A large displacement and finite rotation thin-walled beam formulation including cross-section...

Comput. Methods Appl. Mech. Engrg. 199 (2010) 1627–1643

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Comput. Methods Appl. Mech. Engrg.

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A large displacement and finite rotation thin-walled beam formulationincluding cross-section deformation

Rodrigo Gonçalves a,*, Manuel Ritto-Corrêa b, Dinar Camotim b

a UNIC, Departamento de Engenharia Civil, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, 2829-516 Caparica, Portugalb ICIST/IST, Civil Eng. and Architecture Dept., Universidade Técnica de Lisboa, Av. Rovisco Pais, 1049-001 Lisbon, Portugal

a r t i c l e i n f o

Article history:Received 1 April 2009Received in revised form 28 November 2009Accepted 18 January 2010Available online 1 February 2010

Keywords:Thin-walled membersLarge displacementsFinite rotationsBeam finite elements

0045-7825/$ - see front matter � 2010 Elsevier B.V. Adoi:10.1016/j.cma.2010.01.006

* Corresponding author. Tel.: +351 21 2948580; faxE-mail address: [email protected] (R. Go

a b s t r a c t

This paper presents a new formulation for thin-walled beams that includes cross-section deformation.The kinematic description of the beam emanates from the geometrically exact Reissner–Simo beam the-ory and is enriched with arbitrary cross-section deformation modes complying with Kirchhoff’s assump-tion. The inclusion of these deformation modes makes it possible to capture the cross-section in-planedistortion, wall (plate) transverse bending and out-of-plane (warping), which leads to a computationallyefficient numerical implementation. Several illustrative numerical examples are presented and discussed,showing that the resulting beam finite element leads to solutions that are in very good agreement withthose obtained with standard shell finite elements, albeit involving much less degrees-of-freedom.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

Structural applications of thin-walled beams occur in practi-cally all areas of engineering practice, from the construction indus-try to offshore and aerospace structures. The fast growingpopularity of thin-walled beams is undoubtedly due to their struc-tural efficiency, but aesthetic issues also play a very significantrole. Unfortunately, thin-walled members are highly susceptibleto various complex geometrically non-linear effects, which canonly be adequately captured with computationally expensivestructural analysis methods, such as shell finite element or finitestrip methods (the ‘‘classic” methods). Alternatively, a thin-walledbeam theory incorporating cross-section in- and out-of-plane(warping) deformation may be employed.

The first thin-walled beam theory including cross-section defor-mation was developed by Vlasov [1], for beams with closed multi-cell cross-sections. In this so-called General Variational Method(GVM), the kinematic description of the beam is based on thesuperposition of several cross-section deformation modes that in-clude warping (with linear varying shape functions between cross-section nodes) and in-plane relative motions of the cross-sectionwalls that generate transverse wall bending. This transverse bend-ing of the walls is obtained through the analysis of the cross-sec-tion as a frame, subjected to imposed wall relative motions,assuming small displacements and the inextensibility of the walls.

ll rights reserved.

: +351 21 2948398.nçalves).

Notably, the GVM is the predecessor of most of the subsequentwork in this field, for small to moderate displacements [21,3–5].

Recent years have witnessed a growing interest in the so-called‘‘geometrically exact beam theory” developed by Reissner [10] andSimo [11] (and therefore also known as ‘‘Reissner–Simo beam the-ory”), which owes its name to the fact that it remains valid inde-pendently of the magnitude of the displacements and rotationsinvolved. In its original form, Reissner–Simo’s beam theory as-sumes that the cross-sections remain plane and undeformed, sincethe kinematic description is exclusively based on their rigid-bodymotions (translation and rotation). Subsequently, several research-ers have included additional deformation modes. Simo and Vu-Quoc [12], Gruttmann et al. [13] and Atluri et al. [14] considereda torsion-related warping mode. Petrov and Géradin [15,16] in-cluded distortional and warping functions associated with the St.Venant solution for each of the six internal forces and moments,but not as independent cross-section degrees-of-freedom (more-over, shear deformation was not considered). Other authors in-cluded the torsion-related warping mode but, simultaneously,reduced the number of kinematic parameters by constrainingdeformation in order to eliminate the bi-shear [17] and shear[18,19] strains. The co-rotational formulation of Battini and Pacos-te [20,21] allows the choice of including or not the shear/bi-sheardeformability. Klinkel and Govindjee [22] employed three func-tions to describe the cross-section warping of anisotropic beams,

1 The Generalized Beam Theory, originally developed by Schardt [2,6], is frequentlyviewed as an extension of Vlasov’s classical prismatic beam theory that accounts forboth cross-section out-of-plane and in-plane deformation (see also [7–9]).

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1628 R. Gonçalves et al. / Comput. Methods Appl. Mech. Engrg. 199 (2010) 1627–1643

associated with the St. Venant solutions for pure bending and tor-sion. Finally, Pimenta and Campello [23] and Ritto-Corrêa [24]independently developed formulations that consider arbitrarythree-dimensional cross-section deformation modes, although nei-ther of them has yet been employed to analyze thin-walled beamsand the issue of finding an appropriate set of deformation modesremains an open question.

In this paper, a new formulation for thin-walled beams includ-ing cross-section deformation is proposed, based on the followingfundamental 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.

Starting from Reissner–Simo beam theory, the configuration ofeach cross-section is described by the position vector r of a refer-ence point (an arbitrary cross-section centre C), a finite rotationabout C, using a rotation tensor K, and additional kinematic param-eters associated with pre-established cross-section shape functions(i.e., deformation modes), that allow for cross-section in-plane andout-of-plane deformation. These deformation modes describe onlythe wall mid-line displacements, since the location of points out-side the wall mid-surface is automatically provided by Kirchhoff’sassumption (A2).

Although several alternatives exist for the parameterization ofrotation tensors [25–28], the rotation vector is adopted in thiswork, since (i) it offers a simple geometric interpretation of therotations, (ii) only three parameters are involved, and (iii) thenumerical examples presented here involve rotations smaller than2p, the first singularity for this parameterization. The applicationof the rotation vector parameterization to the geometrically exactbeam theory (without deformation modes) has been developed byCardona and Géradin [29], Pimenta and Yojo [30], Ibrahimbegovicet al. [31] and Ritto-Corrêa and Camotim [32].

All scalar quantities, including the tensor components, are rep-resented by italic letters. Tensors (first and second order) andmatrices are identified by bold italic letters. Where reference ismade to the components of tensors and matrices, rectangularbrackets are used []. The standard directional derivative is denotedby DF (a)[b], where F is the function, a its argument and b indi-cates the direction of the derivative. For the sake of abbreviation,partial scalar derivatives are indicated by subscripts following acomma, e.g., if F ¼ F ða; bÞ, then F ;a ¼ @F=@a. A virtual variationis denoted by d and an incremental/iterative variation by D. Whereno distinction is necessary, variations may be denoted by d.

2. Kinematic description of the thin-walled beam

With respect to an orthonormal direct reference systemðX1;X2;X3Þ with base vectors fEig ði ¼ 1;2;3Þ, three beam configu-rations are defined:

(i) The reference configuration – an imaginary configurationwhere the beam assumes a right prismatic shape;

(ii) The initial configuration – the beam configuration at k ¼ 0,where k is the scalar loading multiplier;

(iii) The current configuration – the beam configuration at k – 0.

