a geometrically exact approach to lateral-torsional buckling of thin-walled beams with deformable...
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Computers and Structures 106–107 (2012) 9–19
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Computers and Structures
journal homepage: www.elsevier .com/locate /compstruc
A geometrically exact approach to lateral-torsional buckling of thin-walledbeams with deformable cross-section
Rodrigo Gonçalves ⇑UNIC, Departamento de Engenharia Civil, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal
a r t i c l e i n f o
Article history:Received 24 August 2011Accepted 25 March 2012Available online 8 May 2012
Keywords:Thin-walled membersLateral bucklingCross-section deformationBeam finite elements
0045-7949/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.compstruc.2012.03.017
⇑ Tel.: +351 21 2948580; fax: +351 21 2948398.E-mail address: [email protected]
a b s t r a c t
In this paper, a new geometrically exact beam formulation is presented, aiming at calculating buckling(bifurcation) loads of Euler–Bernoulli/Vlasov thin-walled beams with deformable cross-section. Theresulting finite element is particularly efficient for problems involving coupling between lateral-torsionalbuckling and cross-section distortion/local-plate buckling. The kinematic description of the beam is geo-metrically exact and employs rotation tensors associated with both cross-section rotation and the rela-tive rotations of the cross-section walls in the cross-section plane. Moreover, arbitrary deformationmodes, complying with Kirchhoff’s assumption, are also included, which makes it possible to capturelocal/distortional/global buckling phenomena. Load height effects associated with cross-section rota-tion/deformation are also included. The examples presented throughout the paper show that the pro-posed beam finite element leads to accurate solutions with a relatively small number of degrees-of-freedom (deformation modes and finite elements).
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1. Introduction
Lateral-torsional buckling of steel beams is a subject which hasattracted the attention of many researchers and is covered exten-sively in various classic textbooks, e.g. [1–4]. An extensive andup-to-date review is provided in Chapter 5 of the recent editionof the Guide to Stability Design Criteria for Metal Structures [5].In General, lateral-torsional buckling failure governs the designof beams subjected to major axis bending and not properly bracedagainst lateral deflection/twisting, whose cross-sections have lowtorsional stiffness. According to modern steel design codes (e.g.[6,7]), the strength of compact section beams is calculated by mul-tiplying the cross-section plastic bending moment resistance by areduction factor which accounts for lateral-torsional buckling,including the effects of imperfections, residual stresses and partialyielding. This reduction factor depends on the elastic critical bifur-cation moment and therefore its accurate calculation is crucial fora safe and economic design. Consequently, much effort has beendevoted to develop efficient charts, formulae and user-friendlycomputer applications, e.g. [8].
For members with high cross-section wall width-to-thicknessratios, the lateral-torsional buckling behaviour is highly affectedby local-plate and distortional buckling phenomena. For instance,the so-called ‘‘lateral-distortional’’ buckling of beams, where lat-eral-torsional buckling occurs with the transverse bending of the
ll rights reserved.
(slender) web, has been investigated in [9–11]. The most commondesign approaches are based on effective width or direct strengthmethods and, again, the calculation of the elastic critical momentis mandatory. Moreover, the direct strength method requires per-forming an accurate elastic stability analysis of the member (withgross cross-section properties), which can only be accomplished byresorting to numerical methods involving shell finite elements, fi-nite strips or beam theories incorporating cross-section deforma-tion, such as Generalized Beam Theory (GBT) [12,13].
Discussing Euler–Bernoulli beam finite element formulations, Piand Bradford [14] showed that the use of a linearized rotation ten-sor leads to incorrect lateral-torsional buckling loads for somecases. Including a non-linear term associated with the work doneby the shear force in the variational principle resolves most prob-lems, but the most efficient solution consists of employing a qua-dratic approximation of the rotation tensor. Obviously, the shearforce term should vanish in a genuine Euler–Bernoulli formulation.
GBT has been proven to be an extremely versatile, elegant andcomputationally efficient method for the structural analysis ofthin-walled members. However, the underlying kinematic descrip-tion is equivalent to employing the linearized rotation tensor fordescribing cross-section rigid-body movements. This was first rec-ognized by the author [15] and led to adding a non-linear virtualwork term of the external forces associated with cross-sectiontwisting, in order to capture the load height effect on the lateral-torsional buckling of beams, under the Vlasov assumption (nullmembrane shear strains). However, strains were still calculatedfrom the classic GBT relations. Other formulations use Vlasov’s
1 Note that subscripts were employed to indicate the wall numbers, but suchindication will be dropped whenever possible to alleviate the notation, under theconvention that an omitted subscript implicitly indicates the wall to which theexpression refers to. For instance, in (2) R means Rn and, in (3), �lAn could have beendisplayed as �lA .
10 R. Gonçalves / Computers and Structures 106–107 (2012) 9–19
assumption for linear shear strains and preserve the associatednon-linear term [16,17], which constitutes the generalization ofthe method mentioned in the previous paragraph for Euler–Ber-noulli beams. Another alternative consists in allowing for bothshear deformation and transverse extension of the walls (e.g.[18–20]), which makes it also possible to capture the load heighteffect, but this is more appropriate for plate-type rather thanbeam-type problems, since it leads to a number of d.o.f. which isalready comparable to the one employed in finite strip methods.
In a recent paper [21], the author has presented a Timoshenko-type beam formulation for thin-walled members undergoing largedisplacements, finite rotations and cross-section deformation. Thisformulation may be considered an extension of the Reissner–Simogeometrically exact beam theory [22,23], where the configurationof each cross-section is described by means of (i) a position vectorr of an arbitrary cross-section centre C, (ii) a finite rotation about Cusing a rotation tensor K and (iii) pre-determined cross-sectiondeformation modes (shape functions) �v. It was shown that theresulting finite element provided efficient and accurate solutions,but it was also found that an adequate modeling of cross-sectionsundergoing moderate-to-large in-plane distortion requires theinclusion of linear and quadratic (at least) deformation modesallowing for the transverse (cross-section in-plane) extension ofthe walls. This requirement is mostly due to a need to accuratelydescribe relative rotations of the walls rather than to allow fortransverse extension, which is hardly relevant for most beam-typeproblems. Subsequently, the formulation was improved by includ-ing also such rotations, using additional rotation tensors R [24].This made it possible to enforce null transverse extension of thewalls and still obtain accurate solutions with a very small numberof deformation modes. For instance, it was shown that the loadheight effect on distortional buckling of C sections may be capturedwithout accounting for transverse extension of the walls.
