ANALYTICAL AND EXPERIMENTAL STUDY OF LATERAL AND

DISTORTIONAL BUCKLING OF FRP WIDE-FLANGE BEAMS

By Julio F. Davalos l and Pizhong Qiao/ Associate Members, ASCE

ABSTRACT: A combined analytical and experimental evaluation of flexural-torsional and lateral-distortional

buckling of fiber-reinforced plastic (FRP) composite wide-flange (WF) beams is presented. Based on energy
principles, the total potential energy equations for instability of FRP WF sections are derived using the nonlinear
elastic theory. For the analysis of lateral-distortional buckling, a fifth-order polynomial shape function is adopted
to model the deformed shape of web panels. The models are validated by testing two geometrically identical
FRP WF beams but with distinct material architectures produced by the pultrusion process. The beams are tested
under midspan concentrated loads to evaluate their flexural-torsional and lateral-distortional buckling responses.
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To detect rotations of the midspan cross sections and onset of critical buckling loads, horizontal transverse bars
are attached to the beam's flanges, and the bar ends are connected to linear variable differential transducers
(LVDTs). For the same purpose, we use strain gauges bonded to the upper and lower surfaces near to the free
edges of the top flange. A good agreement between the proposed analytical approach and experimental and
finite-element analyses results is obtained, and simplified engineering equations for flexural-torsional buckling
are formulated. The proposed analytical solutions can be used to predict flexural-torsional and lateral-distortional
buckling loads for other FRP shapes and to formulate simplified design equations.

INTRODUCTION structural shapes and have developed some design methodol-

Fiber-reinforced plastic (FRP) composite shapes are increas- ogies for these members. The flexural-torsional buckling be-
ingly used in civil engineering structures, and recent applica- havior of pultruded E-glass FRP I-beams has been investigated
tions have shown the structural and cost efficiency of FRP experimentally by Mottram (1992), and the observed results
structural shapes, such as thin-walled I-beams and box-beams. compared well with numerical predictions using a finite-dif-
Because of the complexity of composite materials, design ference method. Mottram (1992) emphasized that there is a
guidelines developed for members of conventional materials, potential danger in analysis and design of FRP beams without
such as steel, concrete, or wood, cannot be readily applied to including shear deformation. Barbero and Raftoyiannis (1994)
FRP shapes, and therefore, there is a need to develop practical extended the formulation by Roberts and Jhita (1983) to study
engineering formulas for the design analysis of FRP shapes. the lateral and distortional buckling of composite FRP 1-
Because of the relatively low stiffness and thin-walled sec- beams. In their study, the stability equilibrium equation of the
tional geometry of FRP shapes, problems with global buckling system was established based on the vanishing of the second
and excessive local deformations are common in current struc- variation of the total potential energy; plate theory was used
tural beams. In general, buckling and deflection limits tend to to allow for distortion of the cross sections, and shear and
be the governing design criteria for current FRP shapes (Dav- bending-twisting coupling effects were included in their anal-
alos et al. 1996a). Because of the high strength-to-stiffness ysis. With the use of the Galerkin method to solve the equi-
ratio of composites, buckling is the most likely mode of failure librium differential equation, Pandey et al. (1995) presented a
before material failure for FRP shapes. A long slender beam theoretical formulation for flexural-torsional buckling of thin-
under bending loads about the strong axis may buckle by a walled composite I-section beams; a parametric study of op-
timal fiber direction for improving the lateral buckling re-
combined twisting and lateral (sideways) bending of the cross
sponse of pultruded I-beams was performed. Lin et al. (1996)
section. This phenomenon is known as flexural-torsional (lat-
developed a finite-element method to study the stability of
eral) buckling. For intermediate span beams, a combination of
thin-walled FRP structural members; they used an element
lateral and local buckling may result in lateral-distortional
with seven degrees of freedom at each node, and the influence
buckling of the cross section. Numerous analyses (Hancock
of the in-plane shear strain on the stability of the members
1978, 1981; Roberts 1981; Roberts and Jhita 1983; Ma and
was considered; their study indicated that the influence of
Hughes 1996) have been presented for steel beams, where the
shear strain on the buckling capacity of FRP members is sig-
material is homogeneous and isotropic. Ma and Hughes (1996) nificant and must be taken into account in design. Based on
and Hughes and Ma (1996) recently proposed an energy an experimental and theoretical study of the behavior of pul-
method for analyzing the lateral-distortional buckling of truded FRP channel section beams under the influence of grad-
monosymmetric steel I-beams under distributed point loading. ually increasing static loads, Razzaq et al. (1996) recently pre-
A fifth-order polynomial shape function was used in their sented a load and resistance factor design (LRFD) approach
study to describe the web's buckled shape, and the results for lateral-torsional buckling. Single-span members with sev-
compared favorably with finite-element analysis. Several re- eral loading locations and various spans were tested, and the
searchers have carried out studies on theoretical and experi- relationship between the lateral-torsional buckling load and the
mental evaluations of lateral and distortional buckling for FRP minor axis slenderness ratio was established. Using these test
'Assoc. Prof., Dept. of Civ. and Envir. Engrg., West Virginia Univ., results, they proposed an elastic buckling load formula for
Morgantown, WV 26506-6103. analysis and design of channel FRP beams. A series of lateral
'Post-Doctoral Fellow and Lect., Dept. of Civ. and Envir. Engrg., West buckling tests on small-scale pultruded E-glass FRP beams
Virginia Univ., Morgantown, wv. were carried out by Brooks and Turvey (1995), Turvey
Note. Discussion open until April I, 1998. To extend the closing date (1996a,b), and Turvey and Brooks (1996). The effects of load
one month, a written request must be filed with the ASCE Manager of position and boundary condition on the lateral buckling re-
Journals. The manuscript for this paper was submitted for review and
possible publication on February 18, 1997. This paper is part of the Jour-
sponse of FRP I-sections were investigated, and the results
lUll of Composites for Construction, Vol. 1, No.4, November, 1997. were correlated with approximate formulas by Nethercot and
©ASCE, ISSN 1090-0268/97/0004-0150-0159/$4.00 + $.50 per page. Rockey (1971) and finite-element eigenvalue analysis. They
Paper No. 15134. attributed the disparity between experiment and analysis to
150/ JOURNAL OF COMPOSITES FOR CONSTRUCTION / NOVEMBER 1997