Since the initial configuration may be viewed as a particular caseof the current configuration, it will be addressed only very briefly.

The position vectors of a material point for the reference, initialand current configurations are designated X; x0; and x, respec-tively. Vectors related to one configuration (e.g., applied forces)are represented in this fashion. Vectors associated with two config-urations (e.g., displacements) take the representation of the ‘‘mostrecent” configuration, where a hat ‘‘^” is added to the vectors asso-ciated with the initial/current configurations, in order to distin-guish them from those related to the reference/currentconfigurations. Second order tensors follow a similar notation,although only uppercase letters are used.

A separate kinematic description of each beam wall is followedand compatibility between walls must be ensured by the introduc-tion of appropriate deformation modes. In the reference configura-tion, the X3 axis coincides with the beam longitudinal axis, i.e., eachbeam cross-section is parallel to the X3 ¼ 0 plane and the beam ini-tial cross-section is located at X3 ¼ 0 (see Fig. 1). The intersectionof the longitudinal axis with each cross-section defines the cross-section centre C. Although it is usually assumed that C coincideswith the cross-section centroid or shear centre, no such restrictionis made in this work. Each wall reference configuration is mappedby the vector

X ¼ X3E3 þ LA þ RðX1E1 þ X2E2Þ; ð1Þwhere LA is a fixed bi-dimensional vector, in the cross-section plane,that references the origin A of the wall mid-line, and R is the wall‘‘local” rotation tensor about A, along the E3 axis, whose purposeis to rotate the base vectors in such a way that RE1 and RE2 definethe through-thickness and wall mid-line directions, respectively.The rotation vector associated with R is uE3, which is kept fixedfor all configurations (reference, initial and current).

For the current configuration, the position vector is expressed as(see Fig. 1, which illustrates all the vectors defined below, concern-ing the position of the cross-section centre C and a given point B,for an arbitrary beam wall)

x ¼ r þ K ð�LA þ R ðX1E1 þ X2E2 þX

i

pðiÞvðiÞÞ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}L

Þzfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{l

; ð2Þ

where

(i) r ¼ rðX3Þ references the cross-section centre C;(ii) K ¼ KðhÞ is the cross-section rotation tensor about C and

h ¼ hðX3Þ is its corresponding rotation vector;(iii) l references each cross-section point with respect to C in a

co-rotating base KfEig;(iv) L references each wall point with respect to A in a com-

pound co-rotating base KRfEig;(v) pðiÞ ¼ pðiÞðX3Þ are the scalar weight functions associated with

each cross-section deformation mode i = 1 . . .D, where D isthe number of deformation modes considered;

(vi) vðiÞ ¼ vðiÞðX1;X2Þ are vectors containing the displacementshape functions associated with each cross-section deforma-tion mode.

This mapping of the beam is kinematically complete, in thesense that it allows for the exact description of any beam configu-ration, even if it is conceptually necessary to consider an infinitenumber of deformation modes. Although several simplifyingassumptions will be introduced in the following discussion, itshould be stressed that the mapping of the cross-section mid-lineremains always kinematically complete.

The deformation mode shape functions are further expressed as

vðiÞðX1;X2Þ ¼ vðiÞðX2Þ þ X1wðiÞðX2Þ; ð3Þ

where vðiÞðX2Þ ¼ vðiÞ1 ðX2ÞE1 þ vðiÞ2 ðX2ÞE2 þ vðiÞ3 ðX2ÞE3 describes thedisplacement of the cross-section wall mid-line and X1w

ðiÞ allows

Fig. 1. Reference and current configurations of a beam wall: (a) spatial view and (b) cross-section in-plane view.

R. Gonçalves et al. / Comput. Methods Appl. Mech. Engrg. 199 (2010) 1627–1643 1629

for a linear through-thickness variation. According to assumptionA2, the variation of the wall thickness may be discarded and onemay write

wðiÞðX2Þ ¼ wðiÞ2 ðX2ÞE2 þ wðiÞ3 ðX2ÞE3 ð4Þand note that wðiÞ3 allows for a through-thickness warping which, inthe classic theory of thin-walled beams, is known as ‘‘secondarywarping” (e.g., [33]). According to assumption A2, functions wðiÞ

are fully dependent on the configuration of the wall mid-surfaceand therefore do not constitute independent kinematic degrees-of-freedom. This issue will be further addressed in Section 3.2, forthe calculation of the strains.

It is now possible to rewrite (2) as

x ¼ xþ X1n; ð5Þ

where vector x maps the wall mid-surface and n is a trough-thick-ness director, given by

�x ¼ r þ K ð�LA þ R ðX2E2 þX

i

pðiÞ�vðiÞÞÞ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}�L

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{�l

; ð6Þ

n ¼ KRðE1 þX

i

pðiÞwðiÞÞ: ð7Þ

For the initial configuration, one readily writes

x0 ¼ x0 þ X1n0; ð8Þ

x0 ¼ r0 þ K0 ðLA þ R ðX2E2 þX

i

pðiÞ0 vðiÞ|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}ÞÞL0

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{l0

; ð9Þ

n0 ¼ K0RðE1 þX

i

pðiÞ0 wðiÞÞ: ð10Þ

For convenience, all the kinematic parameters are grouped in vector/,

½/� ¼

½r�½h�pð1Þ

..

.

pðDÞ

266666664

377777775 ð11Þ

of dimension 6 + D, where D is the number of deformation modes.


3. Strain

3.1. Deformation gradient

From the kinematic description presented in section 2, thedeformation gradient between the reference and current configu-rations reads

F ¼ rx ¼ rðxþ X1nÞ ¼ rxþ n� E1 þ X1rn; ð12Þ

where � is the standard tensorial product. The first term transformsvectors of the mid-surface and therefore may be viewed as a mem-brane deformation gradient, which can be written as

FM ¼ rx ¼ e2 � E2 þ e3 � E3 ð13Þ

in which e2 and e3 are the push-forwards of the base vectors E2 andE3, given respectively by

e2 ¼ x;2 ¼ KR E2 þX

i

pðiÞvðiÞ;2

!¼ KRg2; ð14Þ

e3 ¼ x;3 ¼ r;3 þ K;3lþ KRX

i

pðiÞ;3 vðiÞ

¼ KRðE3 þ Cþ K � ðRtlÞ þX

i

pðiÞ;3 vðiÞÞ ¼ KRg3; ð15Þ

where g2 and g3 are the back-rotations of e2 and e3 by ðKRÞt . In theprevious expression, � is the standard vectorial product and thematerial strain measures C and K, originally proposed by Simo[11], are given by

C ¼ ðKRÞtr;3 � E3 ¼ C1E1 þ C2E2 þ C3E3; ð16ÞK ¼ axiððKRÞtðKRÞ;3Þ ¼ axiðRt

KtK;3RÞ ¼ RtaxiðKtK;3Þ¼ j1E1 þ j2E2 þ j3E3; ð17Þ

where C is a vector that quantifies the extension/shearing of thebeam, K denotes a material spin vector that measures the beam/wall curvature (see also Appendix A) and the axi() operator yieldsthe axial vector of a skew-symmetric second-order tensor.

