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Influence of boundary conditions and geometric imperfections on
lateral–torsional buckling resistance of a pultruded FRP I-beam by FEA
T.T. Nguyen, T.M. Chan
⇑
, J.T. Mottram
School of Engineering, University of Warwick, Coventry CV4 7AL, UK
article info
Article history:
Available online 11 January 2013
Keywords:
Lateral–torsional buckling
Geometric non-linear finite element analysis
Fibre reinforced polymer
Load height
Warping restraint
Initial geometric imperfections
abstract
Presented are results from geometric non-linear finite element analyses to examine the Lateral Torsional
Buckling (LTB) resistance of a Pultruded Fibre Reinforced Polymer (FRP) I-beam when initial geometric
imperfections associated with the LTB mode shape are introduced. A data reduction method is proposed
to define the limiting buckling load and the method is used to present strength results for a range of beam
slendernesses and geometric imperfections. Prior to reporting on these non-linear analyses, Eigenvalue
FE analyses are used to establish the influence on resistance of changing load height or displacement
boundary conditions. By comparing predictions for the beam with either FRP or steel elastic constants
it is found that the former has a relatively larger effect on buckling strength with changes in load height
and end warping fixity. The developed finite element modelling methodology will enable parametric
studies to be performed for the development of closed form formulae that will be reliable for the design
of FRP beams against LTB failure.
Ó2013 Elsevier Ltd. All rights reserved.
1. Introduction
Shapes and systems of Fibre Reinforced Polymer (FRP) made by
the pultrusion processing method [1] are finding applications in ci-
vil engineering works, alongside construction components of con-
ventional materials such as steel, concrete and aluminium.
Standard Pultruded FRP (PFRP) shapes are thin-walled and have
the same cross-sectional shapes as found in conventional steel-
work. They consist of E-glass fibre reinforcement (layers of unidi-
rectional rovings and continuous mats) in a thermoset resin
based matrix. Layers are not necessarily of constant thickness or
flat. Technical information on the pultrusion process, and PFRP
shapes themselves, is found in American pultruders’ Design Manu-
als [2,3]. Pultruded material has a density at 1900 kg/m
3
that is un-
der one-quarter the density of structural steel. Although these
PFRP shapes are similar to steel sections, their structural behaviour
is different. Direct strengths (in the direction of pultrusion) can be
over 200 N/mm
2
that is comparable with structural steel. However,
the longitudinal (tensile or compressive) modulus of elasticity (E
L
)
is up to 10 times lower at 20–30 kN/mm
2
. The modulus of elastic-
ity perpendicular to the direction of pultrusion (E
T
) is about one-
third of the longitudinal value [2,3]. The in-plane shear modulus
(G
LT
) can be between one-tenth and one-quarter of the value of
E
L
. In terms of the material response to direct stress under static
loading, it is virtually linear elastic to failure and can be taken as
linear elastic under shear to the shear strains experienced in prac-
tice. If loading is over the long-term the viscoelastic nature of FRP
will make the material response non-linear. Worldwide the num-
ber of engineering structures using pultruded standard shapes
has increased in recent years, for the reasons given in Mottram
[1] and Bank [4].
A major factor preventing greater penetration of standard PFRP
shapes in civil engineering works is the lack of recognised guide-
lines for practitioners to have full confidence in their structural de-
signs, say, for framed structures. One source of guidance is from
the pultrusion companies who have Design Manuals [2,3] specific
to their range of pultruded shapes. Further guidance, that has no
legal standing, can be found in the 1996 EUROCOMP Design Code
and Handbook [5] and the more recent Italian publication [6] that
is a guide for the design and construction of structures made of FRP
pultruded elements. The Pultrusion Industry Council of the Amer-
ican Composites Manufacturers Association commenced an action,
in 2007, of preparing a design standard for the ‘Load and Resistance
Factor Design of Pultruded Fibre-Reinforced Polymer Structures’
[7]. A pre-standard was finalised in 2010, and it is expected that
the guidelines can be adopted in 2013 as a national ASCE standard.
To have confidence in the resistance formulae in a design standard
it is desirable to have them evaluated by a combination of compu-
tational modelling and physical testing.
Owing to the high strength-to-stiffness ratio of the material, the
design of PFRP members (in braced frames of simple construction)
is normally governed by elastic deflections and/or elastic buckling
0263-8223/$ - see front matter Ó2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.compstruct.2012.12.023
⇑
Corresponding author. Tel.: +44 (0)24 765 22106; fax: +44 (0)24 76 418922.
E-mail addresses: T.Nguyen-Tien@warwick.ac.uk (T.T. Nguyen), T.M.Chan@
warwick.ac.uk (T.M. Chan), J.T.Mottram@warwick.ac.uk (J.T. Mottram).
Composite Structures 100 (2013) 233–242
Contents lists available at SciVerse ScienceDirect
Composite Structures
journal homepage: www.elsevier.com/locate/compstruct
instabilities and rarely by material strength limitations [5,8].In
other words, to execute the Ultimate Limit State (ULS) design ap-
proach the different instability modes must be quantified by a
combination of rigorous numerical analysis and physical testing.
As a type of global instability, Lateral–Torsional Buckling (LTB) is
commonly observed in laterally unrestrained beams of open
thin-walled sections that are subjected to flexure about their major
axis. The beam loses stability in the LTB mode when the member
bends laterally and twists along its length, without any cross-sec-
tion distortion. Because a beam’s moment of resistance to LTB is
influenced by having relatively small lateral (minor-axis) flexure
and torsional stiffness, it is observed that PFRP I-shapes are suscep-
tible to this failure mode governing in ULS design [9].
Researchers can determine LTB resistance, either by a theoreti-
cal treatment (analytical or computational) or by physical testing.
Because the latter approach is expensive and technically challeng-
ing, for the reasons that are uncovered from assessing the reported
test series [9–13], a theoretical treatment is always required to
populate parameters that cannot be characterised by physical test-
ing [13–15].
Numerical models can be divided into those that account for
geometric imperfections and those that assume the beam’s geom-
etry is perfect. It is the analytical solutions for the problem of a
‘perfect’ isotropic beam with constant moment along length that
gives the well-known closed form formula for the elastic critical
buckling moment of resistance (M
cr
). This M
cr
solution is the theo-
retical upper bound resistance that forms the basic strength value
towards the preparation of guidelines for the reliable and safe de-
sign of structural steel I-beams, as given in design standards BS EN
1993-1-1:2005 and ANSI/AISC 360-10.
For doubly symmetric cross-sectional (steel) beams, the lateral–
torsional buckling resistance moment may be calculated using
[16]:
M
cr
¼C
1
p
2
EI
z
ðkLÞ
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
k
k
w
2
I
w
I
z
þðkLÞ
2
GJ
p
2
EI
z
þðC
2
z
g
Þ
2
sC
2
z
g
8
<
:9
=
;
ð1Þ
The variables in Eq. (1) are defined in the nomenclature section.
This equation can be used to determine M
cr
when calculating LTB
resistance in accordance with the design procedure in Eurocode 3
for steel (BS EN 1993-1-1:2005).
