e. C

logic

Finite elements

Finite rotations

Thin-walled beams

Optimization

sist

he

he

al

ral

ed

ge

d b

derivation is greatly simplied by obtention of the derivatives of the director eld via interpolation of

nodal triads. Several numerical examples are presented to show the accuracy of the formulation and

also its frame invariance and path independence.

odernne hanhand,theorie probof comd.

Commonly, the geometrically linear composite thin-walled

theexity

niteed a

rotations [1].

Contents lists available at SciVerse ScienceDirect

w.e

Thin-Walled

Thin-Walled Structures 52 (2012) 102116geometrically exact formulation for isotropic hyperelastic beams.E-mail address: msaravia@conicet.gov.ar (M.C. Saravia).beam theories produce accurate results when modeling wings Updated and Total Lagrangian formulations valid for largedisplacements and based on a degenerate continuum conceptwere presented by Bathe and Bolourchi [2].

Simo [3] and Simo and Vu-Quoc [4,5] developed the rst 3D

0263-8231/$ - see front matter & 2011 Elsevier Ltd. All rights reserved.

doi:10.1016/j.tws.2011.11.007

n Corresponding author.representing accurately the material and the kinematic behavior ofthe structure as well as feeding the optimization algorithms.

2D exact beam theory capable of describing arbitrary largedisplacements and rotations and a 3D theory for second ordertheories are less resources consuming. In addition, the requisiteof optimization target functions containing analytical expressionsfor the cross section stiffness also represents an advantage of beamformulations. Thus, modern design of high aspect ratio wings couldbe facilitated by a beam nite element formulation capable of

the conguration. Unfortunately, this task is not trivial andconsideration of 3D nite rotations introduces a great complto the kinematic description of a beam.

Several authors have studied geometrically exact beamelement formulations. As a starting point, Reissner providNowadays the scenario is changing, optimization techniques arewidely being applied to the design of modern structures; and ofcourse high aspect ratio wings are not an exception. This turnedthe attention back to beam theories, rst because heuristic opti-mization techniques require massive computations and beam

theory.A geometrically exact beam theory must provide the exact

relations between the conguration and the strains in a fullyconsistent manner with the virtual work principle regardless ofthe magnitude of the kinematic variables chosen to parametrize1. Introduction

The study of the mechanics of minvolves two main difculties; on omaterial behavior and, on the otherdeformations. In the last years, shellover beam theories to address theshelped by the increment in poweropment of the nite element metho& 2011 Elsevier Ltd. All rights reserved.

high aspect ratio wingsd, the modeling of thethe treatment of nitees were often preferredlems. This was greatlyputers and the devel-

that suffer small deformation. High aspect ratio composite wingsnormally suffer nite deformation; this demands a good knowl-edge of geometrical nonlinearities, which considerably compli-cates the formulation of the problem. Because of that, most of thereported composite thin-walled formulations only treat approxi-mately such geometrical nonlinearities. Sometimes such approx-imations are not sufcient and higher order theories are needed.In view of this, the present work presents a geometrically exactbeam nite element based on the composite thin-walled beamA consistent total Lagrangian nite elemwalled beams

Martn C. Saravia n, Sebastian P. Machado, Vctor H

Centro de Investigacion en Mecanica Teorica y Aplicada, CONICETUniversidad Tecno

8000 Baha Blanca, Argentina

a r t i c l e i n f o

Article history:

Received 13 July 2011

Received in revised form

8 November 2011

Accepted 16 November 2011Available online 14 January 2012

Keywords:

Composite beams

a b s t r a c t

This work presents a con

anisotropic beam theory. T

with nonlinear behavior. T

and thus the cross section

written in terms of gene

derivatives. The generaliz

transformation that maps

stiffness matrix is obtaine

journal homepage: wwnt for composite closed section thin

ortnez

a Nacional, Facultad Regional Baha Blanca, 11 de Abril 461,

ent geometrically exact nite element formulation of the thin-walled

present formulation is thought to address problems of composite beams

constitutive formulation is based on the relations of composite laminates

stiffness matrix is obtained analytically. The variational formulation is

ized strains, which are parametrized with the director eld and its

strains and generalized beam forces are obtained by introducing a

neralized components into physical components. A consistent tangent

y parametrizing the nite rotations with the total rotation vector; its

lsevier.com/locate/tws

Structures

Vlasov theory for composite beams based on the variationalasymptotic beam sectional analysis was also presented by Yu et

M.C. Saravia et al. / Thin-Walled Structures 52 (2012) 102116 103They used the Reissner relationships between the variation of therotation tensor and the innitesimal rotations to derive thestrain-conguration relations, maintaining the geometric exact-ness of the theory. Simo [3] parametrized the nite rotations withthe rotation tensor, aided by the quaternion algebra to enhancethe computational efciency of the algorithm. He proposed amultiplicative updating procedure for the rotational changes,obtaining a non-symmetrical tangent stiffness.

Another important contribution to the subject was done byCardona and Geradin [6], who presented a different alternative ofparametrization, they used the incremental Cartesian rotation vectorto update the 3D rotations on the basis of the initial conguration.This approach updates the conguration on the basis of the lastconverged conguration. The additive treatment of the rotationaldegrees of freedom gives rise to a symmetrical tangent stiffness. Anisotropic hyperelastic constitutive law was assumed.

Simo and Vu-Quoc [7] incorporated shear and torsion warpingdeformation effects in his geometrically exact model. An exten-sion of the formulation of Simo to curved beams was presented byIbrahimbegovic [8]. He extended the formulation to arbitrarycurved space beams maintaining some key aspects of Simoformulation but using hierarchical interpolation. He also pro-posed an incremental rotation vector formulation [9] to solve thenonlinear dynamics of space beams.

The use of the GreenLagrange strain measures in a geome-trically exact nite element formulation for 3D beams seems tohave been introduced by Gruttmann [10,11]. He obtained aformation parametrized in terms of directors at the integrationpoint, the formulation was greatly simplied by the eliminationof high order strains. In the same direction, Auricchio [12]reviewed the Simo theory making equivalence between GreenLagrange strain measures and Reissner strain measures.

