GBT formulation to analyse the buckling behaviour of thin-walled members with arbitrarily ‘branched’ open cross-sections P.B. Dinis, D. Camotim * , N. Silvestre Department of Civil Engineering and Architecture, ICIST/IST, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisboa, Portugal Received 16 March 2005; received in revised form 19 August 2005; accepted 28 September 2005 Available online 14 November 2005 Abstract This paper presents the derivation, validates and illustrates the application of a Generalised Beam Theory (GBT) formulation developed to analyse the buckling behaviour of thin-walled members with arbitrarily ‘branched’ open cross-sections. Following a brief overview of the conventional GBT, one addresses in great detail the modifications that must be incorporated into its cross-section analysis procedure, in order to be able to handle the ‘branching’ points — they concern mostly issues related to (i) the choice of the appropriate ‘elementary warping functions’ and (ii) the determination of the ‘initial flexural shape functions’. The derived formulation is then employed to investigate the local-plate, distortional and global buckling behaviour of (i) simply supported and fixed asymmetric E-section columns and (ii) simply supported I-section beams with unequal stiffened flanges. For validation purposes, several GBT-based results are compared with ‘exact’ values, obtained by means of finite strip or shell finite element analyses. q 2005 Elsevier Ltd. All rights reserved. Keywords: Thin-walled members; Generalised beam theory (GBT); ‘Branched’ open cross-sections; Member buckling analysis; Local-plate buckling; Distortional buckling; Global buckling. 1. Introduction The Generalised Beam Theory (GBT) was originally developed by Schardt [1–3] and may be viewed as an extension of Vlasov’s classical bar theory that incorporates genuine folded-plate concepts and, thus, is able to take into account in- plane (local) cross-section deformations. Moreover, the member deformed configuration or buckling/vibration mode is expressed as a linear combination of a set of pre-determined cross-section deformation modes — due to this rather unique modal nature, the application of GBT is considerably more versatile and computationally efficient than similar finite strip or shell finite element analyses. Indeed, it has been recently shown that GBT provides a rather powerful, elegant and clarifying tool to investigate a wealth of structural problems involving thin-walled prismatic members [4,5]. For the last four decades, Schardt and his collaborators, at the Technical University of Darmstadt, have devoted an enormous amount of work to the development and application of GBT formulations. However, this work was carried out almost exclusively in the context of the first-order, buckling and vibration analysis of thin-walled members (i) made of isotropic elastic materials and (ii) displaying ‘unbranched’ (mostly open) cross-sections. 1 Moreover, it was rather poorly disseminated among the English-speaking scientific and technical communities — the vast majority of the publications are available only in German and several of them consist of TU Darmstadt Reports or Ph.D. Theses. These communities only became acquainted with GBT in the 1990s, thanks to the work of Davies and his co-workers [4,6,7], who (i) played a key role in the dissemination of GBT around the world, (ii) applied it extensively to investigate the buckling behaviour of cold- formed steel members (e.g. [8–10]) and (iii) provided strong evidence that GBT is a valid and often advantageous alternative to fully numerical finite element or finite strip analyses. Quite recently (i.e. in the last 4–5 years), GBT has attracted the attention of several researchers, which led to the development of a number of new formulations and appli- cations. In this regard, Silvestre and Camotim deserve to be Thin-Walled Structures 44 (2006) 20–38 www.elsevier.com/locate/tws 0263-8231/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2005.09.005 * Corresponding author. Tel.: C351 21 8418403; fax: C351 21 8497650. E-mail address: [email protected] (D. Camotim). 1 In an unbranched open or closed thin-walled cross-section, no internal longitudinal edge is shared by more than two walls. Fig. 1(a) and (b) provide examples of unbranched and branched cross-sections, respectively.
