Generalised Beam Theory (GBT) for stiffened sections

www.springer.com/journal/13296

International Journal of Steel Structures

June 2014, Vol 14, No 2, 381-397

DOI 10.1007/s13296-014-2017-x

Generalised Beam Theory (GBT) for Stiffened Sections

G. Taig and Gianluca Ranzi*

School of Civil Engineering, The University of Sydney, Australia

Abstract

This paper presents an extension to the Generalised Beam Theory (GBT) approach to describe the response of prismatic thin-walled members stiffened by means of generic plate arrangements at different cross-sections along their length. Theconventional deformation modes to be included in the GBT formulation are obtained as the dynamic modes of a planar frame,which represents the cross-section. Two numerical procedures are implemented to account for the presence of the stiffeners.One approach identifies different sets of deformation modes for the unstiffened and stiffened sections, which are then combinedfor the member analysis. The second procedure relies on the use of constraint equations at the stiffened locations to be includedin the member analysis. For the cross-sectional analysis, a new mixed finite element is presented which incorporates theinextensibility condition usually adopted in the framework of the classical GBT, therefore simplifying the steps required for theevaluation of the conventional deformation modes. The proposed technique is applicable to open, closed and partially-closedstiffened sections. Two numerical examples are provided to highlight the ease of use of the method of analysis considering openand partially-closed sections, and their results are validated against those obtained with the commercial finite element softwareAbaqus.

Keywords: cross-sectional analysis, generalised beam theory, steel structures, thin-walled members

1. Introduction

Generalised Beam Theory (GBT) is an elegant and

powerful tool used to predict the structural behaviour of

thin-walled members, originally proposed by Schardt

(1989, 1994). GBT considers thin-walled members as an

assembly of thin plates that are free to bend in the plane

of the cross-section. Unlike classical Vlasov theory

(Vlasov, 1961), which only considers rigid body motions

of the beam section, GBT is able to capture the cross-

sectional deformations. The basis of the GBT approach

consists in describing the displacement field of a thin-

walled member as a linear combination of assumed

deformation modes (in-plane and warping displacements)

whose amplitudes vary along the length of the member.

This falls under the umbrella of Kantorovich’s semi-

variation method aimed at reducing the dimensionality of

the problem through an approach of partially assumed

modes. In this manner, the GBT simplifies a three-

dimensional continuous problem into a vector valued

one-dimensional problem. The GBT approach can be

subdivided into two stages: (1) a ‘cross-sectional analysis’

where the deformation modes are evaluated based on the

geometric and material properties of the section; and (2)

a ‘member analysis’ where the unknown amplitude

functions associated with each deformation mode are

determined. The fundamental step in the GBT consists in

the determination of a suitable set of deformation modes

for the cross-section.

Since its original formulation (Schardt, 1989; Schardt

1994), the GBT has been extensively used in the literature

to model the response of thin-walled members, e.g.

(Davies and Leach, 1994; Silvestre and Camotim, 2002;

Camotim et al., 2010), and it is now a practical

alternative to other more complex methods of analysis

such as the finite strip method (FSM) (Lau and Hancock,

1986; Mahendran and Murray 1986; Young, 2004; Adany

and Schafer, 2006; Vrcelj and Bradford, 2008; Eccher et

al., 2009), finite element method (FEM) (Bakkera and

Peközb, 2003; Ren et al., 2006; Chen and Young, 2007),

or the perturbation methods (Luongo and Pignataro,

1988; Luongo, 1991, 1992, 2001). In particular, GBT is

applicable to first-order analyses (Davies and Leach,

1994; Silvestre and Camotim, 2002), buckling and post-

buckling (Camotim et al., 2010; Nedelcu, 2012; Basaglia

et al., 2008), and linear dynamic simulations (Rebiano et

Note.-Discussion open until November 1, 2014. This manuscript forthis paper was submitted for review and possible publication on Jan-uary 20, 2013; approved on April 1, 2014.© KSSC and Springer 2014

*Corresponding authorTel: +61293515215; Fax: +61293513343E-mail: [email protected]

382 G. Taig and Gianluca Ranzi / International Journal of Steel Structures, 14(2), 381-397, 2014

al., 2008), and is capable of dealing with open branched

cross-sections (Dinis et al., 2006), closed and partially-

closed ones (Goncalves et al., 2010) and circular sections

(Silvestre, 2007). The simplification in the description of

the displacement field, enabled within the GBT with the

use of cross-sectional deformation modes and amplitude

functions, was applied in recent years to the analyses

carried out with the finite strip method (Adany and

Schafer, 2006) and the finite element method (Casafront

et al., 2009) to reduce the number of freedoms required

in the modelling. In very recent years, attention has been

devoted to generalise Vlasov’s classic hypotheses of

inextensibility and shear undeformability on which GBT

was initially based, e.g. (Goncalves et al., 2010; Jönsson

and Andreassen, 2011; Silvestre et al., 2011; Andreassen

and Jönsson, 2012; Piccardo et al., 2013a).

In this context, this paper presents an extension to the

classical GBT formulation to consider the structural

response of thin-walled members stiffened at different

locations along their length, commonly used in structural

application as, for example shown in Fig. 1.

The particularity of the proposed approach relies on its

ability to capture the response of partially stiffened cross-

sections, for which the plate stiffeners cover only part of

the cross-sectional footprint.

The GBT formulation adopted in this paper identifies

the conventional deformation modes as the dynamic

modes of a planar frame, which represents the cross-

section, extending previous work of the authors (Ranzi

and Luongo, 2011, 2013; Piccardo et al., 2013b) to

stiffened members.

