Engineering Structures 24 (2002) 955–964

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Lateral buckling of I-section composite beams

J. Lee a,∗, S.-E. Kim b, K. Hong c
a
Department of Architectural Engineering, Sejong University, 98 Kunja Dong, Kwangjin Ku, Seoul 143-747, South Korea
b
Department of Civil and Environmental Engineering, Sejong University, 98 Kunja Dong, Kwangjin Ku, Seoul 143-747, South Korea
c
Department of Architectural Engineering, Yonsei University, 134 Shinchon, Seodaemun Ku, Seoul 120-749, South Korea

Received 16 July 2001; received in revised form 9 January 2002; accepted 18 January 2002

Abstract

Lateral buckling of a laminated composite beam with I-section is studied. A general analytical model applicable to the lateral
buckling of an I-section composite beam subjected to various types of loadings is developed. This model is based on the classical
lamination theory, and accounts for the material coupling for arbitrary laminate stacking sequence configuration and various bound-
ary conditions. The effects of the location of applied loading on the buckling capacity are also included in the analysis. A displace-
ment-based one-dimensional finite element model is developed to predict critical loads and corresponding buckling modes for a
thin-walled composite beam with arbitrary boundary conditions. Numerical results are obtained for thin-walled composites under
central point load, uniformly-distributed load, and pure bending with angle-ply laminates. The effects of fiber orientation, location
of applied load, and types of loads on the critical buckling loads are parametrically studied.  2002 Elsevier Science Ltd. All
rights reserved.

Keywords: thin-walled structures; laminated composites; finite element method; lateral bucking

1. Introduction gradient. Recently, Trahair [7] investigate the flexural-

torsional behavior of structures.
Fiber-reinforced plastics (FRP) have been increasingly For composite thin-walled beams, Bauld and Tzeng
used over the past few decades in a variety of structures [8] extended Vlasov’s thin-walled bar theory to sym-
that require high ratio of stiffness and strength to weight. metric fiber-reinforced laminates. Kabir and Sherbourne
In the construction industry, recent applications have [9] studied lateral-torsional buckling of I-section com-
shown the structural and cost efficiency of FRP struc- posite beams based on the Rayleigh–Ritz method. Lin et
tural shapes, such as thin-walled open sections through al. [10] studied buckling problem of thin-walled com-
so-called pultrusion [1] process. It is known that the posite structural members by finite element method.
design of thin-wailed members is governed by stability Davalos and Qiao [11] presented a combined analytical
considerations due to their slenderness. and experimental evaluation of flexural-torsional and lat-
Thin-walled open section members made of isotropic eral-distorsional buckling of fiber-reinforced plastic
materials have been studied by many researchers [2,3]. composite with wide-flange beams. Turvey and his col-
The lateral buckling analysis under various configur- laborator conducted a series of lateral buckling tests on
ations were parametrically studied by Bleich [4]. Clark pultruded GRP I-section cantilever beams [12], and
and Hill [5] presented a, solution for the lateral buckling compared their results with simple theoretical results
of a singly symmetric section under various loading con- [13]. Recently, Lee and Kim [14] presented an analytical
ditions. Kitipornchai et al. [6] presented a buckling model which account for flexural-torsional buckling of
analysis for a monosymmetric beam under moment I-section composite beams. The model was capable of
predicting accurate buckling loads and modes for vari-
ous configurations.
∗
Corresponding author. Tel.: +82-2-3408-3287; fax: +82-3-3408- Most of the work concerning the lateral stability of
3331. thin-walled composite beams are focused on I-section.
E-mail address: jhlee@sejong.ac.kr (J. Lee). Recently, Kabir and Sherbourne [15] proposed an ana-

