Flexural–torsional buckling of fiber-reinforced plastic composite cantilever I-beams
ABSTRACT A combined analytical and experimental study of flexural–torsional buckling of pultruded fiber-reinforced plastic (FRP) composite cantilever I-beams is presented. An energy method based on nonlinear plate theory is developed for instability of FRP I-beam, and the formulation includes shear effect and bending–twisting coupling. Three different types of buckling mode shape functions of transcendental function, polynomial function, and half simply supported beam function, which all satisfy the cantilever beam boundary conditions, are used to obtain the eigenvalue solution, and their accuracy in the analysis are investigated in relation to finite element results. Four different geometries of FRP I-beams with cantilever beam configurations and with varying span lengths are experimentally tested under tip loads to evaluate their flexural–torsional buckling response. The loads are applied at the centroid of the tip cross-sections, and the critical buckling loads are obtained by gradually adding weight onto a loading platform. A good agreement among the proposed analytical solutions, experimental testing, and finite element method is obtained, and simplified explicit formulas for flexural–torsional buckling of cantilever beams with applied load at the centroid of the cross-section are developed. The effects of vertical load position through the cross-section, fiber orientation and fiber volume fraction on buckling behavior are also studied. The proposed analytical solutions can be used to predict the flexural–torsional buckling loads of FRP cantilever beams and to formulate simplified design equations.
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ABSTRACT: The purpose of this study is to predict and investigate the time-dependent creep behavior of composite materials. For this, firstly the evaluation method for the modulus of elasticity of whole fiber and matrix is presented from the limited information on fiber volume fraction using the singular value decomposition method. Then, the effects of fiber volume fraction on modulus of elasticity of GFRP are verified. Also, as a creep model, the nonlinear curve fitting method based on the Marquardt algorithm is proposed. Using the existing Findley's power creep model and the proposed creep model, the effect of fiber volume fraction on the nonlinear creep behavior of composite materials is verified. Then, for the time-dependent analysis of a composite material subjected to uniaxial tension and simple shear loadings, a user-provided subroutine UMAT is developed to run within ABAQUS. Finally, the creep behavior of center loaded beam structure is investigated using the Hermitian beam elements with shear deformation effect and with time-dependent elastic and shear moduli.Steel and Composite Structures 01/2007; 7(5). · 0.72 Impact Factor - SourceAvailable from: Ahmed Godat[Show abstract] [Hide abstract]
ABSTRACT: This study investigates the replacement of traditional materials (steel, wood and concrete) in electricity transmission lines by fiber glass pultruded members. The first part of the study summarizes a comparison between different design approaches to experimental data for glass fiber pultruded sections. For this purpose, a total of fifteen specimens made of E-glass and either polyester or vinylester matrix are tested: (i) angle-section, square-section and rectangular-section specimens are subjected to axial compression; (ii) I-section and W-section specimens are tested under bending. The experimental results are summarized in terms of the failure mode, critical buckling load and load–displacement relationships. Design equations available in FRP design manuals and analytical methods proposed in the literature are used to predict the critical buckling load and compared to the experimental results. Design of various FRP pultruded sections and cost estimate are conducted for 69 kV electricity transmission portal frame and a total distance of 10 km. The significance of the present findings with regard to economic solutions is discussed.Composite Structures 05/2013; 105:408-421. · 3.12 Impact Factor - [Show abstract] [Hide abstract]
ABSTRACT: The super convergent laminated composite beam element is newly derived for the lateral stability analysis. For this, a theoretical model of the laminated composite beams is developed based on the first-order shear deformation beam theory. The present laminated beam takes into account the transverse shear and the restrained warping induced shear deformation. The second-order coupling torque resulting from the geometric nonlinearity is rigorously derived. From the principle of minimum total potential energy, the stability equations and force-displacement relationships are derived and the explicit expressions for the displacement parameters are presented by applying the power series expansions of displacement components to simultaneous ordinary differential equations. Finally, the member stiffness matrix is determined using the force-displacement relationships. In order to show accuracy and superiority of the beam element developed by this study, the critical lateral buckling moments for bisymmetric and monosymmetric I-beams are presented and compared with other results available in the literature, the isoparametric beam elements, and shell elements from ABAQUS.Steel and Composite Structures 01/2013; 15(2). · 0.72 Impact Factor
Page 1
Flexural–torsional buckling of fiber-reinforced plastic
composite cantilever I-beams
Pizhong Qiaoa,*, Guiping Zoua, Julio F. Davalosb
aDepartment of Civil Engineering, Auburn Science and Engineering Center, The University of Akron, Akron, OH 44325-3905, USA
bDepartment of Civil and Environmental Engineering, West Virginia University, Morgantown, WV 26506-6103, USA
Abstract
A combined analytical and experimental study of flexural–torsional buckling of pultruded fiber-reinforced plastic (FRP) com-
posite cantilever I-beams is presented. An energy method based on nonlinear plate theory is developed for instability of FRP
I-beam, and the formulation includes shear effect and bending–twisting coupling. Three different types of buckling mode shape
functions of transcendental function, polynomial function, and half simply supported beam function, which all satisfy the cantilever
beam boundary conditions, are used to obtain the eigenvalue solution, and their accuracy in the analysis are investigated in relation
to finite element results. Four different geometries of FRP I-beams with cantilever beam configurations and with varying span
lengths are experimentally tested under tip loads to evaluate their flexural–torsional buckling response. The loads are applied at the
centroid of the tip cross-sections, and the critical buckling loads are obtained by gradually adding weight onto a loading platform. A
good agreement among the proposed analytical solutions, experimental testing, and finite element method is obtained, and sim-
plified explicit formulas for flexural–torsional buckling of cantilever beams with applied load at the centroid of the cross-section are
developed. The effects of vertical load position through the cross-section, fiber orientation and fiber volume fraction on buckling
behavior are also studied. The proposed analytical solutions can be used to predict the flexural–torsional buckling loads of FRP
cantilever beams and to formulate simplified design equations.
