-
d*,
a N
; a
anexiVagenect
dom is developed to solve the governing equations. Numerical
examples are given to show the importance of the shear exibility on
thestability behavior of this type of structures.
materials are their high strength and stiness to weightratio,
good corrosion resistance, enhanced fatigue life and
theory. This theory incorporates the warping deformation,which
is a very important eect in these types of beams [2].
Tzeng. Massa and Barbero [5] proposed a strength-of-materials
formulation for static analysis of composite
model for I-section composite beams considering sheareects and
cross-sectional distortion for interactive buck-ling analysis.
Omidvar [9] analyzed shear exibility associ-ated to bending and
introduced new formulae for shearcoecients. On the basis of earlier
works of Librescu andSong [10], Bhaskar and Librescu [11]
introduced a model
* Corresponding author. Tel.: +54 291 4555220; fax: +54 291
4555311.E-mail address: [email protected] (V.H.
Cortnez).
Computers and Structures 84low thermal expansion among others
[1]. Many structuralmembers are constructed in the form of
thin-walled beams.Accordingly, many research activities have been
conductedtoward the development of theoretical and
computationalmethods for the analysis of the aforementioned
structuralmembers.
The structural analysis of isotropic thin-walled openbeams is
appropriately performed by means of the Vlasovs
thin-walled beams. A study on the location of shear centerwas
performed by Pollock et al. [6].
Since the middle nineties many authors have focusedtheir
research eorts in proposing mathematical modelsto study the
instability behavior of composite thin-walledbeams. Shearbourne and
Kabir [7] analyzed the shear eectin connection with the lateral
stability of composite I-sec-tion beams. Godoy et al. [8] developed
a mathematical 2006 Elsevier Ltd. All rights reserved.
Keywords: Thin-walled; Composite-beam; Buckling; Shear
exibility; Reissners Principle
1. Introduction
Slender composite structures are increasingly used inmany
applications of aeronautical, mechanical, naval andeven civil
engineering industries. The composite materialshave many advantages
that motivate their use in structuralapplications. The most
well-known features of composite
In the middle 80s, Bauld and Tzeng [3] introduced anextension of
the Vlasovs theory for composite materials.Recently, Ghorbanpoor
and Omidvar [4] introduced newequivalent moduli of elasticity and
rigidity to allow decou-pling (in an approximate form) of Bauld and
Tzeng equa-tions. This simplied approach yields nearly the
samenumerical values obtained with the theory of Bauld andStability
of composite thin-walle
Vctor H. Cortnez
Grupo de Analisis de Sistemas Mecanicos, Universidad
Tecnologic
Received 14 January 2005
Abstract
In this paper, a theoretical model is developed for the
stabilitysections. The present model incorporates, in a full form,
the shear way by means of a linearized formulation based on the
Reissnersplacement eld, whose rotations are based on the rule of
semi-tanlateral stability of composite thin-walled beam with
general cross-s0045-7949/$ - see front matter 2006 Elsevier Ltd.
All rights reserved.doi:10.1016/j.compstruc.2006.02.017beams with
shear deformability
Marcelo T. Piovan
acional (FRBB) 11 de Abril 461, 8000, Baha Blanca, Argentina
ccepted 9 February 2006
alysis of composite thin-walled beams with open or closed
cross-bility (bending and non-uniform warping), featured in a
consistentriational Principle. The model is developed using a
non-linear dis-tial transformation. This model allows to study the
buckling andion. A nite element with two-nodes and
fourteen-degrees-of-free-
www.elsevier.com/locate/compstruc
(2006) 978990
-
The aforementioned non-linear terms play an importantrole in
general buckling problems. In fact, in the context ofisotropic
thin-walled beams, Kim et al. [24,25] demon-strated that the
omission of these terms may lead to inaccu-rate results of buckling
loads for some cases, especiallywhen an o-axis loading is
considered.
A non-locking nite element with fourteen degrees offreedom is
developed for analyzing stability under bothaxial and lateral
loads. Parametric studies with dierentcross-sections and laminate
stacking sequences are carriedout.
2. Theory
2.1. Assumptions
A composite thin-walled beam with an arbitrary cross-section is
considered (Fig. 1a and b). The points of thestructural member are
referred to the Cartesian co-ordinatesystem {x,y,z} located at the
geometric center, where thex-axis is parallel to the longitudinal
axis of the beam, while
C
L
l
xn
sz
y^
^
^
^^
x^
i
Junction
uters and Structures 84 (2006) 978990 979for the buckling
analysis under axial compression of com-posite box-beam with
extension-twisting coupling. In thelast three references, it may be
seen an interesting analysisabout the inuence of secondary warping
on the mechanicsof composite thin-walled beams. Recently, Lee et
al. [1214] developed extended models of the Bauld and Tzengstheory,
in order to perform lateral buckling analysis ofcomposite laminated
I-section and channel-section beams.
The above-mentioned papers neglect the shear exibilityor at
least do not consider the shear exibility in a fullform, i.e. shear
exibility due to bending and shear exibil-ity due to non-uniform
warping. It has to be noted that theaforementioned works of Massa
and Barbero [5], Pollocket al. [6], Librescu and Song [10], Bhaskar
and Librescu[11] and Omidvar [9] consider the shear exibility due
tobending, however none of them take into account the
shearexibility due to non-uniform warping. These two eectsmay play
an important role in the prediction of natural fre-quencies and
buckling loads of thin-walled beams (for bothisotropic and
composite materials as shown by Cortnezet al. [15,16]).
According to authors knowledge there are a few papersthat take
into account the shear exibility eect in a fullform on the
mechanics of composite materials. The rstone is that of Wu and Sun
[17], however in their paperthese authors obtained only natural
frequencies andemphasis was given in showing the eectiveness of
thedeveloped nite element. Recently, Kollar [18] and Sapkasand
Kollar [19] explored the buckling behavior of compos-ite columns by
means of an analytical study including full-shear exibility.
However, in these last works, no generalloading conditions were
involved and the governing equa-tions were obtained by means of
classical (equilibrium)approaches [20,21].
Recently, the authors have developed a new model ofcomposite
thin-walled beams, based on the use of theHellingerReissner
principle, that considers shear exibilityin a full form, general
cross-section shapes and symmetricbalanced or especially
orthotropic laminates. On the otherhand, it was formulated taking
into account the existenceof a uniform distribution of initial
axial force and bendingmoments.
The model was applied for analyzing vibration andbuckling
problems.
This paper introduces a generalization of the aforemen-tioned
model in order to consider the existence of an arbi-trary state of
initial stresses, including general o-axisloadings.
To do this, a generalized displacement eld is employedwhich
includes non-linear terms based on the rule of semi-tangential
rotations introduced by Argyris et al. [22,23].
This displacement eld enhances the one employed bythe authors in
a previous work [16]. In these circumstancesthe functional is
extended to account for the new displace-ment terms as well as
generalized initial volume forces and
V.H. Cortnez, M.T. Piovan / Compinitial o-axis forces that were
not considered in the previ-ous work.C
z
y
s
AB
n
(+)
s
(-)
l(S)
r(S)
Cross-section branches
(b)
(a)Fig. 1. Geometry of the beam.
