr = 147 MPa)
PE1-3 (Fcr = 274 MPa)
PE2-3 (Fcr = 274 MPa)
PE2-2 (Fcr = 210 MPa)
PE3-3 (Fcr = 270 MPa)
PE3-2 (Fcr = 194 MPa)
PE1-1 (Fcr = 147 MPa)
PE3-1 (Fcr = 149 MPa)
PE1-2 (Fcr = 200 MPa)
121
a) VE sections
b) PE sections
Figure 6.11 - Compressive stress vs. flexure-induced strain of stubs extracted from VE
and PE 76.2x76.2x6.4 sections.
Ultimate stresses were, in general, lower than the local buckling critical stress.
This is expected due to the presence of wall imperfections, as explained in Chapter 4.
More pronounced difference between both stresses was observed for sections with η ~
0.5, where the critical stress is nearer the material compressive strength. In a few cases,
the ultimate stress was greater than the critical stress, which can be attributed to some
post-buckling reserve of strength. Ultimate failure modes were typically characterized
by a combination of web transverse and junction longitudinal cracks, followed by
0
50
100
150
200
250
300
350
400
450
-2,000 -1,000 0 1,000 2,000 3,000 4,000 5,000 6,000 7,000
Axi
al S
tress
σ(M
Pa
)
Flexure-induced strain εf (µε)
VE4-1 (Fcr = 286 MPa)
VE6-1 (Fcr = 246 MPa)
VE6-2 (Fcr = 364 MPa)
VE5-2 (Fcr = 410 MPa)
VE5-3(Fcr = 385 MPa)
VE6-3 (Fcr = 393 MPa)
VE5-1 (Fcr = 279 MPa)
VE4-2(Fcr = 358 MPa)
VE4-3(Fcr = 475 MPa)
0
50
100
150
200
250
300
350
400
-2,000 0 2,000 4,000 6,000 8,000
Axi
al S
tress
σ(M
Pa
)
Flexure-induced strain εf (µε)
PE4-1 (Fcr = 236 MPa)
PE6-1 (Fcr = 217 MPa)
PE5-2(Fcr = 481 MPa)
PE4-3 (Fcr = 408 MPa)
PE4-2(Fcr = 286 MPa)
PE5-3 (Fcr = 402 MPa)
PE6-2 (Fcr = 275 MPa)PE6-3(Fcr = 445 MPa)
PE5-1 (Fcr = 227 MPa)
122
severe delamination, as shown in Figure 6.12. This is similar to the mode observed by
TURVEY and ZHANG (2006).
Figure 6.12 – Typical failure mode observed for I-sections stub columns.
In Figure 6.13, the experimentally determined buckling coefficients, k (Table
6.3), are plotted against the ratio bf/bw. The buckling coefficients obtained for the
simply-supported flange-to-web junction condition (Eqs. 3.15 and 3.16) and from the
equation proposed in this work (Eq. 3.36), calculated using the average properties of PE
and VE, are also plotted in Figs. 6.13a and 6.13b, respectively. As expected, the
experimental results are usually greater than the proposed expression, which can be
explained by the influence of length and boundary conditions (see discussion associated
with Figure 6.1) and by the increased restraint stiffness due to the presence of fillets at
the flange-to-web junction. On average, the experimental buckling coefficients were
19.7% greater than kcr calculated from Eq. 3.36. Two exceptions were observed for
columns having bf/bw ~ 0.5 (VE5-3 and VE6-3, shown in Fig. 6.13a). Possible
explanations for these exceptions are: i) inaccuracy of Southwell plots due to the
occurrence of the instrumented section crushing prior to transverse deflection growth
stabilization (resulting from the inability to locate the strain gage at the peak of an a
priori unknown buckled shape); ii) reduction in the elastic properties due to the applied
stresses having the same order of magnitude as the crushing stress; or iii) critical
stresses may be affected by transverse shear which has been shown to be important
123
when b/t < 13 (BARBERO, 2011). In one case (PE5-2 shown in Fig. 6.13b), the critical
load was much greater than the lower bound (Eq. 3.36), which is explained by the lack
of a buckling plateau in this case (i.e., deflection growth stabilization did not occur and
resulting in an accurate Southwell plot not being achieved.)
a) VE sections
b) PE sections
Figure 6.13 – Experimental and theoretical buckling coefficients k vs. bf/bw.
Due to difference in the material properties of PE and VE specimens, buckling
coefficients up to 7% greater were expected for VE sections. This expected difference is
very small (falling within one standard deviation of results) and was not be observable
in the present study.
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
0.00 0.20 0.40 0.60 0.80 1.00 1.20
Buc
klin
g c
oef
ficie
nt k
η = bf/bw
VE 102
VE 76
EL/ET = 1.98EL/GLT = 5.14νLT = 0.32
Eq. 3.36
Eqs. 3.15 & 3.16
VE5-3 VE6-3
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
0.00 0.20 0.40 0.60 0.80 1.00 1.20
Bu
cklin
g c
oef
ficie
nt k
η = bf/bw
PE 102
PE 76
EL/ET = 2.37EL/GLT = 5.51νLT = 0.32
Eqs. 3.15 & 3.16
Eq. 3.36
PE5-2
124
6.6. Validation of Approach to Capacity Prediction Proposed in
Chapter 3
In order to validate the approach proposed in this work, the ratios of predicted-
to-experimental critical stresses (Fcr/Fcr,exp) gathered from literature (TOMBLIN and
BARBERO, 1994, TURVEY and ZHANG, 2004) and from the present study are
presented in Figure 6.14. Predicted capacities are based on the simplified Equations
3.17 and 3.18 which assume simply-supported junctions between the web and flange
plates, and the approach proposed in the present work utilizing Eq. 3.29 with k= kcr
defined by Eq. 3.36. The proposed equations indicate better correlation with
experimental results having an average ratio Fcr/Fcr,exp = 0.86 ± 0.11 (one standard
deviation), whereas the use of Eqs. 3.17 and 3.18 leads to an average of 0.53 ± 0.16.
Previous studies have only considered ‘wide flange’ (WF) I-sections (i.e.: those having
bf/d ≈ 1 as pultruded). Considering only such WF shapes, the average ratios of
predicted-to-experimental critical stresses calculated using Equations 3.17 and 3.18, fall
to Fcr/Fcr,exp= 0.48 ± 0.15, while the ratio based on Equations 3.29 and 3.36 remains
unchanged. Thus it is demonstrated that considering only simply-supported junctions
between web and flange plates, while simple, results in underestimation of the buckling
capacity of pultruded I-section by approximately a factor of two. Additionally, the
degree of underestimation increases as the flange slenderness bf/d increases, i.e., Eqs.
3.17 and 3.18 are more conservative. These facts, taken together, have an implication
for the ‘calibration’ of building code material resistance factors (so called φ factors).
Using Equations 3.15 and 3.16, reliability calibrations will result in relatively high φ
factors which a) misrepresent the material reliability; and b) may be appropriate for only
WF shapes. Calibrations using Equations 3.29 and 3.36 will result in more
‘representative’ and uniformly applicable φ factors.