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GBT formulation to analyse the buckling behaviour of thin-walled
members with arbitrarily ‘branched’ open cross-sections
P.B. Dinis, D. Camotim *, N. Silvestre
Department of Civil Engineering and Architecture, ICIST/IST, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
Received 16 March 2005; received in revised form 19 August 2005; accepted 28 September 2005
Available online 14 November 2005
Abstract
This paper presents the derivation, validates and illustrates the application of a Generalised Beam Theory (GBT) formulation developed to
analyse the buckling behaviour of thin-walled members with arbitrarily ‘branched’ open cross-sections. Following a brief overview of the
conventional GBT, one addresses in great detail the modifications that must be incorporated into its cross-section analysis procedure, in order to be
able to handle the ‘branching’ points — they concern mostly issues related to (i) the choice of the appropriate ‘elementary warping functions’ and
(ii) the determination of the ‘initial flexural shape functions’. The derived formulation is then employed to investigate the local-plate, distortional
and global buckling behaviour of (i) simply supported and fixed asymmetric E-section columns and (ii) simply supported I-section beams with
unequal stiffened flanges. For validation purposes, several GBT-based results are compared with ‘exact’ values, obtained by means of finite strip
or shell finite element analyses.
q 2005 Elsevier Ltd. All rights reserved.
Keywords: Thin-walled members; Generalised beam theory (GBT); ‘Branched’ open cross-sections; Member buckling analysis; Local-plate buckling; Distortional
buckling; Global buckling.
1. Introduction
The Generalised Beam Theory (GBT) was originally
developed by Schardt [1–3] and may be viewed as an extension
of Vlasov’s classical bar theory that incorporates genuine
folded-plate concepts and, thus, is able to take into account in-
plane (local) cross-section deformations. Moreover, the
member deformed configuration or buckling/vibration mode
is expressed as a linear combination of a set of pre-determined
cross-section deformation modes — due to this rather unique
modal nature, the application of GBT is considerably more
versatile and computationally efficient than similar finite strip
or shell finite element analyses. Indeed, it has been recently
shown that GBT provides a rather powerful, elegant and
clarifying tool to investigate a wealth of structural problems
involving thin-walled prismatic members [4,5].
For the last four decades, Schardt and his collaborators, at
the Technical University of Darmstadt, have devoted an
enormous amount of work to the development and application
0263-8231/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.
Fig. 3. Cross-section (a) discretisation (natural and intermediate nodes), (b) elementary warping (node r) and flexural (node qCp) functions and (c) base system and
redundant mqCp.
9 The distinction between ‘natural’ and ‘intermediate’ nodes, kept here for
‘historical reasons’, is slightly misleading, as the cross-section free end nodes
are both natural and intermediate — see Fig. 3(a). Indeed, it would be more
logical to classify the cross-section nodes according to the nature of the
imposed elementary functions ‘centred’ on them: (i) warping (‘natural’), (ii)
flexural (‘intermediate’) or (iii) warping and flexural (‘natural and
warping values at the dependent natural nodes that must
be specifically determined — the determination of these
warping values is addressed in the next paragraphs.
The determination of the warping values at the dependent
natural nodes is based on the fact that (i) Vlasov’s null
membrane shear strain assumption has to be satisfied in all the
walls emerging from a given branching node and (ii) the
compatibility between the membrane transverse displacements
must be ensured at that same branching node. Before
addressing the determination of these values, it is worth
pointing out that:
(i) Each dependent node is linked to a particular branching
node, which is (i1) either independent (most cases) or
dependent21 and (i2) always linked to two and only two
independent nodes.
(ii) The warping value at a given dependent node is
obtained on the basis of (ii1) the warping value at the
associated branching node, (ii2) the warping values at
the two corresponding independent nodes and (ii3) the
widths and inclinations of the three (branching) walls
involved.
(iii) When a dependent node is linked to a branching node
that is also dependent, one must begin by determining
the warping value at the latter. This is always ensured if
one determines the dependent node warping values
following an ascending branch order.
Consider now the arbitrary branching node r depicted in
Fig. 6(a) — without any loss of generality, it is assumed that it
belongs to the unbranched sub-section. From this branching
node emerge kC2 walls, linking it (i) to the 2 independent
nodes rK1 and rC1 and also (ii) to k dependent nodes r1,.,
rk (each contained in a first-order branch). All these walls are
oriented as indicated and have widths and inclinations (relative
21 It depends on the particular unbranched cross-section adopted — this is
illustrated in Fig. 5(e), where the branching nodes B1 and B4 are independent in
one case and dependent in the other.
to the horizontal direction and measured clockwise) designated
as brK1, br, br1, .., brk and arK1, ar, ar1, .., ark. Any
dependent node warping value urj(jZ1, .,k) is a function
of only (i) the three independent warping values urK1, ur and
urC1, and (ii) the widths and inclinations of the three walls
involved (WrK1, Wr and Wrj.1). In order to obtain this value, one
must perform the following operations:
(i) Using Vlasov’s assumption and adopting the conven-
tional GBT procedure, determine the transverse
membrane displacements in the walls WrK1 and Wr,
through the expressions
vrK1 ZKur KurK1
brK1
vr ZKurC1Kur
br
; (14)
where a positive value indicates a displacement
‘following’ the respective wall orientation.