J. Compos. Constr., 1997, 1(4): 150-159

 factors not included in the models, such as initial deflections, y (v)
prebuckling displacements, and geometric nonlinearities. Al-
though significant contributions have been provided by pre-
vious investigators, there is a need to develop combined ex-
perimental and analytical studies to characterize the buckling
behaviors of large-scale FRP sections and to propose engi-
neering design analysis equations for lateral and distortional
buckling of FRP beams.

OBJECTIVES AND SCOPE

Based on energy principles and nonlinear elastic plate the-
ory, the present study is concerned with the design analysis of
flexural-torsional buckling and lateral-distortional buckling for
FRP wide-flange (WF) beams. For the case of lateral-distor-
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tional buckling, in which the web can distort, a fifth-order FIG. 1. Displacement Field and Coordinate System of WF
polynomial shape function is adopted to describe the web's Beam
buckled shape. Two large-size FRP WF beams with different
material architectures are tested to study their flexural-torsional strains and curvatures are much less than unity everywhere in
and lateral-distortional buckling responses under midspan con- the plate. For a plate in the x-y-plane, the in-plane finite strains
centrated loads, and to induce global buckling without distor- of the midsurface of the plate are given by Malvern (1969, pp.
tion of the beam cross section (flexural-torsional buckling), 154-161) as
wooden stiffeners are inserted between the flanges on each
side of the web at midspan. Through displacement measure-
ments with linear variable differential transducers (LVDTs)
and strain measurements at the edges of the top flange, bifur-
ex =:: + & [G:r G;r + + (~:r] (5a)

cation response and rotation of the midspan cross sections are

Ey = oy
ov + 21 [(OU)2 (OV)2
oy + oy
+ (:W
uy
)2] (5b)
evaluated. The proposed predictions for both flexural-torsional
and lateral-distortional critical buckling loads correlate closely
with finite-element analyses and experimental results. The an- OV ou ou ou ov ov ow ow
alytical tools presented in this study are proposed to formulate 'Yxy =ox + oy + ox oy + ox oy + ox oy (5e)
engineering equations for design analysis of FRP beams, and
a simplified equation for the flexural-torsional buckling load Based on the von Karman plate theory, only the displacement
of FRP WF beams is given. gradients aw/ax and aw/ay are considered to have significant
values, and, therefore, the nonlinear terms (aw/axi and (aw/
ay)2 are retained in (5). For a stability problem, the displace-
ANALYTICAL STUDY
ment gradients auax and au/ay may become relatively large
Derivation of Total Potential Energy because of in-plane rotations, especially for the flanges,
whereas the terms au/ax and auay, and particularly their qua-
The analyses of flexural-torsional (lateral) and distorsional dratic forms, are significantly smaller than the other terms and
buckling are based on energy considerations, and the total po- can be ignored. Hence, (5) reduces to
tential energy equations governing instability are derived using
plate theory. The total potential energy of the system (e.g., for ex = OU + .!. [(OV)2 + (OW)2] (6a)
a WF beam) is the sum of the strain energy U and potential ax 2 ax ax
energy of the applied loads n
II=U+{} (1) e = av + .!. [(au)2 + (OW)2] (6b)
y oy 2 oy ay
To establish equilibrium using the total potential energy of a
displaced buckling mode, the prebuckling work, which is the OV ou owaw
'YXY =-ox + -oy + - -
ox ay
(6c)
product of the applied loads and their corresponding displace-
ments, can be ignored in stability analysis. Therefore, the total
potential energy n becomes The curvatures of the midplane are defined as
02W a2w 02
W
II=U (2) Kx =-2;
ox
Ky =-2;
ay
Kxy =•2 - -
axoy
(7)
In conformance with the basic approximations for thin-plate
theory, the strain energy in a deformed plate is For buckling analysis of WF beams under bending loads, the
deformation before buckling is ignored. Based on the coordi-
U =& JJIv (<Txex + <Tyey + Txy'Yxy) dV (3)
nate system shown in Fig. 1, the buckled displacement fields
are expressed as follows (Ma and Hughes 1996):
For I-beam sections consisting of two flanges and one web,
the total strain energy in a buckled beam is given by
for the web (in the x-y-plane) (8a)
(4)
where the superscripts if, w, and hi refer to the top flange,
web, and bottom flange, respectively. for the top flange (in the x-z-plane) (8b)
Because the displacement-gradient components are not
small compared with unity, the strains for the buckling prob-
lem are expressed in nonlinear terms. It is assumed that the for the bottom flange (in the x-z-plane) (8e)

JOURNAL OF COMPOSITES FOR CONSTRUCTION / NOVEMBER 1997/151

J. Compos. Constr., 1997, 1(4): 150-159

 For the top flange (x-z-plane), the strains and curvatures in (6)
and (7) become

aul! 1 awl!)2 (avl!)2]