-
Nxx Z e=2e=2
rxx dn; Mxx Z e=2e=2
rxxndn;
Mxs Z e=2e=2
rxsndn 2a; b; c
Nxs Z e=2e=2
rxs dn; Nxn
Z e=2e=2
rxn dn 2d; eInitial shell stress resultants are denoted with the
super-script 0 and the applied shell stress resultants on
theboundaries are denoted as . T x, T s and T n are appliedforces
per unit area in x, s and n directions, respectively,
ters and Structures 84 (2006) 978990y and z are the axes of the
cross-section, but not necessarilythe principal ones. The
co-ordinates corresponding topoints lying on the middle line are
denoted with Y andZ. In addition, a circumferential co-ordinate s
and a nor-mal co-ordinate n are introduced on the middle contourof
the cross-section.
The present structural model is based on the
followingassumptions: (1) the cross-section contour is rigid in
itsown plane; (2) the warping distribution is assumed to begiven by
the Saint-Venant function for isotropic beams;(3) shell force (Nss)
and moment (Mss) resultants corre-sponding to the circumferential
stress rss and the inter-lam-inar force resultant (Nsn)
corresponding to inter-laminarstrain cns are neglected; (4) the
curvature at any point ofthe shell is neglected; (5) twisting
curvature of the shell isexpressed according to the classical plate
theory, but bend-ing curvature is expressed according to the
rst-order sheardeformation theory; in fact, bending shear strain of
thewall is incorporated; (6) the laminate stacking sequence
isassumed to be symmetric and balanced, or specially ortho-tropic
[1] (the corresponding constitutive equations for theshell stress
resultants are given in Appendix I); (7) the dis-placement eld is
considered to be represented by a linearcomponent and a
second-order component based on thesemi-tangential rotations
introduced by Argyris [22,23];(8) higher-order strain components
due to second-orderdisplacements are neglected in the GreenLagrange
straintensor; (9) higher-order shell stress resultants (i.e.
depend-ing on nk, where k > 1) are neglected.
2.2. Variational formulation
Taking into account the adopted assumptions,
theHellingerReissner principle for a composite shell may
bepresented in the following form: [26]:Z Z
NxxdeLxx MxxdjLxx NxsdcLxs MxsdjLxs NxndcLxn
dsdx
Z Z
N 0xxdeNLxx M0xxdjNLxx N 0xsdcNLxs M0xsdjNLxs N 0xndcNLxn
dsdx
Z Z
T xduLx T sduLs T nduLn
dsdx
Z Z Z
XxduLx X sduLs XnduLn
dsdndx
Z Z
T 0xduNLx T 0sduNLs T 0nduNLn
dsdx
Z Z Z
X 0xduNLx X 0sduNLs X 0nduNLn
dsdndx
Z
NxxdULx Mxxd/xx NxsdULs
Mxsd/ss NxndULn dsxLx0
0 1aZ ZeLxx
NxxA11
dNxx cLxs
NxsA66
dNxs kLxs
MxsD66
dMxs
dsdx
Z Z
kLxx MxxD11
dMxx cLxn
NxnAH55
!dNxn
" #dsdx 0 1b
980 V.H. Cortnez, M.T. Piovan / Compuwhere Nxx, Nxs,Mxx,Mxs and
Nxn are shell stress resultantsdened according to the following
expressions:while X x, X s and Xn are forces per unit volume in x,
sand n directions, respectively. The strain componentseLxx; c
Lxs; c
Lxn; j
Lxx; j
Lxs and e
NLxx ; c
NLxs ; c
NLxn ; j
NLxx ; j
NLxs are the rst-
and second-order shell strains which are dened later interms of
the shell displacements ULx ;U
Ls ;U
Ln and shell rota-
tions /xx;/ss, see Fig. 2. Finally, uLx ; uLs ; u
Ln and u
NLx ; u
NLs ; u
NLn
are linear and non-linear components of the displacementeld with
respect to the contour co-ordinate system. Inthe following pages,
superscripts L and NL identiesthe linear and second-order
components of the displace-ment eld and shell strains,
respectively.
It should be noted that, in Eqs. (1), the stress resultantsand
the displacements are variationally independent quan-tities.
Expressions (1a) and (1b) represent the variationalforms of the
equilibrium and compatibility equations,respectively.
2.3. Kinematic expressions
The displacement eld, compatible with assumptions(1), (2) and
(7), is adopted in the form [27]:
uLx x; t uxcx; t yhzx; t zhyx; t xhxx; t 3auLy x; t uycx; t
z/xx; t 3buLz x; t uzcx; t y/xx; t 3cuNLx x; t
1
2z/xx; thzx; t y/xx; thyx; t 3d
uNLy x; t y2
/xx; t 2 hzx; t2h i
z2hzx; thyx; t
3euNLz x; t
z2
/xx; t 2 hyx; t 2h i y
2hzx; thyx; t
3f
ss
_
x^
U_
_
x
xx
_
s^s
A
n^_
Un
e
U
Fig. 2. Displacements dened with respect to the wall co-ordinate
system.
-
uterwhere
ys; n Y s ndZds
; zs; n Zs n dYds
4
In expressions (3), uxc, uyc, uzc are the displacements of
thecentroid, /x is the torsional rotation, hy and hz are thebending
rotations. The variable hx measures the warpingintensity. When the
non-linear components (3d)(3f) areneglected, the displacement eld
is equivalent to that pro-posed in reference [16].
The warping function can be written in the followingform:
xs; n xps xss; n 5where xp and xs are the contour warping
function and thethickness warping function, respectively. They are
denedin the form [27]:
xps 1SZ S0
Z ss0
rs ws ds
ds
Z ss0
rs ws ds
xss; n nls6a;b
with
rs Zs dYds
Y s dZds
;
ls Y s dYds
Zs dZds
7a; b
In expression (6a), w is the shear strain in the middle
line,obtained by means of the Saint-Venant theory of pure tor-sion,
and normalized with respect to d/x/dx, as can be seenin reference
[27] for composite beams or in Krenk andGunneskov [28] for
isotropic beams. For the case of opensections, it can be proved
that w = 0.
The displacements with respect to the curvilinear systemare
obtained by means of the following geometrictransformation:
ULx uLx x; s; 0;UNLx uNLx x; s; 0 8a; b
ULs uLy x; s; 0dYds
uLz x; s; 0dZds
;
UNLs uNLy x; s; 0dYds
uNLz x; s; 0dZds
8c; d
ULn uLy x; s; 0dZds
uLz x; s; 0dYds
;
UNLn uNLy x; s; 0dZds
uNLz x; s; 0dYds
8e; d
/xx ouxcon ; /ss oon
uLydYds
uLzdZds
8f ; g
By introducing the displacements (8) into the denitions ofstrain
components [16], and taking into account hypotheses(1)(9), one can
obtain the following expressions for the
V.H. Cortnez, M.T. Piovan / Comprst (exx,cxs,cxn) and second
(gxx,gxs,gxn) order straincomponents:exx eLxx njLxx; cxs cLxs
njLxs; cxn cLxn 9a; b; cgxx eNLxx njNLxx ; gxs cNLxs njNLxs ; gxn
cNLxn
9d; e; fwhere
eLxx u0xc Y sh0z Zsh0y xP sh0x;
jLxx h0zdZds
h0ydYds
h0xls 10a;b
cLxs u0yc hzdYds
u0zc hydZds
/0x hxrw/0xw;jLxs 2/0z 10c;d
cLxn u0yc hzdZds
u0zc hydYds
/0x hxls 10e
eNLxx 1
2u02yc u02yc /02ycY 2Z2n
Z2/0xu0yc /xhz 0 Y 2/0xu0zc /xhy 0h io 10f
jNLxx 1
2/xhy 0 dZ
ds /xhz 0
dYds
2/0x u0ycdYds
u0zcdZds
/0xrs
10g
cNLxs 1
2/x hz
dZds
hy dYds
rs h0yhz h0zhy
2 rw hx h0zY h0yZ
/x u0zcdYds
u0ycdZds
u0xc xph0x
hydZds
hz dYds
hx rw
10h
cNLxn 1
2/x hy
dZds
hzdYds
ls h0yhz h0zhy
2lshx h0zY h0yZ
/x u0zcdZds
u0ycdYds
u0xc xph0x
hzdZds
hy dYds
hxls
10i
jNLxs h0yhz h0zhy
rw hx h0ydYds
h0zdZds
h0xls
lsh0x hydZds
hzdYds
10j
In the above expressions () 0 denotes derivation with re-spect
to the x variable.
