Southern Cross University
ePublications@SCU
23rd Australasian Conference on the Mechanics of Structures and Materials
2014
A dynamic approach to the generalised beam
theory including conventional, extension and shear
modes
G Ranzi
University of Sydney
G Piccardo
Universita' di Genova
D Dias-da-Costa
University of Sydney
A Luongo
Universita' dell'Aquila
Publication details
Ranzi, G, Piccardo, G, Dias-da-Costa, D, Luongo, A 2014, 'A dynamic approach to the generalised beam theory including
conventional, extension and shear modes', in ST Smith (ed.), 23rd Australasian Conference on the Mechanics of Structures and Materials
(ACMSM23), vol. II, Byron Bay, NSW, 9-12 December, Southern Cross University, Lismore, NSW, pp. 789-794. ISBN:
9780994152008.
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23rd Australasian Conference on the Mechanics of Structures and Materials (ACMSM23)
Byron Bay, Australia, 9-12 December 2014, S.T. Smith (Ed.)
A DYNAMIC APPROACH TO THE GENERALISED BEAM THEORY
INCLUDING CONVENTIONAL, EXTENSION AND SHEAR MODES
G. Ranzi*
School of Civil Engineering, The University of Sydney
Sydney, NSW, 2006, Australia. [email protected] (Corresponding Author)
G. Piccardo
DICCA, Scuola Politecnica, Universita‟ di Genova
Genova, 16145, Italy. [email protected]
D. Dias-da-Costa
School of Civil Engineering, The University of Sydney
Sydney, NSW, 2006, Australia. [email protected]
A. Luongo
Dipartimento d‟Ingegneria Civil, Edile-Architettura ed Ambientale, Universita‟ dell‟Aquila
L‟Aquila, 67100, Italy.
International Research Center M&MoCS, Cisterna di Latina, Italy. [email protected]
ABSTRACT
This paper presents a new approach for the evaluation of the deformation modes within the framework
of the Generalised Beam Theory (GBT) and it is applicable to open, closed or partially-closed thinwalled cross-sections. The proposed procedure requires two dynamic analyses to be carried out on an
unrestrained frame defined by the geometry of the thin-walled section. The calculated in-plane and
out-of-plane dynamic modes are then taken as the basis of the deformation modes to be used in the
GBT member analysis. A numerical example is presented to outline the ease of use of the proposed
method of analysis considering a partially-closed thin-walled cross-section subjected to a longitudinal
distributed load and arranged in a simply-supported configuration.
KEYWORDS
Generalised beam theory, steel members, thin-walled members.
INTRODUCTION
The generalised beam theory (GBT) is a method of analysis widely used for thin-walled members.
With this approach, thin-walled members are considered as an assembly of thin plates, free to bend in
the plane orthogonal to the member axis. The GBT differs from the classical Vlasov theory (Vlasov,
1961) because it is able to account for the deformability of the cross-section. In the spirit of the
Kantorovich‟s semi-variational method, GBT transforms a three-dimensional continuous problem into
a one-dimensional problem. This is achieved by representing the displacement field of the TWMs as a
linear combination of assumed deformation modes, defined at the cross-section in terms of in-plane
and warping components, and amplitude functions describing how these modes vary along the
member length. In this context, the use of GBT can be subdivided into two stages: a cross-sectional
analysis where the deformation modes relevant to a particular thin-walled section are evaluated and
selected, and a member analysis which makes use of these modes to determine the overall structural
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789
response defined in terms of the amplitude functions. The fundamental task to be performed for
adequate modelling with the GBT consists of the identification of an appropriate and suitable set of
deformation modes. Among the different approaches available in the literature, the GBT seems to have
attracted the attention of several researchers in recent years, e.g. (Abambres et al., 2014; Davies and
Leach, 1994; De Miranda et al., 2013; Jonsson and Andreassen, 2011; Nedelcu, 2010; Schardt, 1989;
Silvestre and Camotim, 2002; Silvestre et al., 2011), and it is outside the scope of this paper to provide
an exhaustive state of the art on GBT.
In this context, this paper presents a new approach for the GBT cross-sectional analysis in which the
whole set of conventional, extension and shear modes are evaluated based on the dynamic modes
obtained from a dynamic analysis of an unrestrained frame representing the thin-walled cross-section.
Because of its dynamic nature, this procedure is referred to as GBT-D throughout the paper. After a
brief overview of the proposed formulation, its ease of use is outlined considering the analysis of a
simply-supported beam with a partially-closed section.
GBT-D FORMULATION
A thin-walled member is considered as a prismatic shell, made of flat plates connected along edges,
and possibly constrained at the end cross-sections. The displacement field of the middle surface S of
the shell is expressed as:
u(s, z)  u(s, z)es (s)  v(s, z)e y ( s)  w( s, z)e z ( s)
(1)
where s is the curvilinear abscissa along the middle-line of the transverse profile C ; z is the abscissa
along the beam axis; es(s), ey(s), ez(s) are the tangential, normal and binormal unit-vectors, respectively;
u(s,z), v(s,z), w(s,z) the relevant scalar components of displacement. The displacement field at a point
P = (s,y,z) external to S is expressed by the Kirchhoff hypothesis as:
 d s ( s, y, z )   u  yv,s 