0141-0296/02/$ - see front matter  2002 Elsevier Science Ltd. All rights reserved.
PII: S 0 1 4 1 - 0 2 9 6 ( 0 2 ) 0 0 0 1 6 - 0
956 J. Lee et al. / Engineering Structures 24 (2002) 955–964

lytical solution for predicting the lateral bulking capacity along the contour line of the cross section. The (n, s, z)
of composite channel-section beams. More recently, Lee and (x, y, z) coordinate systems are related through an
[16] gave a closed-form expression for the location of angle of orientation ⍜ as defined in Fig. 1. The third
center of gravity and shear center as a function of lami- coordinate set is the contour coordinate s along the pro-
nation stacking sequence as well as sectional properties. file of the section with its origin at any point O on the
In the present study, a general analytical model appli- profile section. Point P is called the pole axis, through
cable to the lateral buckling of a I-section composite which the axis parallel to the z axis is called the pole
beam subjected to various types of loadings is axis.
developed. This model is based on the classical lami- To derive the analytical model for a thin-walled com-
nation theory [17], and accounts for the material coup- posite beam; the following assumptions are made:
ling for arbitrary laminate stacking sequence configur-
ation, i.e. unsymmetric as well as symmetric, and various 1. The contour of the thin wall does not deform in its
boundary conditions. The effects of the location of own plane.
applied loading on the buckling capacity are also 2. The shear strain of the middle surface is zero in
included in the analysis. A displacement-based one- each clement.
dimensional finite element, model is developed to predict 3. The Kirchhoff–Love assumption in classical plate
critical loads and corresponding buckling modes for a theory remains valid for laminated composite thin-
thin-walled composite beam with arbitrary boundary walled beams.
conditions. Governing buckling equations are derived 4. The time-dependent behavior is neglected.
from the principle of the stationary value of total poten-
tial energy. Numerical results are obtained for thin- According to assumption 1, the midsurface displacement
walled composites under central point load, uniformly- components ū, v̄ at a point A in the contour coordinate
distributed load, and pure bending with angle-ply lami- system can be expressed in terms of a displacements U,
nates. The effects of fiber orientation, location of applied V of the pole P in the x, y directions, respectively, and
load, and types of loads on the critical buckling loads the rotation angle ⌽ about the pole axis,
are parametrically studied. u(s,z) ⫽ U(z)sin⌰(s)⫺V(z)cos⌰(s)⫺⌽(z)q(s) (1a)
u(s,z) ⫽ U(z)cos⌰(s) ⫹ V(z)sin⌰(s)⫺⌽(z)r(s) (1b)
2. Kinematics These equations apply to the whole contour. The out-of-
plane shell displacement ␼ can now be found from the
The theoretical developments presented in this paper assumption 2 (also known as Vlasov assumption) as
require three sets of coordinate systems which are mutu-
ally interrelated. The first coordinate system is the w(s,z) ⫽ W(z)⫺U’(z)x(s)⫺V’(z)y(s)⫺⌽’(z)w(s) (2)
orthogonal Cartesian coordinate system (x, y, z), for where differentiation with respect to the axial coordinate
which the x and y axes lie in the plane of the cross sec- z is denoted by primes (‘); W represents the average axial
tion and the z axis parallel to the longitudinal axis of the displacement of the beam in the z direction; x and y are
beam. The second coordinate system is the local plate the coordinates of the contour in the (x, y, z) coordinate
coordinate (n, s, z) as shown in Fig. 1: wherein the n system; and w is the so-called sectorial coordinate or
axis is normal to the middle surface of a plate element, warping function given by
the s axis is tangent to the middle surface and is directed

冕
w(s)⫺ r(s)ds (3)

The displacement components u, v, w representing the

deformation of any generic point on the profile section
are given with respect to the midsurface displacements
ū, v̄, w̄ by the assumption 3.
u(s,z,n) ⫽ u(s,z) (4a)
∂u(s,z)
v(s,z,n) ⫽ v(s,z)⫺n (4b)
∂s
∂u(s,z)
w(s,z,n) ⫽ w(s,z)⫺n (4c)
∂z
The strains associated with the small-displacement
Fig. 1. Definition of coordinates in thin-walled open section. theory of elasticity are given by
 J. Lee et al. / Engineering Structures 24 (2002) 955–964 957