In the specific case, when loading is a vertical point load (P)at
mid-span, and beam ends are free to warp (k
w
= 1.0) and allowed
to rotate about major and minor axes (k= 1.0), the buckling resis-
tance load P
cr
is given by:
P
cr
¼5:39
p
2
EI
z
L
3
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
I
w
I
z
þL
2
GJ
p
2
EI
z
þð0:63z
g
Þ
2
s0:63z
g
8
<
:9
=
;
ð2Þ
In Eq. (2) z
g
is the height of the load from the shear centre. z
g
is 0.0
at the shear centre and positive when above and negative when
below.
Nomenclature
e
y
load eccentricity displacement normal to the minor axis
plane of the beam cross-section (mm)
bbreadth of I-beam (mm)
hheight of I-beam (mm)
krestraint factor for lateral flexural bending at end sup-
ports; 0.5 for full restraint to 1.0 for fully unrestrained
k
w
restraint factor for warping at end supports; 0.5 for full
restraint to 1.0 for fully unrestrained
C
1
factor to account for the type of moment distribution
and support condition
C
2
factor to account for the vertical position of the load
with respect to the shear centre (centroidal axis) of
the I-beam
C
b
moment modification factor for non-uniform moment
distribution for laterally unsupported span when both
ends of the beam are braced
Emodulus of elasticity (kN/mm
2
)
E
L
longitudinal modulus of elasticity (kN/mm
2
)
E
T
transverse modulus of elasticity (kN/mm
2
)
Gshear modulus (kN/mm
2
)
G
LT
in-plane shear modulus (kN/mm
2
)
m
Poisson’s ratio (for steel)
m
LT
major Poisson’s ratio
I
w
warping rigidity (mm
6
)
I
y
second moment of area for flexure about the beam’s
major axis (mm
4
)
I
z
second moment of area for flexure about the beam’s
minor axis (mm
4
)
Jtorsional rigidity (mm
4
)
Lspan of beam or height of column for defining magni-
tude of out-of-straightness (mm)
wvertical deflection of shear centre at mid-span (mm)
M
cr
elastic critical buckling moment of resistance (kN m)
M
crp
elastic critical buckling moment including the influence
of pre-buckling displacements (kN m)
Pcentral point load (kN)
P
cr
elastic critical buckling load of simply supported beam
subjected to central point loading (kN)
P
cr,FEA
elastic critical buckling load obtained from linear
(Eigenvalue) finite element analysis (kN)
P
Limit
limiting buckling load (kN)
P
Limit
/P
cr
normalized buckling load for influence of geometric
imperfections
P
cr,T
elastic critical buckling load when Pis applied on top
flange (kN)
P
cr,S
elastic critical buckling load when Pis applied at shear
centre (kN)
P
cr,B
elastic critical buckling load of beam Pis applied on bot-
tom flange (kN)
P
cr,Fixed
elastic critical buckling load of beam with end warping
fully fixed (k
w
= 0.5) (kN)
P
cr,Free
elastic critical buckling load of beam with end warping
fully Free (k
w
= 1.0) (kN)
xdistance along beam from one end the other end (mm)
z
g
distance (height) from the shear centre to the point of
load application (mm)
D
s
maximum lateral deformation due to out-of-straight-
ness geometric imperfection (mm)
D
sx
initial lateral deformation distribution along the beam
due to out-of-straightness geometric imperfection
(mm)
D
t
maximum angle of twist due to twist geometric imper-
fection (°)
D
tx
initial twist distribution along the beam due to twist
geometric imperfection (°)
U
x
displacement in X-direction (mm)
U
y
displacement in Y-direction (mm)
U
z
displacement in Z-direction (mm)
U
Rx
rotation about X-axis (°)
234 T.T. Nguyen et al. / Composite Structures 100 (2013) 233–242
According to American steel standard ANSI/AISC 360-10 M
cr
may also be determined for I-beams from the simpler formula:
M
cr
¼C
b
p
Lffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
I
w
I
z
þL
2
GJ
p
2
EI
z
sð3Þ
Unlike Eqs. (1) and (2), Eq. (3) is valid only when the level of loading
coincides with the beam’s shear centre (i.e. for z
g
is zero).
Italian guidelines of 2007 [6] recommend that M
cr
be calculated
using Eq. (1) with both restraint factors kand k
w
equal to either 1.0
or 0.5. The Design Manuals from two American pultruders [2,3]
recommend that designers calculate M
cr
using Eq. (1), with kand
k
w
assumed to be equal to 1.0 (for the lowest LTB moment of resis-
tance). Moreover because they take z
g
= 0.0, the allowable load ta-
bles in [2,3] are ignoring the reduction in strength from having
load applied on the top flange. Reliable calculation of the elastic
critical buckling strength does require the two elastic constants
E
L
and G
LT
to be measured, either by testing coupons or full-sec-
tions [4].
The American pre-standard for PFRP structures [7] recommends
M
cr
to be calculated using Eq. (3), with the expression for C
b
formula taken directly from ANSI/AISC 360-10, and with the
‘full-section’ elastic constants of Eand Greplaced, respectively,
by the Longitudinal modulus of elasticity E
L
and the in-plane shear
modulus G
LT
for the PFRP material.
The aim of the work reported in this paper is to use Finite Ele-
ment Analysis (FEA) to study the change in LTB resistance of a, sin-
gle sized, PFRP I-beam at different spans that is influenced by
varying load height, end displacement boundary conditions and
initial geometric imperfections. The shape is chosen from the stan-
dard range of I-beams pultruded in America [2,3]. To be consistent
with the set-up in a series of physical tests at the University of
Warwick, the problem analysed, using ABAQUS
Ò
, is that of a simply
supported beam (for major axis flexure) having a vertical point
load at mid-span. Verification of the correct FE modelling method-
ology to simulate what can exist in practice will be done by com-
bining results from the experimental and finite element work.
Additional numerical results from FEA will be used with Eq. (2)
and the ‘curve fitting’ method of Dutheil [17] to account for the
modal interaction of local and global buckling modes [18,19],to
prepare a calibrated closed form formula for clauses in a future
Eurocode design standard for FRP material elements.
To ensure the FEA results are appropriate when the beam pos-
sesses initial geometric imperfections it is necessary to use the
geometric non-linear solver in ABAQUS
Ò
. It will be shown that
the calculated vertical load against vertical deflection response of
a PFRP beam does not always give a distinct buckling (bifurcation)
point to signal the onset of LTB failure. To overcome the absence in
numerical calculations of a definite elastic critical buckling load
(for the classical bifurcation point) a data reduction method is pre-
sented by the authors that will provide the limiting buckling load
from the Finite Element (FE) results.
2. Finite element modelling methodology
In this study the nominal geometry of the cross-section for the
PFRP I-beam is 101.6 50.8 6.4 mm (or 4 21/4 in. [2,3]).