The rst and second terms in expressions (10c) and (10e)may be
regarded as the shear strains associated to bending,the third terms
correspond to the shear strain due to non-uniform warping and the
last term in expression (10c) is theSaint-Venant (or pure torsion)
shear strain.
2.4. Equations of motion
Substituting expressions (9) and (10) into Eq. (1a)
andintegrating with respect to variable s, one can obtain
theexpression for the virtual work equation given by
s and Structures 84 (2006) 978990 981LK LKG1 LKG2 LP 0 11
-
is the total torsional moment. QX ;QY ;MZ ;B;QZ ;MY ;MX
terwhere
LK ZL
QXdu0xc MY dh0y MZdh0z Bdh0x T SV d/0x
h idx
ZL
QY d u0yc hz
QZd u0zc hy
T W d /0x hx h idx12a
LKG1 ZL
Q0X2
d u02zc u02yc
P0W
2d /02x M 0Z
2d 2u0zc/
0x /xhy 0h i( )
dx
ZL
M 0Y2
d 2u0yc/0x /xhz 0h i(
M0X
2d h0zhy h0yhz
T 0W d u0xchx )
dx
ZL
Q0Y d u0zc/x u0xchz
/xhy2
Q0Z d/xhz2
u0xchy /xu0yc
dx
ZL
Q0YW d h0xhz
Q0ZW d h0xhy T 0WW d h0xhx n
T 0WZd h0yhx
T 0WY d h0zhx o
dx 12bLKG2
ZL
X 03 dhz X 05 dhy X 06 d/xh i
dx
T 03 dhz T 05 dhy T 06 d/xh ixL
x012c
LP ZL
q1x; tduxc q3x; tdhz q5x; tdhy q7x; tdhx
dx
ZL
q2x; tduyc q4x; tduzc q6x; td/x
dx
QXduxc QY duyc MZdhz Bdhx
QZduzc MY dhy MXd/xxLx0 12d
In the previous equations the following denitions, for thebeam
forces have been made:
QX ZSNxx ds; B
ZS
Nxxxps Mxxls
ds 13a; b
QY ZS
NxsdYds
Nxn dZds
ds;
QZ ZS
NxsdZds
Nxn dYds
ds 13c; d
MY ZS
NxxZs Mxx dYds
ds;
MZ ZS
NxxY s Mxx dZds
ds 13e; f
TW ZS
Nxs rs ws Nxnlsf gds;
T SV ZSNxsws 2Mxs ds 13g; h
982 V.H. Cortnez, M.T. Piovan / CompuMX T SV TW 13icorrespond to
external generalized forces acting at theends.
The functions qi(x, t), i = 1, . . . , 7, are the
generalizedapplied forces per unit length. X 03 ;X
05 ;X
06 and
T 03 ; T05 ; T
06 are functions which condense the initial vol-
ume and surface (at the ends) forces, respectively. Q0X ;
Q0Y ;Q0Z ;M
0X ;M
0Y ;M
0Z ; T
0W and T
0WW ; T
0WY ; T
0WZ ;Q
0YW ;Q
0ZW
are initial beam stress resultants and generalized beamstress
resultants, respectively. All these entities are exten-sively
described in Appendix II.
One may notice that LK, LKG1, LKG2 and LP in Eqs. (12)represent
the virtual work contributions due to incremen-tal, initial and
external forces respectively. It has to bepointed out that the
virtual work due to initial externalforces, LKG2, would have no
meaning if the displacementeld were reduced to the linear form.
However, this contri-bution is fundamental in order to solve
stability problemswith o-axis loadings or general loadings.
It is important to point out that in Eq. (12b) the
termcorresponding to the initial torsional moment M 0X ,
wasobtained naturally without amending the functionalexpression, as
it was performed in other works (for exam-ple in reference [29]).
This fact is due to the adoption ofan enhanced displacement eld in
connection with a morecomprehensive strain eld, in order to
describe the virtualwork contribution of initial stresses and
forces (see refer-ence [27] for a detailed discussion of this
subject, in the con-text of composite beams).
2.5. Constitutive equations for the beam stress resultants
The shell stress resultants can be derived with a
similarapproach to the one employed previously by Cortnezand Piovan
[16], where the reference axes were parallel tothe principal ones
and two-poles where employed. How-ever, with the aim to generalize
that conception (remember,in this article the axes are located at
the geometric centerbut not necessarily parallel to the principal
ones), it isimportant to use matrix representation for the strain
andstress components, in order to avoid excessive
algebraicmanipulation. Accordingly, the eld of shell stress
resul-tants can be assumed in the following way:
Nxx eCTN1 JN 1
QN ; Mxx e3
12CTN2 J
N
1QN 14a; b
Nxs eCTN3 JT 1
QT ; Nxn e3
12CTN2 J
T
1QT ;In the above expressions the integration is carried out
overthe middle contour perimeter. QX is the axial force,MY andMZ
are the bending moments, B is the bimoment, QY andQZ are the shear
forces, TW is the exural torsional mo-ment, TSV is the Saint-Venant
torsional moment and MX
s and Structures 84 (2006) 978990Mxs e3
12CTN5 J
T
1QT 14c; d; e
-
In case of using a principal-axes and two-pole referencesystem,
expressions (14) can be reduced to simplied forms(see Appendix
III). The shell strains can be expressed in thefollowing form:
eLxx CTN1eN ; jLxx CTN2eN 15a; b
cLxn CTN2eT ; cLxs CTN4eT ; jLxs CTN5eT 15c; d; e
In expressions (14) and (15) the following vectors andmatrices
are introduced:
QN QX ;MY ;MZ ;Bh i ;
J ij eZS
gai gaj
ds e
3
12
ZS
gdi gdj
ds 17a
ky Z s0
Zsds aS
IdsZ s0
Zsds 17b
kz Z s0
Y sds aS
IdsZ s0
Y sds 17c
kx Z s0
xP sds aSI
dsZ s0
xP sds 17d
with
ga 1; Zs; Y s;xP s;wsh i;
gd 0; dYds
; dZds
; ls; 2
18
In expressions (17b)(17d) the coecient a can have thevalue 0 or
1 depending on whether the cross-section con-tour is open or
closed, respectively. S denotes the contourperimeter.