In this paper, a new beam finite element is presented, aiming atcalculating efficiently (i.e., accurately and with a reduced numberof d.o.f.) bifurcation loads (linear stability analyses) of open thin-walled beams with slender cross-section. In particular, the pro-posed formulation makes it possible to capture the complex loadheight effects associated with the rotation and deformation ofthe cross-section. It is founded on the following assumptions:
(A1) The thickness is small when compared with the cross-sec-tion dimensions and is constant in each cross-section wall;
(A2) Kirchhoff’s assumption holds, i.e., fibers perpendicular to thewall mid-surface remain undeformed and perpendicular tothe mid-surface;
(A3) The shell-like bending strains (i.e., strain terms varyingacross the wall thickness) are small;
(A4) Vlasov’s null membrane shear strain assumption holds,which also implies the Euler–Bernoulli assumption;
(A5) Transverse membrane strains/stresses are negligible.
The kinematic description stems from the previous formulation[24], which already includes assumptions A1–A3. The newassumptions A4–A5 require non-trivial changes to the formulation,but make it possible to enhance its computational efficiency. It isemphasized that the geometrically exact nature of the proposedformulation means that exact rotation tensors are employed, i.e.,no approximations are made concerning the treatment of rota-tions. Most of the examples presented throughout the paper dealwith lateral-torsional buckling due to the fact that the advantagesof employing the resulting finite element are most evident in thistype of problem.
Concerning the notation, symbols denoting scalars and vec-tors/tensors (second order) are written in italic and bold, respec-tively. The matrix forms associated with the latter are identified
by square brackets []. If one is referring to the beam initial con-figuration, the subscript 0 is employed. The standard directionalderivative is denoted by DF(a)[b], where F is the function, a itsargument and b the direction. Partial scalar derivatives with re-spect to the coordinates Xi are indicated by subscripts followinga comma (e.g. F,i = @F/@Xi), although the prime
0is employed to
designate the derivative with respect to X3, to abbreviate thenotation. A virtual variation is denoted by d and an incrementalvariation by D.
2. Formulation of the thin-walled beam
2.1. Kinematic description
An orthonormal direct reference system (X1,X2,X3), with basevectors {Ei} (i = 1,2,3), is employed. The X3 axis coincides with thebeam initial longitudinal axis and defines the position of the arbi-trary cross-section centre C. For each wall, a co-rotational baseKR{Ei} is defined, where K is the cross-section rotation tensor andR is the wall rotation tensor, along the KE3 axis, describing the rota-tion of the wall chord with respect to the cross-section co-rotatingbase K{Ei} in such a way that, at the initial configuration, R0E1
defines the trough-thickness direction.The deformed configuration of each wall is mapped through
vector x, which is written as a function of the mid-surface mappingvector �x and the trough-thickness director n, i.e.,
x ¼ �xþ X1n; ð1Þ
where, according to assumption A2, n is perpendicular to the wallmid-surface — see Fig. 1(a), where e2 ¼ �x;2 and e3 ¼ �x0. The mid-sur-face vector �x is given by
�l
�x ¼ r þ K �lA þ R0
X2
jK3 �x
264375þX
i
pðiÞvðiÞ1
0p0ðiÞvðiÞ3
264375
0B@1CA
0B@1CA
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{;
ð2Þ
where r(X3) references the beam axis, �l references each mid-surfacepoint with respect to C in a co-rotating base KfEig; �lA references thewall mid-line origin A (see Remark 1 below), jK3(X3) is the torsionalcurvature (associated with K), �xðX2Þ is the corresponding warpingfunction and p(i)(X3) are the scalar weight functions associated witheach cross-section deformation mode i, with shape functions�vðiÞj ðX2Þ (see Remark 3).
Remark 1. The rotation tensor R of each wall is parameterizedusing a rotation vector uE3. In order to avoid a duplication of d.o.f.,at least one of the cross-section walls — the ‘‘reference wall’’—mustrotate with K and thus satisfies u = 0 and R = 1. With the exceptionof the reference wall, �lA depends on the rotation of other walls.According to Fig. 1(b), where wall 1 is taken for reference, �lA1 ¼ �LA1
is fixed, R1 = 1 and, for n > 1,
�lAn ¼ �LA1 þXn�1
k¼1
RkE2Bk; ð3Þ
were Bk is the width of wall k.1
(a) (b)Fig. 1. Deformed configuration of a beam wall midsurface (a) spatial view and (b) cross-section view.
R. Gonçalves / Computers and Structures 106–107 (2012) 9–19 11
Remark 2. One deformation mode is assigned to each wall relativerotation, in order to restore compatibility through transverse wallbending. These modes are designated as ‘‘dummy’’, sincepðiÞ ¼ ui � ðuiÞ0 ¼ ui,2 and the corresponding shape functions �vðiÞ1
are obtained through Vlasov’s method, i.e., the cross-section is ana-lyzed as a plane frame under imposed displacements, assumingsmall displacements and wall inextensibility [2,21,25].