The height has the notation h, which is used to define beam slen-
derness through non-dimensional ratio L/h.InFig. 1 the global
Cartesian (XYZ) co-ordinate system is shown. The Y-axis is for
the section’s major axis, the Z-axis is for the minor axis and the
X-axis is along the beam’s centroidal axis. The nominal values for
the sectional properties are: I
z
is 1.41 10
5
mm
4
,I
w
is
3.16 10
8
mm
6
and Jis 1.59 10
4
mm
4
. These are required later
when Eq. (2) is used to calculate the elastic critical buckling load,
P
cr
. The flanges and web panels are given orthotropic properties,
defined by the four elastic constants of E
L
,E
T
,G
LT
and
m
LT
[4,9]. ABA-
QUS
Ò
[20] offers several options to input mechanical properties
including: a ‘microscopic’ approach (micromechanical modelling
requiring constituent matrix and fibre reinforcement properties);
a ‘macroscopic’ approach (where a layer or panel is taken to be a
single orthotropic material); a ‘mixed’ approach (the panel is mod-
elled as a lamination comprising a number of discrete ‘macro-
scopic’ orthotropic layers). Because the four elastic constants
have previously been determined by coupon testing and micro-
mechanical modelling [19,21] the ‘macroscopic’ approach is appro-
priate for this FE work.
It is observed that mechanical properties of PFRP shapes can
change with pultruder. This change is due to differences in: how
a property is measured; processing conditions; the matrix; the
number of reinforcement layers; the fibre arrangement and the fi-
bre volume fractions. To illustrate the likely differences, Table 1
collates test data for the in-plane shear modulus (G
LT
). Listed in
column (2) are ranges of values taken from the nine sources
[2,3,19,22–27] defined in column (1). Shear modulus is difficult
to measure because it is challenging to have a representative vol-
ume of PFRP material subject to pure shearing. Using knowledge
of measurements of the elastic constants and the findings of the re-
view by Mottram [28], for the determination of in-plane shear
modulus, engineering judgement has been used to establish that,
for this FE work, E
L
is 24 kN/mm
2
,E
T
is 8 kN/mm
2
,G
LT
is 4 kN/
mm
2
and
m
LT
is 0.3. It is noted from the literature that the mechan-
ical properties of flange and web material are actually slightly dif-
ferent. Since flanges and web thin-walled panels are assumed to be
of the same orthotropic material, any influence this secondary
Ux= Uy= Uz= URx= 0
Rigid end plate
Shear centre node
Shear centre node
Uy = Uz= URx = 0
Rigid end plate
Fig. 1. FE Cartesian coordinate system, and mesh with displacement end boundary conditions BC2 (for k= 1 and k
w
= 0.5).
T.T. Nguyen et al. / Composite Structures 100 (2013) 233–242 235
difference might have on FE result has been ignored. Moreover, E
L
and E
T
are for direct tension (or direct compression) and, because
of the layered construction, are not the moduli if the panel is sub-
jected to flexure. By correctly assuming that the material response
is linear elastic to a strain that will exceed the strain at LTB failure,
the FEA does not need to consider a material failure criterion.
Moreover, the influence, if any, of residual stresses throughout
the cross-section, which are unknown in pultruded FRP shapes, is
neglected too.
The choice of element to create a mesh is between solid and
shell types of elements. Linear elastic flat shell elements have suc-
cessfully been adopted in previous FE studies with thin-walled FRP
structures [10,11,29,30]. The element chosen is the second-order
ABAQUS
Ò
/Standard thick shell element S8R, having 8-nodes and
six degrees of freedom per node. The formulation for the element
stiffness matrix adopts the Mindlin plate theory (for first-order
shear deformation) and so this thick shell element has displace-
ment compatibility that avoids there being any discontinuities be-
tween element sides. This FE modelling attributes is known to give
a more accurate shell element in a coarser mesh [31].
Following the boundary conditions defined by Trahair [32] for a
simply supported beam, both ends are fully restrained for Yand Z
translational displacements (U
y
and U
z
) and rotation about the X-
axis (U
Rx
). Ends are free to rotate about major and minor axes
(for k= 1.0) and allowed freely warp (k
w
= 1.0). Because effective
length factors kand k
w
may vary from 0.5 to 1.0, a parametric
study could consider a number of distinct displacement boundary
conditions. FEA results will be presented later for the two end con-
ditions of either having both kand k
w
= 1.0, or having k= 1.0 and
k
w
= 0.5. In what follows these two displacement boundary condi-
tions are given labels BC1 and BC2.
BC1 is specified in the FE modelling by setting U
y
=U
z
=U
Rx
=0
for all the nodes in the cross-section, at both ends. At one end,
the node located at the cross-section’s shear centre has its U
x
nodal
displacement set to zero to remove a rigid-body movement. To
model the BC2 end condition the ends are fixed against warping.
As Fig. 1 shows a vertical plate, comprising of R3D4 rigid surface
elements, is added at the ends. Because the beam flanges must de-
form in accordance with the movement of the ‘rigid’ plate this
modelling feature fully restrains the warping. As the movement
of the rigid plate is controlled by a single reference node at each
end, translational and rotational displacement restraints are im-
posed to the node at the I-beam’s centroid. Fig. 1 shows that to ob-
tain BC2 one end requires this node has U
x
=U
y
=U=U
Rx
= 0, while
at the other end the equivalent node has U
y
=U
z
=U
Rx
=0.
Mesh specification has the shell elements constructed with an
aspect ratio close to one, as this can eliminate any loss in numerical
reliability due to computation for the inclusion of shear flexibility.
To create the beam’s mesh the shell elements are placed at the
mid-planes of the two flanges and web panels. Equally spaced,
there are five nodes (four shell elements) across the 50.8 mm
flange width and nine nodes (eight elements) over the web’s height
of 95.6 mm. This mesh specification gives an element with side
lengths of 12.7 by 12.7 mm for flanges and 11.9 by 12.7 mm for
web. As shown in Fig. 1 1 m length of the 101.6 50.8 6.4 mm
I-shape requires 1108 elements.
Table 2 reports the elastic critical buckling load (P
cr,FEA
) as the
mesh size is refined. These FE results are from Eigenvalue analyses
with a perfectly straight beam, BC1 end conditions and shear cen-
tre point load (P) at mid-span. It can be seen that there is insignif-
icant change in P
cr,FEA
when the element side length is <12.7 mm.
However, it is seen that on doubling the side length to 25.4 mm
the calculated bifurcation load is increased by 10%. Shell elements
with side lengths of 12.7 mm or less are used in the mesh specifi-
cation for the FE results presented next.
The parametric studies have numerical simulations for both lin-
ear Eigenvalue analyses and geometric non-linear analyses. Eigen-
value buckling analysis predicts the elastic critical buckling load
(P
cr,FEA
) of a linear elastic beam where the change in beam geome-
try is neglected on increasing loading, up to the bifurcation. By
applying perturbations to the mesh geometry of the unloaded
beam, and looking for local and global deflections that could pro-
mote the onset of instability due to second-order effects, the FEA
gives load factors (the Eigenvalues) for buckling failures. The input-
ted load in the FE model is multiplied by the outputted load factor
to obtain the elastic critical buckling load. The associated eigenvec-
tor to each Eigenvalue establishes the corresponding mode shape.
The mode shape describes how the structure buckles, but gives no
information for actual load–deflection response. It can be expected
that the lowest Eigenvalue is for the mode having the lowest P
cr,FEA
.