125
Figure 6.14 – Ratios of predicted-to-experimental critical stresses from literature and
present work.
0.00
0.20
0.40
0.60
0.80
1.00
1.20
Fcr/F
cr,e
xp
Specimens
Tomblin andBarbero (1994)
Turvey andZhang (2004)
Present Studywide flange data
(i.e.: bf/d ≈ 1)
Eqs. 3.29 & 3.36WF average = 0.83
Eqs. 3.17 & 3.18WF average = 0.42
Present Studyreduced flange data
(i.e.: bf/d < 1)
126
7. Conclusions
In this study, the performance and strength of pultruded GFRP members subject
to short term concentric compression in room temperature environments was addressed.
Extensive literature review was presented throughout the work and gaps in the
knowledge were identified. In order to fill in these gaps, experimental programs were
conducted and simple, novel and effective design guidelines were developed. The next
paragraphs present conclusions, final considerations and recommendations for future
works for each chapter.
In Chapter 2, characteristics and particularities of pultruded GFRP were
presented, along with an extensive review of the experimental and theoretical
determination of elastic properties. It was shown that current ASTM standard tests for
characterization of pultruded GFRP sections are not always practical due to
requirements for special apparatus or required coupon dimensions being too large to be
accommodated by pultruded shapes. The latter is especially true for transverse material
properties which are often simply neglected in a test program. A brief review of non-
standard methods was presented and all, save perhaps the Timoshenko Beam test, lead
to good estimates of material properties. The proposals made in Chapter 2, such as the
transverse bending test developed in this work and described in Chapter 5, should
encourage other authors to develop and report novel methods, instead of estimating
properties based on ‘experience’; thus enriching experimental investigations. The
effectiveness of such methods should be confirmed by optical techniques such as digital
image correlation and/or by numerical analyzes.
For theoretical determination of elastic properties, the method presented by
DAVALOS et al. (1996) is able to account for the combination of unidirectional and
randomly distributed fiber layers, but it requires data (such as material characterization
and layer thickness) for each layer to be known; such data is often not readily available.
In addition, this rigorous method does not result in a closed-form solution and must be
adjusted for each situation. Recognizing these difficulties, simple empirical closed-form
equations obtained from regression of experimental results were proposed. These
127
equations are suitable for initial design and or estimation of unknown properties in
existing pultrusions. Nonetheless, more tests must be conducted, especially for
transverse properties. To establish good correlations, fiber volume ratios and the
number of roving layers must be reported.
In Chapter 3, the behavior of a perfect column was addressed. The failure modes
of crushing, global buckling and local buckling were discussed and design equations
were developed and presented. For theoretical determination of longitudinal
compressive strength, the method presented by BARBERO et al. (1999a) seems suitable
for pultruded GFRP members although it also requires data for each layer to be known
and does not result in a closed-form solution. Again, a closed-form equation obtained
from regression of experimental results was proposed. The equation is suitable for
initial design, but more tests must be conducted for material supplied by different
manufacturers in order to obtain a reliable equation valid for typical sections.
For global buckling, equations for typical buckling modes were presented.
Emphasis was given to the flexural mode for which in-plane shear may significantly
affect the critical stresses for short to intermediate columns. For local buckling, an
extensive review of existing methods was made and the development of equations to
determine the local buckling critical stress of typical pultruded GFRP sections (angles,
I-shapes, channels and rectangular tubes) was presented. The expressions were derived
using the Rayleigh energy method assuming approximate deflected shapes that account
for both end conditions and continuity at plate intersections and allow for different
section geometries (bf/bw and bw/t) and orthotropic material properties (EL,f/ET,f and
EL,f/GLT). A summary of the resulting equations, their range of applicability in terms of
bf/bw and the observed relative differences of the various values of the derived critical
buckling coefficient, kcr, from those obtained using finite strip method (FSM) was
presented in Table 3.6. The observed differences between the proposed equations and
those promulgated in current standards and guidelines were also shown.The following
conclusions were drawn from the local buckling study:
a. the proposed critical buckling coefficients, kcr, applied in Eq.3.29 represent a
more realistic and accurate representation of local buckling behavior of
pultruded GFRP shapes than those presently adopted and proposed in current
standards and guidelines;
128
b. from the perspective of design, the conventional approach of assuming
simply supported conditions at all plate interfaces (i.e.: Eqs.3.15 and 3.16) is
shown to be very conservative in all cases although the degree of
conservativeness varies with the section shape (Table 3.6). While safe for
design, this leads to inefficient material use, potential variation of reliability
assumed in design, and does not reflect the fundamental kinematics of the
local buckling problem;
c. the approach adopted for determining values of kcr is very simple although
the results are highly dependent on the quality of the assumed deflected
shape. With the exception of the double-sinusoidal approximation for
rectangular tubes (Eqs. 3.25 and 3.26), all functions adopted led to good
agreement within the range of typically available cross sections, with
observed differences from FSM-calculated solutions of less than 10% in all
cases (Table 3.6);
d. the poor agreement obtained for approximate functions for box sections
derived using the double-sinusoidal functions (Eqs. 3.25 and 3.26) is related
to the fact that the double-sinusoidal functions cannot represent the change
of curvature that characterizes the deflected shape of wider walls as bf/bw
decreases, resulting in more significant rotational restraint imposed by the
narrower walls;
e. The approach used to generate the proposed functions can be extended to
other shapes and sections with non-uniform thickness and material
properties. This approach can also be applied to evaluate local buckling of
sections subject to flexure, as well as distortional buckling modes for cross-
section shapes other than those studied.
In Chapter 4, the behavior of a real column comprised of imperfect thin-walled
plates was addressed. In addition to the well-known column slenderness referred to in
GFRP column studies, a plate slenderness parameter, λp, analogous to a similar
definition used for cold-formed steel sections, was introduced (Eq. 4.1). A classification
of sections was proposed in which component plates are compact (λp ≤ 0.70),
intermediate (0.70 < λp < 1.30) or slender (λp ≥ 1.30). Finally, the factors affecting the
strength were described and a design equation for compressive strength of columns was
developed and presented. The equation is a function of column and wall slenderness, λc
129
(Eq. 4.2) and λp (Eq. 4.1), respectively, as well as column and wall imperfections, αc
(Eq. 4.34) and αp,L (Eq. 4.18), respectively. The approach presented in Chapter 4
focused on sections comprised of simply supported walls, such as square tubes.
However, it can be extended for other shapes and similar strength equations (Eq. 4.35)
can be obtained, having different imperfection factors: e.g. different column strength
curves could be proposed. Analysis of Equation 4.35 showed that the reduction of
capacity is a function of both plate and column slenderness, with the greatest reduction
of capacity occurring for λc = λp = 1.0; a column of intermediate slenderness comprised
of intermediate plates.