Two procedures are implemented in this study to

account for the presence of the stiffeners. In the first

approach, a planar frame model, representing the plate

segments of the cross-section placed at their mid-lines, is

used for the unstiffened and each of the stiffened sections

of the member. To prevent the loss of the initial shape of

part of the section due to the presence of the stiffener,

constraints are applied to the planar frames in the stiffened

areas of the sections. A linear dynamic analysis is carried

out on each frame with the resulting eigenmodes chosen

as a family of in-plane deformation modes for that

section. The warping displacements are then calculated in

a post-processing phase by enforcing conditions on the

shear strain. In this context, a mixed finite element is

presented which satisfies the inextensibility condition

Figure 1. Typical stiffened section applications.

Figure 2. Generic thin-walled stiffened cross-sections.

Generalised Beam Theory (GBT) for Stiffened Sections 383

required by the classical GBT formulation and, because

of this, simplifies the cross-sectional analysis aimed at

determining the conventional deformation modes. Details

of this approach are presented in the following when

outlining the GBT cross-sectional analysis. The family of

conventional deformation modes obtained from the cross-

sectional analyses are then combined in the member

analysis to capture the overall structural response accounting

for the additional rigidity provided by the presence of the

stiffeners in the cross-section.

The second approach adopted in this study to simulate

the presence of the stiffeners relies on the use of constraint

equations specified in the member analysis to ensure the

parts of a cross-section in contact with a stiffener do not

deform in the plane of the stiffener. In this case, the GBT

cross-sectional analysis is carried out for the unstiffened

section only following the procedure already available in

the literature, e.g. (Ranzi and Luongo, 2011). Because of

this, the description of this second approach is only

provided in the following when outlining the member

analysis.

The accuracy of the proposed GBT approach is validated

against the results obtained with a finite element model

developed with Abaqus (ABAQUS User’s Manual, 2008).

2. Basis of the GBT Approach

A generic open, closed and partially closed cross-

section is considered, as shown in Fig. 2(a)-(c). In the

approach proposed in this paper, these sections can be

stiffened by any arrangements of plates placed within the

plane of the cross-section. Some particular examples are

illustrated in Fig. 2(d)-(f) to highlight the flexibility of the

formulation to account for any stiffeners’ layout.

The displacement of an arbitrary point P (s, 0, z), as

shown in Fig. 3, on the mid-surface can be described as:

u(s, z)=u(s, z)i+v(s, z)j+w(s, z)k (1)

where s is the curvilinear abscissa along the section mid-

line C, y is the ordinate and z is the coordinate along the

member axis with unit vectors i, j and k and the

displacement components u(s, z), v(s, z), and w(s, z) in the

same triad (Fig, 4).

2.1. Displacement and strain fields

Assuming Kirchhoff theory for thin plates, the thickness

of the plate does not change during deformation and lines

perpendicular to the mid-surface remain straight and

perpendicular after deformation. The displacement of an

arbitrary point Q(s, y, z) lying within the thickness of the

plate segment can therefore be expressed as:

(2)

where the comma denotes differentiation with respect to

the variable that follows. In the context of GBT and using

Kantorovich’s semi-variational method, the displacement

components can be expressed as a linear combination of

K assumed deformation modes:

(3)

where Uk, Vk and Wk are the assumed displacement

components which vary as a function of s, while ϕk are

unknown amplitude functions which vary along the

member axis z. Substituting Eqs. (3) into (2), the

displacement field can be rewritten as:

d s y z, ,( )

ds s y z, ,( )

dy s y z, ,( )

dz s y z, ,( )

u s z,( ) yvs s z,( )–

v s z,( )

w s z,( ) yvz s z,( )–

= =

u s z,( )

Uk s( )ϕk z( )k 1=

K

∑

Vk s( )ϕk z( )k 1=

K

∑

Wk s( )ϕk z, z( )k 1=

K

∑

=

Figure 4. Local and global coordinate systems.

Figure 3. Displacement field.


(4)

As typically done in GBT formulations, the strain field

is subdivided as follows:

(5)

with superscripts M and F denoting membrane and

flexural strain components, respectively. From Eq. (4)

and assuming infinitesimal strain theory, the relevant

strain field can be expressed as:

(6)

=

where the membrane strains are associated with the plane

y = 0, while the flexural components are proportional to

the ordinate y.

According to the two fundamental Vlasov hypothesis

valid for open cross-sections, the plate segments are

assumed to be (i) inextensible along their abscissa (i.e.

=0) and (ii1) shear indeformable in their mid-plane

(i.e. =0). For closed sections or more generally plate

segments belongs to closed loops of partially closed

sections this latter condition is relaxed to (ii2) =

constant stepwise with respect to the abscissa according

to Bredt theory for torsion. Therefore, the displacement

components from Eq. (3) are assumed to be selected in a

way to identically satisfy (i) and (ii1), or (ii2) where

appropriate.

2.2. Stress fields

As a consequence of the internal constraints imposed

on the plate segments by enforcing Vlasov’s hypotheses,

the stress field is composed of two parts, which consist of

active stresses σa and reactive stresses σr , i.e. σ =σa+σr.

The active stresses σa are associated to non-zero strains,

while the reactive ones σr are related to the internal

constraints and spend zero-work on the strain field, i.e.