In contrast, a non-linear geometric analysis predicts the actual
load–deflection response by applying the load in small increments
and evaluating the current (static equilibrium) deformation state
at each increment. The load follows the deformation of the linear
elastic beam until instability occurs, and this corresponds to what
will happen in practice. There is no material non-linearity as it is
assumed the PFRP material behaves perfectly linear elastic. This
modelling assumption is acceptable, providing loading (to failure)
is short-term and deformations from material viscoelasticity
Table 1
Previous test or micromechanical values for the in-plane shear modulus (G
LT
) of PFRP material in standard shapes.
Author(s) or pultruder’s Design Manual In-plane shear modulus, G
LT
kN/mm
2
Test or prediction method Pultruder
(1) (2) (3) (4)
Strongwell [2] Null
Creative Pultrusion Inc. [3] 2.9 Full-section Creative Pultrusions Inc.
Bank [22] 2.4–2.8 Isopescu Creative Pultrusions Inc.
Sonti and Barbero [23] 3.9–4.5 Isopescu and Torsion Creative Pultrusions Inc.
Zureick and Scott [24] 4.1–4.8 Isopescu Strongwell
Steffen [25] 3.5–4.5 Modified (Isopescu) Strongwell
V-notched beam
Turvey [26] 3.0–3.6 Torsion Strongwell
Roberts and Al-Ubaidi [27] 4.4–4.9 Torsion Fiberforce Composites
Lane [19] 3.2–3.7 Resin burn-off and micromechanical modelling Creative Pultrusions Inc.
Table 2
Elastic critical buckling load for lateral–torsional buckling with mesh refinement.
Shell element side
lengths (mm)
Number of elements
per metre
Elastic critical buckling
load, P
cr,FEA
(kN)
25.4 280 11.0
12.7 1108 9.96
8.5 2478 9.95
6.4 4368 9.95
5.1 6860 9.95
236 T.T. Nguyen et al. / Composite Structures 100 (2013) 233–242
remain small. ABAQUS
Ò
[33] will solve the problem of a geometric
non-linear structural problem, having a falling load–deflection
branch, following instability failure, by employing a modified Riks
method. This commonly used non-linear numerical method, also
known as the arc-length method, was originally derived by Riks
[34], following on from pioneering work by Wempner [35]. Later
the arc length method was improved for computational efficiency
by Crisfield [36].
As the post-buckling response is not the main topic under con-
sideration, the Riks analysis was terminated a few increments after
the beam had become unstable and its deformation was progress-
ing into the post-buckling region.
Initial geometric imperfections are introduced into the beam’s
FE mesh by modifying the nodal coordinates through the adoption
of a vector field. The out-of-straightness imperfection was
obtained by scaling the first Eigenvalue buckling mode shape for
Euler (flexural) buckling of a perfectly straight concentrically
loaded column. The deformed shape (exaggerated) from the
Eigenvalue analysis is shown in Fig. 2. The geometric definition
for the twist imperfection was acquired from the deformed shape
of an ‘imperfection-free’ beam subjected to a pure ‘twisting’
moment, that was generated by a torque at the free end, created
by applying there a couple of magnitude Pb (Fig. 3). The static
analysis deformation for twisting along the length of the I-shape
is shown in Fig. 4.
3. Influence of load height
LTB resistance is influenced by the vertical distances of load (z
g
)
from the shear centre due to the additional torque about the longi-
tudinal (centroidal) axis that is generated from the lateral move-
ment of the vertical point load when instability happens. Because
the torque acts in the opposite sense to the LTB twist rotation
when the load is applied below the shear centre the buckling resis-
tance will increase. Likewise, when load acts above the shear cen-
tre, the additional torque acts with the beam’s rotation to decrease
buckling resistance. This behaviour has been confirmed by Sapkás
and Kollár [37] and Machado and Cortínez [38] in their numerical
studies on the same instability mode of failure.
In this study, the effect on resistance of changing load height z
g
is established by Eigenvalue analyses that obtain the bifurcation
load with displacement boundary conditions BC1. The span (L) var-
ies from 1.0 to 5.0 m; which is for slenderness (L/h) ratios from
about 10 to 50. In the FE model the mid-span point load is posi-
tioned along the Z–Zaxis at the top flange and web junction
(z
g
= 47.6 mm), shear centre (z
g
= 0.0 mm) and at bottom flange
and web junction (z
g
=47.6 mm). Note that these values of z
g
are 3.2 mm below what they will be when the actual beam section
is loaded. By changing the elastic constants to those for structural
steel (Eis 210 kN/mm
2
and
m
is 0.3) the effect on P
cr,FEA
for a beam
of this isotropic material is established. To distinguish between the
P
cr,FEA
s for the three load heights they are given the notation P
cr,T
,
P
cr,B
and P
cr,S
for load at Top flange, at Shear centre and at Bottom
flange, respectively. The subscript FEA has been removed from the
notation for these three critical elastic buckling loads.
Plotted in Fig. 5 are LTB load ratios P
cr,T
/P
cr,S
and P
cr,B
/P
cr,S
against slenderness for the beam of either FRP or steel. It is obvious
that for the same beam configuration, the closer the buckling load
ratios are to 1.0, the less significant is the load height influence. The
higher P
cr,B
/P
cr,S
is, and the lower P
cr,T
/P
cr,S
is for PFRP compare to
steel shows that the change of load height is more significant for
the composite material. The maximum difference between the
steel and PFRP curves is 11% at the lowest slenderness ratio of
9.8, and the minimum difference is 4%, when L/his 49.2.
Plotted in Fig. 6 is the normalised LTB load P
cr,FEA
/P
cr
versus L/h
for the PFRP beam. P
cr
was calculated using Eq. (2). Because the
normalised load does not deviate significantly from 1.0, the three
curves in the figure confirm that Eq. (2) gives P
cr,FEA
s similar to
those from the Eigenvalue FEA with the same parameter values.
When slenderness L/his <15 the reliability of Eq. (2) in predicting
LTB resistance is seen to decrease.
Fig. 7 is for plots of mid-span vertical point load (P) with mid-
span vertical deflection (w) for a virtually straight beam. The beam
Fig. 2. Eigenvalue analysis for initial minor axis out-of-straightness geometric
imperfection shape along the I-beam.
Fig. 3. Model for ‘twisting’ couple applied at free ends of I-beam.
Fig. 4. Initial geometric imperfection shape from pure torsion and linear elastic
small displacement FEA.
T.T. Nguyen et al. / Composite Structures 100 (2013) 233–242 237
has a span of 2 m and the out-of-straightness geometric imperfec-
tion has maximum amplitude of span/20,000. From the three load–
deflection curves in the figure it can be seen that P
cr,FEA
(for the
onset of LTB failure) depends on z
g
. Instability happens when the
beam’s vertical stiffness (P/w) dramatically reduces. It is observed
that until the instability starts to develop, the three P–wcurves are
identical, and so initial vertical stiffness is not influenced by the
load height. Mohri et al. [39] recorded a similar observation when
analysing the buckling behaviour of steel beams. It is also worth
noting that, from a designer’s point of view, when the load is ap-
plied below shear centre (i.e. z
g
is negative), the governing limit
state for the PFRP beam section can be a serviceability deflection
limit.