In Chapter 5, experimental studies were carried out to investigate the behavior of
square GFRP tube columns having varying global and section slenderness ratios subject
to concentric compressive load. Additionally simple methods to determine the column
and plate apparent initial imperfections were presented. In order to obtain an accurate
correlation between theory and experiment, the cross-section geometries were
measured, the most important material properties were obtained from specific tests, the
theoretical critical loads were obtained by expressions that include the effects of shear,
and the effective length factor, Ke, was adjusted according to the geometry of the test
fixtures used. The following conclusions are drawn:
a. it was shown that the proposed column curve (Eq. 4.35) represents
adequately a lower bound for the compressive strength of GFRP tubes.
However, additional column and stub tests must be carried out in order to
determine reliable and representative imperfection factors that can be used
for design purposes;
b. short and intermediate columns exhibited a significant reduction of capacity
with respect to the perfect column analogue (Figure 5.15). The greatest
reduction in capacity (ρc = 0.60) was observed for intermediate 76.2x6.4
tubes (λp = 0.94) of intermediate length (λc = 0.78 and λc = 1.04). This
reduction is indicative of an interaction between crushing, local buckling and
global buckling, as the predicted loads corresponding to each failure mode
are similar. Considerable reduction was also observed for short columns
resulting from first order bending moments caused by initial eccentricities
and the inability of the brittle GFRP material to effectively redistribute
stress.
130
c. Eq. 4.35 was plotted in Figure 5.15 for the case of αc = 0.34 (maximum for
76.2x6.4 tubes) and ρp = 0.92 (obtained from Eq. 4.25 for λp = 0.94 and
experimentally determined αp,L = 0.015), along with experimental data and
strength equations proposed by other authors. It can be seen that the
proposed equation captures the experimental data quite well, while the
equations proposed in the literature are unconservative in many cases. The
proposed equation, developed for a strength criterion, did not capture data
for long columns particularly well. This is explained by the fact that testing
of columns exhibiting large lateral deflections was stopped at a lateral
deflection of L/50 and therefore, the columns did not experience their
ultimate failure. Using the same assumptions and a similar approach, an
equation can be calibrated for this or any lateral deflection criterion.
d. the majority of previous studies have focused on I-shaped columns
comprised of slender plates; these are not as critical and do not present the
greatest reductions in theoretical capacity. Thus, it is recommended that
additional tests be carried out focusing on tubular column sections with both
column and plate slendernesses, λc and λp, classified as intermediate.
Reported in Chapter 6, an experimental investigation of the compressive
behavior of I-shaped GFRP stub columns having different cross sections (bf/d and bf/t)
and material properties (EL/ET and EL/GLT) was carried out to validate the proposed
equation 3.36. Finally, sample calculations were presented and compared with
numerical and experimental data extracted from available literature. The following
conclusions are drawn from the study:
a. based on material characterization, extensional and flexural moduli of
elasticity in the longitudinal direction may vary. Ratios EL,f/EL,c of 0.85, for
vinyl ester (VE), and 0.76, for polyester (PE) GFRP pultruded sections were
observed. Since local buckling is a phenomenon related to plate bending,
flexural moduli must be used.
b. Equation 3.36 was developed for an infinitely long plate that is able to
accommodate the critical half-wave length. The present experimental study
as well as the results of others (TOMBLIN and BARBERO, 1994, TURVEY
and ZHANG, 2004, 2006) reported greater critical loads for short columns,
131
which is attributed to the influence of specimen length and boundary
conditions.
c. Vinyl ester (VE) sections exhibited slightly higher elastic properties than
polyester (PE), but no apparent differences were observed in the stub tests.
d. Concerning local buckling of I-shaped GFRP columns, to validate the
proposed Equation 3.36, more tests must be conducted: critical loads and
experimental material properties must be reported. Testing longer columns
with continuous lateral restraint to prevent global buckling would minimize
the influence of end conditions resulting in reduced overstrength and closer
agreement with the proposed equation.
Overall, the conclusions of this study are directly applicable to approaches
promulgated by various international codes/standards. In particular it is demonstrated
that considering only simply-supported junctions between web and flange plates (as is
commonly recommended), while simple, results in underestimation of the buckling
capacity of pultruded I-section by approximately a factor of two. Additionally, this
degree of underestimation increases as the flange slenderness bf/d increases. These facts,
taken together, have an implication for the ‘calibration’ of building code material
resistance factors (so called φ factors). Using the simply supported assumptions (Eqs
3.15 and 3.16), reliability calibrations will result in relatively high φ factors which a)
misrepresent the material reliability; and b) may be appropriate for only wide flange
shapes. Calibrations using the more rigorous approach proposed in this work (Eqs 3.29
and 3.36) will result in more ‘representative’ and uniformly applicable φ factors.
Finally The following conference and journal papers were originated from this
study:
Cardoso, D., Batista, E. and Harries, K.A. (2014), “Projeto de Colunas em PRFV
Submetidas à Compressão Centrada”, XXXVI Jornadas Sul Americanas de Engenharia
Estrutural, Montevideo, November.
Cardoso, D., Harries, K.A. and Batista, E.M. (2014), “Local Buckling of Pultruded
GFRP I-Sections Columns”, Proceedings of the 7thInternational Conference on FRP
Composites in Civil Engineering (CICE 2014), Vancouver, August.
132
Cardoso, D., Harries, K.A. and Batista, E.M. (2014), “On The Determination of
Material Properties for Pultruded GFRP Sections”, Proceedings of the 7thInternational
Conference on FRP Composites in Civil Engineering (CICE 2014), Vancouver, August.
Cardoso, D., Harries, K.A. and Batista, E.M. (in revision), “Compressive Local
Buckling of Pultruded GFRP I-Sections: Development and Numerical/Experimental
Evaluation of an Explicit Equation”, ASCE Journal of Composites for Construction.
Cardoso, D., Harries, K.A. and Batista, E.M. (2014), “Closed-Form Equations for Local
Buckling of Pultruded Thin-Walled Sections”, Thin-Walled Structures. Vol. 79, June
2014, http://dx.doi.org/10.1016/j.tws.2014.01.013
Cardoso, D., Harries, K.A. and Batista, E.M. (2014), “Compressive Strength Equation
for GFRP Square Tube Columns”, Composites Part B: Engineering, Vol 59, pp 1-11.
http://dx.doi.org/10.1016/j.compositesb.2013.10.057
Cardoso, D.T., Harries, K.A. and Batista, E. (2013), “Behaviour of Pultruded GFRP
Tubes Subject to Concentric Compression”, Proceedings of Advanced Composites in
Construction (ACIC 2013), Belfast, September 2013.