? 2003 Elsevier Science Ltd. All rights reserved.
Keywords: Fiber-reinforced plastic composites; FRP structural shapes; Flexural–torsional buckling; Buckling mode shape functions; Structural
stability
1. Introduction
Fiber-reinforced plastic (FRP) structural shapes
(beams and columns) have shown to provide efficient
and economical applications for civil engineering con-
struction (e.g., in bridges, piers, retaining walls, airport
facilities, storage structures exposed to salts and chem-
icals, and others). Most FRP shapes are thin-walled
structures and manufactured by the pultrusion process.
The material constituents for low-cost pultruded FRP
shapes commonly consist of high-strength E-glass fiber
and vinylester or polyester polymer resins, and due to
this choice of materials, the structures usually exhibit
relatively large deformations and tend to buckle globally
or locally. Consequently, buckling is the most likely
mode of failure before the ultimate load reaches the
material failure [1–4].
A long slender beam under bending about the strong
axis may buckle by a combined twisting and lateral
(sideways) bending of the cross-section. This pheno-
menon is known as flexural–torsional (lateral) buckling.
Numerous analytical [5–13] and theoretical investiga-
tions [14–17] have been presented for steel beams, of
which the material is homogeneous and isotropic. Sev-
eral analytical and experimental evaluations of lateral
buckling of FRP structural shapes have been reported,
and some design methodologies for these members have
been proposed. The flexural–torsional buckling behavior
of pultruded E-glass FRP I-beams has been investigated
experimentally by Mottram [18], and the observed results
compared well with numerical prediction using a finite-
difference method. Mottram [18] emphasized that there is
a potential danger in analysis and design of FRP beams
without including shear deformation. A series of lateral
buckling tests on small-scale pultruded E-glass FRP
*Corresponding author. Tel.: +1-330-972-5226; fax: +1-330-972-
6020.
E-mail address: qiao@uakron.edu (P. Qiao).
0263-8223/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved.
doi:10.1016/S0263-8223(02)00304-5
Composite Structures 60 (2003) 205–217
www.elsevier.com/locate/compstruct
Page 2
beams was carried out by Brooks and Turvey [19] and
Turvey [20]; the effects of load position on the lateral
buckling response of FRP I-sections were investigated,
and the results were correlated with an approximate
formula by Nethercot and Rockey [21] and finite element
eigenvalue analysis. With the use of Galerkin method to
solve the equilibrium differential equation, Pandey et al.
[22] presented a theoretical formulation for flexure-
torsional buckling of thin-walled composite I-section
beams, and simplified formulas for several different
loading and boundary conditions were developed. Uti-
lizing the assumed stress functions, the approximate
lateral buckling solutions for anisotropic beams were
given by Murakami and Yamakawa [23]. Using a seven-
degree-of-freedom element, a parametric study of opti-
mal fiber direction for improving the lateral buckling
response of pultruded I-beams was performed by Lin
et al. [24]. A finite element method (FEM) based on
moderate rotation theory for the simulation of thin-
walled composite beams was developed by Fraternal and
Feo [25]. Recently, a displacement-based one-dimen-
sional finite element model for flexural–torsional buck-
ling of composite I-beams was developed by Lee and
Kim [26]. Barbero and Raftoyiannis [1] extended the
formulation of Roberts and Jhita [8] to study the lateral
and distortional buckling of simply supported composite
FRP I-beams under central concentrated loads. In their
study, the stability equilibrium equation of the system
was established based on vanishing of the second varia-
tion of the total potential energy; they used plate theory
to allow for distortion of cross-sections, and beam shear
and bending–twisting coupling effects were included in
the analysis. The flexural–torsional and lateral-distortion
buckling of composite FRP I-beams were studied both
experimentally and analytically by Davalos and Qiao [3];
but in their studies, only simply supported beams loaded
with midspan concentrated loads were studied.
In this paper, a combined analytical and experimental
study on flexural–torsional buckling of FRP cantilever
I-beams is presented, and a simplified equation for
flexural–torsional buckling design is developed. Three
different types of shape functions (exact transcendental
function, polynomial function, and half simply sup-
ported beam function), which all satisfy the cantilever
beam boundary conditions, are used to obtain eigen-
value solutions, and their numerical results are com-
pared and discussed. The position of applied load
through the cross-section at the loading tip is also con-
sidered in the formulation. Four different geometries of
FRP I-beams with varying span lengths are tested under
tip loadings and cantilevered restrained at the other
ends. The analytical solutions are compared with finite
element studies and experimental tests. Parametric
studies are further performed to study the fiber angle
and fiber volume fraction effects on the flexural–
torsional buckling behavior.
2. Analytical algorithm
The analysis of flexural–torsional buckling is based
on total potential energy governing instability and de-
rived using the plate theory. The total potential energy
for a plate structure under external force P is
P ¼ U þ W
where
ð1Þ
U ¼1
W ¼ ?