4. Influence of end warping fixity
When a thin-walled cross-section has an open shape its stiff-
ness under torsion, acting about the centroidal axis, is the sum of
torsional stiffnesses from uniform (St. Venant) torsion (governed
by torsional rigidity G
LT
Jor GJ) and from non-uniform torsion (con-
trolled by warping rigidity E
L
I
w
or EI
w
). When the ends of the beam
have warping fully fixed the state of non-uniform torsion will be
dominant and because the stiffness to twisting deformation in-
creases, so does the resistance to LTB failure. On the other hand
if warping at the ends is free, the state of non-uniform torsion re-
duces (it will be present either side of the mid-span when a torque
is generated by P) and the LTB resistance is the lower bound for
this end displacement boundary condition. It is because the free
warping condition is for lowest strength that k
w
is specified to be
1.0 in design, such as given by the closed form Eqs. (2) and (3).
Minghini et al. [40] investigated by analysis the critical load of a
PFRP portal frame, where the column ends were either free
warping or fully warping fixity. They found that there was a 40%
increase in buckling load between these two bounds on a displace-
ment boundary condition.
Plotted in Fig. 8 is the ratio of elastic critical buckling loads with
BC2 for end warping fully restrained (P
cr,Fixed
) and BC1 for warping
fully free (P
cr,Free
), for slenderness ratios from 9.8 to 49.2. For con-
venience the subscript FEA has been removed from these two FE
Eigenvalue solutions. The top curve is for FRP and the lower curve
is for steel elastic constants. It is noted that the contribution of
non-uniform torsion stiffness to the total torsional stiffness (de-
fined by E
L
I
w
/(E
L
I
w
+G
LT
J)orEI
w
/(EI
w
+GJ)) is bigger for FRP, and
so the effect of changing the warping restraint from free to fixed
is greater for the same beam geometry and displacement and load-
ing boundary conditions. It can be seen from the figure that
P
cr,Fixed
/P
cr,Free
for FRP varies from 1.13 to 1.66 and for steel from
1.07 to 1.48. The maximum difference between steel and FRP is
0.0
0.5
1.0
1.5
2.0
01020304050
Pcr,T/Pcr,S or Pc,rB/Pcr,S
Slenderness ratio L/h
FRP-Bottom/shear
FRP Top/Shear
Steel-Bottom/Shear
Steel-Top/Shear
Effect of load height with FRP beam
Effect of load height with Steel beam
Fig. 5. Effect of load height on elastic critical buckling load when beam is of FRP or
steel.
0.85
0.90
0.95
1.00
1.05
1.10
1.15
0 1020304050
Normalized buckling load Pcr,FEA/Pcr
Slenderness ratio L/h
Fig. 6. Normalised buckling load (P
cr,FEA
/P
cr
) versus slenderness ratio (L/h) for a
simply supported I-beam subjected to a vertical point load at mid-span and
positioned either on top flange, at the shear centre or on bottom flange.
0
1
2
3
4
5
0 1020304050607080
Load, P(kN)
Vertical deflection, w(mm)
Fig. 7. Vertical load (P) with vertical deflection (w) for the PFRP beam, having span
of two metres and load applied at either top flange, shear centre or bottom flange.
0.0
0.5
1.0
1.5
2.0
0 1020304050
Pcr,Fixed /Pcr,Free
Slenderness Ratio L/h
STEEL
FRP
Fig. 8. Influence of end warping fixity boundary condition on LTB resistance.
238 T.T. Nguyen et al. / Composite Structures 100 (2013) 233–242
13%, when L/his 14.8. The minimum difference of 4.3% is found to
be at the slenderness ratio of 44.3.
It is well-known [32] that the effect of warping torsion reduces
as the span gets longer, such that its influence on buckling resis-
tance reduces too. As the contribution of warping torsion (on total
torsion) is highest with end warping fixed, the reduction in buck-
ling resistance is biggest when k
w
= 0.5. As consequence the ratio
P
cr,Fixed
/P
cr,Free
is higher for stockier beams. The result of interaction
of the two torsional stiffnesses on LTB strength is seen from the
curves plotted in Fig. 8.
5. Influence of initial geometric imperfections
In this study only the influence of expected manufacturing
imperfections on the geometry of the PFRP I-shape are considered
in a FE parametric study. Another ‘imperfection’ that is inherent
will be the eccentricity of load from having the vertical load offset
a distance, e
y
, from the Z–Xplane. This imperfection introduces a
‘secondary’ moment of magnitude Pe
y
that will either act with, or
against the beam deformation induced by the presence of geomet-
ric imperfections [41]. In other words the presence of a larger geo-
metric imperfection can be employed to account for the load
eccentricity that is due to tolerances found on-site or in laboratory
testing. As a result this particular form of imperfection is not in-
cluded in the FE modelling. Structural PFRP shapes are pultruded
to possess acceptable geometric imperfections in the form of out-
of-straightness, flatness, twist, angularity, etc. It is imperative to
include these geometric imperfections in the FE modelling meth-
odology as their existence in practice will lower the LTB buckling
resistance. It could be argued that an acceptable approach in FEA
is to scope geometric imperfections, residual stresses and load
eccentricity by modelling the dominant type of imperfection with
a relatively high magnitude. Investigated next is the influence on
LTB resistance of changing the two key geometric imperfections
of minor axis out-of-straightness and (longitudinal) twist. These
two geometric imperfections are directly linked to the two (gov-
erning) LTB deformations of lateral flexural and cross-section twist,
respectively.
The question to next address is how large are the two geometric
imperfections to be? Maximum allowable magnitudes can be
based on ASTM D3917-11 for Standard Specification for Dimen-
sional Tolerance of Thermosetting Glass-Reinforced Plastic Pul-
truded Shapes [42].Table 3 reports information taken from
Tables 3 and 4 in ASTM D3917-11 to give the ‘allowable deviation
from straight’ per unit length. Column (1) lists the type of imper-
fection and column (2) provides an illustration. In column (3) the
values of the allowable deviation from straightness (
D
s
) or twist
(
D
t
) is reproduced. The standard has made the assumption that
the initial lateral out-of-straightness (
D
sx
), along a beam of length
L, to be defined by the geometric relationship
D
sx
¼
D
s
sin
p
x
Lð4Þ
For the initial twist imperfection the assumed distribution
along the length is given by
D
tx
¼
D
t
sin
p
x
2Lð5Þ
In Eqs. (4) and (5) xis the distance along the beam from one end to
the other end.
It is noted that the maximum twisting allowance in 2011 ver-
sion of ASTM D3917 is three times larger than in the previous ver-
sion of 2002. The reason for this significant change has not been
given provenance by the drafting committee. A maximum limit
on the twist angle is not given in ASTM D3917-11; it was stated
to be 3°in 2002.
Table 4 reports measured values for the magnitude of geometric
imperfections, in terms of L, for a range of standard pultruded stan-
dard shapes [2,3]. In column (1) the source to the data is given. Col-
Table 3
Allowable deviation from straightness and twist according to Tables 3 and 4 in standard ASTM D3917-11.
Type of imperfection Illustration of geometric tolerance Allowable deviation from straight
(1) (2) (3)
Out-of-straightness D= 4.167 L(length in m)
Twist Y/L= 3.281°/m
Table 4
Measured initial geometric imperfections for pultruded standard shapes.