[( -
if if
ef=-+- 1 (aw
- +aU -)
ax 2 ax + -ax (9a) Nfz=- ax aza66

., aw if 1 [(au
E;=-+-
if
- )2 + (av
if
- )2] (9b) The total strain energy of the bottom flange can be obtained
az 2 az az in a similar way as
iJw if au if avl! avl!
"1'[.=-+-+----
ax az ax az (9c)
UbI = ~ f La {N~ [e::'Y (a;;/YJ 2N~ aa:1a::/} + + dx dz
1f J. {I (aua;)b'\2 1 [(awa;}
and
b b
,\ (au ,\]2
(10) + "2 .... Olll + 0106 + -;;;}
2 bl 2bl
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In the present study, the foregoing equations are applied in the + 1.- (a v2 )2 + 1.- (a v )2} dx dz
buckling analysis of pultruded FRP WF beams. Most pul- 811 ax 866 axaz (16)
truded FRP sections are produced as symmetric laminated Considering the web as a plate and the deformation field of
structures (Le., there is no bending-extension coupling, BI} = (8a) (Ma and Hughes 1996), the total strain energy of the web
0), and the off-axis piles of pultruded panels are balanced sym- panel is expressed as
metric (Le., there is neither shear-extension nor bending-twist-
ing coupling: A 16 = A 26 = D 16 = D 26 = 0) (Davalos et aI. 1996b).
The panel mechanical properties are independently obtained
W

2....
f
U = ! J. [N; (aw )2 + N; (aw )2 + 2N';,. aw awwJ dx dy
ax ay
W

ax ay
W W

~ ff.. . [D~, e;~wy

either from experimental coupon tests or from analytical pre-
dictions using micro/macromechanics models (Davalos et aI.
1996b).
+ + m2 (a;;wy
For a laminate in the x-y-plane, the midsurface in-plane
strains and curvatures are expressed in terms of the compliance
coefficients and panel resultant forces as (Jones 1975) (17)

Ex all al2 al 6 ~11 ~12 (316 Nx The equilibrium equation in (2) (ll = 0) in terms of the total
Ey a12 a22 a26 ~12 ~22 ~26 Ny potential energy is then solved by the Rayleigh-Ritz method.
"Ixy al 6 a26 a66 ~16 ~26 ~66 Nxy
(11)
= Stress Resultants in WF Beam Panels
Kx ~11 ~12 ~16 811 8 12 816 Mx
Ky (312 (322 (326 812 822 826 My For a simply supported WF beam subjected to a midspan
Kxy (316 ~26 (366 816 826 866 Mxy point load, simplified stress resultant distributions in the cor-
responding web and flange panels are assumed based on beam
and considering the top flange of a WF beam (Fig. 1) to act theory, and the location or height of the applied load is ac-
as a beam element, the transverse resultant forces are neglected counted for in the analysis. For FRP WF beams of uniform
thickness, the membrane forces are expressed in terms of the
(12) midspan point load P. The expressions for the flanges are
Referring to the coordinate system of Fig. 1 and applying (11)
to the x-z-plane, we obtain Nif - PbWt
x - 4/ x,
(0 ~x =S; L/2)

Nx
if _..s.if (13) PbWt
-
all
,
Nif=_(L-x)
x 4/ '
(L/2 ~ x ~ L)

where Nf and N'fz = membrane forces per unit length; and

Mf and M'fz = bending and twisting moments per unit length. Nbl = -PbWt
x
-- X
4/'
(0 ~ x ~ L/2)
The compliance coefficients [a] and [8] are obtained by in-
version of the stiffness matrices [A] and [D]. Then the total PbWt
strain energy of the flange becomes N"! = - - (L - x), (L/2 ~ x ~ L)
4/

Uif = ~ fL. (NrEr + N'[."I'[. + MrKr + M'[.K'[.) dx dz (14)

Similarly, for the web
(18a)

Ignoring fourth-order terms, the total strain energy of the top

flange simplifies to Pt
if tv'; = 2/ xy, (0 ~ x ~ L/2)

Uif =~ fLea {Nr [ (~~Y + (a;ifY] + 2N'[. ~~ aa: } dx dz

tv'; = Pt
2/ (L x)y, (L/2 ~ x ~ L)
1f J.
-
if
+ "2 Ble.
{Iall (aua;)2 + -;;:1 [(awl!)
if

a:;- + a; )]2
(au tv; = 0, (0 ~ x < L/2; L/2 < x ~ L)

2 if 2if P
+ 2. (a v )2 + 2. (a v )2} dx dz N;" = - 2b w' (0 ~ x ~ L/2)
8 tlax 8 axaz
2
66 (15)
where the simplified forms of the membrane stress resultants P
are
N;" = 2bw ' (L/2 ~ x ~ L) (18b)

152/ JOURNAL OF COMPOSITES FOR CONSTRUCTION / NOVEMBER 1997

J. Compos. Constr., 1997, 1(4): 150-159

 where Displacement Fields for Panels of a Buckled
WFBeam
1= (.!.2 JIb"" + ..!.
12
b"")t Assuming that the top and bottom flanges do not distort
(Le., the displacements are linear in the z-direction) and con-
sidering the compatibility conditions at the flange-web junc-
To account for the location of the applied load on the y-axis
tions, the buckled displacement fields (Fig. 1) for the web and
of the beam midspan cross section, the transverse stress re-
the top and bottom flanges of WF section are derived. For the
sultant on the web panel is represented as
web (in the x-y-plane)

N;= P( Y + yp) (x = Ll2 and -b"'/2:S y :s b"'J2) (lSe) u'" = 0, v'" = 0, w'" = w"'(x, y) (19a)
bW ' ,