The selected eld of shell stress resultants (14)
veriesexpressions (13) in addition to the following shell
equilib-rium equations:
Table 2Comparison of buckling loads for a pinnedpinned closed
section beam
Sequence Model Method h/L
0.05 0.10 0.15 0.20
0/0/0/0 No shearexible
FEM 5.23 20.86 46.96 81.45Analytic 5.21 20.84 46.89 81.36
Shear FEM 4.41 12.00 17.59 21.02
V.H. Cortnez, M.T. Piovan / Computers and Structures 84 (2006)
978990 983QT T SV ;QZ ;QY ; TWh iT 16a; b
eN u0xc;h0y ;h0z;h0xD ET
;
eT /0x; u0zc hy
; u0yc hz
; /0x hx D ET 16c; d
CN1 1; Zs; Y s;xP sh iT;
CN2 0; dYds
; dZds
; ls T
16e; f
CN3 w;ky ;kz;kx T
;
CN4 w; dZds
;dYds
; r w T
;
CN5 2; 0; 0; 0h iT 16g; h; i
JN
J 11 0 0 0
0 J 22 J 23 J 24
0 J 23 J 33 J 34
0 J 24 J 34 J 44
266664
377775;
JT
J 55 0 0 0
0 J 22 J 23 J 24
0 J 23 J 33 J 34
0 J 24 J 34 J 44
266664
377775 16j; k
where the following denitions are employed:
Table 1Convergence of the element for buckling and lateral
buckling loads ofclampedfree closed section beam
Number ofelements
Axial bucklingload QX [N]
Lateral bucklingload QZ [N]
1 454,890 243,9222 444,268 186,9785 441,634 165,99810 441,271
163,47315 441,204 162,996T20 441,181 162,82430 441,164
162,700exible Analytic 4.39 11.98 17.54 20.97
0/90/90/0 No shearexible
FEM 2.81 11.27 25.20 44.43Analytic 2.79 11.17 25.14 44.39
Shearexible
FEM 2.57 8.06 13.33 17.30Analytic 2.54 7.99 13.25 17.23
45/45/45/45 No shearexible
FEM 0.55 2.17 4.85 8.58Analytic 0.54 2.17 4.88 8.59
Shearexible
FEM 0.54 2.16 4.77 8.30Analytic 0.54 2.15 4.77 8.30Fig. 3.
Cross-sections analyzed. (a) b = h = 0.1 m, (b) b = h = 0.1 m,(c) b
= h/2 = 0.1 m. The thickness is e = 0.01 m in the three cases.
-
oNxxox
oNxsos
0 19a
oMxxox
oMxsos
Nxn 0 19b
Substituting expressions (14) and (15) into (1b),
integratingwith respect to variable s and taking variations with
re-spect to QX,MY,MZ, B, TSV, QZ, QY and TW one obtains,after some
algebraic manipulation, the following constitu-tive equations for
the beam stress resultants:
QN EJN eN 20aQT G JT
TCQ 1
JT eT GSeT 20bwhere S is the shear stiness matrix and
CQ CN3CTN3 e4A66144AH55
CN2CTN2
G
12GCN5C
TN5 21
and E*, G* and G** are expressed in the form:
E A11e
; G A66e
22a; b
G G for closed sections12D66e3
for open sections
(22c
The constitutive form (20) is a generalization of that ob-tained
by Cortnez and Piovan [16]. The present beammodel is governed by
expressions (12) and (20) along withcorresponding boundary
conditions.
3. Finite element analysis
In order to obtain the buckling loads of the thin-walledshear
deformable beams, a nite element is formulatedbased on the present
theory. The element has two nodeswith seven degrees of freedom in
each one, and constitutesan extension of the element developed by
Cortnez andRossi [15] for isotropic materials.
Table 3Buckling loads (QX [N]) for a U-section beam with dierent
stacking sequences, slenderness ratios and boundary conditions
Boundary conditions Stacking sequence Model h/L = 0.05 h/L =
0.10 h/L = 0.15 h/L = 0.20
SSSS 0/0/0/0 [I] 2.674(XZ) 9.501(XZ) 20.828(XZ) 36.588(XZ)[II]
2.491(XZ) 7.323(XZ) 12.451(XZ) 16.655(XZ)[III] 6.833 22.925 40.219
54.480
0/90/90/0 [I] 1.635(XZ) 5.370(XZ) 11.559(XZ) 20.169(XZ)[II]
1.572(XZ) 4.625(XZ) 8.468(XZ) 12.223(XZ)[III] 3.835 13.881 26.742
39.398
45/45/45/45 [I] 1.235(Y) 2.750(XZ) 4.129(XZ) 5.843(XZ)[II]
1.232(Y) 2.736(XZ) 4.097(XZ) 5.770(XZ)[III] 0.197 0.526 0.795
1.249
CC 0/0/0/0 [I] 9.503(XZ) 36.595(XZ) 81.028(XZ) 141.769(XZ)[II]
7.453(XZ) 16.891(XZ) 22.237(XZ) 25.004(XZ)[III] 21.572 53.844
72.557 82.363
0/90/90/0 [I] 5.371(XZ) 20.173(XZ) 44.444(XZ) 77.619(XZ)[II]
4.704(XZ) 12.429(XZ) 18.348(XZ) 22.048(XZ)[III] 12.426 38.388
58.717 71.595
2.752.740.19
5.054.363.612.942.611.152.001.990.39
0.940.922.230.670.651.680.300.300.00
984 V.H. Cortnez, M.T. Piovan / Computers and Structures 84
(2006) 97899045/45/45/45 [I][II][III]
CSS 0/0/0/0 [I][II][III] 1
0/90/90/0 [I][II][III] 1
45/45/45/45 [I][II][III]
CF 0/0/0/0 [I][II][III]
0/90/90/0 [I][II][III]
45/45/45/45 [I][II][III][I] Model neglecting shear exibility.