2
Z Z
X
X
Pkqk
Z
rijeijdX
k
ð2Þ
and Pkare the externally applied forces and qkare the
corresponding displacements.
For I-beam sections consisting of two flanges and one
web, the strain energy can be expressed as
U ¼ UTFþ UWþ UBF
where the superscripts TF, W and BF refer to top flange,
web, and bottom flange, respectively. The instability
state is characterized by the vanishing of the second
variation of the total potential energy [1,27]
ð3Þ
d2P ¼ d2U ?
X
k
Pkd2qk¼ 0
ð4Þ
where
d2U ¼
Z Z
X
Z
rijd2eij
?
þ1
2drijdeij
?
dX
ð5Þ
Also, since qk can usually be expressed as linear func-
tions of displacement variables, d2qk vanishes and
P
not small compared with unity, the strains for the
buckling problem are expressed in nonlinear terms. It is
assumed that the strains and curvatures are much less
than unity everywhere in the plate. For a plate in the xy-
plane, the in-plane finite strains of the mid-surface of the
plate are given by Malvern [28] as
kPkd2qkcan be omitted in Eq. (4).
Because the displacement-gradient components are
ex¼ou
oxþ1
2
ou
ox
?
?
?2
?2
"
"
oxþou
þ
ov
ox
?
?
?2
?2
ov
oyþow
þ
ow
ox
?
?
?2#
?2#
ey¼ov
oyþ1
oyþov
2
ou
oy
þ
ov
oy
þ
ow
oy
ow
oy
cxy¼ou
ox
ou
oyþov
ox
ox
ð6Þ
Based on the von Karman plate theory, only the dis-
placement gradients ow=ox and ow=oy are considered to
have significant values; therefore, the nonlinear terms
ðow=oxÞ2and ðow=oyÞ2are retained in Eq. (6). For a
prismatic structure, the displacement gradients ðov=oxÞ
and ðou=oyÞ may become relatively large because of in-
plane rotations, especially for the flanges; whereas
206
P. Qiao et al. / Composite Structures 60 (2003) 205–217
Page 3
ðou=oxÞ, ðov=oyÞ, and their quadratic forms are signifi-
cantly smaller than the other terms and can be ignored.
Hence, Eq. (6) reduces to
ex¼ou
oxþ1
2
ov
ox
??2
"
þ
ow
ox
??2#
ey¼ov
oyþ1
2
ou
oy
??2
"
þ
ow
oy
??2#
cxy¼ou
oyþov
oxþow
ox
ow
oy
ð7Þ
The curvatures of the mid-plane are defined as
jx¼ ?o2w
ox2;
jy¼ ?o2w
oy2;
jxy¼ ?2o2w
oxoy
ð8Þ
For buckling analysis of I-beams under bending, the
deformation before buckling is ignored. Based on the
coordinate system shown in Fig. 1, the buckled dis-
placement fields are expressed as follows [12]:
For the web (in the xy-plane):
uw¼ 0;
vw¼ 0;
ww¼ wwðx;yÞð9aÞ
For the top flange (in the xz-plane):
uTF¼ uTFðx;zÞ;
vTF¼ vTFðx;zÞ;
wTF¼ wTFðxÞð9bÞ
For the bottom flange (in the xz-plane):
uBF¼ uBFðx;zÞ;
vBF¼ vBFðx;zÞ;
wBF¼ wBFðxÞð9cÞ
For the top flange (in the xz-plane), the strains and
curvatures in Eqs. (7) and (8) become
eTF
x
¼ouTF
ox
þ1
2
ovTF
ox
?
?
?2
?2
"
"
þ
owTF
ox
?
?
?2#
?2#
eTF
y
¼ovTF
oy
þ1
2
ouTF
oy
þ
owTF
oy
cTF
xy¼ouTF
oy
þovTF
ox
þowTF
ox
owTF
oy
ð10Þ
and
jTF
x
¼ ?o2wTF
ox2;
jTF
y
¼ ?o2wTF
oy2;
jTF
xy¼ ?2o2wTF
oxoy
ð11Þ
In the present study, the above equations are applied in
the buckling analysis of pultruded FRP I-beams. The
panels (e.g., web and flange) of most pultruded FRP
sections [29] are simulated as symmetric laminated
structures (no stretching–bending coupling, Bij¼ 0),
and the off-axis plies of the pultruded panels are bal-
anced symmetric (no extension-shear and bending–twist
coupling, A16¼ A26¼ D16¼ D26¼ 0). The panel me-
chanical properties are obtained either from experi-
mental coupon tests or theoretical prediction using
micro/macromechanics models [4,29].
For a laminated panel in the xy plane, the in-plane
mid-surface strains and curvatures are expressed in
terms of the compliance coefficients and panel resultant
forces as [30]
ex
ey
cxy
jx
jy
jxy
2
6666664
3
7777775
¼
a11
a12
a16
b11
b12
b16
a12
a22
a26
b12
b22
b26
a16
a26
a66
b16
b26
b66
b11
b12
b16
d11
d12
d16
b12
b22
b26
d12
d22
d26
b16
b26
b66
d16
d26
d66
2
6666664
3
7777775
Nx
Ny
Nxy
Mx
My
Mxy
2
6666664
3
7777775
ð12Þ
tf
y
bw
bf
tw
z
x
z( wb f)
y( vb f)
x( ub f)
z( w)
y( v)
z( wtf)
x( utf)
y( vtf)
x( u)
Fig. 1. Coordinate system and geometry of I-beam.