Remark 3. The shape functions of the deformation modes areassumed to satisfy, in each wall,
�vðiÞ1 ð0Þ ¼ �vðiÞ1 ðBÞ ¼ 0; ð4Þ�vðiÞ2 ¼ 0; ð5Þ�vðiÞ3nðBnÞ ¼ �vðiÞ3nþ1
ð0Þ: ð6Þ
Eqs. (4) and (5) ensure that only wall relative rotations cause vari-ations of �lA. Moreover, Eq. (5) is related to assumption A5 and notethat �vðiÞ2 was already omitted from (2). Finally, Eq. (6) enforces warp-ing continuity between connecting walls.
The initial configuration of each beam wall is recovered fromthe previous expressions, by setting r = X3E3, K = 1, u = u0 andp(i) = 0, leading to
x0 ¼ �x0 þ X1n0; ð7Þ
�x0 ¼ X3E3 þ ð�lAÞ0 þ R0X2E2
zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{�l0
; ð8Þn0 ¼ R0E1: ð9Þ
In order to comply with assumption A4, K is expressed as acomposition of two rotations: a twist U (along X3) and a rotationh that transforms E3 into the tangent vector t ¼ r0
kr0k (see Fig. 1(a))along the associated geodesic, i.e.,
K ¼ KhKU; ð10Þ
KU ¼ sin UfE3 þ cos Uð1� E3 � E3Þ þ E3 � E3; ð11Þ
Kh ¼ t31þ gE3�t þ 11þ t3
ðE3 � tÞ � ðE3 � tÞ; ð12Þ
where ea denotes the skew-symmetric tensor whose axial vector isa. This parameterization of the rotation tensor was originally
2 The hat ^ is employed to designate a variation between the initial and deformedconfigurations.
employed by Boyer and Primault [26]. Fig. 2 shows the rotationsassociated with two distinct tangent vector positions, A and B (Uis assumed equal for both cases).
For further developments, all independent kinematic parame-ters are grouped in vector /, reading
½/�t ¼ ½r1 r2 r3 U pð1Þ � � �pðDÞ�; ð13Þ
of dimension 4 + D, where D is the number of deformation modesand the first M weight functions are ‘‘dummy’’.
2.2. Strains
Invoking the small bending strains assumption (A3), the Green–Lagrange strain tensor E may be subdivided into membrane andbending parts, reading [21]
E ¼ EM þ EB; ð14Þ
EM ¼ 12ðFMÞtFM � E2 � E2 � E3 � E3
� �; ð15Þ
EB ¼ X1sym ðFMÞtrn� �
; ð16Þ
Fig. 2. Rotations U and h.
12 R. Gonçalves / Computers and Structures 106–107 (2012) 9–19
where FM = e2 � E2 + e3 � E3 is the membrane deformation gradient.The membrane strains may be expressed in terms of the back-rota-tions of e2 and e3, g2 and g3, as
EM22 ¼
12ðg2 � g2 � 1Þ;
EM23 ¼
12
g3 � g2;
EM33 ¼
12ðg3 � g3 � 1Þ;
ð17Þ
with
g2 ¼ ðKRÞte2 ¼
PpðiÞ�vðiÞ1;2
1jK3 �x;2 þ
Pp0ðiÞ�vðiÞ3;2
264375;
g3 ¼ ðKRÞte3 ¼ KRK ��lR þ�l0RA þ
�u0X2 þP
p0ðiÞ�vðiÞ1
u0P
pðiÞ�vðiÞ1
1þ �þ j0K3 �xþP
p00ðiÞ�vðiÞ3
26643775;ð18Þ
where aR = Rta is the back-rotation, by Rt, of vector a, � = kr0k � 1 isthe extension of the beam axis and the material beam/wall curva-tures read
KKR ¼ axiððKRÞtðKRÞ0Þ ¼ KRK þ KR;
KK ¼ axiðKtK0Þ ¼ jK1E1 þ jK2E2 þ jK3E3;
KR ¼ axiðRtR0Þ ¼ u0E3:
ð19Þ
According to assumptions A4 and A5, EM23 and EM
22 must vanish.From (18), discarding multiplications of strain measures, oneimmediately obtains EM
22 ¼ 0 and
2EM23 ¼
Xi
p0ðiÞ�vðiÞ3;2 þ ð �x;2 þ �dÞjK3 þX
k
sinððunÞ0 � ðukÞ0ÞBku0k;
ð20Þ
where �d ¼ ðR0E1Þ ��l0 measures the distance between the wall mid-line and C. In order to satisfy assumption A4, �x;2 ¼ ��d, which cor-responds to the classic solution, and each ‘‘dummy’’ mode must sat-isfy, besides pðiÞ ¼ ui,
�vðiÞ3;2 ¼ � sinððunÞ0 � ðuiÞ0ÞBi: ð21Þ
This constraint is equivalent to the one employed in GBT-typeformulations. Note also that, since the independent deformationmodes must satisfy Eq. (5), then �vðiÞ3 ¼ 0, otherwise Vlasov’sassumption is violated.
Following the procedure described in [21,24], the bendingstrains are given by
EB22 ¼ �X1
Xi
pðiÞ�vðiÞ1;22;
EB33 ¼ X1 �jR
K2 þ ð�dA þ X2Þj0K3 þ X2u00 þX
k
�dku00k �X
i
p00ðiÞ�vðiÞ1
!;
2EB23 ¼ 2X1 jK3 þ u0 �
Xi
p0ðiÞ�vðiÞ1;2
!;
ð22Þ
where, as in Eq. (3), the summation on k only includes the wall rota-tions that change �lA of the wall under consideration and
�dk ¼ Bk cosððukÞ0 � ðunÞ0Þ;�dA ¼ �LA1 � R0E2 þ
Xk
�dk;ð23Þ
where �dk is the projection of wall k on the direction of the currentwall and �dA is the projection of ð�lAÞ0 on the wall direction.