133
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144
9. Appendix A – Column Tests Reports
-
Date: May/2012 Section: 25.4x3.2 mm Specimen length: 168 mm Fcr = -- Fu = 220 MPa
Visual observations: No local buckling observed No global buckling observed Stop criteria: crushing at column end
0
50
100
150
200
250
0.00 0.50 1.00 1.50 2.00
Str
ess σ
(MP
a)
Lateral Deflection δ (mm)
145
Date: June/2012 Section: 25.4x3.2 mm Specimen length: 175 mm Fcr = 345 MPa Fu = 203 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: crushing at column end
Date: June/2012 Section: 25.4x3.2 mm Specimen length: 178 mm Fcr = 354 MPa Fu = 216 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: crushing at column midheight
0
50
100
150
200
250
0.00 0.50 1.00 1.50 2.00 2.50
Str
ess σ
(MP
a)
Lateral Deflection δ (mm)
y = 344.5x - 1.6
0
0.5
1
1.5
2
2.5
0.00 0.01 0.01 0.02 0.02
δ(m
m)
δ/P (mm/MPa)
0
50
100
150
200
250
0.00 0.50 1.00 1.50 2.00
Str
ess σ
(MP
a)
Lateral Deflection δ (mm)
y = 354.3x - 1.1
00.20.40.60.8
11.21.41.61.8
0.000 0.002 0.004 0.006 0.008
δ(m
m)
δ/σ (mm/MPa)
146
Date: June/2012 Section: 25.4x3.2 mm Specimen length: 201 mm Fcr = 275 MPa Fu = 170 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: crushing at column midheight
Date: June/2012 Section: 25.4x3.2 mm Specimen length: 201 mm Fcr = 304 MPa Fu = 197 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: crushing at column end
020406080
100120140160180
0.00 1.00 2.00 3.00 4.00 5.00 6.00
Str
ess σ
(MP
a)
Lateral Deflection δ (mm)
y = 275.3x - 3.3
0
1
2
3
4
5
6
0.00 0.01 0.02 0.03 0.04
δ(m
m)
δ/σ (mm/MPa)
0
50
100
150
200
250
0.00 1.00 2.00 3.00 4.00
Str
ess σ
(kN
)
Lateral Deflection δ (mm)
y = 304.4x - 2.0
0
0.5
1
1.5
2
2.5
3
3.5
4
0.000 0.005 0.010 0.015 0.020
δ(m
m)
δ/σ (mm/MPa)
147
Date: June/2012 Section: 25.4x3.2 mm Specimen length: 220 mm Fcr = 240 MPa Fu = 189 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: crushing at column midheight
Date: May/2012 Section: 25.4x3.2 mm Specimen length: 227 mm Fcr = 257 MPa Fu = 196 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: crushing at column end
020406080
100120140160180200
0.00 2.00 4.00 6.00 8.00
Str
ess σ
(MP
a)
Lateral Deflection δ (mm)
y = 240.3x - 2.1
0
1
2
3
4
5
6
7
8
0.00 0.01 0.02 0.03 0.04 0.05
δ(m
m)
δ/σ (mm/MPa)
0
50
100
150
200
250
0.00 1.00 2.00 3.00 4.00 5.00 6.00
Str
ess σ
(MP
a)
Lateral Deflection δ (mm)
y = 257.4x - 1.6
0
1
2
3
4
5
6
0.00 0.01 0.01 0.02 0.02 0.03 0.03
δ(m
m)
δ/σ (mm/MPa)
148
Date: May/2012 Section: 25.4x3.2 mm Specimen length: 228 mm Fcr = 240 MPa Fu = 181 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: crushing at column midheight
Date: May/2012 Section: 25.4x3.2 mm Specimen length: 240 mm Fcr = 233 MPa Fu = 176 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: crushing at column midheight
020406080
100120140160180200
0.00 1.00 2.00 3.00 4.00 5.00
Str
ess σ
(MP
a)
Lateral Deflection δ (mm)
y = 240.3x - 1.6
00.5
11.5
22.5
33.5
44.5
5
0.000 0.005 0.010 0.015 0.020 0.025 0.030
δ(m
m)
δ/σ (mm/MPa)
020406080
100120140160180200
0.00 2.00 4.00 6.00 8.00
Str
ess σ
(MP
a)
Lateral Deflection δ (mm)
y = 232.8x - 2.2
0
1
2
3
4
5
6
7
0.00 0.01 0.02 0.03 0.04 0.05
δ(m
m)
δ/σ (mm/MPa)
149
Date: May/2012 Section: 25.4x3.2 mm Specimen length: 248 mm Fcr = 224 MPa Fu = 179 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: δ > L/50
Date: May/2012 Section: 25.4x3.2 mm Specimen length: 280 mm Fcr = 189 MPa Fu = 181 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: δ > L/50
020406080
100120140160180200
0.00 2.00 4.00 6.00 8.00
Str
ess σ
(MP
a)
Lateral Deflection δ (mm)
y = 224.2x - 1.8
0
1
2
3
4
5
6
7
8
0.00 0.01 0.02 0.03 0.04 0.05
δ(m
m)
δ/σ (mm/MPa)
020406080
100120140160180200
0.00 2.00 4.00 6.00 8.00
Str
ess σ
(MP
a)
Lateral Deflection δ (mm)
y = 189.2x - 0.3
0123456789
0.00 0.01 0.02 0.03 0.04 0.05
δ(m
m)
δ/σ (mm/MPa)
150
Date: May/2012 Section: 25.4x3.2 mm Specimen length: 300 mm Fcr = 145 MPa Fu = 136 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: crushing at column midheight
Date: May/2012 Section: 25.4x3.2 mm Specimen length: 350 mm Fcr = 109 MPa Fu = 96 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: δ > L/50
0
20
40
60
80
100
120
140
160
0.00 5.00 10.00 15.00 20.00
Str
ess σ
(MP
a)
Lateral Deflection δ (mm)
y = 144.7x - 0.4
0
2
4
6
8
10
12
14
16
0.00 0.05 0.10 0.15
δ(m
m)
δ/σ (mm/MPa)
0
20
40
60
80
100
120
0.00 2.00 4.00 6.00 8.00 10.00
Str
ess σ
(MP
a)
Lateral Deflection δ (mm)
y = 108.8x - 1.2
0123456789
0.00 0.02 0.04 0.06 0.08 0.10
δ(m
m)
δ/σ (mm/MPa)
151
Date: May/2012 Section: 25.4x3.2 mm Specimen length: 391 mm Fcr = 90 MPa Fu = 84 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: δ > L/50
Date: May/2012 Section: 25.4x3.2 mm Specimen length: 420 mm Fcr = 82 MPa Fu = 64 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: δ > L/50
0102030405060708090
0.00 2.00 4.00 6.00 8.00 10.00 12.00
Str
ess σ
(MP
a)
Lateral Deflection δ (mm)
y = 90.3x - 0.7
0
2
4
6
8
10
12
0.00 0.05 0.10 0.15
δ(m
m)
δ/σ (mm/MPa)
0
10
20
30
40
50
60
70
0.00 2.00 4.00 6.00 8.00 10.00 12.00
Str
ess σ
(MP
a)
Lateral Deflection δ (mm)
y = 82.2x - 3.0
0
2
4
6
8
10
12
0.00 0.05 0.10 0.15 0.20
δ(m
m)