(ii) On the basis of the inclinations of walls WrK1 and Wr
and the values of vrK1 and vr, determine the final
location of the branching node r-identified by a black
circle in Fig. 6(b).
(iii) Based, on the final location of the branching node rand the inclination of wall Wrj.1, evaluate the value of
the transverse membrane displacement vrj.1 required to
ensure that walls WrK1, Wr and Wrj.1 continue to share
node r — see again Fig. 6(b). This can be done by
means of the general expression (valid for vrs022)
vrj:1 Z vr
�cosðarj:1KarÞKsinðarj:1KarÞ!
! KvrK1
vrsinðarK1KarÞC
1
tanðarK1KarÞ
�:
(15)
(iv) Using the values of ur, vrj.1 and brj, evaluate the sought
urj via the expression
22 If one has vrZ0, one just has to ‘switch’ the roles of the walls WrK1 and Wr,
i.e. to reverse their orientations.
Fig. 7. Determination of initial flexural shape functions at a wall and a wall segment: (a) wall flexural displacements due to an elementary warping function, (b) fixed-
end transverse bending moments and deformed configurations and (c) contributions to the initial flexural shape functions.
P.B. Dinis et al. / Thin-Walled Structures 44 (2006) 20–3830
urj Z ur Kvrj:1brj:1: (16)
After performing the above operations for the k dependent
nodes, it is possible to completely define the elementary
warping functions stemming from the imposition of unit
warping values at nodes rK1, r and rC1 — the first two are
displayed in Fig. 6(c).
23 Recall that the wall membrane transverse displacements can be expressed
in terms of (i) the warping values at nodes rK1, r and rC1 and (ii) the widths
and inclinations of the three walls WrK1, Wr and Wrj.1.
3.1.2. Determination of the initial flexural shape functions
As mentioned earlier, the determination of the initial flexural
shape functions requires the solution of a statically and
through finite strip and GBT-based analyses, which
fully validates the latter.
(ii) The critical buckling curve exhibits three distinct
zones, corresponding to (ii1) 1–4 wave local-plate
buckling, (ii2) 1–3 wave distortional buckling and (ii3)
single-wave global (flexural-torsional) buckling. It only
differs from its single-wave counterpart for (ii1) 9!L!27 cm (2–4 wave local-plate buckling) and (ii2) 65!L!180 cm (2–3 wave distortional buckling).
(iii) Since only 10 deformation modes (2–9, 11 and 13)
participate in column buckling modes, a GBT buckling
analysis including only those modes yields exact
results.
(iv) The single-wave buckling curve exhibits local minima
at Lz6 cm and Lz50 cm, (iv1) the former correspond-
ing to a local-plate buckling mode that combines modes
8, 9 (clearly predominant), 11 and 13, and (iv2) the
latter associated with a distortional buckling mode
combining modes 5, 6 (predominant) and 7.
(v) The final descending branch, common to the single-
wave and critical buckling curves, is associated with
global flexural-torsional buckling modes that always
combines modes 2, 3 and 4 (recall that the cross-section
is asymmetric) — while mode 3 is highly predominant
in the longer columns, modes 2 and 4 are prevalent in
the ‘not so long’ columns.25 Note also that a flexural–
25 Because the cross-section is ‘almost symmetric’, the participation of modes
2 and 4 in the buckling modes of very long columns is negligible — see
Fig. 12(b).
torsional–distortional buckling mode, which includes
small participations of modes 6 and 7, is critical for
200!L!300 cm.
(vi) The modal participation diagram presented in
Fig. 12(b) readily shows that several portions of the
single-wave buckling curve can be very accurately
approximated by means of analyses including only a
few (selected) deformation modes. This statement is
fully backed by the three dashed curves, which (vi1)
involve only modes 8C9C11C13, 5C6C7 or 2C3C4 and (vi2) practically coincide with the ‘exact’
curve for L!12 cm, 50!L!120 cm and LO150 cm.