2.3. Bifurcation equation
The virtual work equation is written in terms of the work-con-jugate Green–Lagrange strains E and second Piola–Kirchhoff stres-ses S. Due to assumptions (A4) and (A5), the only membrane stressterm to be included in the virtual work equation is SM
33. As for thebending terms, a plane stress state is assumed. Accordingly, thestresses are obtained from
SM33 ¼ EEM
33; SBv ¼ C EB
v ; ð24Þ
with
SBv
h i¼
SB22
SB33
SB23
264375; EB
v
h i¼
EB22
EB33
2EB23
264375; ð25Þ
½C� ¼
E1�m2
mE1�m2 0
mE1�m2
E1�m2 0
0 0 G
264375; ð26Þ
where E and G are Young’s and shear moduli, m is Poisson’s coeffi-cient and vector forms of the membrane stress and strain tensors,denoted by a subscript ‘‘v’’, have been employed.
The virtual work equation reads
dWint þ dWext ¼ 0() �Z
VSv � dEvdV þ
ZA
d�u � QdA ¼ 0; ð27Þ
where V is the beam initial volume, Q are surface forces acting onthe beam initial mid-surface A and �u is the mid-surface displace-ment vector.
In a linear stability analysis, the bifurcation equation corre-sponds to the linearization of the former equation at the initialconfiguration (/ = /0), under neutral equilibrium (DQ = 0). Retain-ing only the membrane stresses along the pre-buckling path, one isled to
Dd0Wint ¼�Z
VSM
33 Dd0EM33dV
�Z
Vd0EB
v � C D0EBv þ Ed0EM
33 D0EM33
� �dV ;
Dd0Wext ¼Z
ADd0 �u � QdA; ð28Þ
where the subscript 0 was employed in the virtual and incrementalvariation symbols to emphasize that the variations are calculated atthe initial configuration. The pre-buckling stresses SM
33 are calculatedfrom a linear analysis, which amounts to discarding the non-linearterms in the previous equation. The variations of the strains and �uare obtained from (17), (22) and (2), respectively. These are ex-pressed in terms of the independent kinematic parameters (in /)using the auxiliary matrices derived in Appendix A. Therefore, forimplementation purposes, the bifurcation equation is given by Eq.(51) in Appendix A.
2.4. Finite element implementation
The finite element implementation of the proposed formulationinvolves the direct approximation of the independent kinematicparameters through
D/ ¼ WDd; ð29Þ
R. Gonçalves / Computers and Structures 106–107 (2012) 9–19 13
where matrix W contains the approximation functions and Dd arethe associated amplitudes. The bifurcation Eq. (51) in Appendix Abecomes
ðKþ GðkÞÞDd ¼ 0; ð30Þ
where K is the linear stiffness matrix, obtained by substituting (29)in the material part of Eq. (51), and G is the geometric matrix, whichdepends on the scalar load multiplyer k and stems from the geomet-ric part of Eq. (51). The SM
33 stresses along the fundamental path areobtained by solving Kd = F(k), where F is the external load vector,and the bifurcation loads/modes are the eigenvalues/eigenvectorsof (30).
In most examples presented in Section 4, a 2-node Hermiteanbeam finite element is employed, having 4 � (4 + D) d.o.f., whereD is the number of cross-section deformation modes included inthe analysis. However, in example 4.4, single half-wave sinusoidalamplitude functions constitute exact solutions and are thereforeused. In this case, the number of d.o.f. equals 4 + D.
In the next section, the particular case of compact doubly sym-metric cross-sections is analyzed and it is shown that the proposedformulation leads to the classic solutions. In particular, simply sup-ported (i) columns under uniform compression and (ii) beams un-der uniform moment are analyzed, where once again single half-wave sinusoidal amplitude functions constitute exact solutions.
3. Particular case: compact doubly symmetric cross-sections
For compact (non-slender) doubly symmetric cross-sections,one sets p(i) = 0, making r and U the only independent kinematicparameters. In order to retrieve the classic expressions, C mustcoincide with the cross-section centroid, which also coincides withthe shear centre. Principal axes must be employed and X2 is takenfor the major axis. No wall transverse bending occurs SB
22 ¼ 0� �
and thus, in order to retrieve the classic beam theory, the constitu-tive matrix for the bending terms (26) is substituted by
½C� ¼0 0 00 E 00 0 G
264375: ð31Þ
Integrating the material part of the bifurcation equation (Eq.(51) of Appendix A) along the wall mid-line S, one obtains
�Z
L
dr03dU0
dr001dr002dU00
26666664
37777775
t EA 0 0 0 00 GJ 0 0 00 0 EI2 0 00 0 0 EI1 00 0 0 0 EIx
26666664
37777775Dr03DU0
Dr001Dr002DU00
26666664
37777775dX3; ð32Þ
where L is the beam length and the (standard) cross-section geo-metric parameters are given by
A ¼ tS;
J ¼ t3
3 S;
I2 ¼Z
S
t3
12cos2 u0 þ t �l2
1
� �0
� �dX2;
I1 ¼Z
S
t3
12sin2 u0 þ t �l2
2
� �0
� �dX2;
Ix ¼Z
S
t3
12�dA þ X2� �2 þ t �x2
� �dX2:
ð33Þ
As in the classic theory, the former matrix is diagonal, uncou-pling axial force, bending moment and torsion. In the present for-mulation, this was achieved by making C coincident with the cross-section centroid and shear centre (and using principal axes). Forassymetric cross-sections, the shear centre and centroid do not
generally coincide and therefore coupling will occur. However, ifC coincides with the centroid (shear centre), axial force and bend-ing moment (torsion) become uncoupled.