=0. In particular, these can be expressed as σa=

{0, ,0, , , }T on plates lying on open branches

and σa={0, , , , , }T on plates on closed

branches, while σr={ ,0, ,0,0,0}T on all plates. It

should be noted that is made of two contributions: an

active one due to twist of closed branches and a reactive

one due to shear forces applied to any kind of section.

Assuming linear (plane stress) elastic material properties,

the constitutive relationship is expressed as:

σ =Eε+σr (7)

with the elastic matrix E defined as:

(8)

in which E and G are the elastic and shear moduli,

respectively. Substituting Eqs. (6) and (8) into Eq. (7), the

active stress field σa can be rewritten as:

d s y z, ,( )

Uk s( ) yVk s, s( )–[ ]ϕk z( )k 1=

K

∑

Vk s( )ϕk z( )k 1=

K

∑

Wk s( ) yVk s( )–[ ]ϕk z, z( )k 1=

K

∑

=

ε s y z, ,( )εMs 0 z, ,( )

εFs y z, ,( )

εsMs 0 z, ,( )

εzMs 0 z, ,( )

εszMs 0 z, ,( )

εsFs y z, ,( )

εzFs y z, ,( )

εszFs y z, ,( )

= =

ε s y z, ,( )

u,s s z,( )

w,z s z,( )

u,z s z,( ) w,s s z,( )+

yv,ss s z,( )–

yv,zz s z,( )–

2yv,sz s z,( )–

=

Σk 1=

KUk s, s( )ϕk z( )

Σk 1=

KWk s( )ϕk zz, z( )

Σk 1=

KWk s, s( ) Uk s( )+[ ]ϕk z, z( )

y– Σk 1=

KUk ss, s( )ϕk z( )

y– Σk 1=

KVk s( )ϕk zz, z( )

2y– Σk 1=

KVk s, s( )ϕk z( )

εsM

γszM

γszM

σr

Tε

σz

Mσs

Fσz

FτszM

σz

MτszM

σs

Fσz

FτszM

σs

MτszM

τszM

E

0 0 0 0 0 0

E 0 0 0 0

G 0 0 0

E

1 v2

–

-----------vE

1 v2

–

----------- 0

symE

1 v2

–

----------- 0

G

=


(9)

It is worth noting that the normal flexural stress

components are a superposition of two parts whose

magnitude at z depends on ϕk and ϕk,zz . As a

consequence the shape of these stress distributions will in

general change along the member length.

Reactive stresses are induced throughout the member to

ensure the kinematic constraints imposed on the plate

segments are maintained when load is applied. From

equilibrium considerations of an infinitesimal plate

element, the following two differential equations can be

derived:

+ =0 (10)

+ =0 (11)

which are used to determine the reactive stresses and

, respectively. Substituting the expression for

specified in Eq. (9) into Eqs. (10), it is possible to

evaluate the reactive shear stress as:

(12)

Similarly, from Eqs. (11) and (12), can be determined

as follows:

(13)

The calculation of the reactive stresses is further

outlined in Section 3.4.

2.3. Weak form of the problem

The weak form of the problem is derived by means of

the principle of virtual work, which can be expressed as

follows:

(14)

where p(s, z)=ps(s, z)i +py(s, z)j +pz(s, z)k are surface loads

applied to the member and δ (·) denotes a virtual quantity.

Substituting Eqs. (4), (6) and (9) into Eq. (14), the

weak formulation can be rewritten in terms of the

amplitude functions ϕk(z) as:

(15)

in which A and B are differential operators, defined as

A=[1 ∂ ∂2]T and B=[1 ∂]T, while T and q collect the

stiffness and member loading terms, respectively, and are

defined in Appendix.

3. Cross-sectional Analysis

The objective of the classical GBT cross-sectional analysis

is to identify a suitable set of conventional deformation

modes, described by Uk(s), Vk(s) and Wk(s) in Eq. (3), that

can adequately capture the structural response of stiffened

thin-walled members. Two or more families of modes will

be used, one for the unstiffened section and the remaining

ones for each of the different stiffened sections specified

along the member length. Each set of deformation modes

will be chosen as the dynamic eigenmodes of a planar

frame depicting the respective cross-section and the steps

involved in their evaluation is outlined in the following.

3.1. Discrete model and inextensibility element

The linear dynamic analysis of an arbitrary thin-walled

cross-section, as represented in Fig. 2, is performed

employing a discrete model of a planar frame representing

the plate segments forming the cross-section placed at

their mid-lines. Each plate is sub-divided into a discrete

number of finite elements joined together at the nodes as

shown in Fig. 5(a). Using typical finite element procedures,

the free dynamics of the cross-section is transformed into

the following algebraic eigenvalue problem:

(K−λ M)q =0 (16)

where K is the stiffness matrix, M is the mass matrix, q

are vectors of generalised nodal displacements and λ are

the squared natural frequencies of vibration.