During the last years, great efforts were made to shed light tothe problem of loss of objectivity introduced by the interpolationof rotations variables, a problem rst noted by Criseld andJelenic [13]. Jelenic and Criseld [14] implemented the ideasproposed in [13] to complete the development of a strain-invariant and path independent geometrically exact 3D beamelement. Also Ibrahimbegovic and Taylor [15] re-examine thegeometrically exact models to clarify the frame invariance issuesconcerning multiplicative and additive updates of rotations.Betsch and Steinmann [16], Armero and Romero [17] and Romeroand Armero [18] further contributed to the subject presentingframe-invariant formulations for geometrically exact beams usingthe director eld to parametrize the equations of motion. In theseworks, the directors were obtained through parametrization withspatial spins; thus, the obtained tangent stiffness matrices werenon-consistent. Additional treatment of frame invariance can befound in Refs. [19,20]. Makinen [21] developed a Total Lagrangianformulation for isotropic materials. Besides obtaining a consistentstiffness matrix formulated in terms Reissner strains and totalrotations, he demonstrated that some conclusions presented in[14] regarding the frame-invariance of Total Lagrangian formula-tions were incorrect. This misconception was caused by thewrong assumption that linear interpolation of total rotations ispreserved under rigid body rotations.

All the aforementioned formulations deal with isotropic beamswith solid cross section. As a consequence, its extension tocomposite thin-walled beams is not trivial. The advantage ofthin-walled beam formulations is that the inclusion of materialanisotropy is greatly facilitated. The inclusion of anisotropicmaterials to thin-walled and also solid beam nite elementformulations was extensively studied by Hodges [22]. His workis based on the Variational Asymptotic Method (VAM) anddeserves special attention. Besides several interesting develop-

ments, he and his coworkers developed a geometrically exact,al. [23]. These developments were helped by the VariationalAsymptotic Beam Sectional Analysis software (VABS) [24], a toolfor obtaining thin-walled composite beams sectional properties.VABS is based on a 2D nite element analysis of the cross sectionto obtain the stiffness matrix of the underlying 1D theory.

An extensive review on analytical methods for solving geome-trically nonlinear problems of composite thin-walled beams wasdone by Librescu [25]. He used different analytical approaches totreat composite beams undergoing moderate rotations, treatingrotation variables in a vectorial fashion. Piovan and Cortnez [26]and Machado [27] presented a formulation for composite beamsundergoing moderate rotations. Both formulations rely on anassumed displacement eld and treat rotations as vectors, whichconfuses the actual meaning of these variables and also intro-duces uncertainty to the formulation.

In the context of thin-walled composite beams, Saravia et al.[28] presented a geometrically exact formulation for thin-walledcomposite beams using a parametrization in terms of directorvectors. This formulation used spins as rotation variables, thusobtaining an unsymmetrical tangent stiffness. The resulting niteelement implementation was path dependent and non-invariant.

This work presents a frame invariant and path independentnite element formulation of the thin-walled anisotropic beamtheory. The obtention of the cross sectional stiffness matrices isbased on the classical lamination theory and thus can handle anytype of composite material. The cross section stiffness is thusobtained analytically and without the necessity of performing a2D nite element cross sectional analysis. This opens the possi-bility of addressing optimization problems where it is desired toinclude the cross sectional shape in the target functions.

The parametrization of the nite rotation is done with the totalrotation vector. In the present formulation we use interpolation toobtain the derivatives of the director eld, thus avoiding the need of thederivatives of the rotation variables. This greatly simplies the derivationof the linearization of the GreenLagrange strains. Since the variationalformulation is expressed in terms of director eld there is no need ofreparametrization, this is in contrast to the works in [11,1618] wherethe Reissner strain measures must be reparametrized.

Regarding the objectivity and path independence of geome-trically exact formulations, it has been shown that in the presenceof nite three dimensional rotations the concept of objectivity ofstrain measures does not extend naturally from the theory to thenite element formulation [13]. Hence, despite being someformulations frame indifferent, they suffer from interpolationinduced non-objectivity. We demonstrate that in the presentformulation the discrete generalized strains satisfy the frameinvariance property and that the implementation is path inde-pendent. Also, it is shown that although other director parame-trized formulations resulted to be frame invariant and pathindependent [1618], the obtained stiffness matrices were notconsistent. We also show that it is not possible to obtain aconsistent geometrical stiffness matrix completely avoiding theuse of interpolation of the rotation. Finally, several examplesshow the present implementation has a very good correlationagainst 3D anisotropic shell theory.

2. Kinematics

The kinematic description of the beam is extracted from thefully intrinsic theory for the dynamics of curved and twistedcomposite beams, having neither displacements nor rotationsappearing in the formulation. Using the VAM, a generalizedrelations between two states of a beam, an undeformed reference

E 1 x0 Ue X0 UE x e0 Ue E0 UE ,

M.C. Saravia et al. / Thin-Walled Structures 52 (2012) 102116104state (denoted as B0) and a deformed state (denoted as B), as it isshown in Fig. 1. Being ai a spatial frame of reference, we dene twoorthonormal frames: a reference frame Ei and a current frame ei.

The displacement of a point in the deformed beam measuredwith respect to the undeformed reference state can be expressedin the global coordinate system ai in terms of a vector u(u1,u2,u3).

The current frame ei is a function of a running lengthcoordinate along the reference line of the beam, denoted as x,and is xed to the beam cross-section. For convenience, wechoose the reference curve C to be the locus of cross-sectionalinertia centroids. The origin of ei is located on the reference line ofthe beam and is called pole. The cross-section of the beam isarbitrary and initially normal to the reference line.

The relations between the orthonormal frames are given bythe linear transformations:

Ei K0xai, ei KxEi, 1

whereK0x and K(x) are two-point tensor elds ASO(3); thespecial orthogonal (Lie) group. Thus, it is satised that KT0K0 I,KTK I. We will consider that the beam element is straight, sowe set K0 I.