es ⫽ es ⫹ nks
ez ⫽ ez ⫹ nkZ
(5a)
(5b)
冕
dU ⫽ {sz[d苸zo ⫹ (x ⫹ nsin⌰)d␬y ⫹ (y
v

gsz ⫽ nksz (5c) ⫺ncos⌰)d␬x ⫹ (w⫺nq)d␬w)] ⫹ ssznd␬sz其dv (12)

where
冕
l

∂u ∂w ⫽ {Nzd苸zo ⫹ Myd␬y ⫹ Mxd␬x ⫹ Mwd␬w

ēs ⫽ ,ez ⫽ (6a)
∂s ∂z 0

∂u 2
∂u
2
∂u
2
⫹ Mtd␬sz其dz
ks ⫽ ⫺ 2 ,kz ⫽ ⫺ 2 ,ksz ⫽ ⫺2 (6b)
∂s ∂z ∂s∂z Nz, Mx, My and Mw are axial force, bending moments in
All the other strains are identically zero. In eq. (6), ēs the x and y directions, and warping moment (bimoment)
and ␬¯ s are assumed to be zero, and ēz, ␬¯ z, and ␬¯ sz are with respect to the centroid, respectively, defined by
midsurface axial strain and biaxial curvatures of the integrating over the cross-sectional area A as:

冕
shell, respectively. The above shell strains can be con-
verted to beam strain components by substituting eqs. Nz ⫽ szdsdn (13a)
(1) and(2) into eq. (6)
A
ez ⫽ eoz ⫹ xKy ⫹ yKx ⫹ wKw
冕
(7a)
Kz ⫽ Kysin⌰⫺Kxcos⌰⫺Kwq (7b) My ⫽ sz(x ⫹ nsin⌰)dsdn (13b)
Ksz ⫽ Ksz (7c) A

where eoz,␬x,␬y,␬w and ␬sz are axial strain, biaxial curva-

tures in the x and y direction, warping curvature with
冕
Mx ⫽ sz(y⫺ncos⌰)dsdn
A
(13c)
respect to the shear center, and twisting curvature in the
beam, respectively defined as
eoz ⫽ W’ (8a) 冕
Mw ⫽ sz(w⫺nq)dsdn
A
(13d)

␬x ⫽ ⫺V⬙
冕
(8b)
␬y ⫽ ⫺U⬙ (8c) Mt ⫽ szsndsdn (13e)
␬w ⫽ ⫺⌽⬙ (8d) A

␬sz ⫽ 2⌽’ (8e) In eq. (10), V is the potential of transverse load acting
on the cross section at a point a distance ā above the
The resulting strains can be obtained from eqs. (5) and shear center; and the variation of the potential of trans-
(7) as verse loads at shear center becomes [4]
ez ⫽ eoz ⫹ (x ⫹ nsin⌰)␬y ⫹ (y⫺ncos⌰)␬x ⫹ (w
冕
(9a) l

⫺nq)␬w dV ⫽ ⫺ [Mb(⌽dU⬙ ⫹ U⬙d⌽) ⫹ ap⌽d⌽]dz (14)

gsz ⫽ n␬sz (9b) o

where Mb is not the actual bending moment in the beam,

but the simple beam moment due to the loads p.
3. Variational formulation The principle of total potential energy can be stated as

The total potential energy of the system can be stated, d⌸ ⫽ d(u ⫹ v) ⫽ 0 (15)
in its buckled shape, as Substituting eqs. (12) and (14) into eq. (15), the follow-
⌸⫽U⫹V (10) ing weak statement is obtained:

冕
l
where U is the strain energy

冕
0 ⫽ {NzdW’⫺MydU⬙⫺MxdV⬙⫺Mwd⌽⬙ (16)
1
U⫽ (s 苸 ⫹ szsgzs)dv, (11)
2 v z z 0

⫹ 2Mtd⌽’⫺Mb(⌽dU⬙ ⫹ U⬙d⌽)⫺ap⌽d⌽}dz
The variation of the strain energy is circulated by substi-
tuting eq. (7) into eq. (11) In eq. (16), Mb and p are the buckling moment and trans-
958 J. Lee et al. / Engineering Structures 24 (2002) 955–964

verse load, and can be written for various types of load- b3(3) 3
ing as E33 ⫽ [A11aya(2)⫺2B11aya ⫹ D11a]ba ⫹ A (21j)
12 11
Mb ⫽ lf(z) (17a) b3(3) 3
E34 ⫽ ⫺ B (21k)
p ⫽ lg(z) (17b) 12 11
where λ is a buckling parameter and f (z) and g(z) are E35 ⫽ Ba16yaba⫺Da16ba (21l)
polynomial functions which depend on the loading pat-
ba(3)
tern. E44 ⫽ [A11aya(2)⫺2B11aya ⫹ D11a] (21m)
12
b3(3) 3
⫹ D
4. Constitutive equations 12 11

The constitutive equations of a kth orthotropic lamina E45 ⫽ 0 (21n)

in the laminate co-ordinate system are given by E55 ⫽ Dk66bk (21o)
{szssz其 ⫽ [Q11 Q16Q16 Q66] {ezgsz其
k k
(18) where, Aij, Bij and Dij matrices are extensional, coupling
where Q̄ij are transformed reduced stifnesses [17]. Axial and bending stiffness, respectively, defined by

冕
force Nz can now be expressed with respect to the gen-
eralized strains by combining eqs. (13a), (18) and (9) (Aij,Bij,Dij) ⫽ Qij(1,n,n2)dn (22)

冕
Nz ⫽ {Q11[ezo ⫹ (x ⫹ nsin⌰)␬y ⫹ (y
A
(19) In eq. (21), the superscript in the parenthesis () denotes
the power, and repeated index denotes summation. Index
k varies from 1 to 3 whereas a varies from 1 to 2
⫺ncos⌰)␬x ⫹ (w⫺nq)␬w] ⫹ Q16n␬sz其 implying 4 top and bottom flanges (1,2) and web (3) as
Similarly, the other stress resultants (My, Mx, Mw, Mt) shown in Fig. 2. bk denotes width of flanges and web.
can also be written in terms of the generalized strains.
Consequently, the constitutive equations for a thin-
walled laminated composite are obtained as
{NzMyMxMwMt其 ⫽ [E11E12E13E14E15E22E23E24E25 (20)
E33E34E35E44E45sym.E55]{e ␬ ␬ ␬ ␬ 其
z
o
y x w sz

where Eij are stiffnesses of the thin-walled composite,

and can be found in Ref. [16]. It appears that the lami-
nate stiffnesses Eij depends on the cross section of the
composites, and the explicit expressions for I-section are
given as follows:
E11 ⫽ AK11bk (21a)
E12 ⫽ B311b3 (21b)

E14 ⫽ Aa11baya⫺Ba11ba (21c)

E14 ⫽ 0 (21d)
E15 ⫽ B16
k
bk (21e)
ba(3)
E22 ⫽ Aa11 ⫹ D311b3 (21f)
12
E23 ⫽ 0 (21g)
ba(3)
E24 ⫽ [A11aya⫺B11a] (21h)
12
E25 ⫽ D316b3 (21i) Fig. 2. Geometry of a thin-walled composite.
 J. Lee et al. / Engineering Structures 24 (2002) 955–964 959

5. Governing equations ity, (EIx)com and (EIY)com represent flexural rigidities with
respect to x and y axis, (EIw)com and (GJ)com represent
The lateral buckling equations of the present study can warping and torsional rigidities of the thin-walled com-
be derived by integrating the derivatives of the varied posite, respectively, written as
quantities by parts and collecting the coefficients of dU,
(EA)com ⫽ E11 (27a)
dV, dW and d⌽:
(EIy)com ⫽ E22 (27b)
N⬘z ⫽ 0 (23a)
(EIx)com ⫽ E33 (27c)
M⬙y ⫹ (Mb⌽)⬙ ⫽ 0 (23b)
(EIw)com ⫽ E44 (27d)
M⬙x ⫽ 0 (23c)
(GJ)com ⫽ 4E55 (27e)
For the above equations, it is well known that the criti-
Mw ⫹ 2M’l ⫹ MbU⬙ ⫹ ap⌽ ⫽ 0 (23d) cal buckling moment for pure bending is given by the
The natural boundary conditions are of the form: closed form solution for simply-supported boundary
conditions[1]:

冋 册
dW:Nz (24a)

冪
p2(EIy)com p2(EIw)com
dU:M1y ⫹ (Mb⌽)’ (24b) Mcr ⫽ ⫹ (GJ)com (28a)
␫2 ␫2
dU’:My ⫹ Mb⌽ (24c)
’
dV:M x (24d)
6. Finite element model
dV’:Mx (24e)
The present theory for thin-walled composite beams
d⌽:M1w ⫹ 2Mt (24f) described in the previous section was implemented via
d⌽’:Mw (24g) a displacement based finite element method. The gen-
eralized displacements are expressed over each element
By substituting eqs. (20) and (8) into eq. (23), the as a linear combination of the one-dimensional Lagrange
explicit form of the governing equations yield: interpolation function ⌿j and Hermite-cubic interp-
E11W⬙⫺E12U⵮⫺E13V⵮ ⫹ 2E15⌽⬙ ⫽ 0 (25a) olation function yj associated with node j and the
nodal values;
E12W⵮⫺E22U ⫺E24⌽ ⫹ 2E25⌽⵮ ⫹ (Mb⌽)⬙
iv iv
(25b)

冘
n
⫽0 W⫽ wj⌿j (29a)
E13W⵮⫺E23U ⫺E33V ⫺E34⌽ ⫹ 2E35⌽⵮ ⫽ 0
iv iv iv
(25c) j⫽1

冘
n
⫺E24Uiv⫺E34Viv⫺E44⌽iv ⫹ 2E15W⬙⫺2E25U⵮ (25d) U⫽ ujyj (29b)
⫺2E35V⵮ ⫹ 4E55⌽⬙ ⫹ MbU⬙ ⫹ ap⌽ ⫽ 0 j⫽1

冘
n
The above equations are the most general form for lateral V⫽ vjyj (29c)
buckling of a thin-walled laminated composite with an j⫽1
I-section, and the dependent variables, U, V, W and ⌽
冘
n
are fully-coupled.
⌽⫽ fjyj (29d)
If the stacking sequence of the web is symmetric: E12
j⫽1
= E34 = 0, and the thin-walled composite is symmetric
with respect to z axis, E13 = E15 = E24 = 0. Further, if Substituting these expressions into the weak statement
both the web and flange are balanced laminates (±q pairs in eq. (16), the finite element model of a. typical clement
of layers), Aki6 ⫹ Dki6 ⫽ 0, and thus, E25 = E35 = 0. can be expressed as the standard eigenvalue problem:
Finally, eq. (23) can be simplified to the uncoupled dif- ([K]⫺l[G]){⌬} ⫽ {0} (30)
ferential equations using eq. (17) as:
where [K] is the element stiffness matrix
(EA)comW⬙ ⫽ 0 (26a)

冤 冥
⫺(EIy)comUiv ⫹ l(f⌽)⬙ ⫽ 0 (26b) K11 K12K13K14
⫺(EIx)comViv ⫽ 0 (26c) K22K23K24
[K] ⫽ (31)
K33K34
⫺(EIw)com⌽ ⫹ (GJ)com ⫹ l(fU⬙ag⌽) ⫽ 0
iv
(26d)
sym. K44
From the above equations, (EA)com represents axial rigid-
960 J. Lee et al. / Engineering Structures 24 (2002) 955–964

and [G] is the element geometric stiffness matrix All others are zero.
In eq. (30), {⌬} is the eigenvector of nodal displace-

冤 冥
G11 G12G13G14 ments corresponding to an eigenvalue
G22G23G24 {⌬其 ⫽ {w u v f其T (35)
[G] ⫽ (32)
G33G34
sym. G44 7. Numerical results and discussion

The explicit form of [K] and [G] are given by For verification purpose, an isotropic simply-sup-
ported I-section beam with l = 8 m is considered (Fig.