Finally, the development of the third term in (12) requires thecalculation of rn which, from (7), reads

rn¼KRðK�ðE1þX

i

pðiÞwðiÞÞ�E3þX

i

ðpðiÞ;3 wðiÞ �E3þpðiÞwðiÞ;2 �E2ÞÞ:

ð18Þ

3.2. Enforcing Kirchhoff’s assumption

As mentioned earlier, in Section 2, Kirchhoff’s assumption (A2)renders the through-thickness shape functions wðiÞ and associatedweight functions fully dependent on the deformation of the wallmid-surface. The appropriate constraint equations read

n � e2 ¼ 0 ^ n � e3 ¼ 0 ð19Þ

and it should be understood that they hold for all beam configu-rations (reference, initial and current). Since the manipulation ofthese equations (particularly the second one) is far from trivial ina geometrically exact setting, one may resort to a linearization,by invoking the small bending strains assumption (A3) betweenall configurations (i.e., the strain measures (16) and (17) andthe weight functions of the deformation modes are small).Although some of the omitted non-linear terms may be relevantfor curved or pre-twisted/distorted beams, one must bear in mindthat this simplification affects only the bending strain terms andnot the membrane terms. The linearization of the left hand termsyields

n � e2 ¼ E1 þX

i

pðiÞwðiÞ !

� g2 �X

i

pðiÞ wðiÞ2 þ vðiÞ1;2

� �; ð20Þ

n � e3 ¼ E1 þX

i

pðiÞwðiÞ !

� g3

� C1 þX

i

pðiÞ;3 vðiÞ1 þ pðiÞwðiÞ3

� �� dA þ X2

� �j3; ð21Þ

where the scalar dA ¼ LA � RE2 corresponds to the projection of LA onthe wall direction.

Together with (20) and (21), the constraint Eq. (19) lead to thefollowing results:

(i) Each deformation mode with non-null vðiÞ1 must be supple-mented by the through-thickness shape function

wðiÞ2 ¼ �vðiÞ1;2: ð22Þ

(ii) For each deformation mode i with non-null vðiÞ1 , there mustexist another mode j satisfying

pðjÞwðjÞ3 ¼ �pðiÞ;3 vðiÞ1 ði – jÞ ð23Þ

which may be accomplished by adopting

pðjÞ ¼ pðiÞ;3 ^ wðjÞ3 ¼ �vðiÞ1 ði – jÞ: ð24Þ

(iii) The torsional curvature j3 must be accompanied by a defor-mation mode satisfying

pðj3Þ ¼ j3; vðj3Þ ¼ 0; wðj3Þ2 ¼ 0; wðj3Þ

3 ¼ dA þ X2 ð25Þwhich coincides with the St. Venant secondary warping solu-tion for torsion about C.

(iv) Concerning the shearing in the through-thickness directionC1, the constraint equation is

pðC1ÞwðC1Þ3 ¼ �C1 ð26Þ

and may be fulfilled by adopting the following deformationmode

pðC1Þ ¼ �C1 ^ wðC1Þ3 ¼ 1: ð27Þ

However, this approach is quite cumbersome, since pðC1Þ thendepends on both r, K and R (see Eq. (16)) and, in order tosatisfy continuity requirements, the associated shape func-tion must also include membrane warping ðvðC1Þ

3 – 0Þ inadjacent walls – Fig. 2 illustrates the loss of compatibilitythat occurs between the walls of an angle beam if themembrane warping is not included. This situation is clearlydue to a ‘‘conflict” between Kirchhoff’s assumption and theallowance for membrane shear deformation ðC2 – 0Þ. Inorder to overcome this situation, since shear deformation isnegligible for most thin-walled beam problems, one simplyassumes C1 ¼ 0 and therefore the need to introduce thedeformation mode (27) is removed.

The previous results make it possible to write n without thethrough-thickness shape functions

n ¼ KRðE1 þ j3ðdA þ X2ÞE3 �X

i

ðpðiÞvðiÞ1;2E2 þ pðiÞ;3 vðiÞ1 E3ÞÞ ð28Þ

and its gradient (18) becomes

rn�KR �j2þ dAþX2

� �j3;3�

Xi

pðiÞ;33vðiÞ1

!E3�E3

�X

i

pðiÞvðiÞ1;22E2�E2þ j3�X

i

pðiÞ;3 vðiÞ1;2

!E2�E3þE3�E2ð Þ

!;

ð29Þwhere the non-linear terms have once more been discarded.

Fig. 2. Incompatibility between adjacent walls due to membrane shearing ðC2 – 0Þ in one leg of an angle beam.


3.3. Green–Lagrange strains

In order to obtain the Green–Lagrange strains, one first calcu-lates F tF from (12), which yields

F tF ¼ðFMÞtFMþ2X1symððFMÞtrnÞþ2symððn�E1ÞtðFMþX1rnÞÞþX2

1ðrnÞtrnþE1�E1 ð30Þ

an expression that may be further simplified:

(i) The first term is associated with membrane strain and reads

ðFMÞtFM ¼ ðg2 � g2ÞE2 � E2 þ ðg3 � g3ÞE3 � E3

þ ðg2 � g3ÞðE2 � E3 þ E3 � E2Þ: ð31Þ

(ii) The second term couples membrane and bending terms andmay be simplified by using assumption A3. From (14), (15)and (29), dropping all non-linear terms, one obtains

FM� �t

rn � �j2 þ dA þ X2

� �j3;3 �

Xi

pðiÞ;33vðiÞ1

!E3 � E3

�X

i

pðiÞvðiÞ1;22E2 � E2

þ j3 �X

i

pðiÞ;3 vðiÞ1;2

!E2 � E3 þ E3 � E2ð Þ: ð32Þ

Note that membrane effects are no longer present and, as ex-pected, the bending curvature j2 produces longitudinal exten-sions and the torsion curvature j3 generates shear strains.

(iii) The third term may be rewritten as

ðn�E1ÞtðFMþX1rnÞ¼ ðn �e2ÞE1�E2þðn �e3ÞE1�E3

þX1ðn �n;3ÞE1�E3þX1ðn �n;2ÞE1�E2

ð33Þ

and vanishes according to assumption A2, since n remainsperpendicular to e2; e3;n;3 and n;2.

(iv) The fourth term may be omitted on the basis of the smallbending strains assumption (A3) or the thin-walled natureof the beam (assumption A1), a standard procedure in thecontext of thin shell analysis (e.g., [34]).

The strain may be expressed in terms of arbitrary elementaryvectors of the reference configuration ðdXa and dXbÞ through the(non-conventional) Green–Lagrange strain tensor E, defined as

12ðdxa � dxb � dxa

0 � dxb0Þ ¼ dXa � E dXb ) E ¼ 1

2ðF tF � F t

0F0Þ ð34Þ

and, using the previous results, one obtains

E ¼ EM þ EB; ð35Þ

2EM ¼ ðFMÞtFM � ðFM0 Þ

tFM0

¼ ðg2 � g2 � g20� g20ÞE2 � E2 þ ðg3 � g3 � g30

� g30ÞE3 � E3

þ ðg2 � g3 � g20� g30ÞðE2 � E3 þ E3 � E2Þ; ð36Þ

EB¼X1symððFMÞtrn�ðFM0 Þ

trn0Þ

¼�X1

Xi

ðpðiÞ �pðiÞ0 ÞvðiÞ1;22E2�E2þX1

�ðj2�j20 Þ

þðdAþX2Þðj3;3�j3;30 Þ�X

i

ðpðiÞ;33�pðiÞ0;33ÞvðiÞ1

!E3�E3

þX1ðj3�j30 �X

i

ðpðiÞ;3 �pðiÞ0;3ÞvðiÞ1;2ÞðE2�E3þE3�E2Þ; ð37Þ

where EM and EB concern membrane and bending strains, respec-tively. Note that a plane strain ðE11 ¼ 0Þ is retrieved as a conse-quence of assumption A2 and, as in classical beam theories, j2

and j3;3 produce EB33 and j3 produces EB

23. The j3;3 term, associatedwith the torsion-related secondary warping strains, is discarded inthe examples presented in Section 6 but may be relevant for anglesor cruciform cross-sections.