d( s, y, z )   d y ( s, y, z )    v 
 d ( s, y, z )   w  yv, z 
 z

(2)
where a comma denotes differentiation with respect the following variable. The corresponding
infinitesimal strains, εs = ds,s, εz = dz,z, γzs = dz,s + ds,z, are formed by a membrane component (y = 0) and
a flexural one (y ≠ 0), denoted by superscripts m and f, respectively, are defined as follows:
ms  u,s , mz  w, z , sf   yv,ss ,  zf   yv, zz ,  mzs  u, z  w,s ,  zsf  2 yv,sz
(3)
According to the semi-variational approach, the displacement components are expressed as a linear
combination of several known deformation modes Uk(s), Vk(s), Ωk(s) and Wj(s), defined on C, and
unknown amplitude functions υk(z), ψj(z), defined along the beam axis, as:
K
u ( s , z )   U k ( s ) k ( z )  U T φ
k 1
K
v( s, z )   Vk ( s ) k ( z )  V T φ
(4)
k 1
K
J
k 1
j 1
w( s, z )    k ( s) k ( z )  W j ( s )  j ( z )  ΩT φ  W T ψ
where a dash denotes differentiation with respect the (sole) independent variable, and U, V, Ω, W and
φ, ψ are column vectors collecting the homonymous scalar functions. In this representation, a physical
dimension of length is assigned to φ and ψ, from which U, V and W are pure numbers and Ω has the
dimension of a length. (Piccardo et al., 2013) The strain field can then be obtained by substituting
Eqs. 4 into Eq. 3:
ACMSM23 2014
790

U T φ
  ms  

T
T
 m 
  W ψ 
Ω
φ



 z
T
T
T
m

    U  Ω  φ  W ψ 

ε   zsf   
T




 yV φ
 s


f
 
 yV T φ

 sf  
T

  zs  


2 yV φ

(5)
Assuming linear-elastic material properties, the membrane and flexural stresses σ are related to the
corresponding strains ε by means of the elastic matrix E:
E
0