Author(s) Structural shape Maximum imperfection in terms of length L
(1) (2) (3)
Brooks and Turvey [10] I Out-of-straightness for flanges and web is L/900
Zureick and Scott [24] I and box Out-of-straightness of:
I-beam lies between L/812 and L/1835
Box lies between L/1103 and L/8053
Maximum did not always exist at mid-span
Mottram et al. [41] Wide flange I Maximum out-of-straightness is L/4500
Lane and Mottram [43] Wide flange I Minor-axis out-of-straightness is L/3200
T.T. Nguyen et al. / Composite Structures 100 (2013) 233–242 239
umn (2) gives the form of shapes characterised by an ‘in-house’
measurement method of the research centre, and results are listed
in column (3). It is noted that no measurements for the twist
imperfection have been made. Table 4 reports that the minor axis
out-of-straight straightness imperfection (
D
s
in Eq. (4)) can have a
magnitude in the range of L/800–L/4500. These measurements are
found to be significantly lower than the maximum allowed in
ASTM D3917-11 of L/240.
Fig. 9 presents plots of Pagainst wfor the FRP beam at 2 m span
for three initial minor axis out-of-straightnesses in the range of L/
10,000 to L/240. The smallest is for a virtually straight beam and
the largest corresponds to the maximum allowed in manufacture.
For the non-linear FEA the three-point bending load case has the
displacement boundary conditions BC1 (for free warping and free
lateral flexure at both ends). Note that inspection of the deforma-
tion shape of the beams analysed did not find any signs for the
development of a local instability near the sections with a shear
force concentration. The three curves plotted in the figure show
that, as the geometric imperfection increases, the load for LTB fail-
ure become less distinct. This FE buckling load is not the critical
load it is a limiting value (P
Limit
). It is seen from the P–wcurves
in Fig. 9 that with the FRP material remaining linear elastic in
the post-buckling region, the secondary load path has a positive
slope. The curves show that, following instability, the beam devel-
ops post-buckling strength and so onset of LTB instability does not
result in member collapse. This is an important finding when
developing guidelines to design against LTB failure as a ULS, as
its presence is analogous to the additional reliability given to de-
sign provisions in BE EN 1993-1-1:2005 and ANSI/AISC 360-10
from having the ultimate strength of steel higher than yield
strength; the lower (yield) strength is used in the design calcula-
tions for LTB resistance.
Focusing on the shape of the three P–wcurves in Fig. 9, when P
is close to 2.5 kN and wis about 10 mm, it is observed that there is
no clear buckling bifurcation (for P
cr,FEA
) when the out-of-straight-
ness is L/1000 and larger. Singer et al. [44] explains that there are a
number of data reduction methods that can be employed to esti-
mate what the limiting buckling load (P
Limit
) is. A review of meth-
ods previously employed by researchers studying pultruded
shapes is given next. Lee [45], in his PhD work on the flexural–tor-
sional buckling of T-sections, suggested that buckling load should
be estimated by the intersection point of extrapolating the two
‘linear’ lines for the ‘pre-buckling’ and ‘post-buckling’ parts to the
P–wresponse. In his LTB experiments with I-beams, Stoddard
[12] choose to define the limiting buckling load to be the load
when the mid-span rotation (twist) of the top flange attained 5°.
To determine the LTB resistance of an end-loaded cantilever beam
Brooks and Turvey [10] recorded their buckling load as the load at
which the end-rotation started to grow rapidly. In a series of tests
to determine flexural buckling of concentrically loaded columns
Mottram et al. [41] terminated an increase to axial load when
the mid-span lateral deflection reached height/100. Using mea-
surements for column load and lateral mid-span deflection they
employed the Southwell plot method to get an improved estimate
to the elastic critical buckling load. Based on results from non-lin-
ear FE analyses with ANSYS
Ò
, Afifi [46] obtained column buckling
loads by applying the Southwell plot method. This is known [44]
to be an effective data reduction method when estimation to the
elastic critical buckling load, for the perfect member, is required.
Returning to the FE results in Fig. 9 it is seen that the beam’s
stiffness, given by P/w, is very similar during pre-buckling. The load
and deflection when non-linearity occurs is dependent on the
amplitude of the out-of-straightness geometric imperfection.
Using these observations, the authors define the limiting buckling
load P
Limit
as the point on the load–deflection curve when the se-
cant stiffness has been reduced by X%. Fig. 10 illustrates how P
Limit
can be obtained using this stiffness reduction method. As can be
seen the initial constant stiffness is common to establishing P
Limit
for the two non-linear P–wcurves that give a lower prediction as
the size of the geometric imperfections increases.
The question is, how large a percentage is Xto be? To establish
the answer it will be prudent to determine P
Limit
at or near the
point where the secant stiffness is changing rapidly with a small
increase in w. For the 2 m beam with initial out-of-straightness
of L/10,000, Table 5 presents, in columns (1–3), the values of P,w
and the instantaneous secant stiffness P/w. For this virtually
straight beam the results show that the ‘pre-buckling’ stiffness is
constant at 0.262 kN/mm to Pup to 2.31 kN. Secant stiffness is
then found to start reducing and can be seen to be very rapidly fall-
ing away when Pis 2.47 kN. At this load the ‘post-buckling’ stiff-
ness is 0.138 kN/mm (highlighted in bold font), giving a secant
stiffness reduction of around 50%. Based on this evaluation P
Limit
is determined as the value of Pwhen the secant stiffness reduction
Xis 50%.
Taking a constant 1.0 m span the FRP beam is analysed with
either out-of-straightnesses of L/10,000, L/1000, L/500 and L/240
or twists of 0.1°,1°,2°and 3.28°. Note that L/240 and 3.28°are
the maximum allowable in accordance with ASTM 3917-11 (refer
to Table 3). Presented in Fig. 11 is P
Limit
, presented as a load ratio
in terms of P
cr
from using Eq. (2). FE modelling has boundary con-
ditions BC1 and shear centre load (z
g
= 0.0 mm). For the parameters
chosen P
Limit
/P
cr
lies between 0.95 and 1.05 for the full range of the
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0 1020304050607080
Concentrated load, P(kN)
Mid-span vertical displacement, w(mm)
L/10000
L/1000
L/240
Fig. 9. Load (P) and vertical displacement (w) for the FRP beam at 2.0 m span having
an initial minor axis out-of-straightness deformation from L/10,000 to L/240.
Reduction in initial stiffness of X%
Limiting buckling load
Vertical deflection, w
P-wcurves for different geometric
imperfection(s)
PLimit
PLimit
Load, P
Fig. 10. Illustration to show the secant stiffness reduction method to establish the
limiting buckling load when imperfections negate a bifurcation buckling failure.
240 T.T. Nguyen et al. / Composite Structures 100 (2013) 233–242
minor axis out-of-straightness and from 1.01 to 1.05 for the poten-
tial range in the twist imperfection.
The FE results in Fig. 12 are for a constant out-of-straightness of
L/1000 and a constant initial twist of 1°. It is believed that these
geometric deviations are likely to be typical for pultruded I-shapes.