For implementation purposes, one employs vector forms of thestrain tensors, which are denoted by a subscript ‘‘v” and read

½EMv � ¼

EM22

EM33

2EM23

264375; ½EB

v � ¼EB

22

EB33

2EB23

264375: ð38Þ

4. Stress and equilibrium

The work-conjugate of the Green–Lagrange strain tensor E,associated with the reference configuration, is a ‘‘non-conven-tional” second Piola–Kirchhoff stress tensor S. The relation be-tween S and the conventional second Piola–Kirchhoff stresstensor S0, associated with the initial configuration, is analogousto the one between S0 and the Cauchy stress tensor, since one has

S ¼ F�10 S0F�t

0 J0; ð39Þwhere J0 ¼ detðF0Þ and F0 is the deformation gradient between thereference and initial configurations. Since the pair E–S is associatedwith the reference configuration, attention must be paid to the con-stitutive relation adopted whenever non-negligible straining occursbetween the initial and reference configurations.

Using vector forms, the stresses are calculated from the generaltangent constitutive relation

2 In fact, for linear elastic materials, it is possible to integrate analytically almost allbending terms in the beam volume (the exception are the terms associated with j2

and j3Þ. This approach was employed in all the numerical examples.3 Null normal stresses in the wall mid-line direction, the Navier condition for 2D

beam theory.


dSv ¼ CtdEv ; ð40Þ

where Ct is the tangent constitutive matrix. A plane stress state isassumed in all beam walls ðS1k ¼ Sk1 ¼ 0; k ¼ 1 . . . 3Þ, thus generat-ing a mild inconsistency with the plane strain of (36) and (37).

The principle of virtual work, written in the reference configu-ration, reads

dWint þ dWext ¼ 0() �Z

VSv � dEvdV þ

ZA

du � Q dA ¼ 0; ð41Þ

where V is the beam reference volume, Q are surface forces actingon the beam reference mid-surface A, obtained from the surfaceforces at the initial configuration, q0, through

du � Q dA ¼ du � q0 da0 ) Q ¼ q0da0

dA; ð42Þ

where �a0 is the initial mid-surface and du is the virtual variation ofthe mid-surface displacement vector, which can be directly ob-tained from (6) and reads

du ¼ dx ¼ dr þ dKlþ KRX

i

dpðiÞvðiÞ: ð43Þ

In order to implement a standard incremental/iterative scheme, thelinearization of the equilibrium equations in the direction of anincremental/iterative variation of the configuration D/ and theloading parameter Dk are required, which are given by

DdW ¼DmatdWint½D/�þDgeodWint½D/�þDdWext ½D/�þDdWext ½Dk� ¼0;

ð44Þ

DmatdWint ½D/� ¼ �Z

vdEv � CtDEvdV ;

DgeodWint½D/� ¼ �Z

vSv � DdEvdV ;

DdWext½D/� ¼Z

ADdu � QdA;

DdWext½Dk� ¼Z

Adu � DQdA;

ð45Þ

where DmatdWint and DgeodWint are the so-called material and geo-metric parts of the linearized internal virtual work.

The finite element implementation presented in the next sec-tion involves the direct approximation of the kinematic parametersand it is therefore necessary to write explicitly (41) and (44) interms of the parameters contained in /. The complete expressionsare presented in Appendix B, which involve the expressions of thedirectional derivatives of the rotation tensor K and of the strainmeasures C and K given in Appendix A.

Finally, Appendix C is devoted to showing how Vlasov’s classicthin-walled prismatic beam theory [1] can be recovered from theproposed formulation.

5. Finite element implementation

The finite element implementation of the proposed formulationinterpolates directly the kinematic parameters contained in vector/. Since Kirchhoff’s assumption (A2) is adopted, C1 continuity is re-quired for the parameters causing longitudinal wall (plate) bend-ing and, therefore, Hermite cubic functions are employed tointerpolate all parameters, on the basis of their nodal values andderivatives. One obtains a 2-node beam element with 4� ð6þ DÞd.o.f., where D is the number of deformation modes considered.

The parameters in vector /0, corresponding to the initial config-uration, may be interpolated analogously, although the numericalexamples presented in the next section concern only prismaticbeams and therefore these parameters remain constant along thebeam length.

The equilibrium equations are solved using a standard incre-mental/iterative scheme employing Newton–Raphson’s method.

Load, displacement and arc-length control methods are imple-mented. Numerical integration is performed using an arbitrarynumber of Gauss points along the element length and wall width.No integration points along the wall thickness are required for theexamples presented in the next section, since a linear elastic mate-rial is assumed and therefore analytical integration in the through-thickness direction is performed.2 A 3� 3 integration point grid wasadopted, since preliminary studies showed that it provides satisfac-tory results and prevents locking phenomena.

6. Numerical examples

In all the numerical examples presented here, a St. Venant–Kir-chhoff material law is assumed, making it possible to uncouplemembrane and bending terms. As already mentioned, plane stressis assumed and thus the constitutive matrix reads

½C� ¼

E1�m2

mE1�m2 0

mE1�m2

E1�m2 0

0 0 G

264375 or ½C�� ¼

0 0 00 E 00 0 G

264375; ð46Þ

where C* applies for plane stress states under the S22 ¼ 0 con-straint,3 E and G are the Young’s and shear moduli and m is Poisson’sratio.

The results obtained with the proposed beam formulation arecompared with the ones provided by either (i) large displace-ment/small strain Reissner–Mindlin 9-node shell finite elementanalyses, carried out using the computer program ADINA [35], or(ii) finite strip linear stability analyses, performed using the CUFSMsoftware [36] (Example 6.2).

6.1. Lateral-torsional buckling of I-section cantilevers

The first example aims at illustrating the capabilities of the pro-posed beam formulation when no cross-section in-plane deforma-tion occurs. Therefore, one analyzes the lateral-torsional bucklingbehavior of the compact I-section cantilever shown in Fig. 3a, actedby a vertical load applied at the free end cross-section shear centre.The material is characterized by E = 210 GPa and m ¼ 0:3, typicalvalues for steel.

For the beam model, the cross-section is subdivided into threewalls, as shown in Fig. 3a (web + two flanges), and S22 ¼ 0 is as-sumed in all walls, which means that C* is employed. In order tocapture the torsional behavior accurately, the wall mid-surfacewarping deformation mode corresponding to the St. Venant solu-tion for pure torsion is included (Fig. 3b). Only seven kinematicparameters are involved ðr; h; pð1ÞÞ, leading to a 28 � 28 elementstiffness matrix. The fixed support is modeled by eliminating thenodal values of the parameters (not their derivatives) and thus a14n + 7 d.o.f. problem is obtained, where n is the number of beamfinite elements. Numerical integration is performed with only twopoints along X2, since no transverse wall (plate) bending occurs. Anon-uniform discretization with four elements (63 d.o.f., lengths0.3, 0.3, 0.4, 0.5 m) was employed, but the mesh was refined(119 d.o.f., the element lengths were halved) at u2 ¼ 0:8 m, in orderto assess the mesh dependency of the solution.