For the top flange (in the x-z-plane)

where yp = distance from the centroidal axis to the location of uti =utl(x, z) = -z(w(f).., v tf = vtf(x, z) = -Z6'1, w(f =wtl(x) (19b)
the applied load (Fig. I).
For the bottom flange (in x-z-plane)
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soo . . . . , - - - - - - : : ; - - - - - - - - - - - - - - - , ubf = ubf(x. l,) = _l,(w bl).., Vbl = vbl(X. l,) = -ZOb,. w b' = wb/(x) (1ge)
4S0
Flexural-Torsional (Lateral) Buckling of
400
FRPWFBeams
350
For flexural-torsional (lateral) buckling of WF beams, the
~ 300 '\ cross section of the beam is considered as undistorted. Because
.I! 250
\ P applied at the c:ealroid the web panel is not allowed to distort and remains straight in
200
'\ ,/'; flexural-torsional buckling, the rotation of the cross section
~/ and the lateral (sideways) deflection are coupled. The follow-
ISO _....\ ing displacement functions for the web centroidal axis lateral
100 ,.,,.,"'_" displacement (w) and the beam cross section rotation (9) are
so P applied at the top surface ~ selected as

oJ _-:o~fthe~fl~an~ge~__r- ....-~ ~:!!~~~~~

..... w =C 1 sin(1TxlL); 9 = C2 sin(1TxlL) (20)
o 2 3 4 S 6 7 8 9 10 11
The displacements and rotations [referring to (19)] of panels
L(m) then become
FIG. 2. Critical Flexural-Torsional Buckling Load versus Span
h
forWF-AB.am w'" = w + y9; wlf = w +- e (2Ia)
2
soo -,------,---------------, h
w bl =W - -
2
9; elf = ebl = e (2Ib)
450
400 where h = height of beam. By applying the Rayleigh-Ritz
350
method and solving for the eigenvalues of the potential energy
equilibrium equation, the critical buckling load P cr for a mid-
300 span point load is obtained.
~ 250 As an example, based on the foregoing formulation, a de-
J! 200 sign equation for the critical buckling load Pcr applied at the
centroid of the cross section is obtained as
ISO
100
+ a"h2)[('lfh)2(2D" + 4d" +
3
21T hY(6D" a"h') + 48(2D.. + d..)L'J
3('lf 2 + 4)C
(22)
o 2 3 4 S 6 7 8 9 10 11
where all = lIal1 and [au] = [Aur (or al1 = Ext, where Ex can
l

L(m) be obtained from a coupon test); d l1 = 1/8 11 , 4 = 1/866 and

FIG. 3. Critical Flexural-Torsional Buckling Load versus Span [8ul = [Durt [or d l1 = Ex t 3J12, d 66 = Gxyt 3/12 for the web panel
for WF·AC Beam (G xz t 3/12 for the flange panels), where Ex and Gxy (G xz for the

t=12.7mm
3/40z:. CSM& 17.70z:. +1-45 SF

1
54 rOvinBSf2 yield)
3/4QZ:. C,SM 17)j)~. +/-45 SF
54 rovlngs 62 ytela}
3/~~~o~M F'~la)' +1-45 SF
h=304.8 mm 3/40z:. CS~ r.
I 70z:. +/-45 SF
54 rovings ~2 yield)

1-.........
3/40z:. CSM 17,70,/:. +1-45 SF
54 rovings 62 ytelo}
3/40z. CSM 17.70z. +1-45 SF
54 rovings (62 yield)
3/40z. CSM & 17.70z. +1-45 SF
I . b = 304.8 mm . I 21 layers through the thickness of each panel
1<':----->\ Fiber volume fraction: V f = 44.3%

FIG. 4. Dimensions and Panel Fiber Architectures of WF-A Beams

JOURNAL OF COMPOSITES FOR CONSTRUCTION / NOVEMBER 1997/153

J. Compos. Constr., 1997, 1(4): 150-159

 t= 12.7mm
3/4oz. CSM & 17.791;.0/90 SF
S4 rovings (62 yield)

1
h= 304.8 mm
3/40z CSM & 17.79;\. 0/90 SF
S4 rovingS~2 Y1elo)
3/4Qz. C.SM 17.79];. +/-45 SF
S4 roVlngs 2 ytelO}
3/4oz. CSM& 17.7oz. +1-45 SF
S4 rovings ~2 yield)

l~
3/4Qz. C,SM 17.79;\. +1-45 SF
54 rovmgs 62 ytelo)
3/4oz. CSM 17.7oz. 0/90 SF
54 rovings (62 yield)
3/4oz. CSM& 17.7oz. 0/90 SF
I. b = 304.8 mm . I 21 layers through the thickness of each panel
t<'------;)j
Fiber volume fraction: Vf = 44.3%

FIG. 5. Dimensions and Panel Fiber Architectures of WF-AC Beams

flange panels) can be obtained from coupon tests]; D ll and D 66 The displacement and rotation functions along the beam
are obtained from [Dij] matrix; h = width and height of WF length are selected as
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beam (in the foregoing equation, the width is assumed to be