When using a standard frame element for the GBT

cross-sectional analysis (e.g., Ranzi and Luongo, 2011),

the eigenmodes of the section produced solving Eq. (16)

include the elongation of the plate segments forming the

cross-section. In this case, the inextensibility of the plate

σa s y z, ,( )

0

EWk s( )ϕk zz, z( )k 1=

K

∑

G Wk s, s( ) Uk s( )+{ }ϕk z, z( )k 1=

K

∑

y–E

1 v2

–

----------- Vk ss, s( )ϕk z( ) vVk s( )ϕk zz, z( )+{ }k 1=

K

∑

y–E

1 v2

–

----------- Vk s( )ϕk zz, z( ) vVk ss, s( )ϕk z( )+{ }k 1=

K

∑

y– GVk s, s( )ϕk z, z( )k 1=

K

∑

=

τsz s,M

σz z,

M

σs s,

Mτsz z,M

τszM

σs

Mσz

M

τszMs z,( ) Eϕk zzz,

z( ) Wk s( ) sd

s∫

k 1=

K

∑–=

σs

M

σs

Ms z,( ) Eϕk zzzz,

z( ) Wk s( ) sd sd

s∫

s∫

k 1=

K

∑=

σ δε⋅ VdV∫ p δd⋅ Sd

V∫=

T Aϕ( ) Aδϕ( )⋅ zd0

L

∫ q Bδϕ( )⋅ zd0

L

∫=


segments are usually specified imposing internal constraints

within the model as, for example, outlined in (Casafront

et al., 2009; Goncalves et al., 2010; Ranzi and Luongo,

2011; 2013; Piccardo et al., 2013b). In local coordinates,

the requirement of this internal constraint is expressed by

.

For the purpose of this study, the condition of

inextensibility is already built-in within the proposed one-

dimensional mixed finite element. Its nodal freedoms are

collected in vector qe=

and are depicted in Fig. 5(b). The nodal freedom Re

depicts the axial force resisted by the element.

The stiffness and mass coefficients of the proposed

element are described below separating the terms associated

with the axial response ( and ) from those

related to the flexural behaviour ( and ).

In particular, and , associated to freedoms

, are defined as:

(17)

(18)

The stiffness and mass matrices and ,

associated with freedoms , are

readily available from finite element textbooks (Bathe,

2006; Cook et al., 2002):

(19)

(20)

where ' denotes differentiation with respect to s, (EI)e is

the flexural rigidity of the finite element, me is its mass

per unit length, Ne is a matrix of shape functions and Le

depicts the element length.

In this manner, the proposed 7 dof element makes use

of Hermite cubic and constant shape functions for the

transverse and axial displacements, respectively. The

stiffness and mass coefficients of the 7 dof mixed finite

element are provided in Appendix (Eqs. (A5) and (A6)).

3.2. Family of modes and unilateral contact problem

The behaviour of the stiffened member is described

combining the contribution of the family of conventional

modes obtained with the unstiffened and the stiffened

sections present along the member length. For example,

Fig. 6(a) depicts a member with a partially stiffened cross-

section along its length. In this case, two separate dynamic

analyses are performed, one on the frame representing the

unstiffened section (Fig. 6(b)) and one for the stiffened

one (Fig. 6(c)), to determine the deformation modes for

the unstiffened and stiffened sections.

u2eu1e

– 0=

u1e

v1e

θ1e

Re

u2e

v2e

θ2e, , , , , ,{ }

T

Kaxial

eMaxial

e

Kflexure

eMflexure

e

Kaxial

eMaxial

e

u1e

Re

u2e, ,{ }

T

Kaxial

e0 1 0

1– 0 1

0 1– 0

=

Maxial

e me

6------

201

000

102

=

Kflexure

eMflexure

e

v1e

θ1e

v2e

θ2e, , ,{ }

T

Kflexure

eEI( )e Ne

''( )TNe

''( ) sd

Le

∫=

Mflexure

eme Ne

TNe sd

Le

∫=

Figure 5. (a) Discrete planar frame representing cross-section; (b) Inextensibility 7 dof finite element.

Figure 6. (a) Partially stiffened member; (b) unstiffened section; (c) stiffened section.


At a stiffened cross-section, the presence of the stiffener

prevents significant deformations in its plane. This is

implemented in the cross-sectional analysis by specifying

a distributed spring along the contact perimeter between

the stiffener and the surrounding cross-section. The spring

acts in the direction perpendicular to the stiffened plate

segment, therefore restraining the transverse displacements

and affecting only the stiffness matrix describing the

flexural response. Due to the inextensibility of the plate

segments, the axial restraint imposed on the section does

not influence the identification of the conventional modes

and the terms related to the axial freedoms are identical

to those of Eq. (17). In particular, the stiffness matrix to

be used for elements located in stiffened regions of the

cross-section are calculated as follows:

(21)

where the matrix describes the rigidity provided

by the transverse distributed spring of stiffness ke defined

as:

(22)

The value of the distributed stiffness of the spring kedepends on the rigidity of the stiffeners. For the purpose

of this study, the value for ke specified in the simulations

has been selected to be 105 times greater than the flexural

rigidity of the plate segment which it aims to restrain.

This selection has worked well for all the thin-walled

sections considered in this study.

In the evaluation of the conventional deformation modes,

the use of the distributed springs is necessary to ensure

the shapes of the identified conventional modes comply

with the presence of the stiffeners, while the mass of the

stiffener is not included in the mass matrix, which can be

calculated with Eqs. (18) and (20).

The 7 dof finite element associated with the distributed

spring is depicted in Fig. 7(a). For clarity, possible

discretisations required to be carried out for the cross-

section illustrated in Figs. 6(c) is shown in Fig. 7(b). The

relevant stiffness coefficients for the 7 dof element

accounting for the presence of the distributed spring are

specified in Appendix (Eq. (A7)).

3.3. Dynamic eigenmodes and warping functions

The in-plane deformation modes (i.e. Uk and Vk) are

described by the nodal displacements q and their associated

shape functions, obtained solving the eigenvalue problem

specified in Eq. (16).