Recalling the relations (1), we can express the position vectorsof a point in the beam in the undeformed and deformed cong-uration, respectively, as

Xx,x2,x3 X0xX3i 2

xiEi, xx,x2,x3,t x0x,tX3i 2

xiei: 2

Where in both equations the rst term stands for the position ofthe pole and the second term stands for the position of a point inthe cross section relative to the pole. Note that x is the running

Fig. 1. 3D beam kinematics.length coordinate and x2 and x3 are cross section coordinates. Atthis point we note that since the present formulation is thought tobe used for modeling high aspect ratio composite beams, thewarping displacement is not included. As it is widely known, forsuch type of beams the warping effect is negligible [29].

Also, it is possible to express the displacement eld as

Ux,x2,x3,t xX ux,tKIX32

xiEi, 3

where u represents the displacement of the kinematic center ofreduction, i.e. the pole. The nonlinear manifold of 3D rotationtransformations K(h) (belonging to the special orthogonal LieGroup SO(3)) is described mathematically via the exponentialmap as

Kh cosyI sinyy

H 1cosyy2

h h, 4

12 2 0 2 0 2 3 3 2 3 2E13 1

2x00Ue3X00UE3x2e02Ue3E02UE3: 7

To simplify the derivation of the thin-walled beam strains weintroduce now the generalized strain vector e, a vector thatproperly transformed gives the GL strain vector. This transforma-tion actually extracts from the GL strain vector the variablesrelated to the location of a point in the cross section (i.e. xi).Therefore, the mentioned transformation is written as

EGL De, 8

where the transformation matrix is

D1 x3 x2 0 0 0 12x

22

12x

23 x2x3

0 0 0 1 0 x3 0 0 00 0 0 0 1 x2 0 0 0

26643775: 9where h[y1 y2 y3]T is the rotation vector, y its modulus and H isits skew symmetric matrix (often called spinor). Eulers theoremstates that when a rigid body rotates from one orientation toanother, which may be the result of a series of rotations (with onerotation superposed onto the previous), the total rotation can beseen as single (compound) rotation about some spatial xed axish (see e.g. [30]). Therefore, the rotation vector can be understoodas a compound rotation that globally or totally parametrizes thecompound rotation tensor via Eq. (4).

The set of kinematic variables is formed by three displace-ments and three rotations as

V : ff u,hT : 0,-R3g, u,hT u1,u2,u3,y1,y2,y3T : 5

Considering the effects of transverse shear strains gives, ingeneral, e1Ux,140.

3. Beam mechanics

3.1. Strain eld

In this section we present the strain eld obtained whenfeeding the GreenLagrange (GL) strain tensor with the kine-matics. So, we need to express the GL strain tensor in terms ofreference and current position derivatives. First, we obtain thederivatives of the position vectors of the undeformed anddeformed congurations as

X,1 X00x2E02x3E03, x,1 x00x2e02x3e03,X,2 E2, x,2 e2,

X,3 E3, x,3 e3: 6

Note that we have implicitly made the classical assumption ofbeam theories of plane cross-sections remaining plane. Proceed-ing with the derivation, we operate in a conventional way byinjecting the tangent vectors X,i and x,i into the GL strainexpression EGL(1/2)(x,iUx,jX,iUX,j) [31].

According to the kinematic hypotheses, the non-vanishingcomponents of the GL strain vector are only three. In vectornotation, it gives: EGL[E11 2E12 2E13]T, where

E11 1

2x020 X020 x2x00Ue03X00UE03x3x00Ue02X00UE02

12x22e022 E022

1

2x23e023 E023 x2x3e02Ue03E02UE03,

And the generalized strain vector is

e

Ek2k3g2g3k1w2w3w23

266666666666666664

377777777777777775

12x00Ux00X00UX00x00Ue

03X00UE03

x00Ue02X00UE02

x00Ue2X00UE2x00Ue3X00UE3e02Ue3E02UE3e02Ue

02E02UE02

e03Ue03E03UE03

e02Ue03E02UE03

2666666666666666664

3777777777777777775

: 10

As it can be observed, the generalized strain vector e containsnine generalized beam strains which belong to a material descrip-tion and are expressed in a rectangular coordinate system. Thephysical meaning of the generalized strain is: Emeasures the axial

M.C. Saravia et al. / Thin-Walled Structures 52 (2012) 102116 105strain of the reference line of the beam, k2 and k3 are the exuralcurvatures, g2 and g3 are the shear strains and k1 is the rate oftwist or torsional curvature. The meaning of the higher orderstrains is a little more involved: w2, w3, measure both torsional andexural strains and also torsionalexural coupling and exuralexural coupling strains. The last term w23 is a exuralexuraland torsionalexural coupling strain.

The derivation of strain and stress measures is helped by theintroduction of an orthogonal curvilinear coordinate system(x,n,s), see Fig. 2. The cross-section shape will be dened in thiscoordinate system by functions xi(n,s). The coordinate s is mea-sured along the tangent to the middle line of the cross section, inclockwise direction and with origin conveniently chosen. Besides,the thickness coordinate n(e/2re/2) is perpendicular to s andwith origin in the middle line contour.

In order to represent the GL strains in this curvilinear coordi-nate system we make use of the transformation tensor

P 1 0 0

0 dx2dsdx3ds

0 dx3ds dx2ds

26643775, 11

where the functions xi describe the mid-contour of the crosssection.

Hence, the GL strain vector in the curvilinear coordinatesystem is obtained by transforming the rectangular GL strains as

E^GL Exx 2Exs 2ExnT PEGL, 12The curvilinear GL strain vector can then be expressed as

E^GL PDe 13Fig. 2. Curvilinear transformation schematic.Recalling Eqs. (9) and (10), it is found that the GL strain vector incurvilinear coordinates has a remarkably simple closed expression

E^GL Ex2k3x3k212 x

22w212x

23w3x2x3w23

x02g2x

03g3x2x

03x3x

02k1

x03g2x02g3x2x

02x3x

03k1

2666437775, 14

where the prime symbol has been used to denote derivation withrespect to the s coordinate.