equal to the height h). The stretching stiffness coefficients [Aij] Wbf(X)
and bending stiffness coefficients [Dij] used in this study are (lbf(X)
obtained from micro/macromechanics models (Davalos et al.
1996b). In Figs. 2 and 3, the critical flexural-torsional buckling
wlf(x)
=
~
L.J
c .
n SID
[(2n -L 1)1TX] (26)
(llf(x) n_l
loads (Pcr) for WF-A and WF-AC beams (see details about Kbf(X)
material layup for these beams in Figs. 4 and 5) are plotted Kif(X)
versus the span length L. In this study, the material stiffness
coefficients in (22) are computed by micro/macromechanics where wbfC.. ijbfC.. W'fC.. ijlfC.. RbfC.. and RifC.. = constant
models (Qiao et al. 1994; Davalos et al. 1996b) and given in coefficients, which represent the maximum amplitudes of
Table 1 for WF-A and WF-AC beams. The panel stiffnesses buckling deformation. Considering (25) and (26), the displace-
predicted by micro/macromechanics models for the WF-A and ment fields for the top and bottom flanges and the web are
WF-AC beams were compared favorably with coupon test re- obtained from (19). By applying the Rayleigh-Ritz method and
sults, as reported by Davalos et al. (1996c). The predictions solving for the eigenvalues of the potential energy equilibrium
of the proposed formulation will be correlated with experi- equation, which is of order 6 X n, the critical buckling load
mental results presented later. P cr for a midspan point load is obtained. In this study, the
mathematics software MapleV (Char et al. 1991) is used to
Lateral-Distortional Buckling of FRP WF Beams
carry out the numerical computations for both flexural-tor-
For lateral-distortional buckling of WF beams, the cross sec- sional and lateral-distortional buckling of FRP WF beams.
tion of the beam is considered as distorted and the web panel In Figs. 6 and 7, the critical lateral-distortional buckling
is allowed to distort transversely; however, the flange panels loads (Pcr) for WF-A and WF-AC beams (see Figs. 4 and 5)
are still assumed undistorted and the displacements are linear are plotted versus the span length L. The predictions of the
in the z-direction. The deformation of the web, which includes proposed formulation are correlated with experimental results
sideways deflection, is expressed as in the next section.
WW(x, y) = Wbf(X)!,(y) + ebf(x)Ny) + wif(x)h(y)
EXPERIMENTAL EVALUATIONS OF BUCKLING OF
+ e lf(X)!4(y) + Kbf(X)fs(X) + KIf(X)J.(X) (23) FRPWFBEAMS
where 11> h,f3,f4,f~, and 16 = shape functions (Ma and Hughes In this study, two large-scale FRP WF beams, which were
1996), which are obtained by applying compatibility condi- manufactured based on optimum designs (Davalos et aI.
tions at the flange-web junctions as 1996a), are tested to evaluate their flexural-torsional and lat-
eral-distortional buckling responses (Qiao 1997). The beams
elf = (W;)y-hl2; ebf = (w;)y--hI2; w lf = (W W)Y-hl2; tested are 304.8 X 304.8 X 12.7 mm (12 X 12 X 1/2 in.).
W bf WF sections with two different material architectures: (1)
Wbf = (W )y-_hl2; Kif = (W;")Y-hl2; K = (w;")Y--h12 (24)
beam WF-A (Fig. 4) consists of rovings, continuous strand
Substituting (23) into (24), the shape functions can be derived mats (CSM), and +/-45° angle-ply stitched fabrics (SF); (2)
and written as beam WF-AC consists of rovings, CSM, +/-45° angle-ply
SF, and 0°/90° cross-ply SF. The span length considered is L
1 15 Y y' / = 4.42 m (14.5 ft), and both beams are tested under midspan
ft = 2- 8" bW + 5 bwJ - 6 bW'
concentrated loads.
S
W ( 5 7 y 3 / 5 / 1 l y ) Experimental Response for Flexural-Torsional
h=b 32 - 16 bW - 4 bw2 + 2 bwJ + 2 bW' - 3 bw'
Buckling
1 15 y / / For the experimental evaluation of flexural-torsional buck-
!3 = 2 + 8" bW - 5 bw' + 6 bW' ling of the WF beams (Qiao 1997), wooden stiffeners were
inserted between the flanges and web at midspan to prevent

/4 = b
W (5 7 Y 3 / 5 /
4
- 32 - 16 bW + bw' + bw3 -
1
2
bW' - 3 bw' 2 l l) distortions of the beam cross sections. These wood inserts
were needed to force the beam to undergo this buckling mode,
which may not arise naturally for unstiffened WF beams. The
w2(1 1 Y 1/ 1/ Il Ii) beams were restrained laterally at the supports (Fig. 8) by us-
is = b 64 - 32 bW - 8 bw' + 4 bw3 + 4b W' - 2 bw' ing vertical threaded rods fixed at the bottom support and con-

fo = b
w2 ( I 1 Y
64 + 32 bW -
1/
8 bw' -
1 / 1 l 1
4 bw3 + 4 b + 2 bw'
W'
i) (25)
nected to a top roller plate by hand-tighten nuts. This design
prevented out-of-plane twisting of the ends at buckling, but
the beam flanges did not touch the rods. The small roller on
154/ JOURNAL OF COMPOSITES FOR CONSTRUCTION / NOVEMBER 1997

J. Compos. Constr., 1997, 1(4): 150-159

 TABLE 1. Panel Stiffness Coefficients for WF-A and WF-AC Beams
0 11 0'2 0 22 Dee 811 Be. d11 dee
(Nm 3 per m 2 ) (Nm 3 per m 2 ) (Nm 3 per m 2) (Nm 3 per m 2) (X10· Nm per m 2 ) (X10· Nm per m 2 ) (Nm 3 per m2 ) (Nm 3 per m2 )
(1 ) (2) (3) (4) (5) (6) (7) (8) (9)
WF-A beams 4,396.87 991.27 2,430.15 1,068.42 331.13 80.00 3,994.03 1,066.33
WF-AC beams 4,747.26 687.2 2,690.11 761.82 354.66 66.62 4,557.71 763.74

270

240

210

180
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~
150
P applied at the centroid
£ 120

90

60

30
. j''on
P appbed at the top surface of the flange "0- 1J- 0-
0-0....0:::
0 FIG. 8. Support Restraints at Ends of WF Beams
0 2 3 4 5 6 7 8 9 10 11
L(m) experimental physical occurrence of flexural-torsional buck-
ling is shown in Fig. 11, where all attached bars rotated in the
FIG. 6. Critical Lateral-Distortional Buckling Load versus same direction under torsional response of the cross section.
Span for WF-A Beam
For each beam type, two test samples were used and each
270 ~----------------------, sample was tested three times. The LVDTs' readings of the tip
vertical displacements of the transverse bars, described in Fig.
240 9, are shown in Fig. 12 for a typical test of one of the WF-A
beams and in Fig. 13 for a typical test of one of the WF-AC
210
beams, respectively. Typical strain responses along the two
180 'b edges of the top flange at the midspan cross sections also are
\ plotted in Fig. 14 for one of the WF-A beams and in Fig. 15
~ 150 '\ P applied at the centroid for one of the WF-AC beams, respectively. As indicated in
q Figs. 12-15, a bifurcation response point from load-displace-
~ 120