3.1. Columns
For columns (uniformly compressed members), the pre-buck-ling stresses are SM
33 ¼ �P=A, where P is the axial force, and the geo-metric part of Eq. (51) reads
ZL
P
dr01dr02dr03dU0
dr001dr002dU00
26666666664
37777777775
t 1 0 0 0 0 0 00 1 0 0 0 0 00 0 1 0 0 0 00 0 0 ðrCÞ
2 0 0 0
0 0 0 0 ðr�2Þ2 0 0
0 0 0 0 0 ðr�1Þ2 0
0 0 0 0 0 0 ðr �xÞ2
266666666664
377777777775
Dr01Dr02Dr03DU0
Dr001Dr002DU00
26666666664
37777777775dX3;
ð34Þwith the cross-section geometric parameters
ðr�1Þ2 ¼
ZS
tA
�l22
� �0dX2; ðr�2Þ
2 ¼Z
S
tA
�l21
� �0dX2;
ðrCÞ2 ¼ ðr�1Þ
2 þ ðr�2Þ2; ðr �xÞ2 ¼
ZS
tA
�x2dX2:
ð35Þ
Note that all modes are uncoupled. For simply supported members,sinusoidal amplitude functions sin pX3
L constitute exact solutions and(32)–(35) lead to
Pcr ¼minp2EI2
L2
1þ p2ðr�2Þ2
L2
;
p2EI1L2
1þ p2ðr�1Þ2
L2
;
p2EIxL2 þ GJ
ðrCÞ2 þ p2ðr �xÞ2
L2
8<:9=;: ð36Þ
The contribution of the terms with r�1; r�2 and r �x is small and may bediscarded, which recovers the classic expressions. These additionalterms may be also obtained by classic theories if the non-linearstrains are calculated including the term associated with the longi-tudinal displacements of the walls (see, e.g. [27]). It should be alsomentioned that a fourth (useless) solution is retrieved, Pcr = EA.
3.2. Beams
The classic solution for the lateral-torsional buckling of beamsmay be obtained by assuming that the buckling mode involvesonly r2 and U. In this case, SM
33 ¼ �M2ð�l1Þ0
I2, where M2 is the pre-buck-
ling moment along the beam, and the geometric part of Eq. (51)becomes
�Z
L
dUdr002dU00
24 35t 0 M2 0M2 0 M2g0 M2g 0
24 35 DUDr002DU00
24 35dX3; ð37Þ
with the geometric parameter
g ¼Z
S
tI2ð�l1
�l2Þ0 �xdX2: ð38Þ
For eccentric lateral loads, the external work term yields
�Z
L
ZS
Qð�l1Þ0dUDUdX2 dX3; ð39Þ
which coincides with the classic equations [4].For simply supported beams under uniform moment, sinusoidal
amplitude functions may be once more employed and the solution is
Mcr ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2EI1
L2 GJ þ p2EIxL2
� �s1� p2g
L2
�; ð40Þ
which is equivalent to the classic solution, since p2gL2 � 0.
14 R. Gonçalves / Computers and Structures 106–107 (2012) 9–19
4. Numerical examples
All the examples presented in this section concern beams withE = 210 GPa and m = 0.3, typical values for steel. The results ob-tained with the proposed beam finite element are compared withthe ones provided by linear stability anayses carried out using (i)standard beam elements using LTBeam [8], (ii) large displace-ment/small-strain MITC 9-node shell finite elements using ADINA[28] and (iii) finite strips using CUFSM [29].
4.1. Singly symmetric I-section cantilever
The first example aims at showing the accuracy of the proposedfinite element when load height effects must be accounted for inthe calculation of critical buckling moments (Mcr) of compact sin-gly symmetric I-section cantilever beams. The cross-section con-sists of a 40 � 5 top flange, a 150 � 5 web and a 80 � 5 bottomflange (mid-line width � thickness, in mm). The cantilever has alength of 1.5 m, the fixed end eliminates all displacements (includ-ing warping) and rotations, and is acted by a point load at the freeend.
The graph of Fig. 3(a) plots Mcr as a function of the number ofbeam finite elements employed, for three load locations: top flangecentroid, shear centre and bottom flange centroid. These resultswere obtained by choosing the cross-section shear centre as C,but the top and bottom flange centroids were also tested and iden-tical results were retrieved. For comparison purposes, the graphalso shows the Mcr values obtained with the shell models shownin Fig. 3(b), which involve approximately 22000 d.o.f., and LTBeam,using 150 beam elements. Fig. 3(b) shows the buckling modeshapes obtained with the proposed element (with 6 elements)and the shell models. It can be concluded that at least four beamelements are necessary to achieve satisfactory results for the loadacting at the shear centre and bottom flange, whereas only onebeam element already leads to excellent results for the load actingat the top flange. Moreover, employing six finite elements leads toMcr values that are virtually coincident with the ones provided byLTBeam and only up to 1.5% above those of the shell models.
4.2. C-section beams
The second example is extracted from [24] and concerns thebuckling behaviour of the slender C-section cantilever depictedin Fig. 4(a). The fixed end prevents displacements but not the lon-gitudinal bending of the walls. This particular example makes it
(a)Fig. 3. Singly symmetric I-section cantilever (a) crit
possible to assess the efficiency of the proposed element whenthe load height effect influences local-plate/distortional bucklingand no global buckling occurs. Two loading cases are considered(see Fig. 4(d)): (i) forces acting at the flange/web corners and (ii)forces acting at the flange tips.
Wall 2 is taken for reference and only symmetric cross-sectiondeformation modes are included (see Fig. 4(b)): (1) symmetricrotations of the walls 1 and 3 (u1 ¼ �u3 ¼ 1), (2) symmetric rota-tions of the web/flange corners and (3) a local-plate mode involv-ing wall 2 in the form �vð3Þ1 ¼ 1� cosð2pX2=0:2Þ (m). The cross-section centre C is made coincident with the centroid of wall 2.
The shell models take into account the problem symmetry andinvolve 8200 d.o.f., approximately.
The critical bifurcation loads (total load) and associated buck-ling modes are shown in Fig. 4(c) and (d). The beam bucklingmodes correspond to five beam elements. It is observed thatthe mode shape changes with the location of the loads as follows:(i) local-plate buckling near the support for forces at the flange/web corners and (ii) distortional buckling for forces at the flangetips. Concerning the graph in Fig. 4(c), one concludes that theinclusion of only three deformation modes leads to excellentresults with a reduced number of beam elements. Although fiveelements are necessary to capture the local-plate mode accu-rately, a single element already leads to excellent results for thedistortional mode. It should be mentioned that these resultscoincide with the ones obtained in [24] when five elements areemployed, but those results were obtained with a beam elementwhich includes membrane shear deformation — with the presentbeam element, this deformation and the associated d.o.f. are apriori eliminated.