[II] Model allowing for shear exibility. [I(C) clamped, (F)
free.0(XZ) 5.844(XZ) 10.512(XZ) 16.832(XZ)5(XZ) 5.828(XZ)
10.367(XZ) 16.291(XZ)7 0.269 1.376 3.215
8(XZ) 18.982(XZ) 41.995(XZ) 73.819(XZ)9(XZ) 11.436(XZ)
16.858(XZ) 20.314(XZ)7 39.753 59.857 72.4811(XZ) 10.551(XZ)
23.123(XZ) 40.506(XZ)3(XZ) 7.312(XZ) 12.508(XZ) 16.729(XZ)9 30.694
45.905 58.6995(XZ) 3.920(XZ) 6.418(XZ) 9.759(XZ)7(XZ) 3.900(XZ)
6.347(XZ) 9.550(XZ)6 0.535 1.106 2.145
7(XZ) 2.674(XZ) 5.522(XZ) 9.501(XZ)6(XZ) 2.481(XZ) 4.715(XZ)
7.291(XZ)6 7.206 14.620 23.2610(XZ) 1.635(XZ) 3.195(XZ)
5.370(XZ)8(XZ) 1.572(XZ) 2.893(XZ) 4.654(XZ)0 3.827 9.463
13.3359(Y) 1.234(XZ) 2.100(XZ) 2.750(XZ)9(Y) 1.232(XZ) 2.090(XZ)
2.733(XZ)0 0.165 0.485 0.621II] Percentage dierence: 100(QX[II]
QX[I])/QX[II]. (SS) Simply supported,
-
The generalized nodal displacements may be written as
w uxc1 ; uyc1 ; hz1 ; uzc1 ; hy1 ;/x1 ; hx1 ; uxc2 ; uyc2 ; hz2
; uzc2 ; hy2 ;/x2 ; hx2 T
23while the displacement eld in the element is interpolated
inthe form:
uxc a0 a1~x; uyc b0 b1~x b2~x2 b3~x3;
hz b1 b1b32
2b2~x 3b3~x2 24a;b; cuzc c0 c1~x c2~x2 c3~x3; hy c1 b2c3
2 2c2~x 3c3~x2
24d; e/x d0 d1~x d2~x2 d3~x3; hx d1
b3d32
2d2~x 3d3~x2
24f ;gwhere the coecients ais , bis , cis and dis are
indeterminateconstants whereas
~x xle
; b1 12EJ 22GS22l
2e
; b2 12EJ 33GS33l
2e
; b1 12EJ 44GS44l
2e
25a; b; c; dIt must be noted that this interpolation yields:
ouycox
hz b1b32
;ouzcox
hy b2c32
;
o/xox
hx b3d32
26a; b; c; d
It may be easily seen that when the coecients bi (i =1,2,3) are
very small (i.e. slender beams), expressions (26)become zero.
Therefore this element avoids the shear-lock-ing phenomenon. On the
other hand it is possible to use thepresent element as a
Vlasov-type beam element, which maybe obtained from the present one
as a limiting case, by tak-ing large values to the shear coecient
(G*Sii) in the ele-ment stiness matrix, in order to neglect the
shear eect.
Table 4Buckling loads (QX [N]) for a I-section beam with dierent
stacking sequences, slenderness ratios and boundary conditions
Boundary conditions Stacking sequence Model h/L = 0.05 h/L =
0.10 h/L = 0.15 h/L = 0.20
SSSS 0/0/0/0 [I] 5.943(Y) 23.474(Y) 52.910(Y) 93.179(Y)[II]
5.548(Y) 18.454(Y) 32.438(Y) 36.811(Z)[III] 6.651 21.385 38.692
60.494
0/90/90/0 [I] 3.197(Y) 12.734(Y) 28.470(Y) 50.114(Y)[II]
3.098(Y) 11.008(Y) 21.552(Y) 30.890(Y)[III] 3.082 13.553 24.299
38.361
45/45/45/45 [I] 0.620(Y) 2.468(Y) 5.515(Y) 9.713(Y)[II] 0.619(Y)
2.458(Y) 5.468(Y) 9.569(Y)[III] 0.098 0.387 0.855 1.484
CC 0/0/0/0 [I] 23.681(Y) 93.198(Y) 204.215(Y) 350.235(Y)[II]
18.537(Y) 36.900(Z) 39.270(Z) 40.177(Z)[III] 21.721 60.407 80.770
88.529
0/90/90/0 [I] 12.736(Y) 50.125(Y) 109.834(Y) 188.369(Y)[II]
11.156(Y) 31.156(Y) 35.087(Z) 38.152(Z)
12.42.42.40.3
12.110.413.76.56.15.715 22.841 45.298 63.6431.21.20.2
1.41.41.770 6.684 13.831 22.1350.80.71.10.10.10.0
V.H. Cortnez, M.T. Piovan / Computers and Structures 84 (2006)
978990 985[III]45/45/45/45 [I]
[II][III]
CSS 0/0/0/0 [I][II][III]
0/90/90/0 [I][II][III]
45/45/45/45 [I][II][III]
CF 0/0/0/0 [I][II][III]
0/90/90/0 [I][II][III]
45/45/45/45 [I][II][III][I] Model neglecting shear exibility.
[II] Model allowing for shear exibility. [I(C) clamped, (F)
free.00(Y) 3.196(Y) 7.180(Y) 12.734(Y)91(Y) 3.057(Y) 6.512(Y)
10.777(Y)33 4.371 9.301 15.36655(Y) 0.619(Y) 1.391(Y) 2.468(Y)55(Y)
0.619(Y) 1.388(Y) 2.458(Y)00 0.000 0.221 0.39066(Y) 5.020(Y)
11.141(Y) 19.435(Y)63(Y) 4.978(Y) 10.936(Y) 18.829(Y)18 0.851 1.845
3.119
87(Y) 5.943(Y) 13.349(Y) 23.676(Y)61(Y) 5.546(Y) 11.503(Y)
18.435(Y)08 37.843 68.055 79.74668(Y) 9.715(Y) 21.287(Y)
36.509(Y)59(Y) 9.574(Y) 20.639(Y) 34.702(Y)78 1.449 3.044 4.950
42(Y) 48.163(Y) 106.883(Y) 186.445(Y)71(Y) 29.578(Y) 34.487(Z)
37.534(Z)59 38.587 67.734 79.86930(Y) 25.904(Y) 57.486(Y)
100.277(Y)57(Y) 19.987(Y) 31.446(Z) 36.458(Z)II] Percentage
dierence: 100 (QX[II] QX[I])/QX[II]. (SS) Simply supported,
-
Table 5Buckling loads (QX [N]) for a closed-section beam with
dierent stacking sequences, slenderness ratios and boundary
conditions
Boundary conditions Stacking sequence Model h/L = 0.05 h/L =
0.10 h/L = 0.15 h/L = 0.20
SSSS 0/0/0/0 [I] 5.235(Y) 20.863(Y) 46.664(Y) 81.446(X)[II]
4.651(Y) 13.912(Y) 22.041(Y) 27.708(Y)[III] 11.148 33.318 52.766
65.980
0/90/90/0 [I] 2.810(Y) 11.267(Y) 25.201(Y) 44.432(X)[II]
2.681(Y) 9.152(Y) 15.142(Y) 19.285(Y)[III] 4.591 18.773 39.915
56.597
45/45/45/45 [I] 0.546(Y) 2.175(Y) 4.854(Y) 8.576(Y)[II] 0.544(Y)
2.156(Y) 4.774(Y) 8.301(Y)[III] 0.213 0.846 1.659 3.211
CC 0/0/0/0 [I] 20.867(Y) 81.448(X) 88.478(X) 98.284(X)[II]
13.988(Y) 27.858(Y) 34.066(Y) 36.934(Y)[III] 32.967 65.796 61.498
62.421
0/90/90/0 [I] 11.270(Y) 44.411(Y) 82.877(X) 88.348(X)[II]
9.152(Y) 18.875(Y) 24.821(Y) 27.365(Y)[III] 18.790 57.499 70.051
69.026
45/45/45/45 [I] 2.175(Y) 8.578(Y) 18.857(Y) 32.477(Y)[II]
2.157(Y) 8.309(Y) 17.627(Y) 29.071(Y)[III] 0.825 3.138 6.523
10.489
CSS 0/0/0/0 [I] 10.696(Y) 42.471(Y) 82.295(X) 87.329(X)[II]
8.356(Y) 20.118(Y) 27.301(Y) 31.360(Y)[III] 21.874 52.631 66.826
64.090
0/90/90/0 [I] 5.776(Y) 22.936(Y) 50.987(Y) 82.235(X)[II]
4.972(Y) 14.250(Y) 19.874(Y) 26.125(Y)[III] 13.923 37.872 61.022
68.231
45/45/45/45 [I] 1.115(Y) 4.427(Y) 9.842(Y) 17.207(Y)[II]
1.110(Y) 4.345(Y) 9.450(Y) 16.060(Y)[III] 0.475 1.850 3.980
6.665
CF 0/0/0/0 [I] 1.310(Y) 5.234(Y) 11.760(Y) 20.863(Y)[II]
1.269(Y) 4.648(Y) 9.168(Y) 13.893(Y)[III] 3.115 11.204 22.041
33.408
0/90/90/0 [I] 0.707(Y) 2.827(Y) 6.351(Y) 11.297(Y)[II] 0.695(Y)
2.684(Y) 5.359(Y) 8.984(Y)[III] 1.747 5.055 15.620 20.474
45/45/45/45 [I] 0.137(Y) 0.546(Y) 1.226(Y) 2.175(Y)[II] 0.136(Y)
0.544(Y) 1.220(Y) 2.156(Y)[III] 0.051 0.216 0.482 0.850
[I] Model neglecting shear exibility. [II] Model allowing for
shear exibility. [III] Percentage dierence: 100 (QX[II]
QX[I])/QX[II]. (SS) Simply supported,(C) clamped, (F) free.