The parameter changed in the FE study is the span, which varies
from 1.0 m (L/hof 9.84) to 4.0 m (L/hof 39.4). Ratio P
Limit
/P
cr
is seen
to lie within the range of 1.02–1.08. Being always >1.0, the results
show that P
Limit
is higher than that calculated by Eq. (2). A feasible
explanation for why the limiting value is found to be higher is that
the closed form formula ignores the increase in instability resis-
tance due to beam curvature [32]. Roberts [47] found this influence
is significant giving 4% increase with narrow-flange beam (i.e.
height twice the flange breadth). If this influence factor is taken
into account, the buckling resistance can be determined as [47]:
M
crp
¼M
cr
ffiffiffiffiffiffiffiffiffiffiffiffi
1
I
z
I
y
qð6Þ
For the single beam section in this FE study Eq. (6) gives
M
crp
¼1:04M
cr
. This means that a resistance increase of 4% can be
from the pre-buckling deformation.
It is also observed that P
Limit
/P
cr
, for ‘slender’ beams with L/
h> 20, is fairly constant at 1.07–1.10. When the beam becomes
‘stocky’ (with slenderness ratio <15), P
Limit
/P
cr
is now found to be
in range 1.02–1.05. The transition in buckling resistance can be ex-
plained by the presence of a growing contribution from shear
deformation as the beam gets shorter. The model for the develop-
ment of Eq. (2) does not account for shear flexibility.
6. Concluding remarks
Linear Eigenvalue and geometric non-linear Finite Element (FE)
analyses have been performed to study the Lateral–Torsional Buck-
ling (LTB) resistance of the I-beam (of size 101.6 50.8 6.4 mm)
under the influence of load height, displacement boundary condi-
tions and the geometric imperfections of minor axis out-of-
straightness and longitudinal twist. It is found that changing the
load height relative to the shear centre is more significant for a
beam of Pultruded Fibre Reinforced Polymer (PFRP) than of struc-
tural steel. It is found that the change in resistance is significant
when the beam is ‘stockier’, as given by span/height ratio <15.
When the Eigenvalue results are normalised using the prediction
for the elastic critical buckling load from a closed form formula it
is found that the ratio remains close to 1.0 (between 0.94 and
1.02). The FE study on influence of load height with the PFRP beam
does show that the post-buckling load path is significantly affected
by this parameter.
By studying the influence of the degree of end warping fixity on
LTB resistance it is found to be more significant when the beam is
of FRP material. The level of influence is reduced as the beam span
increases and an explanation for the finding is developed in the
paper.
To simulate pre-buckling response in the presence of geometric
imperfections, emanating during the pultrusion process, it is nec-
essary to perform a geometric non-linear analysis. It is found that
the vertical load against vertical deflection curve does not show a
clear buckling (bifurcation) point for LTB failure. A data reduction
method is proposed to allow an acceptable prediction of the limit-
ing buckling load. It is given by the point on the load–deflection
curve at which the secant stiffness has reduced by 50% from its ini-
tial constant value. By adopting this method, limiting buckling
loads are obtained for a 1.0 m long FRP beam having initial geo-
metric imperfections of out-of-straightness from span/10,000 to
span/240 and twist along span from 0.1°to 3.28°. Normalised
buckling load is shown to be in the range of 0.95–1.05 and 1.01–
Table 5
Vertical deflection (w) with vertical load (P) at 2 m span with out-of-straightness
geometric imperfection of L/10,000 (see Fig. 9).
Load P(kN) Deflection w(mm) Secant stiffness P/w(kN/mm)
(1) (2) (3)
0.070 0.267 0.262
0.140 0.534 0.262
0.245 0.934 0.262
0.403 1.53 0.262
0.639 2.44 0.262
0.993 3.79 0.262
1.53 5.81 0.262
2.31 8.80 0.262
2.35 9.18 0.256
2.36 9.53 0.248
2.37 10.0 0.236
2.38 11.1 0.215
2.41 13.3 0.182
2.47 17.8 0.138
2.54 23.7 0.107
2.63 30.7 0.086
2.73 38.5 0.071
2.85 47.0 0.061
2.98 56.0 0.053
0.94
0.96
0.98
1.00
1.02
1.04
1.06
Normalized buckling load PLimit/P
cr
Imperfection
Beam with initial out of
straightness imperfection
Beam with initial twist
imperfection
L/10000
L/1000
L/500
L/240
Max- ASTM 3917-11
0.1o
1o
2o
3.28o
Max-ASTM 3917-11
Fig. 11. Influence of magnitude of initial geometric imperfections on LTB resistance
of a pultruded beam of size 101.6 50.8 6.4 mm at span of 1.0 m.
1.02
1.04
1.06
1.08
1.10
0 1020304050
Normalized buckling load PLimit /Pcr
Slenderness ratio L/h
Initial out of straightness
imperfection of
Initial twist imperfection of 1 Deg
L/1000
Fig. 12. Influence of having constant geometric imperfections and varying beam
span from 1.0 to 4.0 m.
T.T. Nguyen et al. / Composite Structures 100 (2013) 233–242 241
1.05, respectively. A value of 1.0 means the FE result is identical to
that from a closed form equation developed for the design of iso-
tropic (steel) members in bending. By setting the two imperfec-
tions constant at span/1000 and 1°the buckling load ratio, from
increasing spans from 1.0 to 4.0 m, is found to be 1.03 to 1.08
and 1.04 to 1.1, respectively. The lowest ratios are for slenderness
ratio <15 and can be explained by the growing influence of shear
deformation. The physical explanation for why the FE results for
the FRP I-beam are above that calculated by the closed form equa-
tion is given.
The authors recommend their FE modelling methodology for
PFRP beam problems with geometric imperfection failing with
the lateral–torsional buckling mode.
For the development and calibration of a closed form formula
for inclusion in a design standard other geometric imperfections
such as flatness, angularity and residual stress might have an
important role to play in reducing LTB resistance. Based on the rel-
evance of the results from the work reported in this paper the
authors believe that the inclusion of all ’field’ imperfections can
be scoped by modelling a single (dominant) imperfection with
appropriate magnitude. The most practical to work with is that
of out-of-straightness about the minor-axis, having maximum
amplitude of, say, span/200.
Acknowledgements
The first author gratefully acknowledges his scholarships from
the Vietnamese International Education Development and the
School of Engineering at the University of Warwick, UK.
References
[1] Mottram JT. Does performance based design with fibre reinforced polymer
components and structures provide any new benefits and challenges? Struct
Eng 2011;89(6):23–7.
[2] Anonymous. Strongwell design manual. Bristol, VA: Strongwell Composites
Inc. (accessed 01.08.12). <www.strongwell.com/>.
[3] Anonynous. The new and improved pultrex pultrusion design manual
(imperial version), vol. 4 Rev. 6. Alum Bank, PA: Creative Pultrusion Inc.
(accessed 01.08.12). <www.creativepultrusions.com/library.html>.
[4] Bank LC. Composites for construction – structural design with FRP material. New
Jersey: John Wiley & Sons; 2006.
[5] Clarke JL, editor. Structural design of polymer composites: eurocomp design
code and handbook. London: S & FN Spon; 1996.