4.3. Tee-section beams
In this example, the local/global buckling coupling caused bythe load height effect is examined. A cantilever with the thin-walled tee cross-section shown in Fig. 5(a) is selected. For this par-ticular cross-section, torsion induces secondary warping only andtherefore �x ¼ 0. As in the previous example, the fixed end doesnot prevent the longitudinal bending of the walls and two loadingcases are examined (see Fig. 5(d)): (i) force acting at the wall inter-sections and (ii) force acting at the web tip.
For the beam model, walls 1 and 2 are taken for reference, sinceit is not expected that the relative rotation of these walls will influ-ence the beam behaviour significantly, and C is made coincidentwith the point where all walls intersect. Five deformation modesare included in the analysis (Fig. 5(b)): (1) rotation of wall 3
(b)ical moments and (b) associated mode shapes.
(a)(c)
(b) (d)Fig. 4. C-section cantilever (a) geometry, (b) deformation mode shapes, (c) critical loads and (d) associated mode shapes.
R. Gonçalves / Computers and Structures 106–107 (2012) 9–19 15
(u3 ¼ 1), (2) rotation at C, (3) rotation of the web tip and no rota-tion at C and (4–5) two local modes in the web given by�vð4Þ1 ¼ 1� cosð2pX2=0:1Þ and �vð5Þ1 ¼ X2
2ðX2 � 0:1Þ2ðX2 � 0:05Þ (m).The shell models are depicted in Fig. 5(d) and involve 4600
d.o.f., approximately.Fig. 5(c) shows the critical loads as a function of the number of
beam finite elements and modes included in the analysis. Fig. 5(d)depicts the associated critical buckling modes (the beam resultswere obtained with five modes and elements). Two distinct buck-ling modes are detected, depending on the location of the loads: (i)local buckling near the support for the force acting at C and (ii) glo-bal torsional/distortional buckling for the force at the web tip. Thegraph makes it possible to conclude that all five deformationmodes and at least five beam elements are necessary to capturethe local mode accurately, but a single element and single defor-mation mode (mode 1) already leads to excellent results for theglobal mode (if all five modes are included the results are un-changed and are not plotted in the graph). The results for the globalmode show that the relative rotation mode 1 participates in thebeam buckling behaviour and thus that the coupled load height ef-fect must be taken into account. With the present beam element,this coupling is taken into account with Eq. (50) of Appendix A.
3 The rotation of walls 1 and 5 is not restrained and is calculated using Vlasov’smethod [2,21,25].
4.4. Hat-section beams
The final example concerns beams with the narrow hat-shapedcross-section shown in Fig. 6(a) and aims at providing evidence ofthe capabilities of the proposed beam element when local/distor-tional/global buckling phenomena interact and when the cross-section deformation modes include warping of the mid-line.
The upper flange (wall 3) is taken for reference and C is posi-tioned at the middle of this wall. In this case torsion induces warp-ing, which is calculated according to the classical Vlasov theory,using C as the pole — the configuration of this mode is shown in
Fig. 6(b). Only two wall relative rotation modes are considered(see Fig. 6(b)): symmetric (1) and anti-symmetric (2) rotations ofthe webs/lips (u2 ¼ �u4 ¼ 1 and u2 ¼ u4 ¼ 13). Note that, sincethe lips undergo lateral movements, the associated ‘‘dummy’’ modesinclude warping (in the lips), which is calculated according to Eq.(21).
Several local-plate modes are also included, which are obtainedby subdividing the webs into four walls each and the upper flangeinto two walls and imposing unit displacements at each new nodeand at each lip end, along the local X1 axes, totalling 9 modes. Amode orthogonalization is subsequently employed to hierarchizethese modes [25], which makes it possible to obtain modes 3–11in Fig. 6(b).
First, one analyzes simply supported beams under uniform hog-ging moment and a thin-walled cross-section is selected, withthickness t = 0.5 mm. As discussed in Section 3.2, single half-wavesinusoidal amplitude functions constitute exact solutions for thecritical bifurcation load and are therefore employed. A single finiteelement is obviously sufficient and thus the beam model has anumber of d.o.f. which equals 4 + D, where D is the number ofdeformation modes included in the analysis. For comparison, a fi-nite strip analysis is performed using CUFSM [29], which is espe-cially suited for simply supported elements subjected to uniformloading. The cross-section is subdivided according to the discreti-zation employed for the beam model (13 nodes), which leads to4 � 13 = 52 d.o.f.
Fig. 7(b) shows the critical loads as a function of the bucklingmode half-wavelength and the number of deformation modes in-cluded in the analysis. It can be concluded that excellent resultsare obtained with the beam model if at least the first fivemodes are included, although less modes are necessary as the
(a) (c)
(b) (d)
Fig. 5. Tee-section cantilever (a) geometry, (b) deformation mode shapes, (c) critical loads and (d) associated mode shapes.
(a) (b)Fig. 6. Hat-section (a) geometry and (b) deformation mode shapes.
16 R. Gonçalves / Computers and Structures 106–107 (2012) 9–19
half-wavelength increases. This is also attested by the shapesof the critical buckling modes shown in Fig. 7(a) (the beamresults were obtained using all 11 deformation modes): local-plate buckling occurs for the smaller wavelengths, which can onlybe captured by including several deformation modes, followed by
lateral/torsional/distortional buckling with decreasing participa-tion of the distortional mode as the wavelength increases.