δ/σ (mm/MPa)
152
Date: May/2012 Section: 25.4x3.2 mm Specimen length: 439 mm Fcr = 74 MPa Fu = 61 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: δ > L/50
Date: May/2012 Section: 25.4x3.2 mm Specimen length: 470 mm Fcr = 67 MPa Fu = 56 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: δ > L/50
0
10
20
30
40
50
60
70
0.00 2.00 4.00 6.00 8.00 10.00 12.00
Str
ess σ
(MP
a)
Lateral Deflection δ (mm)
y = 74.2x - 2.3
0
2
4
6
8
10
12
0.00 0.05 0.10 0.15 0.20
δ(m
m)
δ/σ (mm/MPa)
0
10
20
30
40
50
60
0.00 2.00 4.00 6.00 8.00 10.00 12.00
Str
ess σ
(MP
a)
Lateral Deflection δ (mm)
y = 67.4x - 2.4
0
2
4
6
8
10
12
0.00 0.05 0.10 0.15 0.20 0.25
δ(m
m)
δ/σ (mm/MPa)
153
Date: May/2012 Section: 25.4x3.2 mm Specimen length: 539 mm Fcr = 52 MPa Fu = 43 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: δ > L/50
Date: May/2012 Section: 25.4x3.2 mm Specimen length: 629 mm Fcr = 46 MPa Fu = 40 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: δ > L/50
05
101520253035404550
0.00 2.00 4.00 6.00 8.00 10.00 12.00
Str
ess σ
(MP
a)
Lateral Deflection δ (mm)
y = 51.9x - 2.3
0
2
4
6
8
10
12
0.00 0.05 0.10 0.15 0.20 0.25 0.30
δ(m
m)
δ/σ (mm/MPa)
05
1015202530354045
0.00 5.00 10.00 15.00 20.00
Str
ess σ
(MP
a)
Lateral Deflection δ (mm)
y = 46.3x - 2.3
0
2
4
6
8
10
12
14
16
0.00 0.10 0.20 0.30 0.40
δ(m
m)
δ/σ (mm/MPa)
154
Date: May/2012 Section: 50.8x3.2 mm Specimen length: 256 mm Fcr = -- Fcrℓ = 117 MPa Fu = 119 MPa
Visual observations: Local buckling observed No global buckling observed Stop criteria: crushing at midheight
Date: May/2012 Section: 50.8x3.2 mm Specimen length: 254 mm Fcr = -- Fcrℓ = 121 MPa Fu = 119 MPa
Visual observations: Local buckling observed No global buckling observed Stop criteria: crushing at midheight
0
20
40
60
80
100
120
140
0.00 0.50 1.00 1.50
Str
ess σ
(MP
a)
Mid-height wall deflection ∆ (mm)
y = 116.59x + 0.531
y = 67.6x + 0.00
100
200
300
400
500
600
0.00 1.00 2.00 3.00 4.00 5.00
∆(m
m)
∆/σ (mm/MPa)
0
20
40
60
80
100
120
140
0.00 0.50 1.00 1.50
Str
ess σ
(MP
a)
Mid-height wall deflection ∆ (mm)
y = 121.4x - 0.0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0.000 0.005 0.010 0.015
∆(m
m)
∆/σ (mm/MPa)
155
-
Date: June/2012 Section: 50.8x3.2 mm Specimen length: 292 mm Fcr = -- Fcrℓ = -- Fu = 107 MPa
Visual observations: Local buckling observed No global buckling observed Stop criteria: crushing at midheight Note: data recorded after 30 kN
-
Date: October/2012 Section: 50.8x3.2 mm Specimen length: 294 mm Fcr = -- Fcrℓ = -- Fu = 109 MPa
Visual observations: Local buckling observed No global buckling observed Stop criteria: crushing at midheight
0
20
40
60
80
100
120
0.00 0.50 1.00 1.50 2.00
Str
ess σ
(MP
a)
Mid-height wall deflection ∆ (mm)
0
20
40
60
80
100
120
0.00 0.50 1.00 1.50
Str
ess σ
(MP
a)
Mid-height wall deflection ∆ (mm)
156
Date: May/2012 Section: 50.8x3.2 mm Specimen length: 330 mm Fcr = -- Fcrℓ = 117 MPa Fu = 109 MPa
Visual observations: Local buckling observed No global buckling observed Stop criteria: crushing at midheight
Date: June/2012 Section: 50.8x3.2 mm Specimen length: 546 mm Fcr = -- Fcrℓ = 102 MPa Fu = 109 MPa
Visual observations: Local buckling observed No global buckling observed Stop criteria: crushing at midheight
0
20
40
60
80
100
120
0.00 0.50 1.00 1.50 2.00
Str
ess σ
(MP
a)
Mid-height wall deflection ∆ (mm)
y = 116.9x - 0.1
00.20.40.60.8
11.21.41.61.8
2
0.00 0.01 0.01 0.02 0.02
∆(m
m)
∆/σ (mm/MPa)
0
20
40
60
80
100
120
0.00 1.00 2.00 3.00 4.00
Str
ess σ
(MP
a)
Mid-height wall deflection ∆ (mm)
y = 101.7x - 0.0
0
0.5
1
1.5
2
2.5
3
3.5
0.00 0.01 0.02 0.03 0.04
∆(m
m)
∆/σ (mm/MPa)
157
Date: October/2012 Section: 50.8x3.2 mm Specimen length: 546 mm Fcr = -- Fcrℓ = -- Fu = 106 MPa
Visual observations: Local buckling observed No global buckling observed Stop criteria: crushing at midheight
Date: May/2012 Section: 50.8x3.2 mm Specimen length: 768 mm Fcr = 131MPa (initial) Fcrℓ = -- Fu = 92 MPa
Visual observations: Local buckling observed Global buckling observed Stop criteria: crushing at midheight
0
20
40
60
80
100
120
0.00 1.00 2.00 3.00 4.00
Str
ess σ
(MP
a)
Mid-height wall deflection ∆ (mm)
0
0.5
1
1.5
2
2.5
3
3.5
0.00 0.01 0.02 0.03 0.04
∆(m
m)
∆/σ (mm/MPa)
0102030405060708090
100
0.00 2.00 4.00 6.00 8.00 10.00 12.00
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
y = 131.0x - 1.4
y = 77.1x + 0.6
0
2
4
6
8
10
12
14
0.00 0.05 0.10 0.15 0.20
δ(m
m)
δ/σ (mm/MPa)
158
Date: May/2012 Section: 50.8x3.2 mm Specimen length: 824 mm Fcr = 129MPa (initial) Fcrℓ = -- Fu = 81 MPa
Visual observations: Local buckling observed Global buckling observed Stop criteria: crushing at midheight
Date: May/2012 Section: 50.8x3.2 mm Specimen length: 826 mm Fcr = 124MPa (initial) Fcrℓ = -- Fu = 105 MPa
Visual observations: Local buckling observed Global buckling observed Stop criteria: crushing at midheight
0102030405060708090
0.00 5.00 10.00 15.00
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
y = 128.6x - 2.9
0
2
4
6
8
10
12
14
0.00 0.05 0.10 0.15 0.20
δ(m
m)
δ/σ (mm/MPa)
0
20
40
60
80
100
120
0.00 2.00 4.00 6.00 8.00
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
y = 124.4x - 1.0
0
1
2
3
4
5
6
7
0.00 0.02 0.04 0.06 0.08
δ(m