P. Qiao et al. / Composite Structures 60 (2003) 205–217
207
Page 4
and considering the top flange that can bend and twist
as a plate and also bend laterally as a beam and also
neglecting the transverse resultant forces (i.e., NTF
NTF
z
¼ 0), then the total potential energy is
z
¼
xz¼ MTF
d2UTF¼
Z Z
þ1
NTF
xd2eTF
x
?
þ1
2dNTF
xdeTF
x
2dMTF
xdjTF
x þ1
2dMTF
xzdjTF
xz
?
dxdz
ð13Þ
Substituting Eqs. (10)–(12) into Eq. (13), we can obtain
d2UTF¼1
2
Z Z
þ1
NTF
x
odvTF
ox
??2
"(
þ
odwTF
ox
??2#
a11
oduTF
ox
?
?
?2
?2)
þ1
d11
o2dvTF
ox2
??2
þ4
d66
o2dvTF
oxoz
dxdz
ð14Þ
The second variation of the total strain energy of the
bottom flange can be obtained in a similar way as
d2UBF¼1
2
Z Z
þ1
NBF
x
odvBF
ox
??2
"(
þ
odwBF
ox
??2#
a11
oduBF
ox
?
?
?2
?2)
þ1
d11
o2dvBF
ox2
??2
þ4
d66
o2dvBF
oxoz
dxdz
ð15Þ
Considering the web as a plate and the deformation
field of Eq. (9a), the second variation of the total strain
energy of the web panel is expressed as
d2UW
¼1
2
Z Z
NW
x
odwW
ox
??2
#
"
þNW
y
odwW
oy
??2
"
þ2NW
xy
odwW
ox
odwW
oy
dxdy þ1
2
Z Z
o2dwW
oy2
DW
11
o2dwW
ox2
??2
þDW
22
o2dwW
oy2
??2
?2#
þ2DW
12
o2dwW
ox2
þ4DW
66
o2dwW
oxoy
?
dxdy
ð16Þ
The buckling equilibrium equation d2U ¼ 0 in terms of
the lateral potential energy is then solved by the Ray-
leigh-Ritz method.
2.1. Stress resultants in I-beam panels
For a cantilever beam subjected to a tip concentrated
vertical load, simplified stress resultant distributions on
the corresponding panel are obtained from beam theory,
and the location or height of the applied load is ac-
counted for in the analysis. For FRP I-beams of uni-
form thickness, the membrane forces are expressed in
terms of the tip applied concentrated load P. The ex-
pressions for the flanges are
NTF
x
¼bwtf
¼ NTF
¼ ?bwtf
¼ NBF
2I
xz¼ 0
PðL ? xÞ
NTF
z
ð17aÞ
NBF
x
2I
PðL ? xÞ
NBF
zxz¼ 0
ð17bÞ
Similarly for the web
NW
x¼tw
xy¼ ?Ptw
IPðL ? xÞy
"
NW
2I
bw
2
??2
? y2
#
ð17cÞ
The transverse normal stress resultant ðNW
from the concentrated load is zero. Denoting ypas the
distance from the centroidal axis to the location of
the applied load, the transverse normal stress near the
concentrated load, for the case of yp6¼ ?ðbw=2Þ, is
y¼ Py þ bw=2
ypþ bw=2
NW
yp? bw=2
and for the case of yp¼ ?bw=2 or yp¼ bw=2
y¼ ?Py þ yp
bw
yÞ away
NW
? bw=26y 6yp
y¼ Pbw=2 ? y
yp6y 6bw=2
ð17dÞ
NW
? bw=26y 6bw=2
ð17eÞ
where
I ¼
1
2bftfbw2
?
þ1
12bw3tw
?
2.2. Displacement field of buckled I-beam panels
Assuming that the top and bottom flanges do not
distort (i.e., the displacements are linear in the z direc-
tion) and considering compatibility conditions at the
flange–web intersections, the buckled displacement field
for the web, top and bottom flange panels (Fig. 1) of the
I-section are derived. For the web (in the xy-plane)
uW¼ 0;
vW¼ 0;
wW¼ wðx;yÞð18aÞ
208
P. Qiao et al. / Composite Structures 60 (2003) 205–217
Page 5
For the top flange (in the xz-plane)
uTF¼ uTFðx;zÞ ¼ ?zdwTF
vTF¼ vTFðx;zÞ ¼ ?zhTF;
For the bottom flange (in the xz-plane)
uBF¼ uBFðx;zÞ ¼ ?zdwBF
vBF¼ vBFðx;zÞ ¼ ?zhBF;
For flexural–torsional (lateral) buckling of I-section
beams, the cross-section of the beam is considered as
undistorted. As the web panel is not allowed to distort
and remains straight in flexural–torsional buckling, the
sideway deflection and rotation of the web are coupled.
The shape functions of buckling deformation for both
the sideway deflection and rotation of the web, which
satisfy the cantilever beam boundary conditions, can be
selected as exact transcendental function, polynomial
function or half of the simply supported beam function
[31–33]. These three shape functions are all considered
in this paper. The exact transcendental functions are
dx
;
wTF¼ wTFðxÞð18bÞ
dx
;
wBF¼ wBFðxÞð18cÞ
w
h
??
¼
? w w
?h h
()
X
m¼1;2;3;...