冕
l
3). The material properties are assumed to be:
ij ⫽
K11 E11⌿’i ⌿’j dz (33a)
E ⫽ 1GPa,n ⫽ 0.34 (36)
0
The results by the present approach are compared to

冕
l
those of closed-form solutions in Table 1 for various
ij ⫽ ⫺ E12⌿i yj dz
K12 ’ ’’
(33b) types of loading conditions.
0
The functions f(z) and g(z) in eq. (26) are given as
follows for various types of loading:

冕
l

K ⫽ ⫺ E13⌿’iy’’j dz
13
ij (33c) 앫 pure bending
0 f(z) ⫽ 1; g(z) ⫽ 0 (37)

冕
l
앫 uniformly distributed load
ij ⫽
K14 2E15⌿’iyjdz (33d) 1 l2
f(z) ⫽ ( ⫺z2); g(z) ⫽ 1 (38)
0 24
앫 central point load

冕
l
1 l
f(z) ⫽ ( ⫺z); g(z) ⫽ 0 (39)
K ⫽ E22y⬙jy⬙jdz
22
ij (33e) 22
0

冕
l

ij ⫽ ⫺ (E24y⬙iy⬙j⫺2E25y⬙iy⬘j)dz
K24 (33f)
0

冕
l

ij ⫽
K33 E33y⬙iy⬙jdz (33g)
0

冕
l

ij ⫽
K34 (E34y⬙iy⬙j⫺2E35y⬙iy⬘j)dz (33h)
0

冕
l

K ⫽ (E44y⬙iy⬙j ⫹ 4E55y⬙iy⬙j)dz
44
ij (33i)
0

冕
l

G ⫽ fy⬙iyjdz
24
ij (34a)
0

冕
l

ij ⫽
G44 agyiyjdz (34b)
Fig. 3. Geometry of a thin-walled composite beam [10].
0
 J. Lee et al. / Engineering Structures 24 (2002) 955–964 961

Table 1
Lateral buckling moments and loads of a simply-supported I-section beam for various loading cases

Load acts at Pure bending Uniformly distributed load Central point load

Present Ref. [1] Present Ref. [1] Present Ref. [1]

Shear center 33.429 33.429 4.723 4.774 22.716 22.989

Top flange ... ... 3.875 3.876 18.887 17.721
Bottom flange ... ... 5.753 5.777 27.248 29.335

⫽ 1 at the loading point

It is seen that the results by present finite element

analysis are in good agreements with theclosed form
solution in Ref. [4].
The next example shows the verification of the present
approach for orthotropic beams. The buckling para-
meters of a simply-supported I-shaped composite beam
with a span l = 10 m in Ref. [10] are compared to those
of the present theory. Three cases of loading are con-
sidered: pure bending, central point load, and uniformly
distributed load. The geometry of the beam is shown
in Fig. 3. The critical buckling loads and moments are
computed and shown in Table 2 for various ratios of
E1/G12 (E1 = 17.225GPa), where subscript 1 and 2 indi-
cate fiber direction and perpendicular to fiber direction,
respectively. As shown in Table 2, the present results
agree well with the results in Ref. [10] for all the ranges
of E1/G12.
As a next example, a thin-walled I-section composite
beam with length l = 8 m is considered in order to inves-
tigate the effects of fiber orientation and span-to-height
ratio on the lateral buckling capacity. The geometry of Fig. 4. Thin-walled I-section composite beam.
the I-section is shown in Fig. 4, and the following engin-
eering constants are used;E1/E2 = 25, G12/E2 = 0.6, v12 Pcrl2
= 0.25 P̄cr ⫽ (41)
E2t3b33
For convenience, the following nondimensional buck-
ling moment and load are used: qcrl3
q̄cr ⫽ (42)
Mcrl E2t3b33
M̄cr ⫽ (40)
E2t3b33 The top and bottom flanges are considered as unidi-

Table 2
Comparison of Lateral buckling moments and loads of a simply-supported I-section beam

E1/G12 Pure bending (kNm) Uniformly distributed load(kN) Central point load( kN / m)