m)
δ/σ (mm/MPa)
159
Date: May/2012 Section: 50.8x3.2 mm Specimen length: 851 mm Fcr = 119 MPa (initial) Fcrℓ = -- Fu = 96 MPa
Visual observations: Local buckling observed Global buckling observed Stop criteria: crushing at midheight
Date: May/2012 Section: 50.8x3.2 mm Specimen length: 910 mm Fcr = 90 MPa Fcrℓ = -- Fu = 78 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: δ > L/50
0
20
40
60
80
100
120
0.00 1.00 2.00 3.00 4.00 5.00 6.00
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
y = 118.5x - 0.8
0
1
2
3
4
5
6
0.00 0.02 0.04 0.06 0.08
δ(m
m)
δ/σ (mm/MPa)
0102030405060708090
0.00 5.00 10.00 15.00 20.00 25.00
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
y = 90.4x - 3.1
0
5
10
15
20
25
0.00 0.05 0.10 0.15 0.20 0.25 0.30
δ(m
m)
δ/σ (mm/MPa)
160
Date: May/2012 Section: 50.8x3.2 mm Specimen length: 1156 mm Fcr = 58 MPa Fcrℓ = -- Fu = 58 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: δ > L/50
Date: June/2012 Section: 50.8x3.2 mm Specimen length: 1699 mm Fcr = 27 MPa Fcrℓ = -- Fu = 25 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: δ > L/50
0
10
20
30
40
50
60
70
0.00 10.00 20.00 30.00 40.00
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
y = 58.1x - 0.1
0
5
10
15
20
25
30
35
0.00 0.10 0.20 0.30 0.40 0.50 0.60
δ(m
m)
δ/σ (mm/MPa)
0
5
10
15
20
25
30
0.00 10.00 20.00 30.00 40.00
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
y = 26.5x - 2.4
0
5
10
15
20
25
30
35
40
0.00 0.50 1.00 1.50
δ(m
m)
δ/σ (mm/MPa)
161
-
Date: August/2012 Section: 76.2x6.4 mm Specimen length: 340 mm Fcr = -- Fcrℓ = -- Fu = 235 MPa
Visual observations: No local buckling observed No global buckling observed Stop criteria: crushing at column end
-
Date: August/2012 Section: 76.2x6.4 mm Specimen length: 340 mm Fcr = -- Fcrℓ = -- Fu = 259 MPa
Visual observations: No local buckling observed No global buckling observed Stop criteria: crushing at column end
0
50
100
150
200
250
0.00 0.05 0.10 0.15 0.20
Str
ess σ
(MP
a)
Lateral deflection ∆ (mm)
0
50
100
150
200
250
300
0.00 0.50 1.00 1.50 2.00 2.50
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
162
-
Date: August/2012 Section: 76.2x6.4 mm Specimen length: 594 mm Fcr = -- Fcrℓ = -- Fu = 199 MPa
Visual observations: No local buckling observed No global buckling observed Stop criteria: crushing at column end
-
Date: August/2012 Section: 76.2x6.4 mm Specimen length: 638 mm Fcr = -- Fcrℓ = -- Fu = 218 MPa
Visual observations: No local buckling observed No global buckling observed Stop criteria: crushing at column end
0
50
100
150
200
250
0.00 0.10 0.20 0.30 0.40 0.50 0.60
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
0
50
100
150
200
250
0.00 0.05 0.10 0.15 0.20
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
163
Date: August/2012 Section: 76.2x6.4 mm Specimen length: 692 mm Fcr = 382 MPa Fcrℓ = -- Fu = 259 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: crushing at column end
Date: August/2012 Section: 76.2x6.4 mm Specimen length: 737 mm Fcr = 368 MPa Fcrℓ = -- Fu = 250 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: crushing at column end
0
50
100
150
200
250
300
0.00 1.00 2.00 3.00 4.00 5.00
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
y = 382.4x - 2.2
00.5
11.5
22.5
33.5
44.5
5
0.000 0.005 0.010 0.015 0.020
δ(m
m)
δ/σ (mm/MPa)
0
50
100
150
200
250
300
0.00 2.00 4.00 6.00 8.00 10.00 12.00
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
y = 367.8x - 4.7
0
2
4
6
8
10
12
0.00 0.01 0.02 0.03 0.04 0.05
δ(m
m)
δ/σ (mm/MPa)
164
Date: August/2012 Section: 76.2x6.4 mm Specimen length: 838 mm Fcr = 275 MPa Fcrℓ = -- Fu = 185 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: crushing at column end
Date: August/2012 Section: 76.2x6.4 mm Specimen length: 918 mm Fcr = 293 MPa Fcrℓ = -- Fu = 239 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: crushing at column end
020406080
100120140160180200
0.00 1.00 2.00 3.00 4.00
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
y = 275.2x - 1.7
0
0.5
1
1.5
2
2.5
3
3.5
4
0.000 0.005 0.010 0.015 0.020
δ(m
m)
δ/σ (mm/MPa)
0
50
100
150
200
250
300
0.00 5.00 10.00 15.00 20.00
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
y = 293.4x - 3.2
0
2
4
6
8
10
12
14
16
0.00 0.02 0.04 0.06 0.08
δ(m
m)
δ/σ (mm/MPa)
165
Date: August/2012 Section: 76.2x6.4 mm Specimen length: 1013 mm Fcr = 210 MPa Fcrℓ = -- Fu = 162 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: δ > L/50
Date: September/2012 Section: 76.2x6.4 mm Specimen length: 1264 mm Fcr = 139 MPa Fcrℓ = -- Fu = 113 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: δ > L/50
020406080
100120140160180
0.00 5.00 10.00 15.00 20.00 25.00
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
y = 210.2x - 6.3
0
5
10
15
20
25
0.00 0.05 0.10 0.15
δ(m
m)
δ/σ (mm/MPa)
0
20
40
60
80
100
120
0.00 5.00 10.00 15.00 20.00 25.00 30.00
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
y = 139.0x - 5.8
0
5
10
15
20
25
30
0.00 0.05 0.10 0.15 0.20 0.25
δ(m
m)
δ/σ (mm/MPa)
166
Date: August/2012 Section: 76.2x6.4 mm Specimen length: 1278 mm Fcr = 147 MPa Fcrℓ = -- Fu = 133 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: δ > L/50 Note: DWT angle of 7.5 deg. Results were corrected.