Since the unstiffened cross-section is modelled by

means of a free planar frame (i.e. not externally

constrained) the eigenvalue λ =0 is found to have an

algebraic multiplicity of 3. To form an eigenbasis of the

system, the associated eigenvectors are taken as the three

in-plane rigid body deformation modes, which include

horizontal and vertical translations as well as a rotation

about an arbitrary point (but not necessarily the shear

centre) of the cross-section. The stiffened cross-section

on the other hand is modelled as an externally constrained

planar frame with no rigid body modes.

The warping displacements Wk are obtained by applying

nil (ii1) or constant (ii2) shear membrane strain assumption

for each deformation mode. This is implemented as

outlined in the following, recalling the expression for the

shear membrane strain of Eq. (6), as:

(23)

where:

(24)

and is a piecewise constant function equal to

zero for plates on open branches or unknown for plates

on closed branches. Due to the inextensibility condition,

will also be piecewise constant for each plate.

Therefore, Wk is a piecewise linear function and can be

expressed as:

(25)

Kflexure S⋅

eEI( )e Ne

''( )TNe

''( ) sd Kspring

e+

Le

∫=

Kspring

e

Kspring

eke Ne

TNe sd

Le

∫=γszM

γszM

Γk s( )ϕk z, z( )k 1=

K

∑=

Γk s( ) Uk s( ) Wk s, s( )+=

Γk s( ) Γk

e=

Uk s( ) Uk

e=

Wk s( ) Wk s, s( )s Wk

i1

+Wk

i2

Wk

i1

–

Le

-------------------s= = Wk

i1

+

Figure 7. (a) FE with transverse spring stiffness; (b) stiffened section.


where Le is the length of the element and and

are the nodal warping displacements at the near (s=0) and

far (s=Le) nodes of element e. Eq. (24) can now be re-

written based on the known displacements and

unknown nodal warping displacements and for

each element within the cross-section.

(26)

For elements on open branches it is necessary and

sufficient that the following constraint be enforced for

each element.

(27)

On the other hand for elements belonging to closed

branches, the nodal warping displacements are related to

the tangential shear flow acting on the j-th loop X0j,

through the following constitutive relationship.

(28)

A positive shear flow is defined by the arbitrary

positive rotation assigned to each loop. When the

arbitrary location vector of an element aligns (differs)

with the positive rotation for a given loop a positive

(negative) shear flow results.

One of the unknowns remains arbitrary and is related to

the uniform extension of the member. To ensure that the

uniform extension mode is orthogonal to each of the

deformation modes warping displacements it is

convenient to enforce that the average warping of each

mode to be equal to zero

(29)

Now that the warping displacements have been

determined, it is now possible to evaluate the reactive

stresses, stiffness matrix and load vector.

3.4. Reactive stresses

The expressions for the reactive stresses and ,

previously introduced in Eqs. (12) and (13), can be

rewritten as:

(30)

(31)

where the functions and depend on the

warping modes:

(32)

(33)

Since is a piecewise linear function,

and are piecewise parabolic and cubic

functions, respectively. Substituting Eq. (25) into Eqs.

(32) and (33), functions and can be

expressed in terms of the nodal warping displacements

and for each element (determined in Section

3.3) and the unknown constants of integration and

:

(34)

(35)

The reactive stress constants of integration and

are obtained by enforcing equilibrium at the nodes

of the section for each deformation mode as:

(36)

(37)

For elements included in closed loops, the following

additional conditions apply:

(38)

(39)

4. Member Analysis

The member analysis is performed to describe the

structural response of thin-walled structural systems and

is implemented in the following by means of the finite

element method. This is carried out approximating the

amplitude functions ϕ(z) in the weak formulation of the

problem (Eq. (15)) with:

(40)

where Ne and ϕ e are the shape functions and nodal

displacements respectfully.

Wk

i1

Wk

i2

Uk

e

Wk

i2

Wk

i1

Γk

eUk

e=

Wk

i2

Wk

i1

–

Le

-------------------+

Γk

eUk

e=

Wk

i2

Wk

i1

–

Le

------------------- 0=+

Qk

j

GteΓk

eGte Uk

e Wk

i2

Wk

i1

–

be-------------------+

⎝ ⎠⎜ ⎟⎛ ⎞

± )Qk

j(j IC

e∈

∑= =

Wk s( ) sd

C∫ 0=

Wk s( )

τszM

σs

M

τszM

ETk s( )ϕk zzz,z( )

k 1=

K

∑–=

σs

MEYk s( )ϕk zzzz,

z( )k 1=

K

∑=

Tk s( ) Yk s( )

Tk s( ) Wk s( ) sd

s∫=

Yk s( ) Wk s( ) sd sd

s∫

s∫=

Wk s( ) Tk s( ) Tk

e=

Yk s( ) Yk

e=

Tk s( ) Yk s( )

Wk

i2

Wk

i1

Cτ k,

Cσ k,

Tk

e Wk

i2

Wk

i1

–

Le

-------------------s2

= Wk

i1

s Cτ k,

e+ +

Tk

e Wk

i2

Wk

i1

–

Le

-------------------s3

= Wk

i1

s2

Cτ k,

es C

σ k,

e++ +

Cτ k,

Cσ k,

τk sz,

i M,

i In∈

∑ ϕk zzz,ETk

i

i In∈

∑– 0= =

σk s,

i M,

i In∈

∑ ϕk zzzz,EYk

i

i In∈

∑ 0= =

τk sz,

Mds

Xj

∫° ϕk zzz,ETk s( )ds

Xj

∫°– 0= =

σk s,

Mds

Xj

∫° ϕk zzzz,EYk s( )ds

Xj

∫° 0= =

ϕ Ne

ϕe≅


Based on this approximation, the weak form of the

problem (Eq. (15)) can be rewritten, applying the linear

algebra identity Aa · Bb=BTAa · b, as:

(41)

from which it is possible to identify the element stiffness

matrix Ke and its loading vector pe dealing with

distributed loads:

(42)

(43)

The numerical solution is then sought based on standard

finite element procedures (Bathe, 2006; Cook et al.,

2002). Once the unknown nodal displacements are

evaluated, the displacement field can be calculated using

Eq. (4), the strain field using Eq. (6) and stress field with

Eq. (9).