90 \ ment and load-strain curves can be observed when the applied

load reached the critical buckling load for each individual

/"'"'~
beam.
60 As discussed in the foregoing sections, the displacement
functions for flexural-torsional buckling for the web central
30
P applied at the top surface of the flange "-1J --0 -0 ;]- -0><--0---<
displacement (w) and rotation of the beam (6) are given in
O-t---.---.------r--.---,---,--..,--.-----,--,---j (20). By solving for the eigenvalues of the energy equation,
023 4 5 6 7 8 9 10 11
the critical buckling load Per for a midspan point load applied
at the centroid of the cross section of a WF beam is given by
L(m)
(22) (as shown in Fig. 2 for the WF-A section and Fig. 3 for
FIG. 7. Critical Lateral-Distortional Buckling Load versus the WF-AC section). To verify the prediction accuracy of the
Span for WF-AC Beam explicit solutions, the test beams also were analyzed with the
commercial finite-element program ANSYS (ANSYS User's
the top flange did not prevent free rotation of the beam at the Manual 1992), using Mindlin 8-node isoparametric layered
support. Using transverse bars attached to the beam midspan shell elements (SHELL 99). The mesh used consisted of four
cross section (as shown in Fig. 9), LVDTs were installed to elements along the flange width, four elements along the web
measure the rotation angles of the cross section at critical height, and 60 elements along the beam length. Boundary con-
buckling loads. Also, strain gauges were installed on the com- ditions were imposed at the centroid of the beam ends to sim-
pression flanges to detect the onset of lateral buckling. The ulate a simply supported beam. The beam ends were laterally
load was applied to the top flange and lateral supports were restrained at the top-flange nodes. The wood stiffeners at mid-
placed close (but not touch) to the beam to prevent cata- span were modeled with 2 X 4 elements. The eigenvalue anal-
strophic failures (see Fig. 10). The hydraulic ram was rigidly ysis provided a critical buckling load corresponding to the lat-
attached to a supporting franle (Fig. 10), and the load applied eral-torsional buckling mode. The results are given in Tables
through a loading block was not capable of acting as a "fol- 2 and 3. The analytical values correlate with the FE predictions
lower load" when flexural-torsional buckling occurred and the within a 5% difference. We also evaluate the critical buckling
whole cross section rotated suddenly. At the onset of buckling, loads applied at the centroid using the equation proposed by
this loading arrangement had a restraining effect against ro- Pandey et al. (1995), and the results are 172.13 kN for the
tation of the cross section as flexural-torsional buckling pro- WF-A beam [5.5% difference with the present study in (22)]
ceeded and induced a restoring torque making the critical load and 182.06 kN for WF-AC beam [5.9% difference with the
slightly higher than for the case of a true follower load. The present study in (22)]. The experimental average values are
JOURNAL OF COMPOSITES FOR CONSTRUCTION / NOVEMBER 1997/155

J. Compos. Constr., 1997, 1(4): 150-159

 joEoE'-- 0.:..:..8;.:
..:.;m:...:.(3_2_ln.:..) .~IE-E---_O-.8-1
. m-'-(32_I....;n) ~~1

Strain 112 Strain III

Strain 114 Strain 113

Attached ban
WFbeam
t:=======.:~.2.~-(s.:.6..~) .:.:.:.:.:_.=:i==::::;t]rc--

0.76 m (30 In)

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FIG. 9. Arrangement of LVDTs and Strain Gauges at Midspan Cross Section of WF Beams

150
0.- -<l
135 o LVDTIII
a LVDTII2
120
t>. LVDTII3
105 'V LVDTII4
¢ LVDTIIS

i
l'l..
90

75

!
'8-
60

45

30 LVIm14 LVDTt3
LYOTlIS
15

0
0 10 20 30 40 SO 60 70 80

FIG. 10. Experimental Setup for Buckling Tests on WF Beams Displacement, II (mm)

FIG. 12. Typical Load-Displacement Curves for Flexural-Tor-

sional Buckling of WF-A Beam

150
11
135 o LVDTIlI
120 a LVDTII2
t>. LVDT#3
105 'V LVDT#4

90 ¢ LVDTIIS
i
l'l.. 75
"1
j 60

FIG. 11. Flexural-Torsional Buckling of WF Beam

45

30

15

0
0 10 20 30
'8'
40
LVImI4

50
LVDTtS

60
LVDT1I3

70 80
approximately 20% lower and 8% higher than the explicit so- Displacement, II (mm)
lutions for the loads applied, respectively, at the centroid and
FIG. 13. Typical Load-Displacement Curves for Flexural-Tor-
at the midpoint between the top flange and the centroid. It is sional Buckling of WF-AC Beam
interesting to note that the effect of the restoring moment
caused by the experimental loading arrangement is similar to
Experimental Response for Lateral-Distortional
that of applying the load at a point halfway between the top
Buckling
flange and the centroid. More importantly, when the load is
applied at the centroid, the analytical solution predicts a crit- Experimental tests on the WF beams, without placing wood
ical load 70% higher than when the load is applied at the top stiffeners at the midspan section, also were carried out to eval-
flange, which is the case in engineering practice. uate lateral-distortional buckling responses (Qiao 1997). The
156/ JOURNAL OF COMPOSITES FOR CONSTRUCTION / NOVEMBER 1997

J. Compos. Constr., 1997, 1(4): 150-159

 150..,.----------------------, TABLE 3. Comparison of Flexural-Torsional Buckling Loads
for WF-AC Beams
135
Pcr
120
(kN)
105 Analytical Finite Average
Load application solution element experimental
~ 90 Strain gage #3 (1 ) (2) (3) (4)
~ 75
Centroid of section 171.95 166.86 -
] 60 Halfway between top 130.59 128.97 139.3 (COY = 1.6%)
flange" and centroid
45 Top of flange" 101.02 102.17 -
30 Note: COY, coefficient of variation.
"In the explicit and FE analyses, this corresponds to the midplane.
15
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O-l---,--,---,---,.-----r---,r---r---,.------I
-4500 -4000 -3500 -3000 -2500 -2000 -1500 -1000 -500 0
Strain (jill)