Now we can refer to Fig. 2 (see also [27]) to easily verify thatthe location of a point anywhere in the cross-section can beexpressed as

x2n,s x2sndx3ds

, x3n,s x3sndx2ds

, 15

where xi locates the points lying in the middle-line contour.As it will be further claried in the next section, we will use

ve independent curvilinear strain measures (collected in thevector e s) to describe the strain state of the thin-walled beamlaminate (see [32]) as

e s exx gxs gxn Kxx Kxsh iT

: 16

Pursuing the mentioned objective of describing the strain stateof the beam in terms of the generalized strain vector, we rstmove to an intermediate step and introduce Eq. (15) into Eq. (14)to express the GL strains as a function of the mid-surfacecoordinates xi and its derivatives. After doing that we found thata matrix T establishes the relationship between the GL curvi-linear strains and the generalized strains as

e s T e: 17Substituting Eq. (15) into Eq. (14) and neglecting higher order

terms in the thickness (terms in n2) we obtain

T s

1 x3 x2 0 0 0 12x2

212x

2

3 x2x30 0 0 x

02 x

03 x2x

03x3x

02 0 0 0

0 0 0 x03 x2

3 x2x02x3x

03 0 0 0

0 x02 x

03 0 0 0 x2x

03 x3x

02 x2x

02x3x

03

0 0 0 0 0 x022 x023 0 0 0

26666666664

37777777775:

18It is interesting to note that the matrix T plays the role of a

double transformation matrix that directly maps the generalizedstrains e into the curvilinear GL strain e s without the necessity ofan intermediate transformation.

Now, it is straightforward to obtain the curvilinear strains as afunction of mid-contour quantities and the generalized strains as

Es

Ek3x2k2x30:5w2x2

2w23x2x30:5w3x2

3

g2x02g3x

03k1x

03x2x

02x3

g3x02g2x

03k1x

02x2x

03x3

k2x02k3x

03w2x

03x2w3x

02x3w23x

02x2x

03x3

k1x022 x

023

26666666664

3777777777519

3.2. Constitutive relations

The most interesting capability of the present formulation is tohandle composite materials in a geometrically exact frameworkwithout modifying the classical thin-walled beam approach. Inthis section we present the equations that describe the mechanicsof the composite material. The reduction to the isotropic case isstraightforward.

For an orthotropic lamina, the relationship between the

second PiolaKirchhoff stress tensor and its energetic conjugate;

3.3. Beam forces

M.C. Saravia et al. / Thin-Walled Structures 52 (2012) 102116106the GL strain tensor, can be expressed in curvilinear coordinatesas a matrix of stiffness coefcients Qij [3233]

sxxssssnnssnsxnsxs

26666666664

37777777775

Q11 Q12 Q13 0 0 Q16Q12 Q22 Q23 0 0 Q26Q13 Q23 Q33 0 0 Q360 0 0 Q44 Q45 0

0 0 0 Q45 Q55 0

Q16 Q26 Q36 0 0 Q66

26666666664

37777777775

ExxEssEnngsngxngxs

26666666664

37777777775: 20

In matrix form

rQe s: 21In the above equation Qij are components of the transformed

constitutive (or stiffness) matrix dened in terms of the elasticproperties (elasticity moduli and Poisson coefcients) and berorientation of the ply [32].

The shell stress resultants in a lamina result from the integra-tion of 3D stresses in the thickness, and are thus dened as

Nij Z e=2e=2

sijdn, Mij Z e=2e=2

sijndn: 22

Employing Eqs. (20) and (22) and neglecting the normal stressin the thickness (i.e. snn0) it is possible to obtain a constitutiverelation between the shell forces and strains as

Nxx

Nss

Nxs

Nsn

Nxn

Mxx

Mss

Mxs

2666666666666664

3777777777777775

A11 A12 A13 0 0 B11 B12 B16A12 A22 A23 0 0 B12 B22 B26A13 A23 A33 0 0 B16 B26 B66

0 0 0 AH44 AH45 0 0 0

0 0 0 AH45 AH55 0 0 0

B11 B12 B16 0 0 D11 D12 D16

B12 B22 B26 0 0 D12 D22 D26

B16 B26 B66 0 0 D16 D26 D66

2666666666666664

3777777777777775

ExxEssgxsgsngxnkxxksskxs

2666666666666664

3777777777777775,

23where Nxx, Nss, and Nxs are axial, hoop and shear-membrane shellforces, respectively, and Nxn and Nsn are transverse shear shellforces. Also Mxx, Mss and Mxs are axial bending, hoop bending andtwisting shell moments, respectively. The same nomenclature isextended to the shell strain resultants, thus exx and ess are axialand hoop normal strains, respectively, gxs, gsn and gxn are shearshell strains and Kxx, Kss and Kxs are axial, hoop and twistingcurvatures, respectively. The coefcients Aij, A

Hij , Bij and Dij in the

constitutive matrix are shell stiffness-coefcients that result fromthe integration of Qij in the thickness [32].

Although the last relationships were derived for a singlelamina, we can obtain the constitutive relations for a laminateby spanning the integrals in the thickness of the lamina over thedifferent layers of the laminate (each layer being a single lamina).Therefore, using the hypotheses of plane stress in the laminateand rigid cross section the relations 0 simplify to

Nxx

Nxs

Nxn

Mxx

Mxs

26666664

37777775A11 A16 0 B11 B16

A16 A66 0 B16 B66

0 0 AH

55 0 0

B11 B16 0 D11 D16B16 B66 0 D16 D66

266666664

377777775

exxgxsgxnKxxKxs

26666664

37777775, 24

where Aij are components of the laminate reduced in-planestiffness matrix, Bij are components of the reduced bending-extension coupling matrix, Dij are components of the reducedbending stiffness matrix and A

H

55 is the component of the reducedtransverse shear stiffness matrix.