The second approach considered in this study to

simulate the presence of the stiffener is implemented in

the member analysis by means of constraint equations. In

this case, the conventional deformation modes are those

calculated for the unstiffened cross-section following the

procedure already available in the literature, e.g. (Ranzi

and Luongo, 2011). The adopted constraint equations

ensure that the perimeter of the stiffener in contact with

the structural section does not deform in the plane of the

stiffener. These are implemented following standard finite

element procedures (Bathe, 2006; Cook et al., 2002),

already applied to thin-walled members in references

(Adany and Schafer, 2008; Casafront et al., 2009; Ranzi

and Luongo, 2011). With this approach the stiffener is

modelled as an infinitely rigid component.

4.1. Abaqus shell finite element model

A shell finite element model has been developed in this

study to validate the accuracy of the results obtained with

the GBT approach. This has been carried out using the

commercial finite element software Abaqus (ABAQUS

User’s Manual, 2008). The 4-node linear shell element

(S4R5) with reduced integration and hourglass control

available from the Abaqus library has been used to

describe the geometry of the thin-walled members. The

member is meshed using the structured technique with

quad shaped elements. The properties of the thin-walled

section are defined using the general shell stiffness

procedure available in the property module, allowing the

membrane stiffnesses to be uncoupled as seen in Eq. (8).

The FE models used in the simulations of the lipped

channel and box girder sections are illustrated in Figure 8

along with the adopted three dimensional cartesian

coordinate system, in which the cross-section is defined

in the X-Y plane and the member axis aligns with the Z-

direction. Simply supported boundary conditions are used

where the section's in-plane displacements are restrained

but left free to warp, i.e. U1=U2=UR3=0, at either end of

the member while at mid-span the warping of the section

is restrained, i.e. U3=UR1=UR2=0. At the location of each

of the stiffeners, face partitioning is used to define the

domain of the stiffened regions which is subsequently

converted into a rigid body. A reference point is defined

on this rigid body to enforce the required constraints.

External loads are modelled by means of uniformly

distributed pressures applied to the plate segments. The

linear elastic analysis is carried out using a general static

step adopting the direct equation solver method.

5. Applications

The ability of the GBT approach to predict the response

of stiffened thin-walled members is outlined in the

following by means of two worked examples, one dealing

with an open section and one with a partially-closed one.

ANe( )TTAN

e

zd0

L

∫⎩ ⎭⎨ ⎬⎧ ⎫

ϕe

BNe( )Tq zd

0

L

∫=

Ke

ANe( )TTAN

e

zd0

L

∫=

pe

BNe( )Tq zd

0

L

∫=

Figure 8. Abaqus model.


The members considered are simply supported (i.e. the

in-plane displacements of the member at each end are

restrained but are free to warp while at mid-span warping

is restrained due to symmetry, i.e. ϕk=0 at each end and

ϕk,z=0 at mid-span) with a total span of 1 m and three

stiffeners spaced evenly at 0.25 m centres. To better

highlight the particularity of the proposed method of

analysis, the specified stiffeners cover only part of the

cross-sections considered. The two approaches described

to account for the contribution of the stiffeners, one based

on the use of the family of conventional modes (see

Section 3.2) and one based on the constraint equations

(see Section 4) have been considered in the following

and, for clarity, have been referred to approach A and B,

respectively.

5.1. Lipped channel section

The geometry of the stiffened lipped channel section is

illustrated in Fig. 9. The section thickness is 1 mm and

the material properties include an elastic modulus E of

200 GPa and a Poisson’s ratio υ of 0.3. The load consists

of a uniform pressure of 0.01 MPa applied to the web.

The stiffener is welded to both web and flanges, and

extends only for half of the flange width.

Figure 9. Lipped channel section with stiffeners.

Figure 10. Lipped channel section. (a) rigid modes; (b) modes related to unstiffened section S1; and (c) modes related to

stiffened section S1.


When using approach A, two dynamic analyses are

carried out to evaluate the sets of conventional deformation

modes required for the analysis of this member, one

related to the unstiffened section and one dealing with the

stiffened section. For ease of reference, the two sets of

conventional modes have been denoted as S1 and S2. For

illustrative purposes, Fig. 10 depicts the in-plane and out-

of-plane components of the four rigid modes (1-4), and

the first eight conventional deformation modes obtained

for S1 (5-12) and for S2 (13-20). In the case of approach

B, the conventional modes considered in the analysis

consist of the rigid modes and of those included in S1.

The accuracy of the results obtained for the member

analysis is validated against the values calculated with a

shell finite element model developed in Abaqus. For

clarity, the in-plane and warping displacements have been

scaled to provide a clear overview of the cross-sectional

deformations, while the stress components have been left

unscaled to enable a graphical comparison among their

magnitudes. Because the two approaches A and B lead to

identical results, only one set of results has been plotted

in the figures for clarity and simply labelled as GBT. Very

good agreement is observed between the GBT and

Abaqus values. In particular, Fig. 11 depicts the variations

of the displacements calculated at a cross-section located

half depth away from mid-span (i.e. 50 mm away from

mid-span). The comparisons between stresses are provided

in Fig. 12 at the same location along the member length.