Present Ref. [10] Present Ref. [10] Present Ref. [10]

2.6 13.50 13.30 7.35 7.19 1.22 1.25

5.2 10.15 9.94 5.52 5.38 0.92 0.93
10 8.06 7.90 4.39 4.27 0.73 0.74
20 6.66 6.38 3.63 3.46 0.60 0.60
40 5.83 5.50 3.28 2.97 0.53 0.52
962 J. Lee et al. / Engineering Structures 24 (2002) 955–964

rectional while the web laminate is assumed to be angle-

ply laminates [q/⫺q]. The coupling stiffnesses E12, E13,
E24, E25, E34 and E35 become zero as seen in the previous
section, but E15 does not vanish due to unsymmetric
stacking sequence of the web. The critical moment of a
simply-supported beam under pure bending by the finite
element analysis and the closed-form solution, which
neglects the coupling effect of E15 from eq. (28a) are
given in Fig. 5. As the fiber orientation is rotated off-
axis, the discrepancy between the present finite element
analysis and the closed-form solution becomes no more
negligible. For example; more than 10% error exists
between two results at q = 30°. That is, the closed-form
solution is no more valid for off-axis fiber orientation
due to coupling stiffnesses.
The next example presents a simply-supported beam
under central point load. It is considered that the beam
is loaded at the shear center, top flange and the bottom
flange. The critical buckling loads for the three cases are
plotted with respect to the fiber angle variation in Fig.
6. All the three cases show similar trends. That is, as
the fiber orientation is rotated off-axis, lateral buckling Fig. 6. Lateral buckling load with respect to the fiber angle of a sim-
ply-supported I-section composite beam under central point load.
load increases to a maximum value at q = 45° and then
decreases to the minimum value at q = 90°. The critical
buckling load for a simply-supported beam under uni-
formly-distributed load is illustrated with respect to the
fiber angle variation in Fig. 7. In general, the behavior
of buckling load is similar to that of the beam with cen-
tral point load.
In order to investigate the effect of span-to-height
ratio on the buckling load, a simply-supported beam
under pure bending is considered for four different span

Fig. 7. Lateral buckling load with respect to the fiber angle of a sim-
ply-supported I-section composite beam under uniformly distributed
load.

lengths (l = 2,4,8, 16 m) in Fig. 8. For all the cases

considered, the maximum buckling load occurs near 45°
implying that the lateral buckling strength is governed
by torsional ridigity (GJ)com.
The next example is the same as before except that
Fig. 5. Lateral buckling moment with respect to the fiber angle of a in this case, boundary condition is clamped-free, i.e. can-
simply-supported I-section composite beam under pure bending. tilever beam under point load at free end. Fig. 9 shows
 J. Lee et al. / Engineering Structures 24 (2002) 955–964 963

8. Concluding remarks

A one-dimensional finite element model was

developed to study the lateral buckling of a composite
with I-section. The model is capable of predicting accur-
ate lateral buckling loads and moments for various con-
figurations. The effects of loading condition, location of
applied load and the fiber angle of web on buckling loads
and moments of composite are studied.
Based on the above analytical developments and
numerical results, the following conclusions are made:

앫 For the beam under pure bending with off-axis fiber

orientation, orthotropic closed-form solution is not
appropriate for predicting lateral buckling loads due
to the existence of coupling stiffness.
앫 Lateral buckling capacity of beams with transverse
loads are affected by the location of applied load as
well as the fiber orientation.
앫 For all the cases of span-to-height ratio of the com-
Fig. 8. Lateral buckling moment with respect to the fiber angle of posite beams considered, the maximum buckling load
a simply-supported I-section composite beam under pure bending for occurs near 45” fiber angle implying that the lateral
various span lengths. buckling strength is governed by torsional ridigity
(GJ)com.

Acknowledgements

The work presented in this paper was supported by

funds of National Research Laboratory program (2000-
N-NL-01-C-162) from Ministry of Science and Tech-
nology in Korea. Authors wish to appreciate the finan-
cial support.

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