It must be noted that according to the plane stress hypothesis

essgns0, but in order to avoid overstiffening effects we setThe objective of this subsection is to reduce the 2D formula-tion to a 1D formulation. In order to do that, it is rst necessary toexpress the shell forces as a function of the generalized strains.Replacing Eq. (17) into Eq. (25) we obtain

Ns CT e: 26Since we are pursuing to formulate the theory in terms of

generalized quantities, we need to nd a one dimensional stress(or force) entity such as to be work conjugate with the general-ized strains. To that purpose, we rst transform the shell forces inEq. (26) back to the generalized space using the doubletransformation matrix T . Hence, we obtain the transformed backshell strain as

NGs T TNs T TCT e: 27We see that NGs is a vector of generalized shell stresses dened inthe global coordinate system. It is a function of the cross sectionmid-contour and thus integration over the contour gives thevector of generalized beam forces (work conjugate with thegeneralized strains) as

Sx ZSNGs ds

ZST TCT ds

ex, 28

Sx Dex: 29Note that since the generalized strain vector e is not a function

of the curvilinear coordinate s, (see Eq. (10)) it was taken out ofthe integral over the contour. So, the new matrix D was denedsuch that

DZST TCT ds: 30

It is good to note that D contains functions xi that dene thecross section mid-contour and also all the anisotropic materialconstants. Besides, it contains not only all geometrical couplingsbut also all material couplings. Commonly, the functions xi aredened as piecewise functions, and so the integral to evaluate Dneeds to be performed in a piecewise manner (see e.g. [25]).

The evaluation of beam constitutive matrixD does not involvea 2D nite element analysis of the cross section (as, for example,in the VABS approach [24]). Although the constitutive constantsare not as accurate as that the ones obtained with the mentionedmethod, the present approach is simpler, faster and it also opensthe possibility of addressing optimization problems of largedeformation of thin-walled composite beams. A detailed studyof the performance of both methods can be found in [29].

4. Variational formulation

The weak form of equilibrium of a three dimensional body B isgiven by [34,35]

G/,d/ ZB0rUdedV

ZB0q0bUd/dV

ZpUdumUdhdx, 31

where b, p and m are body forces, prescribed external forces andNssgns0 [32]. This generates a mild inconsistency typical ofthin-walled beam formulations

We can express the above relation in matrix form as

Ns Ce s, 25where C is the composite shell constitutive matrix and e s is thecurvilinear shell strain vector dened in Eq. (17).prescribed external moments respectively. e is the 3D GL strain

Again, dW is a skew symmetric matrix such that dWadwa.Therefore, we can rewrite Eq. (32) as

dei dw ei: 41

Now, recalling Eq. (38), we can write the last equation as afunction of the total rotation vector like

dei Tdh ei: 42The set of kinematically admissible variations can now be

dened as

dV : fd/ du,dhT : 0,-R39d/ 0 on Sg, 43

where S describes the boundaries with prescribed displacementsand rotations.

To obtain the virtual generalized strains we will also needto nd the variation of the derivative of the director eld.

M.C. Saravia et al. / Thin-Walled Structures 52 (2012) 102116 107tensor, work conjugate to the second PiolaKirchhoff stress tensorr. We note that r could be dened in either a rectangular or acurvilinear coordinate system (such a distinction is, at least at thispoint, unnecessary).

To maintain the variational formulation parametrized in termsof the director eld, its admissible variation must be found. Thenthe generalized virtual strains can be obtained; so the virtualwork of the internal and external forces can be derived. Therefore,we aim to express the virtual work principle as a function of thegeneralized virtual strain vector and its work conjugate beamforces vector.

4.1. Finite rotations and director variations

There are various ways to parametrize nite rotations: Eulerangles, a four parameter quaternion intrinsic representation [3,8],a three parameter rotational vector [6], etc. These parametriza-tions can be total or incremental, as well as their combinations,and they lead to multiplicative or additive updating procedures.

It is known that the parametrization of nite rotations withspins leads to a non-symmetric tangent matrix [4], althoughsymmetry is recovered at equilibrium. This kind of parametriza-tion has the advantage of giving very simple expressions for thetangent matrix but, as a consequence of the interpolation of spins,it has the drawback of being path dependent and non frameinvariant [14]. On the other hand, using the rotational vector toparametrize nite rotations leads to a symmetric tangent matrixbut its derivation can be more complicated due to the complexityof the linearization of the virtual strains.

In this work we choose to describe the nite rotation with therotation vector. It will be shown that the properties of frameindifference and path independency are satised and some com-mon difculties arising from the linearization are easily overcome.

To obtain the generalized strains variations, the admissiblevariation of the director eld is required. Remembering that weset K0I and recalling Eq. (1), we can writedei dKEi dKEi: 32

The admissible variation of the rotation tensor (Lie variation)can be obtained introducing an innitesimal virtual rotationsuperposed onto the existing nite rotation, see e.g. [36, 37]. Thisvirtual rotation lies in the tangent space at K (spatial virtualrotation), or in the tangent space at I (material virtual rotation),and is represented by a skew symmetric matrix dW, or dW,respectively (see Fig. 3). These variables are called spins [38].

To nd the variation of the rotation tensor we rst construct theperturbed rotation tensor by exponentiating the spatial spin as

KE expEdWK: 33At this point we note that K is a two point tensor, it takes

vectors from the tangent space in the initial conguration to thetangent space in the current conguration. Thus, we can use it torelate spatial and material spins as

dWKTdWK, dW KdWKT : 34From which we can write the material version of the kinema-

tically admissible perturbed nite rotation tensor as

KE KexpEdW: 35Enforcing the additive property to hold, it can be devised yet

another way of constructing the perturbed nite rotation tensor.Making use of the rotation vector, it is proposed

KE expHEdH: 36Recalling Eq. (33) and remembering that Kexp(H) we nd

thatexpHEdH expEdWexpH, 37where we are trying to nd an incremental rotation tensor, i.e. thevirtual rotation tensor dH, such that it belongs to the sametangent space as the rotation tensor H, i.e. TISO(3). The vector hwhose skew matrix is H is the total rotation vector.

Taking derivatives with respect to the parameter E we obtain(see e.g. [21,39])

dw Tdh, 38

where T is the spatial tangential transformation

Th sinyy

I 1cosyy2

H ysinyy3

h h: 39

These different choices for the construction of a kinematicallyadmissible representation of the perturbed rotation tensor,together with the type of algorithm chosen to perform the cong-uration update, lead to different nite element formulations: TotalLagrangian, Updated Lagrangian and Eulerian formulations [6].Since we chose the total rotation vector to parametrize the niterotation, the present formulation is Total Lagrangian.