The contribution of the different modes to the overall

response is measured by means of a participation factor

Pk defined as:

(44)

where Wk is the work done by mode k in the member’s

deformed configuration. From Eq. (15), the internal work

can be calculated based on the amplitude functions as:

(45)

In this manner, the participation factor accounts for the

actual work done by a deformation mode in relation to

the total work done by all modes in the final deformed

configuration.

For the case study considered using approach A, the

modal participation associated with the conventional

deformation modes of the unstiffened section S1 accounts

for the majority of the work done with major contribution,

ordered for decreasing values of Pk, from modes 9 (with

P9=31.5%), 3 (P3=23.8%), 7 (P7=9.6%), 5 (P5=5.3%) and

11 (P11=4.8%). The deformation modes associated with the

stiffened cross-section S2 account for approximately one-

quarter of the total work done, with contributions from

modes 13 (P13=9.4%), 14 (P14=9%), 15 (P15=3.1%), and

Pk

Wk

Σi 1=

KWi

------------------=

Wk Aϕk( )TTkk Aϕk( ) zd0

L

∫=

Figure 11. Lipped channel section displacements.

Figure 12. Lipped channel section stress distributions.


16 (P16=3.1%). Due to the symmetry of this particular

problem, only symmetric modes from S1 and near equal

participation of consecutive modes from S2 (i.e. modes 13-

14 and modes 15-16) were triggered during the analysis.

For approach B, significant contributions were observed

from modes 3 (P3=65.4%), 7 (P7=28.8%) and 5 (P5=5.5%).

To better highlight the influence of the stiffeners on the

overall response, the proposed results have been compared

to those obtained for an unstiffened lipped channel member.

Figs. 11 and 12 show how the unstiffened member is more

flexible as expected, resulting in a significant overestimation

of the deflections and maximum stresses for each stress

Figure 13. Box girder section with stiffeners.

Figure 14. Box girder section. (a) rigid modes; (b) modes related to unstiffened section S1; and (c) modes related to

stiffened section S2.


component for the case study under consideration.

Differences are also related to the shape of the stress

distributions calculated for the stiffened and unstiffened

members.

Comparing the use of the two approaches (A and B)

considered to account for the presence of the stiffener, it

is observed that, while they are equivalent in terms of

accuracy, the use of approach B (the one based on the

constraint equations applied in the member analysis)

leads to an easier implementation as it follows standard

finite element procedures. It also seems to trigger a smaller

number of modes to describe the structural behaviour.

The use of approach A requires care in the creation of the

different families of modes and in their combinations,

making sure to avoid the use of dependent modes.

5.2. Box girder section

Consider a box girder section (partially closed, branched)

comprised of steel plates segments with a thickness of

1 mm, E=200 GPa and υ=0.3 subjected to a pressure of

0.01 MPa applied uniformly to the left-hand half of the

top flange and stiffeners located on the right-hand side of

the closed box (Fig. 13).

Also in this case, two linear dynamic analyses were

performed to determine suitable deformation modes

families, S1 and S2, for the unstiffened and stiffened

sections respectively, when using approach A. Figure 14

shows the in-plane components of the rigid modes (1-4)

and first 12 deformation modes of S1 (5-16) and S2 (17-

28). The conventional modes included in the analysis

with approach B are those of the rigid modes and those

specified in S1.

Figure 15 shows the sections displacement while Fig.

16 depicts the section stress distributions, with the results

being plotted at (with D being the depth of the

section equal to 50 mm). The results are generally in

good agreement between the Abaqus and GBT models of

the stiffened section with a similar trend being observed

with regards to the differences in flexural stresses in the

region directly adjacent to the stiffeners.

The GBT modal decomposition provides insight into

the structural behaviour of this stiffened member. With

approach A, S1 has significant contributions from modes

2 (P2=23.5%), 9 (P9=11.1%), 5 (P5=9.2%), and 16

(P16=2.9%), while S2 has major contributions from modes

17 (P17=23.7%), and 19 (P19=19.5%). With regards to

approach B, meaningful contributions were found from

modes 9 (P9=56.1%), 2 (P2=36.1%), 6 (P6=2.7%) and 4

(P4=1.2%). In the case of the box girder, both symmetrical

zL

2---

D

2----±=

Figure 16. Box girder section stress distributions.

Figure 15. Box girder section deformations.


and unsymmetrical modes participated in the deformation

of the member.

As previously observed for the lipped channel section,

approaches A and B lead to accurate representations of

the structural response, with approach B being easier to

implement in a computer program.

6. Conclusions

A linear elastic Generalised Beam Theory (GBT)

formulation has been presented for the analysis of

stiffened thin-walled members. The particularity of the

proposed approach relies on its ability to describe the

response of stiffened members in which the plate

stiffeners can be specified over selected parts of open,

closed and partially closed cross-sections.