Finally, a 600 mm length cantilever with a semi-compact cross-section (t = 2 mm), subjected to tip point loads, is analyzed. Twoloading cases are considered: forces located (i) at the top flange/
(a) (b)Fig. 7. Simply supported hat-section beam (a) mode shapes and (b) critical loads.
(a) (b)Fig. 8. Hat-section cantilever beam (a) mode shapes and (b) critical loads.
R. Gonçalves / Computers and Structures 106–107 (2012) 9–19 17
web intersections and (ii) at the bottom flanges/web intersections.In contrast to the previous cantilever examples, the fixed end pre-vents all displacements and rotations.
Fig. 8(a) shows the buckling modes obtained with the shell andbeam models (the beam results were obtained with 8 elements andall 11 modes). Fig. 8(b) makes it possible to assess the variation ofthe (total) critical load with the number of beam finite elementsand deformation modes. The buckling mode varies according tothe location of the loads as follows: (i) lateral/torsional bucklingfor forces acting at the top flange and (ii) lateral/torsional/distor-tional buckling for forces at the bottom flanges. Only the firsttwo modes are necessary to achieve satisfactory results (surpris-ingly, even for the lateral/torsional/distortional buckling case)and the inclusion of all 11 modes only improves these results mar-ginally.4 Although two beam elements already lead to good results,the differences with respect to the shell models only fall below 2% if
4 The Fcr values obtained with 8 elements/11 modes are actually plotted in thegraph, but virtually coincide with the ones corresponding to 8 elements/2 modes.
one uses (i) six elements for top flange loading and (i) eight elementsfor bottom flange loading.
5. Closing remarks
In this paper, the buckling behavior of Euler–Bernoulli/Vlasovthin-walled beams with deformable cross-section was addressedand a new geometrically exact beam element was developed andimplemented. Noteworthy features of the proposed beam elementare:
(i) The kinematic description of the beam is geometrically exactand employs rotation tensors associated with cross-sectionrotation and also the relative rotations of the walls. Mem-brane shear deformation is a priori eliminated and thereforethe element is not susceptible to shear locking. Moreover,transverse membrane extensions and stresses EM
22; SM22
� �,
hardly relevant in most beam-type problems, are alsodiscarded. Arbitrary deformation modes (complying with
18 R. Gonçalves / Computers and Structures 106–107 (2012) 9–19
Kirchhoff’s assumption) are also included, which makes itpossible to capture local/distortional/global bucklingphenomena.
(ii) The load height effect is taken into account through the vir-tual work of the external loads, which is non-null due to theinclusion of the rotation tensors in the kinematic descrip-tion. If linearized rotations are employed, this virtual workterm vanishes.
(iii) The position of the beam reference axis is arbitrary andtherefore eccentricity may be included in a straightforwardmanner. Since both the beam extension and cross-sectionrotation are calculated with respect to this axis, the classicequations are only retrieved for cross-sections whose shearcentre and centroid coincide.
(iv) The examples show that the proposed beam finite elementleads to accurate solutions with a relatively small numberof deformation modes and elements. In particular, it wasdemonstrated that complex phenomena, such as the loadheight effect resulting from the interaction between wall rel-ative rotations and cross-section rotation, are adequatelyand accurately captured.
Appendix A. Bifurcation equation in terms of the independentkinematic parameters
In order to write the bifurcation equation in terms of the param-eters in /, one first requires the explicit expressions of the varia-tions of � and KK, calculated at the initial configuration.Concerning �, one obtains
d0� ¼ ðdr0 � tÞ0 ¼ dr03;
Dd0� ¼ dr0 � ð1� t � tÞ Dr0
kr0k
� �0¼ dr01Dr01 þ dr02Dr02
ð41Þ
and, for the curvature, the required components are
d0jK1 ¼ �dr002;
d0jK2 ¼ dr001;
d0jK3 ¼ dU0;
d0j0K3 ¼ dU00;
Dd0jK1 ¼ dr001DUþ Dr001dUþ dr002Dr03 þ Dr002dr03 þ dr02Dr003 þ Dr02dr003;
Dd0jK2 ¼ dr002DUþ Dr002dU� dr001Dr03 � Dr001dr03 � dr01Dr003 � Dr01dr003;
Dd0jK3 ¼12�dr002Dr01 � Dr002dr01 þ dr001Dr02 þ Dr001dr02� �
;
Dd0j0K3 ¼12�dr0002 Dr01 � Dr0002 dr01 þ dr0001 Dr02 þ Dr0001 dr02� �
:
ð42Þ
The bifurcation equation may be written in a more compactform if the following shape functions for the ‘‘dummy’’ deforma-tion modes are introduced
�v�ðiÞ1 ¼ �vðiÞ1 � X2 if i ¼ n;
�v�ðiÞ1 ¼ �vðiÞ1 � �di if i – n;
�v�ðiÞ2 ¼ sinððunÞ0 � ðuiÞ0ÞBi;
ð43Þ
where n means the wall under consideration and, according to (21),�v�ðiÞ3;2 ¼ ��v�ðiÞ2 . Note that these new terms have a clear physical mean-ing: they correspond to the linearized displacements caused by thelocal rotations uið¼ pðiÞÞ. To achieve a more compact notation, thestarred versions of the shape functions are also extended to theindependent deformation modes, but in this case one has�v�ðiÞ1 ¼ �vðiÞ1 and �v�ðiÞ2 ¼ �v�ðiÞ3 ¼ 0.