0

E  0


0

 0
0
E
0
0
0
0
0
0
0
G
0
0
0
0
0
0
0
0
E
1  2
E
1  2
0
E
1  2
E
1  2
0
0
0 
0

0


0

G 
(6)
where E, G, ν are the longitudinal, tangential and Poisson moduli, respectively.
The GBT deformation modes to be used in the member modeling are determined from a crosssectional analysis. In the proposed GBT-D approach, this is carried by solving two dynamic
eigenvalue problems of an unconstrained planar extensible frame which consist of in-plane oscillations
(in which all points are free to move transversally to the axis) and out-of-plane (or warping)
oscillations (in which case all points move parallel to the z-axis). In particular, the conventional and
extension modes are evaluated from the in-plane dynamic analysis in which the cross-sectional is
simulated as an unrestrained planar frame, while the shear modes are determined from the out-of-plane
(warping) dynamic problem in which the frame representing the cross-section is described by pure
shear beam elements. The detailed description of the derivation of the conventional, extension and
shear based on the GBT-D are described in (Piccardo et al., 2013; Ranzi and Luongo, 2011).
APPLICATION
The ease of use of the proposed GBT-D approach is outlined in the following considering the analysis
of a simply-supported thin-walled member formed by the partially-closed section illustrated in
Figure 1. The applied load consists of an in-plane longitudinal force (per unit area), fs = fy = 0 and
fz = f0 cos(πz/L), f0 = 1 N/mm2, applied over a strip of 20 mm.
The structural response of the member is outlined in Figure 2, while the deformation modes of the
cross-section are illustrated in Figure 3. The latter are shown displaying the U- and V- components by
the final position of the nodes and the variation of the W-components is diagrammed over the crosssection, separating the conventional modes (Figure 3a), the extensional modes (Figure 3b) and shear
modes (Figure 3c). The calculated displacements and stresses are plotted at the abscissa where they
attain the maximum values (i.e. at z = 0 or z = L). Two subsets of modes have been considered in the
analyses: (a) conventional and extensional modes („C+E‟ curves), and (b) the full set of modes („all
modes‟ curves). These results confirm the need to account for the shearing deformation to accurately
predict the structural response in terms of both in-plane and warping displacements (Figures 2a,b) as
well as stresses (Figures 2c-e). This kind of response is particularly important for short members, such
as those considered here, for which the length is comparable to the cross-sectional width.
ACMSM23 2014
791
y
f0 cos ( z / L)
s
z
(a) layout of member and loading arrangement
80
200
80
20
60
Loaded width
of plate
NOTE:
- all dimensions in mm
- member length of 360 mm
(b) cross-sectional geometry
Figure 1. Layout of the box girder and loading arrangement
scaled by x 2000
scaled by x 1000
(b)
(a)
-11.5 MPa
-38.1 MPa
(d)
(c)
GBT C+E
5.3 MPa
GBT all modes
(e)
Figure 2. Results calculated for the box girder: (a) in-plane displacements (z = L/2), (b) warping
displacements (z = 0,L), (c) longitudinal stresses σz (z = L/2), (d) shear stresses τzs (z = 0,L), and (e)
transverse stresses σs (z = L/2)
ACMSM23 2014
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Mode 1
W
U,V
U,V
Mode 3
U,V
Mode 2
W
Mode 4
W
U,V
W
Mode 5
U,V
W
U,V
Mode 6
W
U,V
U,V
Mode 7
W
Mode 9
Mode 8
U,V
W
U,V
W
Mode 10
W
(a)
Mode 1
U,V
Mode 2
U
Mode 3
Mode 4
U,V
U
Mode 5
U,V
U
U,V
U,V
U
Mode 6
U
U,V
U
(b)
Mode 1
Mode 4
Mode 2
Mode 5
Mode 3
Mode 6
(c)
Figure 3. First rigid and deformation modes for the box girder: (a) conventional modes
(diagrams in W), (b) extension modes (diagrams in U alone), and (c) shear modes (diagrams)
CONCLUSIONS
This paper presented a new approach for the cross-sectional analysis within the framework of the
Generalised Beam Theory (GBT). This procedure is applicable to open, closed or partially closed
cross-sections. The proposed procedure determines the GBT deformation modes, including the
conventional, extension and shear modes, as the dynamic modes of an unrestrained frame which
represents the thin-walled cross-section. A numerical example was presented to illustrate the ease of
ACMSM23 2014
793
use of the proposed approach for the case of a partially-closed thin-walled section arranged in a
simply-supported static configuration and subjected to a longitudinal distributed load.
ACKNOWLEDGMENTS
The work in this article was supported by the Australian Research Council through its Discovery
Projects funding scheme (DP1096454) and by an award under the Merit Allocation Scheme on the
NCI National Facility at the ANU. Part of the computational services used in this work was provided
by Intersect Australia Ltd.
REFERENCES
Abambres, M, Camotim, D, Silvestre, N, Rasmussen, KJR (2014) “GBT-based structural analysis of
elastic-plastic thin-walled members”, Computers and Structures, Vol. 136, pp. 1-23.
Davies, JM, Leach, P (1994) “First-order Generalised Beam Theory”, Journal of Constructional Steel
Research, Vol. 31, pp. 187-220.
De Miranda, S, Gutierrez, A, Miletta, R, Ubertini, F (2013) A generalized beam theory with shear
deformation. Thin-Walled Structures, Vol. 67, pp. 88-100.
Jönsson, J, Andreassen, MJ (2011) Distortional eigenmodes and homogeneous solutions for semidiscretized thin-walled beams. Thin-Walled Structures, Vol. 49, No. 6, pp. 691-707.
Nedelcu, M (2010) “GBT formulation to analyse the behaviour of thin-walled members with variable
cross-section”. Thin-Walled Structures, Vol. 48, pp. 629-638.
Piccardo, G, Ranzi, G, Luongo, A (2013) A complete dynamic approach to the GBT cross-section
analysis including extension and shear modes, Mathematics and Mechanics of Solids, published
online 30 July 2013, DOI: 10.1177/1081286513493107.
Ranzi, G, Luongo, A (2011) A new approach for thin-walled member analysis in the framework of
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Schardt, R (1989) “Verallgemeinerte Technicsche Biegetheory”, Berlin: Springler-Verlag.
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