For the shell model, the regular mesh displayed in Fig. 5 wasused, which involves about 3500 d.o.f.. In order to prevent localizeddeformation at the point of load application, the force was halvedand applied at the two flange/web junctions, as shown in the

Fig. 3. I-section cantilever (a) geometry and loading and (b) torsion-related warping mode.

Fig. 4. I-section cantilever: applied load vs. displacement of the shear centre.


bottom of the figure. The boundary conditions consist of eliminat-ing the nodal displacements at the support, not the nodal rotations.

Fig. 4 shows the non-linear equilibrium paths4 obtained withthe beam/shell models, the linear solution and the bifurcation pointprovided by the beam model.5 A virtually perfect match is observedup to u2 ¼ 0:8 m, after which the paths start to diverge due to theoccurrence of cross-section in-plane distortion near the support.Obviously, a mesh refinement does not improve the results signifi-

4 These equilibrium paths were obtained by introducing small imperfections in themodels.

5 The bifurcation loads were detected by inspecting the determinant of thestructural tangent stiffness matrix along the (non-linear) equilibrium path of thebeam without imperfections.

cantly, since distortional deformation modes were not included inthe beam model. Fig. 5 makes it possible to compare the evolutionof the deformed configurations obtained with both models (eachbeam finite element has a distinct shading), evidencing an excellentagreement.

6.2. Bifurcation behavior of lipped channel columns

The second example concerns the local/distortional/globalbifurcation behavior of simply supported and uniformly com-pressed columns with the lipped channel cross-section shown inFig. 6a, previously analyzed by the authors [37]. The material prop-erties are E = 70 GPa and m ¼ 0:33.

Fig. 5. I-section cantilever: deformed configurations.

6 This assumption, known as ‘‘Vlasov’s assumption”, is generally acceptable foropen cross-sections.

7 However, note also that SM23 ¼ 0 is obtained as a consequence of eM

23 ¼ 0.


A simple linear stability analysis is performed, in which the pre-buckling deformations are neglected ð/ ¼ /0Þ and the only non-null stress component along the fundamental path is S33 ¼ P=A,where P is the axial thrust and A is the cross-section area. Thebifurcation loads P and buckling modes D/ correspond to thenon-trivial solutions of the initial stress problem

DdWð/0; S33 ¼ P=AÞ½D/� ¼ 0: ð47Þ

Since the assumption of small-to-moderate displacements holds inlinear stability analyses, one may employ the constraint Eqs. (89) inAppendix C, leading to a Vlasov-type beam formulation witharbitrary deformation modes, where the only independentkinematic parameters are uC (the displacement of the cross-section

centre C), h3 and pðiÞ. Moreover, the following assumptions, typicallyused in GBT-type formulations (e.g., [8,9]), are adopted: (i) vðiÞ;2 ¼ 0,which implies EM

22 ¼ 0, (ii) eM23 ¼ 06 (small strain) which, after linear-

izing g2 � g3 ¼ 0, implies that a given in-plane deformation mode jautomatically generates an associated warping mode i satisfyingvðiÞ3;2 ¼ �vðjÞ2 ^ pðiÞ ¼ pðjÞ;3 , and (iii) SM

22 ¼ 0, which means that C* is usedfor the membrane terms7 and C for the bending terms. The cross-sec-tion deformation modes, shown in Fig. 6(b), are obtained from a‘‘GBT cross-section analysis”, based on the cross-section subdivisioninto 10 walls [37].

Fig. 6. Lipped channel cross-section (a) geometry and (b) deformation modes.


The exact solutions of the problem are given by

uC ¼uC sinpnX3

L

� �; h3¼ h3 sin

pnX3

L

� �; pðiÞ ¼ pðiÞ sin

pnX3

L

� �; ð48Þ

where ð�Þ and n are the buckling mode amplitudes and longitudinalhalf-wave number. One therefore ends up with an equation systemof dimension 4 + D, where the unknowns ð�Þ are calculated from(47), for a given n.

For comparison purposes, finite strip linear stability analysesare also performed, employing the cross-section discretizationadopted to determine the deformation modes. The critical bifurca-tion stresses and associated buckling modes, obtained with bothmethods, are plotted in Fig. 7, as functions of the column length/cross-section height ratio (L/h), and a virtually perfect match isobserved.

6.3. Distortion of a lipped channel beam

In order to assess the accuracy of the proposed formulationwhen moderate-to-large cross-section in-plane deformation oc-curs, the beam shown in Fig. 8a is analyzed, with the lipped chan-nel cross-section considered in the previous example. The endsections are fixed along the X1 and X2 directions, i.e., the cross-sec-tion in-plane displacements are restrained. Due to the symmetry ofthe problem, only 1/4 of the beam is analyzed, as shown in thefigure.

The beam model takes into account the cross-section symme-try by considering only the first five walls in Fig. 6a and enforcingr2 ¼ h1 ¼ h3 ¼ 0, although r1; r3 and h2 may be arbitrarily large.Only the symmetric deformation modes in Fig. 6b are includedin the analysis, namely modes 1, 3, 5, 7 and 9. The warping dis-placements associated with mode 3 are treated as a separate

Fig. 7. Lipped channel column critical bifurcation stress vs. L/h.

Fig. 8. Lipped channel beam (a) geometry and loading and (b) deformation modes 11–17.


mode (mode 10) and 3 additional warping modes are also in-cluded, associated with linear warping functions in each wall(modes 11–13 in Fig. 8b). Two analyses are carried out. The firstone includes the above nine modes and assumes SM

22 ¼ 0 (thus C*is used for the membrane terms). The second one assumes SM

22 – 0only in walls 2–3, making it necessary to include linear and atleast quadratic vðiÞ2 modes in these walls (modes 14–17 in the fig-ure), totaling 13 modes. In both cases, a regular mesh of 4 finiteelements was used, corresponding to 103 and 139 d.o.f.,respectively.

Concerning the shell model, the discretization adopted is de-picted in Fig. 10 and involves about 4200 d.o.f.

Figs. 9 and 10 show the non-linear equilibrium paths andassociated 1/4-beam deformed configurations, obtained withthe beam and shell models. The beam model with nine modesyields fairly accurate results up to F = 0.6 kN, after which the dif-ferences with respect to the shell model exceed 5%. The beammodel with 13 modes is considerably more accurate, since a

5% difference is only attained above F = 23 kN. These resultsare quite remarkable if one takes into consideration thatassumption A3 restricts the field of application of the formula-tion to small bending strains.

6.4. Lateral-torsional buckling of C-shaped cantilevers

Finally, one investigates the lateral-torsional buckling behaviorof the C-section cantilever displayed in Fig. 11a, acted by a verticalload applied at the top flange/web intersection. This example aimsat illustrating the potential of the beam formulation when largedisplacements and finite rotations are combined with cross-sectionin-plane deformation. The material properties are E = 210 GPa andm ¼ 0:3.