Fig. 5. Lateral buckling loads of a composite clampedclamped
I-beamwith stacking sequence {0/90/90/0}. Comparison of models
neglecting andallowing shear exibility.
Fig. 4. Variation of lateral buckling load with respect to
slenderness ratio,for a composite clampedclamped I-beam with
stacking sequence {0/90/90/0}.
986 V.H. Cortnez, M.T. Piovan / Computers and Structures 84
(2006) 978990
-
exibility is performed. It is possible to see a strong inu-ence
of shear eects in the prediction of lateral bucklingloads,
especially at large slenderness ratios (i.e. h/L).
Theaforementioned gures were obtained with models of 30nite
elements.
4.4. Comparisons
In order to check the eciency of the introduced theory,in the
present section comparisons with the theories of Leeand Kim [12],
Sherbourne and Kabir [7] and with shellmodels of COSMOS/M are
performed. In these compari-sons, models with 30 nite elements of
the present beamtheory are employed.
The model of Lee and Kim [12] is compared with thepresent theory
neglecting the shear exibility in theFig. 6. This Figure shows
buckling loads due to compres-
uterSubstituting (23) into the governing variational Eq. (11)and
assembling in the usual way, one arrives to
K kKGW 0 27
where K, KG, and W* are the global stiness, global geo-metric
stiness matrices and the global displacement vectorrespectively. In
order to calculate the eigenvalues, it is nec-essary to obtain the
prebuckling state by solving the self-equilibrating system of
initial stresses and, initial volumeand surface forces (see
references [24,25,27]).
4. Numerical examples and discussion
In order to evaluate the shear eect on the stabilitybehavior of
the analyzed structural members, numericalcomparisons are performed
among the present model pre-dictions and results obtained by
neglecting the shear defor-mability (Ghorbanpoor and Omidvar, [9],
and Lee et al.[1214]). Dierent cross-sectional shapes, laminate
schemesand slenderness ratios are considered. The analyzed
mate-rial is graphite-epoxy (AS4/3501) whose properties areE1 = 144
GPa, E2 = 9.65 GPa, G12 = 4.14 GPa, G13 =4.14 GPa, G23 = 3.45 GPa,
m12 = 0.3, m13 = 0.3, m23 = 0.5,q = 1389 kg/m3. The considered
laminate schemes are: (a){0/0/0/0}, (b) {0/90/90/0} and (c)
{45/45/45/45}. Theanalyzed cross-sections are shown in Fig. 3.
4.1. Convergence analysis and comparisons
In Table 1, it is presented a convergence analysis forbuckling
and lateral buckling of a clampedfree beam witha closed rectangular
section, with h/L = 0.1, {h,b,e} ={0.1,0.5,0.01} m, and stacking
sequence of {0/0/0/0}. Onthe other hand, in Table 2 a comparison
with analyticalresults [16] of buckling loads for a pinnedpinned
beamwith closed section is shown. Dierent stacking sequencesand
slenderness ratios were considered. In Table 2 modelswith 30 nite
elements were employed. From these tables, itis possible to note a
fast convergence as well as good agree-ment with analytical
results.
4.2. Buckling problems
In Tables 35, comparisons of buckling loads, of shearexible and
non-shear exible models for U-section, I-sec-tion and closed
section are presented. In these tables, theeigenvalues (buckling
loads) are normalized with respectto 105 and the capital letters
(Y), (Z), (X) and (XZ) areemployed, in order to indicate the
corresponding bucklingmode. In this way, (Y), (Z), (X) and (XZ)
stand for exuralmode in y-direction, exural mode in z-direction,
tor-sional mode and exural torsional mode (characteristic
ofmonosymmetric cross-sections), respectively.
In Tables 35, it is possible to see the strong inuence of
V.H. Cortnez, M.T. Piovan / Compshear exibility in laminates
{0/0/0/0} and {0/90/90/0},which increases with the slenderness
ratio h/L. In this sensethe percentage dierences can reach 88.529%
for clampedclamped I-beams with a {0/0/0/0} stacking sequence
(seeTable 4, h/L = 0.20). Also, it is possible to note that
buck-ling modes are dominant exural torsional in the U-beamsand
dominant exural in y-direction, for beams with theother two types
of cross-section. The inuence of shear ex-ibility is also observed
in the change of the buckling modeshape, as it can be seen, for
example, in Table 5, in the caseof a clampedclamped boundary
condition, laminates{0/0/0/0} or {0/90/90/0} with h/L = 0.15 or
higher. Thesetables were obtained with models of 30 nite
elements.
4.3. Lateral buckling problems
In Fig. 4, the variation of the lateral buckling load
withrespect to the slenderness ratio is depicted. This case,
per-formed with the shear exible model, corresponds to
aclampedclamped I-beam with stacking sequence {0/90/90/0}, carrying
a point load at the middle of the beam.On the other hand, in Fig.
5, a comparison of lateral buck-ling loads between models allowing
and neglecting shear
0 15 30 45 60 75 90
0.1
0.2
0.3
105 NQX
Authors Shear NeglectedLee and Kim [12]
0
Fig. 6. Buckling loads of a composite simply supported I-beam
withstacking sequence {a/a/a/a}. Comparison of the present model
(reducedneglecting shear exibility) with other models.
s and Structures 84 (2006) 978990 987sive loads (applied at both
ends) for a simple supportedcomposite I-beam with stacking sequence
{a/a/a/a} in
-
As one can see in Table 6, the reduced model of the authors
and orthotropic cantilever I-beam. The materials are steel
terTable 7Comparisons of buckling loads (QX) and lateral
buckling loads (QZ) of a
Table 6Lateral buckling loads (concentrated load QZ [kN] in x =
L/2, applied atthe centroid) for a simply supported composite
I-beam (angesber-angle a = 0, web lamination {a/a/a/a})Web
berangle, a
Allowing shearexibility
Neglecting shear exibility
[I] [II] [III] [I] [II] [III]
0 82.1 83.4 1.57 88.4 89.2 0.9610 84.1 85.5 1.66 90.3 91.3
1.0720 89.0 90.4 1.63 95.2 96.4 1.1730 93.3 94.4 1.15 100.2 101.5
1.3640 96.2 95.1 1.22 103.2 102.2 0.9845 96.5 95.0 1.49 103.4 102.1
1.20
[I] Model of Sherbourne and Kabir [30]. [II] Authors model.
