# 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:**A general analytical model based on shear-deformable beam theory has been developed to study the flexural–torsional coupled buckling of thin-walled composite beams with arbitrary lay-ups under axial load. This model accounts for all the structural coupling coming from the material anisotropy. The seven governing differential equations for coupled flexural–torsional–shearing buckling are derived. The resulting coupling is referred to as sixfold coupled buckling. Numerical results are obtained for thin-walled composite beams to investigate effects of shear deformation, fiber orientation and modulus ratio on the critical buckling loads and corresponding mode shapes.Composite Structures - COMPOS STRUCT. 01/2009; 90(3):295-303. - SourceAvailable from: Ahmed Godat[Show abstract] [Hide abstract]

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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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