For the beam model, the cross-section is subdivided intothree walls and the centre C is chosen as the point of load appli-cation. A total of nine deformation modes are considered (thefirst five of them are shown in Fig. 11b): (i) mode 1 is the tor-

Fig. 9. Lipped channel beam: applied load vs. lip lateral displacement.


sion-related warping mode, calculated with C as the pole, (ii)modes 2–3 are distortional modes, associated with the rotationsof walls 1 and 3,8 (iii) mode 4 is associated with the symmetricrotation of the cross-section fold-lines, (iv) mode 5 is related tothe local bending of wall 2 in the form 1� cosð2pX2=0:2Þ and(v) modes 6–9 are linear and quadratic vðiÞ2 modes in walls 1and 3. Three analyses are carried out, with an increasing numberof modes: 1, 1–3 and 1–9 (with SM

22 – 0 in walls 1 and 3). In allcases, a non-uniform discretization using eight elements was em-ployed, corresponding to 119, 153 and 255 d.o.f., respectively. Likein Example 6.1, the fixed support is modeled by eliminating thenodal values of the kinematic parameters, but not their deriva-tives. In this particular example, the coupling of the bending termsassociated with the beam curvature and the deformation modeswas discarded.

The shell element mesh can be observed in Fig. 13 and involvesapproximately 4400 d.o.f.. The boundary conditions prevent thenodal displacements only.

The non-linear equilibrium paths are displayed in Fig. 12. Thebeam model with 1 mode yields accurate results only up tou2 ¼ 0:7 m, since it does not capture the cross-section distortionoccurring near the support. Although the beam model with 3modes provides an improved accuracy for u2 > 0:7 m, only theinclusion of all nine modes makes it possible to obtain an excellentagreement with the shell model for almost all the displacementrange, which is also illustrated in the deformed configurations dis-played in Fig. 13. Nevertheless, a close inspection of the figure re-veals that the beam model does not capture the torsional rotationaccurately for u2 ¼ 1:3 m. Finally, in the bottom of Fig. 13, anotherview of the u2 ¼ 1:1 m deformed configuration is displayed, in or-der to enable a clearer visualization of the complex torsional–dis-tortional beam behavior.

8 These modes are calculated by imposing wall rotations at the cross-section andanalyzing it as a plane frame, using small displacement theory – this procedure isanalogous to the ‘‘GBT cross-section analysis” [8,9].

7. Closing remarks

Attention is drawn to the following aspects concerning the thin-walled beam formulation presented in this work:

(i) The beam kinematic description involves genuine cross-sec-tion finite rotations and arbitrary deformation modes tomodel cross-section in-plane and out-of-plane deformation(although Kirchhoff’s assumption is deemed valid). Thedeformation modes are written in a co-rotational frameadhering to the cross-section.

(ii) The numerical examples presented in the paper show thatthe results obtained with the finite element implementationof the proposed beam formulation are in good agreementwith the ones yielded by refined meshes of shell finite ele-ments, even though only a relatively small number of defor-mation modes were included in the beam elements. One ofthe most impressive features is the fact that the numbersof degrees-of-freedom involved in the beam finite elementmodels are considerably lower than the ones in the shellmodels.

(iii) Although only elastic beams were analyzed in this work, theformulation is capable of handling other types of small strainconstitutive laws (recall that the bending strains are alwaysassumed to be small). In particular, small strain plasticitymay be implemented in a standard fashion and withouttoo much difficulty. The authors are currently workingtowards achieving this goal.

Appendix A. Directional derivatives of K; K and C

A given rotation tensor K may be obtained from the associatedrotation vector h by using the well-known Rodrigues formula [26]

K ¼ 1þ a1eh þ a2

eh2 ¼ a01þ a1eh þ a2h� h; ð49Þ

where eh denotes the skew-symmetric tensor whose axial vector is h

and the scalar functions ai read

Fig. 10. Lipped channel beam: deformed configurations.


a0 ¼ cos h; a1 ¼sin h

h; a2 ¼

1� cos h

h2 ; a3 ¼h� sin h

h3 ; ð50Þ

where h ¼ khk. For the local rotation tensor R, the rotation vector isgiven by u ¼ uE3 and Eq. (49) yields

R ¼ sin ueE3 þ cos uð1� E3 � E3Þ þ E3 � E3: ð51Þ

In order to write the directional derivatives of rotation tensors, onemay employ the scalar trigonometric functions introduced by Ritto-Corrêa and Camotim [32]

biðhÞ ¼ a0iðhÞ=h; ciðhÞ ¼ b0iðhÞ=h; ði ¼ 0;1;2;3Þ; ð52Þ

where ð�Þ0 ¼ dð�Þ=dh, which make it possible to write

dai ¼ a0idh ¼ a0ih � dh

h¼ bih � dh; dbi ¼ cih � dh: ð53Þ

, , , , , ,

Fig. 12. C-section cantilever: applied load vs. vertical displacement of the point of load application.

Fig. 11. C-section cantilever (a) geometry and loading and (b) deformation modes 1–5.


These trigonometric functions are explicitly given by

b0 ¼ �a1; b1 ¼h cos h� sin h

h3 ; b2 ¼h sin h� 2þ 2 cos h

h4 ;

b3 ¼�2hþ 3 sin h� h cos h

h5 ;

c0 ¼ �b1; c1 ¼ð3� h2Þ sin h� 3h cos h

h5 ;

c2 ¼8� 5h sin hþ ðh2 � 8Þ cos h

h6 ;

c3 ¼8hþ ðh2 � 15Þ sin hþ 7h cos h

h7 :

ð54Þ

As reported in [32], (i) with the exception of a0, the functionsexhibit a ‘‘damping” as h increases and (ii) the maximum ampli-tude decreases for the higher order derivative functions. For arbi-trary vectors a and b, the first and second directional derivativesof K, in arbitrary directions u and v, may now be expressed as[32]

DK½u�a ¼ NDKðaÞu;

DKt ½u�a ¼ NDKt ðaÞu;

D2K½u;v �a � b ¼ ND2K

ða;bÞu � v ;

ð55Þ

Fig. 13. C-section cantilever: deformed configurations.


where the auxiliary operators are explicitly given by

NDKðaÞ ¼ �a1ea � a2ðfeha þ eheaÞ þ b1ðeha� hÞ þ b2ðeh2a� hÞ;

NDKt ðaÞ ¼ a1ea � a2ðfeha þ eheaÞ � b1ðeha� hÞ þ b2ðeh2a� hÞ; ð56Þ

ND2Kða;bÞ ¼ a2ðeaeb þ ebeaÞ þ b1ðeab� hþ h� eabþ ðeha � bÞ1Þ

þ b2ððeaeb þ ebeaÞh� hþ h� ðeaeb þ ebeaÞhþ ðeh2a � bÞ1Þþ ðc1

eha � bþ c2eh2a � bÞh� h:

Note that ND2K is symmetric, but has arguments that do not com-mute. The previous expressions make it possible to write the virtualvariations of the rotation tensors and corresponding incremental/iterative linearizations as

dKa¼NDKðaÞdh; dKta¼NDKt ðaÞdh; DdKa �b¼ND2Kða;bÞDh �dh:

ð57Þ

Concerning vector C, it follows from (16) and the previous resultsthat

dC ¼ dððKRÞtr;3Þ ¼ RtðNDKt ðr;3Þdhþ Ktdr;3Þ; ð58ÞDdC � a ¼ DdððKRÞtr;3Þ � a ¼ dh � ðND2K