The weak form of the equations of motion was parametrizedin terms of the current frame and its derivatives, to ease thederivation of the virtual work we use rotation variables thatbelong to the tangent space at K. Considering the latter, we willuse the spatial virtual rotation tensor (i.e. dW) to obtain thekinematically admissible variation of the rotation tensor. Recal-ling Eq. (33) we can express the variation of the rotation tensorin terms of the spatial spin as

dK ddE

expEdWK9E 0 dWK: 40

Fig. 3. Geometric interpretation of the exponential map.Noting that e0 Th0 we can nd the variation of the directors

M.C. Saravia et al. / Thin-Walled Structures 52 (2012) 102116108derivative as

de0i dTh0 Tdh0 eiTh0 Tdh ei: 44

4.2. Virtual generalized strains

The variations of the directors and its derivatives are nowused to obtain the virtual generalized strains. ConsideringdEi0 and dX00 0 and performing the variation to Eq. (10)we obtain

de

x00Udu0

e03Udu0 x00Ude03

e02Udu0 x00Ude02

e2Udu0 x00Ude2e3Udu0 x00Ude3de02Ue3e02Ude3

2de02Ue022de03Ue03

de02Ue03e02Ude03

2666666666666666664

3777777777777777775

: 45

In order to maintain the compactness of the formulation, itwill be useful to write the last expression as a function of a newset of kinematic variables du as

deHdu: 46where

H

x0T0 0 0 0 0 0

e0T3 0 0 0 0 x0T0

e0T2 0 0 0 x0T0 0

eT2 0 x0T0 0 0 0

eT3 0 0 x0T0 0 0

0 0 0 e0T2 eT3 0

0 0 0 0 2e0T2 0

0 0 0 0 0 2e0T30 0 0 0 e0T3 e

0T2

2666666666666666664

3777777777777777775

, du

du0

dwde2de3de02de03

26666666664

37777777775: 47

4.3. Internal virtual work

Having derived the expressions for the admissible varia-tions of the current basis vectors and the generalized strainswe develop in this section the expressions for the internalvirtual work of the beam. Recalling Eq. (31), the internalvirtual work of a three dimensional body can be written invector form as

Gint/,d/ ZB0deTrdV , 48

which in the curvilinear coordinate system is written as

Gint/,d/ Z

ZS

ZedeTrdndsdx: 49

We can now use the denition of the shell resultant forces inEq. (22) to reduce the 3D formulation to a 2D formulation.Therefore, integration of Eq. (49) in the n direction we can writethe internal virtual work in terms of shell quantities as

Gint/,d/ Z

ZSdeTsNsds dx: 50

The reduction to a one dimensional formulation is now aidedby the deduction of 1D beam forces presented in Eq. (28).

Transforming the virtual curvilinear shell strains into virtualgeneralized strains we can rewrite the last expression as

Gint/,d/ ZdeT

ZST TNs ds

dx 51

In which the term in parentheses is the generalized beamforces vector (see Eq. (28)). Recalling Eq. (27) the beam forcesvector can be found as a function of the shell stresses as

Sx ZST TNs ds: 52

The explicit expression of the beam forces can be found inAppendix A.1.

Finally, we write the one dimensional version of the virtualwork principle in terms of the generalized strains and thegeneralized beam forces

Gint/,d/ ZdeTSdx: 53

4.4. External virtual work

The virtual work of external forces can be written as

Gext/,d/ ZnUdumUdhdx, 54

where n is the external forces vector andm the external momentsvector. These vectors are dened according to

nZS

Zebdnds

ZStds,

mZS

ZeX bdnds

ZSX tds, 55

where b is the distributed body force vector and t is externalstress vector.

4.5. Weak form of equilibrium

The variational equilibrium statement can now be written interms of generalized components of 1D forces and strains. Recal-ling Eqs. (53) and (54) the virtual work of a composite beam ispresented in its one dimensional form as

G/,d/ ZdeTSdx

ZnUdumUdhdx: 56

Using Eq. (46) it is possible to re-write the last expression as

G/,d/ ZHduTSdx

ZnUdumUdhdx: 57

5. Linearization of the weak form

The solution of the nonlinear system of equations requiresthe linearization of these equations with respect to an incre-ment in the congurations variables. The linearization of thevariational equilibrium equations is obtained through the direc-tional derivative and, assuming conservative loading, its applica-tion gives two tangent terms; the material and the geometricstiffness matrices.

Being L[G(/,d/)] the linear part of the functional G(/,d/), wehave

LG/^,d/ G/^,d/DG/^,dfUD/, 58where the rst term G/^,df is the unbalanced force at theconguration /^ (for simplicity, the hat operator c will be

omitted hereafter). The Frechet differential in the second term is

city, we have dropped c. Applying the denition in Eq. (59) andrecalling Eqs. (53) and (45), we obtain the tangent stiffness as

M.C. Saravia et al. / Thin-Walled Structures 52 (2012) 102116 109DGint/,dfUD/ZdeTDDeDdeTSdx, 60

where is the length of the undeformed beam. The integral of therst term gives raise to the material stiffness matrix and from theintegral of the second term evolves the geometric stiffness matrix.

Using Eq. (46) the rst term of the above equation takes theform

D1Gint/,d/UD/ZduTHTDHDudx: 61

On the other hand, from Eq. (59); the general expression of thegeometric stiffness operator gives

D2Gint/,d/UD/ZDdeTSdx: 62

The linearization of the virtual generalized strains gives

Dde

du0UDu0

du0UDe03de03UDu0 x00UDde03du0UDe02de02UDu0 x00UDde02du0UDe2de2UDu0 x00UDde2du0UDe3de3UDu0 x00UDde3

de02UDe3de3UDe02e3UDde02e02UDde32e02UDde02de02UDe022e03UDde03de03UDe03

de02UDe03de03UDe02e03UDde02e02UDde03

2666666666666666664

3777777777777777775

63

To complete the development of the geometric stiffness matrixwe need to nd the linearization of the virtual generalized strainsDdeT, but we rst need to obtain the linearized virtual directors.Using Eq. (42), the linearization of the virtual directors can beobtained as

Ddei DTdh eiTdh TDh ei: 64The linearization of the virtual director derivatives is more

involved, it has a complicated expression that requires thelinearization of both the tangential map (DT) and its variation(DdT). By recalling Eq. (44) we obtain

Dde0i DdTh0 Tdh0 eidTh0 Tdh0 DeiDTh0Tdh eiTh0 DTdh ei

DdTh0 dTDh0DTdh0 TDdh0 eidTh0 Tdh0DeiDTh0 TDh0 Tdh eiTh0DTdh eiTdh Dei 65

To nd Dde in terms of the kinematic variables we would needto inject the expressions in Eqs. (64) and (65) into Eq. (63). As itwill be claried in the next section; in order to avoid the use ofsuch complicated expression for Dde0i, we will use interpolation ofDdei to obtain the discrete form of Eq. (63). So, the geometricstiffness matrix will be directly formulated in its discrete form.