FIG. 14. Typical Load-Strain Curves for Flexural-Torsional

Buckling of WF-A Beam

150

135
120

105

~
90
~ 75 Strain gage # \
] 60
45 STRAIN'2 STRAIN'l

30
SmJNM L~ FIG. 16. Lateral-Distortional Buckling of WF Beam

90 . . , . - - - - - - - - - - - - - - - - - - - - - ,
15
O-l---,---,--,------,--,.---,---,--.--.,..---Jj
-5000 -4500 -4000 -3500 -3000 ·2500 ·2000 -1500 -1000 -500 0
80 ---/3
70
Strain (jill)
o LVDT#\
FIG. 15. Typical Load-Strain Curves for Flexural-Torsional o LVDT#2
Buckling of WF-AC Beam A LVDT#5

Lvo~m
TABLE 2. Comparison of Flexural-Torsional Buckling Loads
for WF-A Beams
Pcr
(kN) 20
LVOCj23
Analytical Finite Average 10 LVOTl'
Load application solution element experimental
(1 ) (2) (3) (4)
Centroid of section 163.13 163.30 - -20 ·10 0 10 20 30 40 50 60 70 80 90
Halfway between top 124.41 125.68 138.2 (COY = 3.7%) Displacement 11 (rom)
flange" and centroid
Top of flange" 96.58 101.77 - FIG. 17. Typical Load-Displacement Curves for Lateral-Diator-
tlonal Buckling of WF-A Beam
Note: COY, coefficient of variation.
"In the explicit and FE analyses, this corresponds to the midplane.
sponses along the two edges of the top flange at the midspan
cross sections also are plotted in Fig. 19 for WF-A and Fig.
beam span is also 4.42 m (14.5 ft), and for each beam type, 20 for WF-AC beams. In Fig. 20, only one strain gauge is
two samples were tested and each sample was tested at least plotted because the remains channels of the data acquisition
three times. The experimental phenomenon of lateral-distor- system were used to collect displacement. As indicated in Figs.
tional buckling is shown in Fig. 16. The right side attached 17- 20, a bifurcation response point from load-displacement and
bars (LVDT 1 and LVDT 3) rotated closer to each other, and load-strain curves can be observed when the applied load
the left side attached bars (LVDT 2 and LVDT 4) rotated fur- reached the critical buckling load for each individual beam. For
ther apart from each other, indicating that distortional response Figs. 18(a and b), we can infer that, initially, the bottom flange
of the cross section took place. The LVDTs' readings of the distorted (see LVDTs 3 and 4), followed by web distortion for
tip vertical displacements of the transverse bars in Fig. 9 are a load range between about 38 and 62 kN [see Fig. 18(b)] and
shown in Fig. 17 for a typical WF-A beam and Figs. 18(a and the top flange was somewhat constrained by the loading block
b) for a typical WF-AC beam, respectively. Typical strain re- used. Then the top flange began to distort slightly [see Fig.
JOURNAL OF COMPOSITES FOR CONSTRUCTION / NOVEMBER 1997/157

J. Compos. Constr., 1997, 1(4): 150-159

 (8) 90
90 .....
80
80

70
o LVD1'JIIl
[] LVD1'JII2
A LVDTN3
70

60
f
Strlin gage N2

~
60 V LVDTN4
o LVDTN5
50
~
'"
50
]'" 40

] 40 30 STRAlN,2 STRAIN'I

30
L~VDDI 20
~r'3
20 10
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LvoC/:Ln
10 LVDD5 O+---,---,.--.,--r--.,----r---,---,.-~

·2700 ·2400 -2100 ·1800 ·1500 ·1200 -900 -600 ·300 0

0
0 5 10 15 20 25 30 35 40 45 50 55 60 65 Strain (J,lS)
Displacement II (mm) FIG. 19. 'TYpical Load-5traln Curves for lateral-Distortional
Buckling of WF-A Beam
(b)
90,-----------------------,
110
LVDTII6 80
100
90 70

80
70
~ 60
'"
] 50

~
40 VDTI16
LYDT'2 LYDT'l
30 20

20 L VDTiU LYDTn 10
LYDT"
10 O+---.----,.----r---.----,.---,-----:>f
0 ·2100 ·1800 ·1500 ·1200 ·900 -600 ·300 o
-45 -40 ·35 -30 ·25 ·20 ·15 ·10 -5 0 5 10 Strain (j.i£)
Transverse displacement II (mm)
FIG. 20. 'TYpical Load-5traln Curve for Lateral-Distortional
FIG. 18. LVDT Readings for a WF-AC Beam: (a) 'TYpical Load- Buckling of WF-AC Beam
Displacement Curves for Lateral-Distortional Buckling of WF-
AC Beam; (b) 'TYpical Displacement of LVDT 6 during Lateral- TABLE 4. Comparison of Lateral-Distortional Buckling Loads
Distortional Buckling of WF-AC Beam for WF-A Beams