ðRa; r;3ÞDhþ NtDKðRaÞDr;3Þ

þ dr;3 � NDKðRaÞDh: ð59Þ

For the curvature K, we begin by noting that a material spin vectordX satisfies

dX ¼ axiðKtdKÞ ð60Þ

and that the relation between dh and dX is given by (e.g., [31,32,38])

dX ¼ T tdh; ð61Þ

where

T ¼ 1þ a2eh þ a3

eh2; T t ¼ 1� a2eh þ a3

eh2: ð62Þ

For arbitrary vectors a and b, the directional derivatives of T read[32]


DT½u�a ¼ NDTðaÞu;DT t ½u�a ¼ NDTt ðaÞu;D2T½u;v �a � b ¼ ND2Tða;bÞu � v

ð63Þ

with the auxiliary operators

NDTðaÞ ¼ �a2ea � a3ðfeha þ eheaÞ þ b2ðeha� hÞ þ b3ðeh2a� hÞ;

NDTt ðaÞ ¼ a2ea � a3ðfeha þ eheaÞ � b2ðeha� hÞ þ b3ðeh2a� hÞ;

ND2T ða;bÞ ¼ a3ðeaeb þ ebeaÞ þ b2ðeab� hþ h� eabþ ðeha � bÞ1Þ

þ b3ððeaeb þ ebeaÞh� hþ h� ðeaeb þ ebeaÞhþ ðeh2a � bÞ1Þþ ðc2

eha � bþ c3eh2a � bÞh� h:

Once more, ND2T is symmetric, but its arguments do not commute.Finally, (17), (60) and (61) lead to K ¼ RtT t

h;3 and one may write

dK ¼ RtdðT th;3Þ ¼ RtðNDT t ðh;3Þdhþ T tdh;3Þ; ð65Þ

DdK � a ¼ DdðT th;3Þ � Ra

¼ dh � ðND2TðRa; h;3ÞDhþ NtDT ðRaÞDh;3Þ þ dh;3 � NDTðRaÞDh:

ð66Þ

(64)

Appendix B. Equations expressed in terms of the kinematicparameters

Since the proposed finite element interpolates directly the kine-matic parameters in /, it is necessary to rewrite (41) and (41) interms of these parameters. The strains involve first and secondderivatives of the parameters with respect to X3 and it is thereforeconvenient to begin by defining the vectors dU and DU, which con-tain the virtual and incremental/iterative variations of /;/;3 and/;33, i.e.,

½dU� ¼½d/�½d/;3�½d/;33�

264375; ½DU� ¼

½D/�½D/;3�½D/;33�

264375; ð67Þ

where / is defined in (11) and has dimension D ¼ 6þ D (D is thenumber of deformation modes). We may then write

½d/� ¼

½dr�½dh�dpð1Þ

..

.

dpðDÞ

266666664

377777775; ½D/� ¼

½Dr�½Dh�Dpð1Þ

..

.

DpðDÞ

266666664

377777775 ð68Þ

and, therefore, vectors dU and DU have dimension 3D ¼ 3ð6þ DÞ.For implementation purposes, it is useful to define the following

auxiliary matrices, associated with first and second variations ofvectors

½da� ¼ ½HDa�½dU�; ð69ÞDda � b ¼ ½dU�t ½HD2aðbÞ�½DU�; ð70Þ

where a and b are vectors and

½HDa� ¼ ½Hð1ÞDa � ½Hð2ÞDa � ½H

ð3ÞDa �

h i; ½HD2aðbÞ�¼

½Hð11ÞD2aðbÞ� ½Hð12Þ

D2aðbÞ� ½Hð13Þ

D2aðbÞ�



D2aðbÞ�



D2aðbÞ�

2666437775

ð71Þ

with each sub-matrix of dimension 3�D and D�D, respectively,i.e.,

ð72Þ

Using these auxiliary matrices, the components of expressions (41)and (44) may be rewritten in a more suitable way for numericalimplementation purposes, i.e.,

dWint ¼�Z

L

ZS

Z t=2

�t=2½dU�t ½HDEv �

t

S22

S33

S23

264375dX1dX2dX3;

dWext ¼Z

L

ZS½dU�t ½HDu�

t ½Q �dX2dX3;

DmatdWint ½D/� ¼�Z

L

ZS

Z t=2

�t=2½dU�t ½HDEv �

t½Ct �½HDEv �½DU�dX1dX2dX3;

DgeodWint ½D/� ¼�Z

L

ZS

Z t=2

�t=2½dU�t ½HD2Ev

ðSv Þ�½DU�dX1dX2dX3;

DdWext ½D/� ¼Z

L

ZS½dU�t ½HD2uðQÞ�½DU�dX2dX3;

DdWext ½Dk� ¼Z

L

ZS½dU�t ½HDu�t ½DQ �dX2dX3;

ð73Þ

where L, S and t are the beam length, cross-section mid-line overallwidth and wall thickness, respectively, at the reference configura-tion, and, from the membrane-bending strain uncoupling (35),one has

½HDEv � ¼ ½HDEMv� þ ½HDEB

v�; ½HD2Ev

ðSvÞ� ¼ ½HD2EMvðSvÞ� þ ½HD2EB

vðSvÞ�:ð74Þ

As it will be shown next, the matrices involved in the linearizedterms are symmetric, which indicates invariance with respect to achange between dU and DU, and therefore a symmetric tangentoperator is obtained.

The membrane strain terms read

½dEMv � ¼

dEM22

dEM33

2dEM23

264375 ¼ g2 � dg2

g3 � dg3

g2 � dg3 þ dg2 � g3

264375 ¼ ½HDEM

v�½dU� ð75Þ

DdEMv � Sv ¼

DdEM22

DdEM33

2DdEM23

264375

tS22

S33

S23

264375

¼Dg2 � dg2

Dg3 � dg3 þ g3 � Ddg3

Dg2 � dg3 þ dg2 � Dg3 þ g2 � Ddg3

264375

t S22

S33

S23

264375

¼ ½dU�t½HD2EMvðSvÞ�½DU� ð76Þ

where advantage was taken from the fact that Ddg2 ¼ 0. After somealgebraic manipulation, the auxiliary matrices for a given wall read


½HDEMv� ¼

½g2�t½HDg2

�

½g3�t½HDg3

�

½g2�t ½HDg3

�þ ½g3�t ½HDg2

�

2666437775; ð77Þ

½HD2EMvðSv Þ� ¼ S22½HDg2

�t½HDg2�þS33ð½HDg3

�t ½HDg3�þ ½HD2g3

ðg3Þ�Þ

þS23ð½HDg2 �t ½HDg3 �þ ½HDg3 �

t ½HDg2 �þ ½HD2g3ðg2Þ�Þ; ð78Þ

½HDg2� ¼ ½Hð1ÞDg2

� ½0� ½0�h i

; ½Hð1ÞDg2� ¼ ½03�3� ½03�3� ½vð1Þ;2 � � � �h i

; ð79Þ

½HDg3 � ¼ ½Hð1ÞDg3� ½Hð2ÞDg3

� ½0�h i

;

½Hð1ÞDg3� ¼ ½½03�3�½RtðNDKt ðr;3Þ�elNDTt ðh;3ÞÞ�½K�vð1Þ� � � ��; ð80Þ

½Hð2ÞDg3� ¼ ½½KR�t ½�RtelT t �½vð1Þ� � � ��;

½HD2g3ðbÞ� ¼

½Hð11ÞD2g3ðbÞ� ½Hð12Þ

D2g3ðbÞ� ½0�

½0� ½0�

Sym: ½0�

2666437775;

½Hð11ÞD2g3ðbÞ� ¼

½03�3� ½03�3� ½03�1� � � �

ND2KðRb;r;3Þ

þ

ND2TðelRb;h;3Þ

26643775 ½Nt

DT t ðh;3ÞRðvð1Þ �bÞ� � � �

0 � � �

Sym: . ..