6. Finite element formulation

The implementation of the proposed nite element is based onlinear interpolation and one point reduced integration (thusavoiding shear locking). The most relevant procedure of the niteobtained in a standard way as

DG/,d/UD/ ddEG/ED/9E 0, 59

where D/ fullls the geometric boundary conditions. For simpli-element implementation is the use of interpolation to obtain thederivatives of the director eld, this greatly simplies the expres-sion of the tangent matrix.

6.1. Interpolations and directors update

We interpolate the position vectors in the undeformed anddeformed conguration as

X Xnnj 1

NjX^j, xXnnj 1

NjX^j u^j, 66

where c will hereon indicate nodal values, Nj is the shapefunction value at node j and nn is the number of nodes perelement (which in the present case is 2). Using Eq. (1) the directoreld at the iteration n1 is found asn1ei Kn1hEi, 67where K is the total rotation tensor.

According to Eq. (67), we could nd the derivative of thedirectors as

e0i K0Ei 68as done in most Total Lagrangian formulations [6,21]; but thisgreatly complicates the expression for the variation of the derivativeof the directors and also requires the calculation of the derivative ofthe rotation tensor. As a consequence, the linearization process iscumbersome and the resulting expressions of the tangent stiffnessmatrices are much more complicated. In order to simplify thederivation we use interpolation to obtain the directors derivatives.So, it will be accepted that

e0iXnnj 1

N0je^ji 69

where e^ji stands for the director i at the node j. Although this

approximation is expected to be accurate enough to be used inalmost every practical situation, we will analyze in the numericalinvestigations section the impact of this approximation in theaccuracy of the solution. As it will be shown later, the use ofinterpolation to obtain the derivative of the director eld leads to apath independent solution.

6.2. Objectivity of the generalized strain measures

Several works have been devoted to demonstrate the preser-vation of the objectivity of the discrete strain measures [1320].The works of Criseld and Jelenic [13,14] shown that geometri-cally exact beam nite element formulations parametrized withiterative spins, incremental rotation vectors and total rotationvector fail to satisfy the objectivity of its discrete strain measures.Recently, Makinen [21] showed that their conclusions regardingthe objectivity of the discrete strain measures of formulationsparametrized with the total and the incremental rotation vectorare incorrect. The misleading conclusions in [13,14] about theTotal and Updated Lagrangian formulations arise from the factthat linear interpolation does not preserve an observer transfor-mation, which in the cited work was assumed.

In virtue of the desire of obtaining a formulation where thediscrete strain measures are objective, interesting works presentedformulations that gained that property by avoiding the interpola-tion of rotation variables [1618]. This was aided by parametrizingthe equation of motion in terms of nodal triads, obtaining thediscrete forms via interpolation of directors. Although the discretestrain measures derived in this works preserve the objectivityproperty, the linearization of the spins was not consistent and thetangent stiffness matrix results to be non-symmetrical (implying

the loss of the quadratic convergence property).

dei Njde^ji , de0i N0jde^ji, 74

M.C. Saravia et al. / Thin-Walled Structures 52 (2012) 102116110In the present formulation we have chosen a mixed approach,parametrizing the nite rotations with the total rotation vectorand the strain measures with the directors and its derivatives. It isinteresting note that the parametrization of the variational for-mulation with the directors greatly simplies the expressions ofthe tangent stiffness matrix (as it is showed in next Section 6.3).However, the linearization of the director variation cannot bewritten exclusively in terms of directors and it is not possible tofully eliminate interpolated rotations from the formulation. Thepropagation of interpolated rotations shall clearly be seen fromthe expression of the matrix B:

To check the objectivity of the generalized discrete strainmeasures we superpose a rigid body motion to the congurationand then test the invariance of the strains. The rigid body motionmodies the current conguration as

xn0 cQx0 eni Qei 70where cAR3 and QASO(3). If, for simplicity, we assume zeroinitial strain and we apply the above transformations to Eq. (10)and consider, for example, its effect over k2 we have

k2 Xnnj 1

N0jxj0

0@ 1AU Xnnj 1

N0je^j3

0@ 1Akn2 cQ

Xnnj 1

Njxj0

0@ 1A24 350U Q Xnnj 1

Nje^j3

0@ 1A24 350

c0 Q 0Xnnj 1

Njxj0

0@ 1AQ Xnnj 1

N0jxj0

0@ 1A24 35U Q 0 Xnnj 1

Nje^j3Q

Xnnj 1

N0je^j3

0@ 1A71

Noting that since the rigid body motion is xed c0 Q0 0, wehave

kn2 QXnnj 1

N0jxj0

0@ 1AU QXnnj 1

N0je^j3

0@ 1A Xnnj 1

N0jxj0

0@ 1AU Q TQXnnj 1

N0je^j3

0@ 1A72

Now, the orthogonality property of the superimposed rotationgives QTQ I, and thus

kn2 k2 Xnnj 1

N0jxj0

0@ 1AU Xnnj 1

N0je^j3

0@ 1A 73From which we observe that the generalized strain measure is

not affected by the superimposed rigid body motion. It is interestingto note that since linear interpolation of vector elds is invariantunder rigid body motion (i.e. Q

Pnnj 1 N

0je^ji

Pnnj 1 N

0jQe^

ji) and the

scalar product is invariant under orthogonal transformations,the above conclusion clearly makes sense. The frame invariance ofthe remaining generalized strains can be proven in a similar manner.We note that the generalized strains can be obtained by interpola-tion of nodal strains as k2

Pnnj 1 N

0jx0jUe^

ji, But although the

frame invariance property is maintained, this form of calculating thediscrete strains is less accurate.