Next, Fig. 13(a) makes it possible to compare the GBT-
based critical buckling curves (Ncr vs. L) relative to simply
supported and fixed columns. While the former, already
presented in Fig. 12(a), was determined through the application
of Galerkin’s method (sinusoidal shape functions), the latter
was obtained by employing the beam finite element formu-
lation addressed in Section 2.2.1 — the column longitudinal
discretisation involved 4–24 beam elements (depending on the
buckling mode wave number) and up to the 12 cross-section
deformation modes were included in the analyses — the modal
participation diagram presented in Fig. 13(b) shows that only
eight of them are relevant for the fixed columns. As for
Fig. 13(c), it displays the configurations of the relevant modal
amplitude functions concerning the simply supported and fixed
columns with LZ200 cm. In order to validate the GBT-based
results, Fig. 13(a) also includes Ncr values obtained by means
of Abaqus finite element analyses — fine meshes of four-node
Fig. 12. Buckling behaviour of simply supported E-section columns: (a) Nb vs. L curves, (b) modal participation diagram pi vs. L (nwZ1) and (c) typical buckling
mode shapes.
0
20
40
60
80
100Ncr (kN)
L (cm)
(b)
(a) (c)
Fixed
Simply Supported
GBT (all modes, any nw )
FEM (ABAQUS )
GBT (FE formulation and selected modes)
6 1000100
LPM DM
LPM DM FTDM + FTM
0
0.25
0.5
0.75
1.0
10 100 1000
9
6
5
6
5
79
10
3
2
4
67
8
10
50
100
150 200
x (cm)
0.2
0.1
–0.2
–0.3
–0.1
–0.4
0.0
0.4
~
0.3
7
5
6
9
3
2
4
7
60.2
0.1
–0.2
–0.3
–0.1
–0.5
–0.6
–0.4
0.050 100 150
200
x (cm)
k (x)
k (x)
6 L (cm)
~
Fixed
Simply Supported
pi
FTDM + FTM
Fig. 13. Critical buckling behaviour of simply supported and fixed E-section columns: (a) Ncr vs. L curves, (b) fixed column modal participation diagram pi vs. L
(fixed columns) and (c) relevant modal amplitude functions for the columns with LZ200 cm.
P.B. Dinis et al. / Thin-Walled Structures 44 (2006) 20–3834
isoparametric shell (S4) elements were adopted to discretise
the columns. Concerning these results, it is worth pointing out
that:
(i) Once again, the GBT-based results (critical buckling
loads) are virtually ‘exact’ — indeed, they coincide
with the ones, now obtained by means of Abaqus shell
finite element analyses.
(ii) For 6%L!110 cm and 110%L!370 cm, the fixed
columns buckle in local-plate (9 plus a bit of 5C6C8)
and distortional (5C6C7 plus a bit of 9C10) modes
displaying 1–12 and 2–6 waves, respectively. Unlike its
simply supported counterpart, the fixed column Ncr vs.
L curve exhibits no local minima — it decreases
monotonically and, in the local-plate and distortional
buckling ranges, tends to the simply supported critical
load values Pcr.LPZ52.0 kN and Pcr.DZ43.5 kN. In
particular, notice that no fixed column buckles in a
single-wave distortional mode — the warping restraint
considerably increase the distortional stiffness near the
column supports.
(iii) For 370%L!500 cm and 500%L!2000 cm, buckling
takes place in single-wave flexural–torsional–distor-
tional (2C3C4 plus a bit of 6C7) and flexural–
torsional (2C3C4) modes — recall that, due to the
cross-section ‘slight asymmetry’, the participation of
mode 3 is rather minute.