26666666666664

37777777777775; ð81Þ

½Hð12ÞD2g3ðbÞ� ¼

½03�3� ½03�3� ½03�1� � � �

½NtDKðRbÞ� ½Nt

DTðelRbÞ� ½03�1� � � �

½01�3� ½ðTRÞðvð1Þ �bÞ�t 0 � � �

..

. ... ..

. . ..

266666664

377777775:

For the bending strain, discarding the j3;3 term in (37), one obtains

½dEBv � ¼

dEB22

dEB33

2dEB23

264375 ¼ ½HDEB

v�½dU�;

DdEBv � Sv ¼

DdEB22

DdEB33

2DdEB23

264375

tS22

S33

S23

264375 ¼ ½dU�t½HD2EB

vðSvÞ�½DU�;

ð82Þ

½HDEBv� ¼ �X1 ½Hð1ÞDEB

v� ½Hð2Þ

DEBv� ½Hð3Þ

DEBv�

h i;

½Hð1ÞDEB

v� ¼

vð1Þ1;22 � � �

½03�3�E2 � E2Rt

þ�2E3 � E3

0B@1CANDTt ðh;3Þ

264375 0 � � �

0 � � �

266666664

377777775;

½Hð2ÞDEB

v� ¼

0 � � �

½03�3�E2 � E2Rt

þ�2E3 � E3

0B@1CAT t

264375 0 � � �

2vð1Þ1;2 � � �

266666664

377777775;

½Hð3ÞDEB

v� ¼

0 � � �½03�3� ½03�3� vð1Þ1 � � �

0 � � �

264375;

ð83Þ

½HD2 EBvðSv Þ� ¼ X1

½Hð11ÞD2 EB

vð2S23E3 � S33RE2Þ� ½Hð12Þ

D2 EBvð2S23E3 � S33RE2Þ� ½0�

½0� ½0�Sym: ½0�

26643775;

½Hð11ÞD2 EB

vðbÞ� ¼

½03�3� ½03�3� ½03�1� � � �½ND2 T ðb;h;3Þ� ½03�1� � � �

0 � � �

Sym: . ..

266664377775;

½Hð12ÞD2 EB

vðbÞ� ¼

½03�3� ½03�3� ½03�1� � � �½03�3� ½Nt

DT ðbÞ� ½03�1� � � �½01�3� ½01�3� 0 � � �

..

. ... ..

. . ..

266664377775:

ð84Þ

Finally, concerning ðuÞ, one has

½du� ¼ ½HDu�½dU�; ð85Þ

Ddu � b ¼ ½dU�t ½HD2uðbÞ�½DU�; ð86Þ

½HDu� ¼ ½½Hð1ÞDu� ½0� ½0��; ½Hð1ÞDu � ¼ ½½1� ½NDKðlÞ� ½KRvð1Þ� � � ��; ð87Þ

½HD2uðbÞ� ¼½Hð11Þ

D2uðbÞ� ½0� ½0�

½0� ½0�Sym: ½0�

264375;

½Hð11ÞD2uðbÞ� ¼

½03�3� ½03�3� ½03�1� � � �½ND2K

ðl;bÞ� ½NtDKðRvð1ÞÞb� � � �

0 � � �

Sym: . ..

266664377775:

ð88Þ

and, when the loading acts on C, then l ¼ 0; du ¼ dr and the auxil-iary matrices are considerably simpler (in particular, ½HD2uðbÞ� ¼ 0).

Appendix C. Recovering Vlasov’s thin-walled prismatic beamtheory

Vlasov’s classical thin-walled prismatic beam theory for opencross-sections [1] may be recovered from the proposed formula-tion. To show it, one must (i) restrict the formulation to straightbeams, small displacements and linear elastic materials, (ii) dis-card cross-section in-plane deformation, (iii) include a warpingmode vð1Þ ¼ xE3, where x ¼ xðX2Þ is the mid-line warping func-tion, and (iv) eliminate membrane shear strains (i.e., shear andbi-shear deformation) through the introduction of the followingconstraint equations

uC1;3 ¼ h2 ^ uC2;3 ¼ �h1 ^ pð1Þ ¼ h3;3 ^x;2 ¼ �l � RE1 ¼ �d; ð89Þ

where uC is the displacement of the cross-section centreC; l ¼ LA þ X2RE2 references the wall mid-line with respect to C andd measures the perpendicular distance between the wall mid-lineand C. The total number of independent kinematic parameters there-fore equals 4 (e.g., the three components of uC and the twist h3).

From (5), (6) and (28), the linearization of the position vectoryields (note that K � 1þ eh)

x ¼ X3E3 þ uC þ ð1þ ehÞlþ pð1ÞxE3; ð90Þ

where

xðX1;X2Þ ¼ xþ X1ðdA þ X2Þ; ð91Þl ¼ LA þ RðX1E1 þ X2E2Þ ¼ lþ X1RE1; ð92Þ

are the cross-section warping function and position vector with re-spect to C, respectively. Aligning the axes with each wall, the dis-placement vector between the initial and current configurationsreads

u ¼ Rtðx� x0Þ ¼ RtðuC þ h� lÞ þ pð1ÞxE3 ð93Þ


and the relevant small strain tensor components are given by

e33 ¼ u3;3 ¼ u03 � u002l2 � u001l1 þ h00x; ð94Þc23 ¼ u2;3 þ u3;2 ¼ 2X1h

0; ð95Þ

where, in order to abbreviate the notation, the subscripts ‘‘C” in uC

and ‘‘3” in h3 were dropped and ð�Þ0 ¼ dð�Þ=dX3. The above resultswere obtained using the constraint Eqs. (89) which, incidentally,may be derived from the condition cM

23 ¼ 0.In order to uncouple the equilibrium equations, the reference

axes must coincide with the cross-section central principal axesand the centre of twist must coincide with the shear centre (i.e.,d and dA must be calculated with respect to the shear centre). Then,the integration of the internal virtual work over the cross-sectionarea A yields

dWint ¼ �Z

VðEe33de33 þ Gc23dc23ÞdV

¼ �Z

LðNdu03 �M1du002 þM2du001 þ Tdh0 þ Bdh00ÞdX3 ð96Þ

which, together with the external virtual work, constitutes the var-iational statement of Vlasov’s equilibrium equations, where thestandard cross-section parameters and stress resultants are givenby

A¼Z

AdA; I1 ¼

ZA

l22dA; I2 ¼

ZA

l21dA; J¼

ZA

4X21dA; Ix¼

ZAx2dA; ð97Þ

N¼u03EA; M1 ¼�u002EI1; M2 ¼u001EI2; T ¼ h0GJ; B¼ h00EIx: ð98Þ

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