[III] Per-centage dierence: 100jQZ[I] QZ[II]j/QZ[I].
988 V.H. Cortnez, M.T. Piovan / Compuboth the web and anges. The
beam has the followingdimensions: length 800 cm, web height 20 cm,
ange width10 cm and thickness 1 cm. The composite material is
agraphite-epoxy with E11 = 133.4 GPa, E22 = 8.78 GPa,G12 = 3.67 GPa
and m12 = 0.26. As one can see, the presenttheory can be reduced to
the Lee and Kim [12] model.
Sherbourne and Kabir, in their theory, considered onlythe shear
exibility due to bending, whereas they neglectedthe shear exibility
due to non-uniform torsion warping.Also the virtual work terms due
to o-axis forces werenot taken into account by Sherbourne and
Kabir. In thiscontext, the authors model can be reduced to the
Sher-bourne and Kabir model, by neglecting the
aforementionedcomponents.
Table 6 oers the critical loads for the lateral buckling ofa
simply supported I-beam, whose anges have a berorientation a = 0
and the web has the stacking sequence{a/a/a/a}. The loading scheme
is composed by a pointload applied in the web direction and located
at themidspan in the cross-sectional centroid, i.e. {x,y,z} =
cantilever I-beam, obtained with the present shear deformable
beamtheory and nite element shell models of COSMOS/M
Type ofload
Material Type of model Value[N]
QZ Steel Beam theory 85,045Shell8T (COSMOS/M) 81,643Dierence [%]
4.16%
QX Steel Beam theory 112,082Shell8T (COSMOS/M) 109,840Dierence
[%] 2.04%
QX AS4/3501-6{0/90/90/0}
Beam theory 41,861Shell8T (COSMOS/M) 40,698Dierence [%]
2.85%
QX AS4/3501-6{45/45/45/45}
Beam theory 8012Shell8T (COSMOS/M) 7958Dierence [%] 0.68%
The dierence [%] is taken with respect to the load values of the
shellmodels.thin-walled composite beams has been presented in
thispaper. This model constitutes a generalization of that
pro-posed by the authors in Ref. [16].
The generalization consists in the adoption of anenhanced
displacement eld with non-linear terms basedon the rule of
semi-tangential rotations. On the other hand,the model takes into
account an arbitrary state of initialstresses.
In order to solve the governing variational equation,
anon-locking nite element with fourteen degrees of free-dom was
employed. The numerical results demonstratethat the shear exibility
has a remarkable eect on the crit-ical loads, especially when one
of the material axes coin-cides with the longitudinal axis of the
beam.
Acknowledgements
The present study was sponsored by Secretara de Cien-cia y
Tecnologa de Universidad Tecnologica Nacional,Research Project
25/B008 and by CONICET (NationalCouncil of Scientic and
Technological Research).
Appendix I
The constitutive equations of symmetric balanced lami-(E = 210
GPa, G = 80.76 GPa) and Graphite-Epoxy AS4/3501-6 (E11 = 144 GPa,
E22 = 9.68 GPa, G12 = 4.14 GPa,G23 = 3.45 GPa, m12 = 0.3, m23 =
0.48). The beam has thefollowing dimensions: length 100 cm, web
height 10 cm,anges width 5 cm and overall thickness 1 cm. Two
typesof loads are considered. The rst is a lateral load QZapplied
in the cross-sectional center at the free end in theweb direction.
The second is a compressive load QX appliedin the cross-sectional
center at the free end. For this com-parison models with more than
12000 shell elements(SHELL8T) were employed to calculate the
buckling loadsin COSMOS/M. In Table 7 one can see a good
agreementof buckling loads obtained with the present theory andwith
nite element shell models.
5. Conclusions
A new model for general stability analysis of shearableagrees
well (with less that 1.70%) with the model of Sher-bourne and
Kabir, both, allowing or neglecting the shearexibility
components.
Finally Table 7 shows a comparison of the presentshear
deformable beam theory with shell models ofthe nite element program
COSMOS/M for an isotropic{L/2,0,0}. The cross-section dimensions
are {h,b,e} ={20.32,10.16,0.953} cm., the beam has a length L =
12h,and the material properties can be found in Reference [7].
s and Structures 84 (2006) 978990nates may be expressed in terms
of shell stress resultants inthe following form [1]:
-
NN
N
M
8>>>>>>>>>>>>>
with
A11
AH55
D11
wheraccor[1, chcause
A:II:90 0 0
n o Z n o
(3)
3 B y 2 2
(4)
Appe
Wcipal
utersequence.
Appendix II
The initial forces expressions due to initial volume forcesand
initial surface forces, which were introduced in (12c),are now
dened extensively in the following form:
(1) Volume initial forces:
X 03 N 01 hz N 03 N 04
2hy N
06
2/x A:II:1
X 05 N 03 N 04
2hz N 02 hy
N 052
/x A:II:2
X 06 N 062
hz N05
2hy N 01 N 02
/x A:II:3
with
N 01 ;N02 ;N
03
n oZA
yX 0y ; zX0z ; yX
0z
n odsdn
A:II:4N 04 ;N
05 ;N
06
n oZA
zX 0y ; yX0x ; zX
0x
n odsdn
A:II:5(2) Surface initial forces:
T 03 H 01 hz H 03 H 04
2hy H
06
2/x A:II:6
T 05 H 03 H 04
2hz H 02 hy
H 052
/x A:II:70 0A:I:1
A11 A212
A22; A66 A66 A
226
A22;
AH55 AH45 2AH44
A:I:2:a; b; c
D11 D212
D22; D66 D66 D
226
D22A:I:2:d; e
e Aij, Dij and AHij are plate stiness coecients dened
ding to the lamination theory presented in referenceapter 6].