?
sin
kmx
L
???
kmx
L
? sinh
kmx
L
??
? bm cos
??
? cosh
kmx
L
? ???
ð19aÞ
where
bm¼sinhðkmÞ þ sinðkmÞ
cosðkmÞ þ coshðkmÞ
and kmsatisfies the following transcendental equation
ð19bÞ
cosðkmÞcoshðkmÞ ? 1 ¼ 0
with k1¼ 1:875104, k2¼ 4:694091, k3¼ 7:854757;...
The polynomial functions are defined as
ð19cÞ
w
h
??
¼
? w w
?h h
()
X
m¼1;2;3;...
1
ðm þ 1Þðm þ 2Þðm þ 3Þðm þ 4ÞCm
x
L
? ?
ð20aÞ
where
Cm
x
L
? ?
¼
x
L
? ?mþ4
þ1
?1
6ðm þ 1Þðm þ 3Þðm þ 4Þ
x
L
? ?3
2ðm þ 1Þðm þ 3Þðm þ 4Þ
x
L
? ?2
ð20bÞ
The half of simply supported beam functions are con-
sidered as
w
h
??
¼
? w w
?h h
??
X
m¼1;2;3;...
1
?
? cos
ð2m ? 1Þpx
2L
? ??
ð21Þ
The displacements and rotations (referring to Eq. (18))
of panels then become
wW¼ w þ yh;
hTF¼ hBF¼ h
wTF¼ w þbw
2h;
wBF¼ w ?bw
2h
ð22Þ
By applying the Rayleigh-Ritz method and solving
for the eigenvalues of the potential energy equilibrium
equation, the flexural–torsional buckling load, Pcr, for a
free-end point load applied at the centroid of the cross-
section is obtained. For different assumed buckling
shape functions, the explicit equations of critical buck-
ling load are given as follows:
The critical buckling load based on exact transcen-
dental shape function is:
Pcr¼ W1fbwLW2þ ð
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
W3þ W4þ W5þ W6þ W7
p
Þ=bwg
ð23Þ
where
W1¼ ð6bfþ bwÞ=½2L3ð76:5bf2? 6:96bfbwþ 0:16bw2Þ?
W2¼ ð123bf? 5:6bwÞD16
W3¼ a11bf3ð279:5bf2? 25:5bfbwþ 0:6bw2Þ
W4¼ 6707:5bf5bw3d11D11? 610:8bf4bw4d11D11
þ bf3bw3D11ð13:9bw2d11þ 30261L2d66Þ
W5¼ bfbw5ð62:7L2d66D11? 305:4bw2D2
? 1377:4L2D2
W6¼ bw6ð7bw2D2
W7¼ a11bf3bw2ð1118bf5d11? 101:8bf4bwd11þ 2:3bf3bw2d11
þ 5043:5bf3L2d66þ 4:64bw5D11þ 20:9bw3L2D66Þ
W8¼ a11bf4bw3½bwð?203:6bw2D11þ 10:5L2d66? 918:5L2D66Þ
þ bfð2235:8bw2D11? 459:5L2d66þ 10087L2D66Þ?
11
16? 5511L2D11D66Þ
11þ 31:4L2D2
16þ 125:5L2D11D66Þ
The critical buckling load based on fifth-order poly-
nomial function is:
Pcr¼ W1 945LbwD16
?
q
þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
17424a2
11bf2bw2þ 176a11bf3W2þ 3bwðW3þ W4Þ
?
ð24Þ
P. Qiao et al. / Composite Structures 60 (2003) 205–217
209
Page 6
where
W1¼
W2¼ 396bf3d11þ 1763L2bfd66þ 792bw3D11þ 3526L2bwD66
W3¼ 139392bf3d11D11þ 620576L2bfd66D11
W4¼ 297675L2D2
The critical buckling load based on half simply sup-
ported beam function is:
?
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
where
30ð6bfþ bwÞ
7L3ð5160bf? 229bwÞ
16þ 704D11ð99bw2D11þ 1763L2D66Þ
Pcr¼ W1 96LbwD16
þ
p4a2
11bf6bw2þ 4p2a11bf3W2þ 12bwW2
q
?
ð25Þ
W1¼
W2¼ p2bf3d11þ 48L2bfd66þ 2p2bw3D11þ 96L2bwD66
W3¼ 2p4bf3d11D11þ 96p2L2bfd66D11
þ bw½p4bw2D2
and in Eqs. (23)–(25), the following material parameters
are defined as:
p2ð6bfþ bwÞ
24L3ð35:22bf? 2:13bwÞ
11þ 192L2ð4D2
16þ p2D11D66Þ?
a11¼ 1=a11;
d11¼ 1=d11;
a66¼ 1=a66;
d66¼ 1=d66
3. Experimental evaluations of buckling of FRP cantilever
I-beams
In this study, four geometries of FRP I-beams, which
were manufactured by the pultrusion process and pro-
vided by Creative Pultrusions, Inc., Alum Bank, PA,
were tested to evaluate their flexural–torsional buckling
responses. The four I-sections (Fig. 2) consisting of (1)
I4 ? 8 ? 3=8 in. ðI4 ? 8Þ; (2) I3 ? 6 ? 3=8 in. ðI3 ? 6Þ;
(3) WF4 ? 4 ? 1=4 in. ðWF4 ? 4Þ; and (4) WF6 ? 6?