18(a)], and, finally, at critical load, the top flange distorted and Pcr
the section can no longer sustain the applied load. (kN)
As discussed in the analysis section, the displacement func- Analytical Finite Average
tions for lateral-distortional buckling of the web at the center Load application solution element experimental
W
(W is selected as a fifth-order polynomial function [see (23)
) (1 ) (2) (3) (4)
and (25)]. By solving for the eigenvalues of the energy equa- Centroid of section 143.91 132.81 -
tion, the critical lateral-distortional buckling loads PeT for a Top of flange" 90.69 91.68 85.1 (COY = 2.3%)
midspan point load applied at the centroid of the cross section Note: COY. coefficient of variation.
and also at the top flange of a WF beam are plotted in Fig. 6 "In the explicit and FE analyses. this corresponds to the midplane.
for the WF-A beam and in Fig. 7 for the WF-AC beam, re-
spectively. To verify the prediction accuracy of the explicit
solution, the test beams also were analyzed with the commer- using nonlinear elastic theory. The flexural-torsional and lat-
cial finite-element program ANSYS (ANSYS User's Manual eral-distortional buckling responses are analyzed using this ap-
1992), using Mindlin 8-node isoparametric layered shell ele- proach and simplified engineering equations for flexural-tor-
ments (SHELL 99). The same mesh was used as for the flex- sional buckling are formulated. For the analysis of
ural-torsional buckling case. The critical eigenvalue distinctly lateral-distortional buckling, a fifth-order polynomial shape
corresponded to a lateral-distortional buckling mode. The re- function is adopted to model the deformed shape of the web
sults given in Tables 4 and 5 indicate that the predicted ana- panels and predict the critical loads. The models exhibit good
lytical values agree within 6 to 10% with the FE and average accuracy in relation to finite-element analysis results and pro-
experimental values. vide acceptable correlations with experimental data, within the
limitations of the experimental work.
CONCLUSIONS Two types of large-scale FRP WF beams are tested under
Based on energy principles, the total potential energy equa- midspan concentrated loads to evaluate their flexural-torsional
tions for instability of pultruded FRP WF sections are derived and lateral-distortional buckling responses. Using transverse
158/ JOURNAL OF COMPOSITES FOR CONSTRUCTION / NOVEMBER 1997

J. Compos. Constr., 1997, 1(4): 150-159

 TABLE 5. Comparison of Lateral·Dlstortlonal Buckling Loads Davalos, J. E, Salim, H. A., Qiao, P., Lopez-Anido, R., and Barbero, E.
for WF·AC Beams J. (1996b). "Analysis and design of pultruded FRP shapes under bend-
ing." Composites. Part B: Engrg. J., 27B(3-4), 295-305.
Pcr Davalos, J. F., Qiao, P.• and Salim, H. A. (1996c). "Characterization of
(kN) pultruded FRP wide-flange beams." Proc. of ASCE 4th Mat. Engrg.
Conf., ASCE, New York, N.Y., 223-232.
Analytical Finite Average
Hancock, G. 1. (1978). "Local, distortional, and lateral buckling of 1-
Load application solution element experimental beams." J. Struct. Div., ASCE, 104(11), 1787-1798.
(1 ) (2) (3) (4) Hancock. G. J. (1981). "Distortional buckling of I-beams." J. Struct.
Centroid of section 150.54 135.03 - Div., ASCE, 107(2), 355-370.
Top of flange" 94.63 100.91 85.1 (COV = 4.4%) Hughes. 0 .• and Ma. M. (1996). "Lateral distortional buckling of mono-
symmetric beams under point load." J. Engrg. Mech., ASCE, 122(10),
Note: COV, coefficient of variation. 1022-1029.
"In the explicit and FE analyses, this corresponds to the midplane. Jones, R. M. (1975). Mechanics ofcomposite materials. Hemisphere Pub-
lishing Corp.• New York, N.Y.
Lin, Z. M., Polyzois, D., and Shah, A. (1996). "Stability of thin-walled
bars attached to the beam midspan cross section, LVDTs are pultruded structural members by finite element method." Thin- Walled
installed to measure the rotations of the cross section and de-
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Struct., 24(1), 1-18.

termine critical buckling loads. Similarly, strain gauges bonded Ma, M., and Hughes, O. (1996). "Lateral distortional buckling of mono-
at the edges of the top flange are used to detect the onset of symmetric I-beams under distributed vertical load." Thin-Walled
buckling. The experimental setup used in this study can be Struct., 26(2), 123-145.
Malvern. L. E. (1969). Introduction to the mechanics of a continuous
applied to other FRP shapes to detect the onset of flexural- medium. Prentice-Hall, Inc., Englewood Cliffs, N.J.
torsional and lateral-distortional buckling. Mottram, J. T. (1992). "Lateral-torsional buckling of a pultruded 1-
A good agreement is obtained in this study between the beam." Composites, 32(2), 81-92.
proposed analytical approach and the finite-element analyses. Nethercot, D. A., and Rockey, K. C. (1971). "A unified approach to the
and the combined analytical and experimental program re- elastic lateral buckling of beams." The Struct. Engr., London, U.K.,
ported in this paper can be further improved to translate the 49(7), 321-330.
present analytical solutions for flexural-torsional and lateral- Pandey, M. D., Kabir, M. Z., and Sherbourne. A. N. (1995). "Flexural-
torsional stability of thin-walled composite I-section beams." Com-
distortional critical buckling loads into simplified design equa- posites Engrg., 5(3), 321-342.
tions for pultruded FRP WF sections. Qiao, P. (1997). "Analysis and design optimization of fiber-reinforced
plastic (FRP) structural beams," PhD dissertation, West Virginia Uni-
ACKNOWLEDGMENTS versity, Morgantown, W.Va.
Qiao, P., Davalos, J. E. and Barbero, E. J. (1994). "FRPBEAM: a com-
The writers thank Creative Pultrusions for producing the WF sections.
puter program for analysis and design of FRP beams." Rep. No. CFC-
This study was partially sponsored by a Construction Productivity Ad-
94-191, Constructed Facilities Center. West Virginia University, Mor-
vancement Research (CPAR) program, U.S. Army Corps of Engineers
gantown, W.Wa.
Construction Engineering Research Laboratories (USACERL), and the
Composites Institute (CI) of the Society of Plastic Industries (SPI). Their Razzaq, Z.• Prabhakaran, R., and Sirjani, M. M. (1996). "Load and re-
financial support is gratefully appreciated. sistance factor design (LRFD) approach for reinforced-plastic channel
beam buckling." Composites. Part B: Engrg. J., 27B(3-4), 361-369.
Roberts. T. M. (1981). "Second order strains and instability of thin-walled
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JOURNAL OF COMPOSITES FOR CONSTRUCTION / NOVEMBER 1997/159

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