The coecient D16 has been neglected, be-of its low value for the
considered laminate stackingMxs: ;
0 0 0 0 D66 jLxs: ;xx
xs
xn
xx
9>>>>>>>=>>>>>>>
A11 0 0 0 0
0 A66 0 0 0
0 0 AH55 0 0
0 0 0 D11 0
266666664
377777775
eLxx
cLxs
cLxn
jLxx
8>>>>>>>>>>>>>
9>>>>>>>=>>>>>>>
V.H. Cortnez, M.T. Piovan / CompT 06 H 62
hz H 52
hy H 01 H 02
/x A:II:8 (16)tantsT 0WW ZS
N 0xsdxds
N 0xndxdn
xpsds A:II:19
ndix III
hen the cross-section axes are assumed to be the prin-ones and a
two-pole reference is employed, the formsT 0WY ZS
N 0xsdxds
N 0xndxdn
Y sds A:II:17
T 0WZ ZS
N 0xsdxds
N 0xndxdn
Zsds A:II:18T 5 2 hz zBF z hy 2 /x
A:II:13
T 06 zBF 0x2
hz yBF0x
2hy yBF 0y zBF 0z
/x
A:II:14The generalized beam stress resultants can be denedin
terms of the shell stress resultants introduced in(A.I.1) by means
of the following expressions:
Q0YW ZS
N 0xsdYds
N 0xndZds
xp ds A:II:15
Q0ZW ZS
N 0xsdZds
N 0xndYds
xp ds A:II:16A:II:12
0 zBF0y yBF 0z 0 yBF 0x4 5 6A
y x x
A:II:10Consider an initial point-load F 0 F 0x ; F 0y ; F 0z
n o,
applied at an o-axis point B(xB,yB,zB), as it is shownin the
Fig. 1b. Now, applying DeltaKronecker mul-tipliers at the point B,
the virtual work term for theo-axis forces can be expressed as
LKG22 T 03 dhz T 05 dhy T 06 d/xh i
xxBA:II:11
where
T 0 y F 0hz zBF 0y yBF 0z hy zBF
0x /xH ;H ;H zT 0; yT 0; zT 0 dsdnwith
H 01 ;H02 ;H
03
n oZA
yT 0y ; zT0z ; yT
0z
n odsdn
s and Structures 84 (2006) 978990 989can be simplied and the eld
of the shell stress resul-can assumed to be of the form:
-
Nxx e NJ 11 MyJ 22
Z MzJ 33
Y BJ 44
xp
A:III:1
Mxx e3
12
MyJ 22
dYds
MzJ 33
dZds
BJ 44
ls
A:III:2
Mxs e3
6J 55T SV A:III:3
Nxs e QzJ 22 kys QyJ 33
kzs T wJ 44 kws
ewJ 55
T SV
A:III:4
[9] Omidvar B. Shear coecient in orthotropic thin-walled
compositebeams. J Compos Construction 1996;2(1):4656.
[10] Librescu L, Song O. On the aeroelastic tailoring of
composite aircraftswept wings modeled as thin walled beam
structures. Compos Eng1992;2(57):497512.
[11] Bhaskar K, Librescu L. A Geometrically non-linear theory
forlaminated anisotropic thin-walled beams. Int J Eng Sci
1995;33(9):133144.
[12] Lee J, Kim SE. Flexural-torsional buckling of thin-walled
I-sectioncomposites. Comput Struct 2001;79:98795.
[13] Lee J, Kim SE. Lateral buckling analysis of thin-walled
laminatedchannel-section beams. Compos Struct 2002;56:3919.
990 V.H. Cortnez, M.T. Piovan / Computers and Structures 84
(2006) 978990Nxn e3
12
QZJ 22
dYds
QYJ 33
dZds
TWJ 44
ls
A:III:5
It is clear that, as dened in (17a) J11, J22, J33, J44 and
J55are the cross-sectional area, second-order area
moments,cross-sectional warping constant and St. Venant
torsionconstant, respectively.
Further explanations of the aforementioned
expressions(A.III.1)(A.III.5) can be found in Cortnez and
Piovan[16] for composite materials or Vlasov [2] for
isotropicmaterials.
References
[1] Barbero EJ. Introduction to composite material design.
Taylor andFrancis Inc; 1999.
[2] Vlasov VV. Thin walled elastic beams. Jerusalem: Israel
Program forScientic Translation; 1961.
[3] Bauld NR, Tzeng LS. A Vlasov theory for ber-reinforced
beamswith thin-walled open cross sections. Int J Solids Struct
1984;20(3):27797.
[4] Ghorbanpoor A, Omidvar B. Simplied analysis of
thin-walledcomposite members. J Struct Eng ASCE
1996;122(11):137983.
[5] Massa JC, Barbero EJ. A strength of materials formulation
for thin-walled composite beams with torsion. J Compos Mater
1998;32(17):156094.
[6] Pollock GD, Zak AR, Hilton HH, Ahmad MF. Shear center
forelastic thin-walled composite beams. Struct Eng Mech
1995;3(1):91103.
[7] Sherbourne AN, Kabir MZ. Shear strains eects in lateral
stability ofthin-walled brous composite beams. J Eng Mech ASCE
1995;121(5):6407.
[8] Godoy LA, Barbero EJ, Raftoyiannis I. Interactive buckling
analysisof ber-reinforced thin-walled columns. J Compos Mater
1995;29(5):591613.[14] Lee J, Kim SE, Hong K. Lateral buckling
I-section beams. Eng Struct2002;24:95564.
[15] Cortnez VH, Rossi RE. Dynamics of shear deformable
thin-walledopen beams subjected to initial stresses. Rev Int Met
Num Calculo yDiseno en Ingeniera 1998;14(3):293316.
[16] Cortnez VH, Piovan MT. Vibration and buckling of composite
thin-walled beams with shear deformability. J Sound Vib 2002;
258(4):70123.
[17] Wu XX, Sun CT. Vibration analysis of laminated composite
thin-walled beams using nite elements. AIAA J 1990;29(5):73642.
[18] Kollar LP. Flexural-torsional buckling of open section
compositecolumns with shear deformation. Int J Solids Struct
2001;38:752541.
[19] Sapkas A, Kollar LP. Lateral-torsional buckling of
composite beams.Int J Solids Struct 2001;39:293963.
[20] Chen WF, Lui EM. Structural stability. Theory and
implementa-tion. New York: Elsevier Science and Publishing;
1987.
[21] Timoshenko SP, Gere JM. Theory of elastic stability. 2nd
ed. NewYork: McGraw Hill; 1961.
[22] Argyris JH, Hilpert O, Malejannakis GA, Scharpf DW. On
thegeometrical stiness of a beam in space. Comput Methods Appl
MechEng 1979;20:10531.
[23] Argyris JH. An excursion into large rotations. Comput
MethodsAppl Mech Eng 1982;32:85155.
[24] Kim MY, Chang SP, Park HG. Spatial postbuckling analysis
ofnonsymmetric thin-walled frames. I: Theoretical considerations
basedon the semitangential property. J Eng Mech
2001;127(8):76978.
[25] Kim MY, Chang SP, Park HG. Spatial postbuckling analysis of
non-symmetric thin-walled frames. I: Geometrically nonlinear FE
proce-dures. J Eng Mech 2001;127(8):77990.
[26] Washizu K. Variational methods in elasticity and
plasticity. NewYork: Pergamon Press; 1968.
[27] Piovan MT. Theoretical and computational study in the
mechanics ofcomposite thin walled curved beams, considering
non-conventioanleects [in spanish]. PhD thesis. Department of
Engineering, Univers-idad Nacional del Sur. Baha Blanca. Argentina,
2002.
[28] Krenk S, Gunneskov O. Statics of thin-walled pretwisted
beams. Int JNumer Methods Eng 1981;17:140726.
[29] Conci A. Large displacements analysis of thin walled beams
withgeneric open section. Int J Numer Methods Eng
1993;33:210927.
[30] Bathe KJ. Finite element procedures. NJ, USA:
Prentice-Hall; 1996.
Stability of composite thin-walled beams with shear
deformabilityIntroductionTheoryAssumptionsVariational
formulationKinematic expressionsEquations of motionConstitutive
equations for the beam stress resultants
Finite element analysisNumerical examples and
discussionConvergence analysis and comparisonsBuckling
problemsLateral buckling problemsComparisons
ConclusionsAcknowledgementsAppendix IAppendix IIAppendix
IIIReferences