3=8 in. ðWF6 ? 6Þ were made of E-glass fibers and
polyester resins. Based on the lay-up information pro-
vided by the manufacturer and a micro/macromechanics
approach [29], the panel material properties of the FRP
I-beams are obtained and given in Table 1. The
clamped-end of the beams was achieved using two steel
angles attached to a vertical steel column (Fig. 3). Using
a loading platform (Fig. 4), the loads were initially ap-
plied by sequentially adding steel angle plates of 111.2 N
(25.0 lbs), and as the critical loads were being reached,
incremental weights of 22.2 N (5.0 lbs) were added until
the beam buckled. The tip load was applied through a
chain attached at the centroid of the cross-section (Fig.
4). Two LVDTs and one level were used to monitor the
rotation of the cross-section, and the sudden sideway
movement of the beam was directly observed in the
experiment. The buckled shapes of four geometries at a
span length of 365.8 cm (12.0 ft) are shown in Figs. 5–8,
and their corresponding critical loads were obtained by
WF 4x4x1/4"
(WF4x4)
WF 6x6x3/8"
(WF6x6)
I 3x6x3/8"
(I3x6)
I 4x8x3/8"
(I4x8)
Fig. 2. Four representative FRP structural shapes.
Table 1
Panel stiffness coefficients for I-sections
Section
D11(Ncm)
D12(Ncm)
D22(Ncm)
D66(Ncm)
a11(N/cm)
a66(N/cm)
d11(Ncm)
d66(Ncm)
I4 ? 8
I3 ? 6
WF4 ? 4
WF6 ? 6
Note: a11¼ 1=a11, a66¼ 1=a66, d11¼ 1=d11, d66¼ 1=d66.
150,200
146,800
45,728
145,700
28,905
28,792
10,749
28,679
69,100
68,648
23,824
68,422
33,082
32,969
12,194
32,856
3,378,000
3,465,000
1,995,000
3,115,000
521,500
539,000
308,000
476,000
208,900
210,000
50,018
196,500
40,195
40,873
12,646
38,502
Fig. 3. Cantilever configuration of FRP beams.
210
P. Qiao et al. / Composite Structures 60 (2003) 205–217
Page 7
summing the weights added during the experiments.
Varying span lengths from 182.9 cm (6.0 ft) to 396.2 cm
(13.0 ft) for each geometry were tested; two beam sam-
ples per geometry were evaluated, and an averaged value
for each pair of beam samples was considered as the
experimental critical load. The measured critical buck-
ling loads and comparisons with analytical solutions and
numerical modeling results are given in the next section.
4. Results and discussion
As discussed in the previous section, the displacement
functions for flexural–torsional buckling consist of the
lateral displacement ðwÞ and rotation ðhÞ of the beam
and are approximated using Eqs. (19)–(21) for tran-
scendental function, polynomial function, and half of
simply supported beam function, respectively. By solv-
ing for the eigenvalues of the energy equation (Eq. (4)),
the critical buckling load, Pcr, can be explicitly obtained
as given in Eqs. (23)–(25) using three different approxi-
mate shape functions. To verify the accuracy of the
proposed analytical approach, the four experimentally
tested FRP I-beam sections are considered (i.e., I4 ? 8,
I3 ? 6, WF4 ? 4 and WF6 ? 6). The analytical solutions
Fig. 4. Load applications at the cantilever beam tip.
Fig. 5. Buckled I4 ? 8 beam.
Fig. 6. Buckled I3 ? 6 beam.
P. Qiao et al. / Composite Structures 60 (2003) 205–217
211
Page 8
and experimental results are also compared with classi-
cal approach based on Vlasov theory [22] and FEM. The
commercial finite element program ANSYS is employed
for modeling of the FRP beams using Mindlin eight-
node isoparametric layered shell elements (SHELL99).
The comparisons of critical buckling loads among
analytical solution using the exact transcendental shape
function, the classical Vlasov theory [22], experimental
data and finite element results are given in Table 2 for
span lengths of L ¼ 304:8 cm (10.0 ft) and L ¼ 365:8 cm
(12.0 ft), and the present analytical solution shows a
good agreement with FEM results and experimental
data. The critical buckling loads versus the lengths (L)
for the four geometries of I4 ? 8, I3 ? 6, WF4 ? 4 and
WF6 ? 6 are shown in Figs. 9–12, respectively, and
these figures indicate that the present analytical predic-
tions for exact transcendental shape function are slightly
higher than the FEM results but close to experimental
Fig. 7. Buckled WF4 ? 4 beam.
Fig. 8. Buckled WF6 ? 6 beam.
10 1520
Length L (cm)
25 30 35
0
Present
Experiment
cr (N)
Flexure-Torsional Buckling Load P
20000
18000
16000
14000
12000
10000
8000
6000
4000
2000
Classical Solution
FEM
Fig. 9. Flexure–torsion buckling load for I4 ? 8 beam.
Table 2
Comparisons for flexural–torsional buckling loads of I-beams
Length L
(cm)
SectionAnalyti-
cal solu-
tion Pcr
(N)
Classical
solution
Pcr(N)
Finite
element
Pcr(N)
Experi-
mental
data Pcr
(N)
304.8I4 ? 8
I3 ? 6
WF4 ? 4
WF6 ? 6
I4 ? 8
I3 ? 6
WF4 ? 4
WF6 ? 6
4765
2338
1498
8526
5201
2360
1783
10,860
4503
2174
1436
8624
4010
2058
1476
–
365.83192
1494
1014
5614
3321
1547
1151
6428
2956
1365
933
5774
2943
1356
920
5476
Note: Analytical solution based on exact transcendental shape func-
tion; classical solution based on the Vlasov theory [22].