(iv) The comparison between the two Ncr vs. L curves
clearly shows that, besides the expected critical load
increase, fixing the column end sections leads to a
change in the buckling mode nature and/or longitudinal
configuration (wave number and modal decomposition)
. In order to illustrate these changes, one compares the
buckling behaviours of simply supported and fixed
columns with LZ200 cm. While the former buckles at
PcrZ36.3 kN in a single-wave flexural–torsional–
distortional mode, which combines deformation
Fig. 14. In-plane deformed configurations of all but the first I-section deform
modes 2 (25%), 3 (5%), 4 (45%), 6 (19%) and 7
(5%), the latter exhibits a distortional buckling mode,
occurring for PcrZ46.2 kN and involving modes 5
(41%), 6 (33%), 7 (23%) and 9 (3%). Moreover,
Fig. 13(c) underlines the differences between the
configurations of the corresponding amplitude func-
tions: (iv1) only single-wave sinusoidal functions in the
simply supported column and (iv2) periodic functions
with three ( ~f5 and ~f9) or five ( ~f6 and ~f7) unequal
waves in the fixed column.
4.2. I-Section beams
Taking into account the discretisation shown in Fig. 10(b) (8
independent natural nodes and 11 intermediate nodes), one is
led to 19 deformation modes — with the exception of the first
one (axial extension), their in-plane deformed configurations
are displayed in Fig. 14. It is worth noting that the inclusion of
the flange end stiffeners is responsible for the existence of 4
distortional modes.
Fig. 15 shows numerical results concerning the buckling
behaviour of simply supported beams subjected to uniform
positive and negative major axis bending — all the
conventions adopted in Fig. 12 are retained. The two sets of
curves depicted in Fig. 15(a) provide the variation of the beam
positive and negative bifurcation moment Mb with the length
L, for (i) single-wave (nwZ1) and (ii) critical (any nw)
buckling. In the first case, the GBT analyses included, once
more, either (i) all deformation modes or (ii) just a few
selected ones. This figures also includes several Mb values,
obtained through Cufsm2.6 finite strip analyses and used to
validate the GBT-based results. As for Fig. 15(b) and (c), they
present (i) the two single-wave buckling modal participation
diagrams (MbO0 and Mb!0) and (ii) FEM-based local-plate
(MbO0 and Mb!0) and distortional (MbO0) buckling mode
shapes — they correspond to beams with lengths equal to 17,
ation modes: rigid-body (2–4), distortional (5–8) and local-plate (9–19).
Fig. 15. Buckling behaviour of simply supported I-section beams: (a) Mb vs. L curves and (b) modal participation diagram pi vs. L (nwZ1) for MbO0 and Mb!0 and
(c) typical local-plate (MbO0 and Mb!0) and distortional (MbO0) buckling mode shapes.
P.B. Dinis et al. / Thin-Walled Structures 44 (2006) 20–3836
24 and 170 cm.26 The observation of these results leads to the
following conclusions:
(i) Once more, a nearly perfect coincidence exists
between the GBT-based and finite strip results.
(ii) The critical buckling curves concerning positive and
negative moments are qualitatively different. While
the MbO0 curve exhibits three distinct zones,
corresponding to 1–6 wave local-plate buckling, 1–
11 distortional buckling wave and single-wave
flexural–torsional buckling, its Mb!0 counterpart
only exhibits two zones, associated with 1–42 wave
local-plate buckling and single-wave flexural–torsional
buckling27 (distortional buckling is never critical).
Moreover, Fig. 15(c) shows that the local-plate
buckling mode is triggered (ii1) by flange buckling
for MbO0 and (ii2) by web buckling for Mb!0 —
26 Note that distortional buckling is never critical for Mb!0 — see Fig. 15(a).27 This beam instability phenomenon is also commonly designated as ‘lateral-
torsional buckling’.
Fig. 15(b) also provides clear evidence of this fact, as
the contribution of deformation mode 9 is considerably
larger in the latter case.
(iii) For MbO0, the single-wave buckling curve exhibits
local minima at Lz17 and Lz170 cm, (iii1) the
former corresponding to a local-plate buckling mode
that combines three dominant modes (9, 10 and 11 —
fairly equal participations) with small contributions
from modes 14 and 16, and (iii2) the latter associated
with a ‘pure’ distortional buckling mode that combines
modes 5 (clearly predominant) and 6.
(iv) For Mb!0, the single-wave buckling curve exhibits
only one local minimum at Lz24 cm, which
corresponds to a local-plate buckling mode that
combines a dominant contribution of mode 9 with
decreasingly important participations of modes 10, 11,
13, 14 and 18. Note also that the negative critical
moment is equal to McrZ14.1 kNm, about 23% below
its positive counterpart (McrZ18.2 kNm) — recall that