Date: August/2012 Section: 76.2x6.4 mm Specimen length: 1338 mm Fcr = 133 MPa Fcrℓ = -- Fu = 109 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: δ > L/50
0
20
40
60
80
100
120
140
0.00 5.00 10.00 15.00 20.00 25.00 30.00
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
y = 146.9x - 2.6
0
5
10
15
20
25
30
0.00 0.05 0.10 0.15 0.20 0.25
δ(m
m)
δ/σ (mm/MPa)
0
20
40
60
80
100
120
0.00 5.00 10.00 15.00 20.00 25.00 30.00
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
y = 133.4x - 6.0
0
5
10
15
20
25
30
35
0.00 0.05 0.10 0.15 0.20 0.25 0.30
δ(m
m)
δ/σ (mm/MPa)
167
Date: August/2012 Section: 76.2x6.4 mm Specimen length: 1437 mm Fcr = 111 MPa Fcrℓ = -- Fu = 96 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: δ > L/50
Date: August/2012 Section: 76.2x6.4 mm Specimen length: 1702 mm Fcr = 83 MPa Fcrℓ = -- Fu = 68 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: δ > L/50
0
20
40
60
80
100
120
0.00 10.00 20.00 30.00 40.00
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
y = 111.3x - 4.5
0
5
10
15
20
25
30
35
0.00 0.10 0.20 0.30 0.40
δ(m
m)
δ/σ (mm/MPa)
0
10
20
30
40
50
60
70
80
0.00 10.00 20.00 30.00 40.00
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
y = 82.6x - 7.3
0
5
10
15
20
25
30
35
40
0.00 0.10 0.20 0.30 0.40 0.50 0.60
δ(m
m)
δ/σ (mm/MPa)
168
-
Date: September/2012 Section: 88.9x6.4 mm Specimen length: 329 mm Fcr = -- Fcrℓ = -- Fu = 168 MPa
Visual observations: No local buckling observed No global buckling observed Stop criteria: crushing at column end
-
Date: September/2012 Section: 88.9x6.4 mm Specimen length: 354 mm Fcr = -- Fcrℓ = -- Fu = 208 MPa
Visual observations: No local buckling observed No global buckling observed Stop criteria: crushing at column end
020406080
100120140160180
0.00 0.05 0.10 0.15 0.20
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
0
50
100
150
200
250
0.00 0.10 0.20 0.30 0.40 0.50
Str
ess σ
(MP
a)
Mid-height wall deflection ∆ (mm)
169
-
Date: September/2012 Section: 88.9x6.4 mm Specimen length: 506 mm Fcr = -- Fcrℓ = -- Fu = 210 MPa
Visual observations: No local buckling observed No global buckling observed Stop criteria: crushing at column midheight
-
Date: September/2012 Section: 88.9x6.4 mm Specimen length: 635 mm Fcr = -- Fcrℓ = -- Fu = 212 MPa
Visual observations: No local buckling observed No global buckling observed Stop criteria: crushing at column midheight
0
50
100
150
200
250
0.00 0.05 0.10 0.15
Str
ess σ
(MP
a)
Mid-height wall deflection ∆ (mm)
0
50
100
150
200
250
0.00 1.00 2.00 3.00 4.00 5.00
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
170
Date: September/2012 Section: 88.9x6.4 mm Specimen length: 800 mm Fcr = 368 MPa Fcrℓ = -- Fu = 193 MPa
Visual observations: No local buckling observed No global buckling observed Stop criteria: crushing at column midheight
Date: September/2012 Section: 88.9x6.4 mm Specimen length: 935 mm Fcr = 197 MPa Fcrℓ = -- Fu = 177 MPa
Visual observations: No local buckling observed No global buckling observed Stop criteria: crushing at column midheight
0
50
100
150
200
250
0.00 1.00 2.00 3.00 4.00 5.00
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
y = 367.6x - 3.6
00.5
11.5
22.5
33.5
44.5
0.00 0.01 0.02 0.03 0.04
δ(m
m)
δ/σ (mm/MPa)
020406080
100120140160180200
0.00 0.10 0.20 0.30 0.40 0.50
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
y = 196.5x - 0.1
00.05
0.10.15
0.20.25
0.30.35
0.40.45
0.5
0.00 0.00 0.00 0.01 0.01
δ(m
m)
δ/σ (mm/MPa)
171
Date: September/2012 Section: 88.9x6.4 mm Specimen length: 949 mm Fcr = 284 MPa Fcrℓ = -- Fu = 185 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: crushing at column midheight
Date: September/2012 Section: 88.9x6.4 mm Specimen length: 1041 mm Fcr = 273 MPa Fcrℓ = -- Fu = 181 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: crushing at column midheight
020406080
100120140160180200
0.00 2.00 4.00 6.00 8.00
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
y = 284.0x - 3.1
0
1
2
3
4
5
6
7
0.00 0.01 0.02 0.03 0.04
δ(m
m)
δ/σ (mm/MPa)
020406080
100120140160180200
0.00 2.00 4.00 6.00 8.00 10.00
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
y = 273.0x - 3.9
0123456789
0.00 0.02 0.04 0.06 0.08
δ(m
m)
δ/σ (mm/MPa)
172
Date: September/2012 Section: 88.9x6.4 mm Specimen length: 1149 mm Fcr = 208 MPa Fcrℓ = -- Fu = 166 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: crushing at column midheight
Date: September/2012 Section: 88.9x6.4 mm Specimen length: 1337 mm Fcr = 163 MPa Fcrℓ = -- Fu = 156 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: crushing at column midheight
020406080
100120140160180
0.00 5.00 10.00 15.00 20.00
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
y = 208.3x - 3.6
02468
1012141618
0.00 0.02 0.04 0.06 0.08 0.10
δ(m
m)
δ/σ (mm/MPa)
020406080
100120140160180
0.00 5.00 10.00 15.00 20.00 25.00 30.00
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
y = 163.0x - 1.1
0
5
10
15
20
25
30
35
0.00 0.05 0.10 0.15 0.20
δ(m
m)
δ/σ (mm/MPa)
173
Date: September/2012 Section: 88.9x6.4 mm Specimen length: 1468 mm Fcr = 137 MPa Fcrℓ = -- Fu = 140 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: δ > L/50
Date: September/2012 Section: 88.9x6.4 mm Specimen length: 1800 mm Fcr = 93 MPa Fcrℓ = -- Fu = 90 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: δ > L/50
0
20
40
60
80
100
120
140
160
0.00 10.00 20.00 30.00 40.00
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
y = 136.9x - 0.3
0
5
10
15
20
25
30
35
0.00 0.05 0.10 0.15 0.20 0.25
δ(m
m)
δ/σ (mm/MPa)
0102030405060708090
100
0.00 10.00 20.00 30.00 40.00
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
y = 92.6x - 1.2
0
5
10
15
20
25
30
35
40