212
P. Qiao et al. / Composite Structures 60 (2003) 205–217
Page 9
values and lower than the classical solution using Vlasov
theory [22].
Using the analytical solution of exact transcendental
shape function, the critical buckling load versus the
beam length (L) is presented in Fig. 13. As expected, the
critical load decreases as the span length increases and
lateral buckling becomes more prominent. The present
predictions show a good agreement with FEM results
and experimental data for long beam spans (see Figs. 9–
12), while for shorter span-lengths the buckling mode is
more prone to lateral distortional instability which is not
considered in the present study. This phenomenon can
also be observed in Figs. 14 and 15, where the critical
buckling mode shapes are shown for the buckled I4 ? 8
beams with the respective span lengths of L ¼ 182:9 cm
10 1520 2530 35
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
Present
Classical Solution
FEM
Experiment
Length L (cm)
cr
Flexural-Torsional Buckling Load P (N)
Fig. 10. Flexure–torsion buckling load for I3 ? 6 beam.
10 15 20
Length L (cm)
25 3035
0
Present
Classical Solution
FEM
Experiment
cr
7000
6000
5000
4000
3000
2000
1000
Flexural-Torsional Buckling Load P (N)
Fig. 11. Flexure–torsion buckling load for WF4 ? 4 beam.
101520
Length L (cm)
2530 35
0
40000
35000
30000
25000
20000
15000
10000
5000
cr
Flexural-Torsional Buckling Load P (N)
Present
Classical Solution
FEM
Experiment
Fig. 12. Flexure–torsion buckling load for WF6 ? 6 beam.
10 15 20253035
0
2000
4000
6000
8000
12000
I4X8
I3X6
WF
WF
4X
6X
4
6
28000
26000
24000
22000
20000
18000
16000
14000
10000
Length L (cm)
cr
Flexural-Torsional Buckling Load P (N)
Fig. 13. Flexural–torsion buckling load for different I-section beams.
Fig. 14. Flexure–torsional buckling mode shape for I4 ? 8 at the
length of 182.9 cm (6.0 ft).
P. Qiao et al. / Composite Structures 60 (2003) 205–217
213
Page 10
(6.0 ft) (lateral distortional) and L ¼ 304:8 cm (10.0 ft)
(flexural–torsional) using finite element modeling (AN-
SYS).
4.1. Effect of mode shape function
The accuracy and convergence of different assumed
buckling shape functions (i.e., exact transcendental
shape function, fifth-order polynomial function and half
simply supported beam function) are also investigated.
The critical buckling loads using the three types of
buckling shape functions are compared with the FEM
results for the span length of L ¼ 304:8 cm (10.0 ft) and
L ¼ 365:8 cm (12.0 ft) (see Table 3) and for varying span
lengths (see Figs. 16–19). As shown in Table 3, the an-
alytical solutions using exact transcendental shape
function and fifth-order polynomial function show rel-
atively good correlations with FEM results; whereas the
solution with half simply supported beam function
which is commonly used in the cantilever beam model-
ing demonstrates a large discrepancy. Therefore, the
exact transcendental function and fifth-order polyno-
mial function should be selected in the flexure-torsional
buckling simulation.
Fig. 15. Flexure–torsional buckling mode shape for I4 ? 8 at the
length of 304.8 cm (10.0 ft).
Table 3
Comparisons for flexural–torsional buckling loads of different assumed buckling shape functions
Length L
(cm)
SectionTranscenden-
tal function
Pcr(N)
Error with FE
(%)
Polynomial
function Pcr
(N)
Error with FE
(%)
Half of the SS
function Pcr
(N)
Error with FE
(%)
Finite element
Pcr(N)
304.8 I4 ? 8
I3 ? 6
WF4 ? 4
WF6 ? 6
I4 ? 8
I3 ? 6
WF4 ? 4
WF6 ? 6
4765
2352
1498
8504
5.8
8.2
4.3
)1.4
4747
2338
1494
8481
5.4
7.6
4.2
)1.6
5334
2743
1600
9348
18.5
26.2
11.5
8.4
4503
2174
1436
8624
365.8 3192
1494
1014
5614
8.0
9.4
8.6
2.8
3178
1485
1009
5619
7.5
8.8
8.1
2.7
3578
1880
1089
6192
21.1
37.8
16.7
7.2
2,956
1365
934
5774
101520253035
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
Transcendental Function
Polynomial Function
Cosine Function
FEM
Length L (cm)
cr
Flexural-Torsional Buckling Load P (N)
Fig. 16. Comparison of flexure–torsional buckling for I4 ? 8 using
different assumed buckling shape functions.
1015 20
Length L (cm)
2530 35
0
9000
8000
7000
6000
5000
4000
3000
2000
1000
cr
Flexural-Torsional Buckling Load P (N)
Transcendental Function
Polynomial Function
Cosine Function
FEM
Fig. 17. Comparison of flexure–torsional buckling for I3 ? 6 using
different assumed buckling shape functions.
214
P. Qiao et al. / Composite Structures 60 (2003) 205–217
Page 11
4.2. Effect of load locations
To account for the location of the applied load along
the vertical direction of the beam tip cross-section, the
transverse stress resultant on the web panel is repre-
sented in Eq. (17e), and the analytical solutions can be
formulated to obtain the critical buckling loads at any
location along the web at the load–tip cross-section. The
comparisons of critical buckling loads among three lo-
cations (centroid, top and bottom) are shown in Figs.