For implementation purposes, the variations of a given vector aare expressed in terms of / through
½da� ¼ ½HDa�
½d/�½d/0�½d/00�½d/000�
266664377775;
Dda � b ¼
½d/�½d/0�½d/00�½d/000�
266664377775
t
½HD2aðbÞ�
½D/�½D/0�½D/00�½D/000�
266664377775;
ð44Þ
where b is a vector and the auxiliary matrices are of the form
½HDa� ¼ Hð1ÞDa
h iHð2ÞDa
h iHð3ÞDa
h iHð4ÞDa
h ih i;
½HD2a� ¼
Hð11ÞD2a
h iHð12Þ
D2a
h iHð13Þ
D2a
h iHð14Þ
D2a
h iHð21Þ
D2a
h iHð22Þ
D2a
h iHð23Þ
D2a
h iHð24Þ
D2a
h iHð31Þ
D2a
h iHð32Þ
D2a
h iHð33Þ
D2a
h iHð34Þ
D2a
h iHð41Þ
D2a
h iHð42Þ
D2a
h iHð43Þ
D2a
h iHð44Þ
D2a
h i
2666666664
3777777775;
ð45Þ
whose submatrices HðiÞDa
h iand HðijÞ
D2a
h ihave dimension 3 � (4 + D)
and (4 + D) � (4 + D), respectively, and HD2a
� is symmetric, i.e.,
HðijÞD2a
h it¼ HðjiÞ
D2a
h i.
With the auxiliary matrices, one may write
d0EBv
h i¼X1 HDEB
� ½d/�½d/0�½d/00�½d/000�
2666437775;
d0EM33¼d0g3 �E3¼½E3�t ½HDg3
�
½d/�½d/0 �½d/00 �½d/000 �
2666437775;
Dd0EM33¼Dd0g3 �E3þd0g3 �D0g3¼
½d/�½d/0�½d/00�½d/000�
2666437775
t
½HD2g3ðE3Þ�þ½HDg3
�t ½HDg3�
� � ½D/�½D/0 �½D/00 �½D/000 �
2666437775;
Dd0 �u �Q ¼
½d/�½d/0�½d/00�½d/000�
2666437775
t
½HD2 �uðQÞ�
½D/�½D/0 �½D/00 �½D/000 �
2666437775;
ð46Þ
where the non-null submatrices are given by
Hð1ÞDEB
h i¼
0 0 0 0 ��v�ð1Þ1;22 � � �0 0 0 0 0 � � �0 0 0 0 0 � � �
264375;
Hð2ÞDEB
h i¼
0 0 0 0 0 � � �0 0 0 0 0 � � �0 0 0 2 �2�v�ð1Þ1;2 � � �
264375;
Hð3ÞDEB
h i¼
0 0 0 0 0 � � �� cosu0 � sinu0 0 �dA þ X2 ��v�ð1Þ1 � � �
0 0 0 0 0 � � �
264375;
ð47Þ
R. Gonçalves / Computers and Structures 106–107 (2012) 9–19 19
Hð2ÞDg3
h i¼
0 0 0 ð�l1 sinu��l2 cos uÞ0 �v�ð1Þ1 � � �0 0 0 ð�l1 cos uþ�l2 sin uÞ0 �v�ð1Þ2 � � �0 0 1 0 0 � � �
264375;
Hð3ÞDg3
h i¼
0 0 0 0 0 � � �0 0 0 0 0 � � �
�ð�l1Þ0 �ð�l2Þ0 0 �x �v�ð1Þ3 � � �
264375;
ð48Þ
Hð13ÞD2g3
h i¼
0 0 0 � � �0 0 0 � � �0 0 0 � � �ð�l2Þ0 �ð�l1Þ0 0 � � �
� cos u0 �v�ð1Þ1
þ sin u0 �v�ð1Þ2
!� sin u0 �v�ð1Þ1
� cos u0 �v�ð1Þ2
!0 � � �
..
. ... ..
. . ..
266666666666664
377777777777775;
Hð22ÞD2g3
h i¼
1 0 0 � � �0 1 0 � � �0 0 0 � � �... ..
. ... . .
.
266664377775;
Hð23ÞD2g3
h i¼
0 0 ð�l1Þ0 0 � � �0 0 ð�l2Þ0 0 � � �ð�l1Þ0 ð�l2Þ0 0 0 � � �
0 0 0 0 � � �... ..
. ... ..
. . ..
266666664
377777775;
Hð24ÞD2g3
h i¼ �x
2
0 �1 0 � � �1 0 0 � � �0 0 0 � � �... ..
. ... . .
.
266664377775:
ð49Þ
The components of ½HD2 �uðQÞ� are not given, since the associatedterm may be written in a much more compact format using directnotation, i.e.,
Dd0 �u � Q ¼ Q��Dr01dr01 � dUDU� �
ð�l1Þ0
þX
k
sinðukÞ0BkðDUduk þ dUDuk þ DukdukÞ!; ð50Þ
where it is assumed that the applied forces only have componentalong X1 (Q = QE1) and are applied at wall mid-line origins (pointsA), in a configuration at each cross-section that is symmetric withrespect to X1 and anti-symmetric with respect to the warpingfunctions.5
Integrating along the wall thickness t, the bifurcation equationmay be finally written in terms of the independent kinematicparameters,
�Z
L
½d/�½d/0�½d/00�½d/000�
264375
t ZS
SM33t ½HD2g3
ðE3Þ� þ ½HDg3�t ½HDg3
�� ��
þ t3
12½HDEB �t ½C�½HDEB � þ Et½HDg3
�t½E3�½E3�t ½HDg3�
� ½HD2 �uðQÞ��dX2
½D/�½D/0�½D/00�½D/000�
264375dX3 ¼ 0; ð51Þ
5 If a single force is applied at A, then �xðAÞ ¼ �vðiÞ3 ðAÞ ¼ 0. If two (equal) forces areapplied at A1 and A2, then �xðA1Þ þ �xðA2Þ ¼ 0 and �vðiÞ3 ðA1Þ þ �vðiÞ3 ðA2Þ ¼ 0.
where L and S are the beam length and cross-section mid-line,respectively. In this equation, the first and last terms are associatedwith the pre-buckling stresses SM
33 and loads Q and are usually des-ignated as ‘‘geometric’’ or ‘‘initial stress’’ terms, whereas theremaining ones are designated as ‘‘material’’ or ‘‘constitutive’’terms.
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