0.00 0.10 0.20 0.30 0.40 0.50
δ(m
m)
δ/σ (mm/MPa)
174
Date: September/2012 Section: 88.9x6.4 mm Specimen length: 1994 mm Fcr = 76 MPa Fcrℓ = -- Fu = 79 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: δ > L/50
Date: September/2012 Section: 88.9x6.4 mm Specimen length: 2103 mm Fcr = 69 MPa Fcrℓ = -- Fu = 71 MPa
Visual observations: No local buckling observed (naked eyes) Global buckling observed Stop criteria: δ > L/50
0102030405060708090
0.00 10.00 20.00 30.00 40.00 50.00
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
y = 75.8x - 0.2
05
1015202530354045
0.00 0.10 0.20 0.30 0.40 0.50 0.60
δ(m
m)
δ/σ (mm/MPa)
0
10
20
30
40
50
60
70
80
0.00 10.00 20.00 30.00 40.00 50.00
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
y = 69.1x + 0.0
05
1015202530354045
0.00 0.20 0.40 0.60 0.80
δ(m
m)
δ/σ (mm/MPa)
175
Date: September/2012 Section: 102x6.4 mm Specimen length: 302 mm Fcr = -- Fcrℓ = 151 MPa Fu = 162 MPa
Visual observations: Local buckling observed No global buckling observed Stop criteria: crushing at column end
Date: October/2012 Section: 102x6.4 mm Specimen length: 352 mm Fcr = -- Fcrℓ = 170 MPa Fu = 175 MPa
Visual observations: No local buckling observed No global buckling observed Stop criteria: crushing at column midheight
020406080
100120140160180
0.00 0.05 0.10 0.15 0.20
Str
ess σ
(MP
a)
Mid-height wall deflection ∆ (mm)
y = 150.5x + 0.0
00.020.040.060.08
0.10.120.140.160.18
0.2
0.000 0.002 0.004 0.006 0.008 0.010 0.012
∆(m
m)
∆/σ (mm/MPa)
020406080
100120140160180200
-0.50 0.00 0.50 1.00 1.50 2.00
Str
ess σ
(MP
a)
Mid-height wall deflection ∆ (mm)
y = 169.7x + 0.0
00.20.40.60.8
11.21.41.61.8
0.00 0.00 0.00 0.01 0.01 0.01
∆(m
m)
∆/σ (mm/MPa)
176
-
Date: September/2012 Section: 102x6.4 mm Specimen length: 506 mm Fcr = -- Fcrℓ = -- Fu = 164 MPa
Visual observations: Local buckling observed No global buckling observed Stop criteria: crushing at column midheight
-
Date: October/2012 Section: 102x6.4 mm Specimen length: 757 mm Fcr = -- Fcrℓ = -- Fu = 167 MPa
Visual observations: Local buckling observed No global buckling observed Stop criteria: crushing at column midheight
020406080
100120140160180
-0.20 0.00 0.20 0.40 0.60
Str
ess σ
(MP
a)
Mid-height wall deflection ∆ (mm)
020406080
100120140160180
-0.80 -0.60 -0.40 -0.20 0.00
Str
ess σ
(MP
a)
Mid-height wall deflection ∆ (mm)
177
-
Date: October/2012 Section: 102x6.4 mm Specimen length: 799 mm Fcr = -- Fcrℓ = -- Fu = 163 MPa
Visual observations: Local buckling observed No global buckling observed Stop criteria: crushing at column midheight
-
Date: October/2012 Section: 102x6.4 mm Specimen length: 1160 mm Fcr = -- Fcrℓ = -- Fu = 161 MPa
Visual observations: Local buckling observed No global buckling observed Stop criteria: crushing at column midheight
020406080
100120140160180
0.00 0.20 0.40 0.60 0.80 1.00 1.20
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
020406080
100120140160180
-0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80
Str
ess σ
(MP
a)
Mid-height wall deflection ∆ (mm)
178
Date: October/2012 Section: 102x6.4 mm Specimen length: 1318 mm Fcr = 181 MPa Fcrℓ = -- Fu = 150 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: crushing at column midheight
Date: October/2012 Section: 102x6.4 mm Specimen length: 1468 mm Fcr = 140 MPa Fcrℓ = -- Fu = 140 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: crushing at column midheight
0
20
40
60
80
100
120
140
160
0.00 5.00 10.00 15.00 20.00
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
y = 181.0x - 3.3
02468
101214161820
0.00 0.05 0.10 0.15
δ(m
m)
δ/σ (mm/MPa)
0
20
40
60
80
100
120
140
160
-5.00 0.00 5.00 10.00 15.00 20.00 25.00 30.00
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
y = 139.8x - 0.9
0
5
10
15
20
25
30
0.00 0.05 0.10 0.15 0.20
δ(m
m)
δ/σ (mm/MPa)
179
Date: October/2012 Section: 102x6.4 mm Specimen length: 1559 mm Fcr = 149 MPa Fcrℓ = -- Fu = 120 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: crushing at column midheight
Date: October/2012 Section: 102x6.4 mm Specimen length: 1726 mm Fcr = 112 MPa Fcrℓ = -- Fu = 94 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: δ > L/50
0
20
40
60
80
100
120
140
-10.00 0.00 10.00 20.00 30.00 40.00
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
y = 148.8x - 4.7
0
5
10
15
20
25
30
35
0.00 0.05 0.10 0.15 0.20 0.25 0.30
δ(m
m)
δ/σ (mm/MPa)
0102030405060708090
100
0.00 10.00 20.00 30.00 40.00
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
y = 111.6x - 5.4
0
5
10
15
20
25
30
35
40
0.00 0.10 0.20 0.30 0.40
δ(m
m)
δ/σ (mm/MPa)
180
Date: October/2012 Section: 102x6.4 mm Specimen length: 1918 mm Fcr = 93 MPa Fcrℓ = -- Fu = 81 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: δ > L/50
Date: October/2012 Section: 102x6.4 mm Specimen length: 1964 mm Fcr = 83 MPa Fcrℓ = -- Fu = 83 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: δ > L/50
0102030405060708090
0.00 10.00 20.00 30.00 40.00 50.00
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
y = 92.8x - 5.2
05
1015202530354045
0.00 0.10 0.20 0.30 0.40 0.50 0.60
δ(m
m)
δ/σ (mm/MPa)
0102030405060708090
-10.00 0.00 10.00 20.00 30.00 40.00 50.00
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
y = 83.0x - 0.6
05
1015202530354045
0.00 0.10 0.20 0.30 0.40 0.50 0.60
δ(m
m)
δ/σ (mm/MPa)
181
Date: October/2012 Section: 102x6.4 mm Specimen length: 2171 mm Fcr = 74 MPa Fcrℓ = -- Fu = 68 MPa
Visual observations: No local buckling observed Global buckling observed Stop criteria: δ > L/50
0
10
20
30
40
50
60
70
80
-10.00 0.00 10.00 20.00 30.00 40.00 50.00
Str
ess σ
(MP
a)
Lateral deflection δ (mm)
y = 73.7x - 3.4
05
101520253035404550
0.00 0.20 0.40 0.60 0.80
δ(m
m)
δ/σ (mm/MPa)