20–23 for the four given FRP sections, and they dem-
onstrate that as the load height increases, the beam is
more vulnerable to buckling.
4.3. Effect of fiber architecture and fiber volume fraction
To investigate the effect of fiber angle orientation on
flexural–torsional buckling behavior, the original stit-
ched ?45? angle layers in the laminated panels of
WF4 ? 4 were substituted by (?h) layers with h as a
design variable. The critical buckling load with respect
to ply angle (h) at fiber volume fraction of 50% is shown
in Fig. 24, where a maximum critical buckling load can
be observed at h ¼ 30? for the tip load applied at the
centroid of the cross-section. Similarly, the effect of fiber
volume fraction on flexural–torsional buckling behavior
is studied, and the lateral buckling load versus fiber
volume fraction is shown in Fig. 25. As anticipated, the
10 1520
Length L (cm)
253035
0
1000
2000
3000
4000
5000
cr
Flexural-Torsional Buckling Load P (N)
Transcendental Function
Polynomial Function
Cosine Function
FEM
Fig. 18. Comparison of flexure–torsional buckling for WF4 ? 4 using
different assumed buckling shape functions.
10 1520
Length L (cm)
25 3035
0
40000
30000
20000
10000
cr
Flexural-Torsional Buckling Load P (N)
P Applied at Top Flange
P Applied at Centroid
P Applied at Bottom Flange
Fig. 20. Flexure–torsion buckling load for I4 ? 8 beam at different
applied load positions.
10 15 20
Length L (cm)
25 30 35
0
2000
4000
6000
8000
10000
12000
14000
16000
cr
Flexural-Torsional Buckling Load P (N)
Load Applied at Top Flange
Load Applied at Centroid
Load Applied at Bottom Flange
Fig. 21. Flexure–torsion buckling load for I3 ? 6 beam at different
applied load positions.
1015 20253035
0
35000
30000
25000
20000
15000
10000
5000
Length L (cm)
Transcendental Function
Polynomial Function
Cosine Function
FEM
cr
Flexural-Torsional Buckling Load P (N)
Fig. 19. Comparison of flexure–torsional buckling for WF6 ? 6 using
different assumed buckling shape functions.
P. Qiao et al. / Composite Structures 60 (2003) 205–217
215
Page 12
fiber volume fraction is of significant importance for
improving the buckling resistance.
5. Conclusions
In this paper, a combined analytical and experimental
study is presented to study the flexural–torsional buck-
ling behavior of pultruded FRP composite cantilever
I-beams. The total potential energy based on nonlinear
plate theory is derived, and shear effects and beam
bending–twisting coupling are accounted for in the
analysis. Three different types of buckling mode shape
functions, namely transcendental function, polynomial
function, and half simply supported beam function,
which all satisfy the cantilever beam boundary condi-
tions, are used to obtain the analytical solutions and
explicit prediction formulas. An experimental study of
four different geometries of FRP cantilever I-beams is
performed, and the critical flexural–torsional buckling
loads for different span lengths are obtained. A good
agreement among the proposed analytical solutions,
experimental testing, and FEM is obtained. The study
on effect of buckling mode shape functions indicate that
the approximations by exact transcendental function
and polynomial function compare well with FEM re-
sults and may be more applicable for the buckling
10 1520 253035
0
10000
8000
6000
4000
2000
cr
Flexural-Torsional Buckling Load P (N)
Length L (cm)
Load Applied at Top Flange
Load Applied at Centroid
Load Applied at Bottom Flange
Fig. 22. Flexure–torsion buckling load for WF4 ? 4 beam at different
applied load positions.
10 1520253035
0
10000
20000
30000
40000
50000
60000
70000
cr
Flexural-Torsional Buckling Load P (N)
Length L (cm)
Load Applied at Top Flange
Load Applied at Centroid
Load Applied at Bottom Flange
Fig. 23. Flexure–torsion buckling load for WF6 ? 6 beam at different
applied load positions.
0
WF 4X4
50% vol. fraction
Beam Length L=243.8 cm
Beam Length L=304.8 cm
Beam Length L=365.8 cm
(+θ/-θ)
3000
2500
2000
1500
1000
500
0
10
20
3040
50
60
70
80 90
Ply Angle
cr
Flexural-Torsional Buckling Load P (N)
Fig. 24. Influence of lamination angle (?h) on flexure–torsional
buckling of WF4 ? 4 beam.
0
1000
2000
3000
4000
5000
0
20
40
60
80
Fiber Volume Fraction (%)
cr
Flexural-Torsional Buckling Load P (N)
Beam Length L=243.8 cm
Beam Length L=304.8 cm
Beam Length L=365.8 cm
Fig. 25. Influence of fiber volume fraction on flexural–torsional
buckling of WF4 ? 4 beam.
216
P. Qiao et al. / Composite Structures 60 (2003) 205–217
Page 13
modeling of cantilever beam configuration. A paramet-
ric study on the effects of load location through the
height of the cross-section, fiber orientation and fiber
volume fraction on buckling behavior is also presented.
The explicit and experimentally validated analytical
formulas for the flexural–torsional buckling prediction
can be effectively used to design and characterize the
buckling behavior of FRP structural shapes.
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