Stability of composite thin-walled beams with shear deformability

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


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


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


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


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

Documents

Stability of composite thin-walled beams with shear deformability