Two approaches have been considered in this study to

account for the presence of the stiffeners. The first one

combines the use of different sets of conventional

deformations modes, one related to the unstiffened

section and the remaining ones dealing with the different

stiffened sections. A linear dynamic analysis is then

performed on a planar frame representing the unstiffened

and stiffened cross-sections, with the resulting eigenmodes

chosen as the in-plane components of the conventional

deformation modes. The warping components are calculated

in a post-processing phase by imposing conditions on the

shear strain. To simplify the cross-sectional analysis, a

mixed finite element has been developed which accounts

for the inextensibility condition included in the classical

GBT formulation. The second approach considered to

model the presence of the stiffener does not affect the

GBT cross-sectional analysis and requires the use of

constraint equations in the implementation of the member

analysis to include the rigidity of the stiffeners. Both

approaches have been shown to provide accurate results,

even if it has been noted that the second one (based on the

use of constraint equations) is easier to implement because

it is based on standard finite element procedures and

appears to describe the structural behaviour with a

smaller number of modes. The first approach (based on

the use of a family of conventional modes derived for

both unstiffened and stiffened sections) requires a higher

level of care in the creation of the different families of

modes and in their combinations, making sure to avoid

the use of dependent modes.

Numerical applications have been presented to illustrate

the ease of use of the proposed GBT approach. For this

purpose, two case studies have been considered, one with

a lipped channel and one with a box girder, both stiffened

with different plate arrangements. The results obtained

with the proposed GBT procedure have been validated

against the values calculated with a finite element model

developed in the commercial software Abaqus. The

results obtained using the two models (GBT and Abaqus)

have shown very good agreement.

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.

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Appendix

The coefficients of the K ×K symmetric stiffness matrices B, C, Da, Db and Dc, and the K ×1 load vectors q1 and q2

are defined as follows:

(A1a)

(A1b)

(A1c)

(A1d)

(A1e)

(A2a)

(A2b)

The 3K ×3K matrix T and the 2K ×1 vector q are determined as:

(A3)

(A4)

Stiffness and mass matrices for the 7 dof finite element of an unstiffened plate segment are:

(A5)

Bhk EtWhWkEt

3

12 1 ν2

–( )------------------------VhVk+ sd

C∫=

Dhk

a νEt3

12 1 ν2

–( )------------------------VhVk ss, sd

C∫=

Dhk

b νEt3

12 1 ν2

–( )------------------------Vh ss, Vk sd

C∫=

Dhk

cGt Wh s, Uh+( ) Wk s, Uk+( ) Gt

3

3--------Vh s, Vk s,+ sd

C∫=

BhkEt

3

12 1 ν2

–( )------------------------Vh ss, Vk ss, sd

C∫=

qh1z( ) pz s z,( )Wh s( ) sd

C∫=

qh2z( ) ps s z,( )Uk s( ) py s z,( )Vh s( )+[ ] sd

C∫=

T

B 0 Da

0 Dc

0

Db

0 C

=

qq2

q1

=

Ke

0 0 0 1 0 0 0

012 EI( )e

Le

3------------------

6 EI( )e

Le

2--------------- 0 0

12 EI( )e

Le

3------------------–

6 EI( )e

Le

2---------------

06 EI( )e

Le

2---------------

4 EI( )e

Le

--------------- 0 06 EI( )e

Le

2---------------–

2 EI( )e

Le

---------------

1– 0 0 0 1 0 0

0 0 0 1– 0 0 0

012 EI( )e

Le

3------------------–

6 EI( )e

Le

2---------------– 0 0

12 EI( )e

Le

3------------------

6 EI( )e

Le

2---------------–

06 EI( )e

Le

2---------------

2 EI( )e

Le

--------------- 0 06 EI( )e

Le

2---------------–

4 EI( )e

Le

---------------

=


(A6)

For plate segments within the stiffened area of the cross-section, the stiffness matrix is:

(A7)

Me

meLe

3----------- 0 0 0

meLe

6----------- 0 0

013meLe

35-----------------

11meLe

2

210-----------------0 0

9meLe

70--------------

13meLe

2

420-----------------–

011meLe

2

210-----------------

meLe

3

105----------- 0 0

13meLe

2

420-----------------

meLe

3

140-----------–

0 0 0 0 0 0 0

meLe

6----------- 0 0 0

meLe

3----------- 0 0

09meLe

70--------------

13meLe

2

420-----------------0 0

13meLe

35-----------------

11meLe

2

210-----------------–

013meLe

2

420-----------------–

meLe

3

140-----------– 0 0

11meLe

2

210-----------------–

meLe

3

105-----------

=

Ks

e

0 0 0 1 0 0 0

013keLe

35----------------

12 EI( )e

Le

3------------------+

11keLe

2

210----------------

6 EI( )e

Le

2---------------+ 0 0

9keLe

70-------------

12 EI( )e

Le

3------------------–

13keLe

2

420----------------

6 EI( )e

Le

2---------------+–

013keLe

210----------------

12 EI( )e

Le

3------------------+

13keLe

105----------------

12 EI( )e

Le

3------------------+ 0 0

13keLe

2

420----------------

6 EI( )e

Le

2---------------–

keLe

3

140----------

2 EI( )e

Le

---------------+–

1– 0 0 0 1 0 0

0 0 0 1– 0 0 0

09keLe

70-------------

12 EI( )e

Le

3------------------–

13keLe

2

420----------------

6 EI( )e

Le

2---------------– 0 0

13keLe

35----------------

12 EI( )e

Le

3------------------+

11keLe

210----------------

6 EI( )e

Le

2---------------––

013keLe

2

420----------------

6 EI( )e

Le

2---------------+–

keLe

3

140----------

2 EI( )e

Le

---------------+– 0 011keLe

2

210----------------

6 EI( )e

Le

2---------------––

keLe

3

105----------

4 EI( )e

Le

---------------+

=

Documents

Generalised Beam Theory (GBT) for stiffened sections