[6] Anonynous. Guide for the design and construction of structures made of thin
FRP pultruded elements. Rome, Italy: National Research Council of Italy (CNR);
2008 (accessed 01.12.12). <http://www.cnr.it/documenti/norme/Istruzioni
CNR_DT205_2007_eng.pdf>.
[7] Anonymous. Pre-standard for load and resistance factor design (LRFD) of
pultruded fiber reinforced polymer (FRP) structures. Arlington, VA: American
Composites manufacturers Association (ACMA); press release (accessed
01.08.12). <http://www.acmanet.org/pressreleases/2011/011711.html>.
[8] Chambers RE. ASCE design standard for pultruded fiber-reinforced-plastic
(FRP) structures. J Compos Constr 1997;1(1):26–38.
[9] Mottram JT. Lateral–torsional buckling of a pultruded I-beam. Composites
1992;23(2):81–92.
[10] Brooks RJ, Turvey GJ. Lateral buckling of pultruded GRP I-section cantilevers.
Compos Struct 1995;32(1–4):203–15.
[11] Turvey GJ. Effects of load position on the lateral buckling response of
pultruded GRP cantilevers – comparisons between theory and experiment.
Compos Struct 1996;35(1):33–47.
[12] Stoddard WP. Lateral–torsional buckling behavior of polymer composite I-
shaped members. PhD Thesis, Georgia Institute of Technology, Atlanta,
Georgia; 1997.
[13] Trumpf H. Local and global stability of plane frame works made of orthotropic
FRP-profiles.PhD Thesis, University of Aachen, Germany, Shaker-Verlag; 2006.
[14] Zhang S. Lateral–torsional buckling of simply supported and cantilvered fiber
reinforced polymeric I-beams. PhD Thesis, Georgia Institute of Technology,
Atlanta, Georgia; 2000.
[15] Ascione L, Giordano A, Spadea S. Lateral buckling of pultruded FRP beams.
Compos Part B: Eng 2011;42(4):819–24.
[16] Clark JW, Hill HN. Lateral buckling of beams. J Struct Div-ASCE 1960;86(ST7):
175–96.
[17] Dutheil J. Verifica Delle aste Compresse: Principi Fondamentali. Constr Metal
1966;18(4):266–75.
[18] Di Tommaso A, Russo S. Shape influence in buckling of GFRP pultruded
column. J Mech Compos Mater 2003;39(4):329–40.
[19] Lane A. An experimental investigation of buckling mode interaction in
pultruded fiber reinforced plastic columns. PhD Thesis, University of
Warwick, UK; 2002.
[20] Abaqus. ‘User Manual, version 6.10’, Hibbitt, Karlsson and Sorensen, Inc.; 2011.
[21] Tomblin J, Barbero E. Local buckling experiments on FRP columns. Thin-Wall
Struct 1994;18(2):97–116.
[22] Bank LC. Shear properties of pultruded glass fiber reinforced plastic materials. J
Mater Civ Eng 1990;2(2):118–22.
[23] Sonti SS, Barbero EJ. Material characterization of pultruded laminates and
shapes. J Reinf Plast Compos 1996;15(7):701–17.
[24] Zureick A, Scott D. Short-term behavior and design of fiber-reinforced
polymeric slender members under axial compression. J Compos Constr
1997;1(4):140–9.
[25] Steffen RE. Behavior and design of fiber-reinforced polymeric composite equal-
leg angle struts. PhD Thesis, Georgia Institute of Technology, Atlanta, Georgia;
1998.
[26] Turvey GJ. Torsion tests on pultruded glass-reinforced plastic sheets. J Compos
Sci Technol 1998;58(8):1343–51.
[27] Roberts TM, Al-Ubaidi H. Flexural and torsional properties of pultruded fiber
reinforced Plastic I – Profiles. J Compos Constr 2002;6(1):28–34.
[28] Mottram JT. Shear modulus of standard pultruded fiber reinforced plastic
material. J Compos Constr 2004;8(2):141–7.
[29] Qiao PZ, Zou GP, Davalos JF. Flexural–torsional buckling of fiber-reinforced
plastic composite Cantilever I – Beams. Compos Struct 2003;60(2):205–17.
[30] Shan LY, Qiao PZ. Flexural–torsional buckling of fiber-reinforced plastic
composite open channel beams. Compos Struct 2005;68(2):211–24.
[31] Mottram JT, Shaw CT. Using finite elements inmechanical design. UK: McGRAW-
HILL Book Company Europe; 1996.
[32] Trahair NS. Flexural–torsional buckling of structures. London, UK: E & FN
Spon; 1993.
[33] Abaqus. Theory manual, version 6.11. Hibbitt, Karlsson and Sorensen, Inc.;
2011.
[34] Riks E. An incremental approach to the solution of snapping and buckling
problems. Int J Solids Struct 1979;15(7):529–51.
[35] Wempner GA. Discrete approximations related to nonlinear theories of solids.
Int J Solids Struct 1971;7(11):1581–99.
[36] Crisfield MAA. Fast incremental/iterative solution procedure that handles
‘‘Snap-through’’. Comput Struct 1981;13(1–3):55–62.
[37] Sapkás Á, Kollár LP. Lateral–torsional buckling of composite beams. Int J Solids
Struct 2002;39(11):2939–63.
[38] Machado SP, Cortínez VH. Lateral buckling of thin-walled composite
bisymmetric beams with prebuckling and shear deformation. Eng Struct
2005;27(8):1185–96.
[39] Mohri F, Azrar L, Potier-Ferry M. Lateral post-buckling analysis of thin-walled
open section beams. Thin-Wall Struct 2002;40(12):1013–36.
[40] Minghini F, Tullini N, Laudiero F. Buckling analysis of FRP pultruded frames
using locking-free finite elements. Thin-Wall Struct 2008;46(2):223–41.
[41] Mottram JT, Brown ND, Anderson D. Physical testing for concentrically loaded
columns of pultruded glass fibre reinforced plastic profile. Struct Build
2003;156(2):205–19.
[42] ASTM D3917-11. Standard specification for dimensional tolerance of
thermosetting glass-reinforced plastic pultruded shapes. West Conshohocken,
PA: American Society for Testing and Materials (ASTM); 2011.
[43] Lane A, Mottram JT. Influence of modal coupling on the buckling of
concentrically loaded pultruded fibre-reinforced plastic columns. Proc Inst
Mech Eng, Part L: J Mater Des Appl 2002;216(2):133–44.
[44] Singer J, Arbocz J, Weller T. Buckling experiments experimental methods in
buckling of thin-walled structures: volume I basic concepts, column, beams
and plates. Chichester: John Wiley & Sons; 1997.
[45] Lee S. Flexural-torsional buckling of pultruded T-sections. PhD Thesis, Georgia
Institute of Technology, Atlanta, Georgia; 2001.
[46] Afifi AAM. Buckling of stiffened pultruded GRP plates and columns. PhD Thesis,
University of Lancaster, UK; 2007.
[47] Roberts TM. Influence of shear deformation on buckling of pultruded fiber
reinforced plastic profiles. J Compos Constr 2002;6(4):241–8.
242 T.T. Nguyen et al. / Composite Structures 100 (2013) 233–242