6.3. Discrete virtual directors

The objective of this section is to obtain the discrete version ofthe virtual generalized strains and its linearization; rst we need toobtain the discrete version of the director variation and itsderivatives. Regarding the director variations, although the expres-sion in Eq. (44) does not complicate substantially the formulation,expression (65) actually does. A simpler way to obtain the directorvariations would help to simplify the expression of the tangent

stiffness very much.j 1 j 1

The obtention of the linearization of the directors and itsderivatives is more involved and requires the linearization ofthe tangential transformation dened in Eq. (39). Observing thelinearization of the variation of the directors appears in the virtualstrains (and also in its linearization) always pre multiplied bysome constant vector a, it is preferable to obtain the expressionfor this product and not only for the second variation. Thus,recalling Eq. (64) we nd that

aUDdei aUfDTdh eiTdh TDh eig 75

Switching to matrix notation, using spinors in place of crossproducts and reordering some terms we can re-write the aboveequation as

aUDdei dhTDTT ~eiadwT ~a ~e iDw, 76

where ~e i is the spinor of the director i and

DTT ~eia DTT ~e iaUDh: 77

The linearization of the term TT ~e ia gives

DTT ~e iaUDh fc1a hc2 ~ha hc3hUah hc4 ~ac5hUaIh agUDh: 78

where

c1 ycosysiny

y3, c2

ysiny2cosy2y4

,

c3 3siny2yycosy

y5, c4

cosy1y2

, c3 ysiny

y379

Now, introducing Eq. (78) into Eq. (76) and recalling Eq. (38) it ispossible to rewrite the discrete form of Eq. (76) as

aUDdeidw^TXnnj 1

NjXa,e^ji ~a ~^ej

i24 35Dw^, 80

where ~^ej

i is the spinor of the director i at node j and

Xa,e^ji T1T DTT ~e iaUDhT1: 81

In the same form, the expression for the second variation ofthe directors derivatives can be found in its discrete form bymaking use of Eq. (74)

aUDde0i dw^TXnnj 1

N0jXa,e^ji ~a ~^e

j

i24 35Dw^ 82

The last expressions show that consistent linearization ofvirtual directors necessarily leads to terms that are conjugate torotations. This precludes the possibility of obtaining a consistenttangent stiffness free of interpolated rotations.

6.4. Discrete virtual strains

Having derived the expressions for the discrete virtual direc-tors, its derivatives and its corresponding linearization, it is nowpossible to nd a discrete expression for the discrete virtualAssuming holonomic constraints we may interchange varia-tions and derivatives, i.e. d(e0)(de)0. Using this property, we canuse Eq. (69) to obtain the variation of the directors and itsderivatives as

Xnn Xnngeneralized strain and its linearization.

yMl=EI we obtain the magnitude of the two moments that,

M.C. Saravia et al. / Thin-Walled Structures 52 (2012) 102116 111We can relate the two kinematic vectors du and d/ by meansof a matrix B as

duXnnj 1

Bjd/^j, 83

where

Bj

N 0j 0

0 NjT j

0 Nj ~ejT2 T j

0 Nj ~ejT3 T j

0 N0j ~ejT2 T j

0 N0j ~ejT3 T j

2666666666664

3777777777775, d/^j

du^jdh^j

" #:

du0

dwde2de3de02de03

26666666664

3777777777584

where ~ indicates the skew symmetric matrix of a vector,c indicates a nodal variable. Thus ~eji is a skew director in thedirection i of the node j and T j is a tangential transformation at thisnode. Henceforth summation over index j will be implicitly dened,so we will omit the summation symbol and the node index i.

Finally, recalling Eq. (46) we can write the virtual generalizedstrains as

deHBd/^: 85The discrete form of the incremental virtual strains, i.e. Dde, is

more difcult to obtain. Using the structure of the geometricstiffness operator of Eq. (62) we can obtain a matrix G as tosatisfy the equality DdeTS duTGDu, a lengthy manipulationgives

G

S1 0 Q2 Q3 M3 M2A 0 0 0 0

0 0 0 0

0 M1 0

Sym 2P2 P23

2P3

26666666664

37777777775: 86

where

AX2j 1

fM2N0jQ3NjXx00,e^j3 ~x 00 ~^e

j

3

M3N0jQ2NjXx00,e^j2 ~x 00 ~^e

j

2

TN0jXe3,e^j2 ~e3 ~^e

j

2NjXe02,e^j3 ~e 02 ~^e

j

3

2P2N0jXe02,e^j2 ~e 02 ~^e

j

22P3N0jXe03,e^j3 ~e 03 ~^e

j

3

P23N0jXe03,e^j2 ~e 03 ~^e

j

2N0jXe02,e^j3 ~e 02 ~^e

j

3g 87

We note that A result to be symmetric and as a consequenceGis also symmetric. Although it is strictly not a necessary condition,the fact that the matrixG is symmetric, guarantees the symmetryof the tangent stiffness matrix.

6.5. Tangent stiffness matrix

Introducing Eq. (83) into Eq. (61) we can obtain the discreteform of the material virtual work as

D1Gint/^,d/^UDf^ZBd/^THTDHBD/^dx: 88

Then, the element material stiffness matrix is

kM ZBTHTDHBdx: 89applied at the tip of the beam, produce a deformed shape of half acircle and a full circle of a BernoulliEuler beam, respectively. Thesemoments are: M13.80761107 and M27.615221107. Fig. 4shows the deformed shapes obtained after application of thesemoments.

Tables 1 and 2 present the numerical results obtained for themaximum tip displacements for both load cases (M1 and M2).

As it can be observed from Tables 1 and 2, the present niteelement has a relatively poor performance when the mesh iscoarse. This is an expected behavior since the obtention of thederivatives of the director eld using interpolation introduces andadditional interpolation error that the formulations based on thederivative of the rotation tensor does not have. However, it isclearly seen that increasing the number of elements the solutionconverges to the solution presented in [28]. Thus, convergence ofthe proposed nite element can simply be adjusted by increasingthe mesh density.

It should be noted that for the present example the Eulerianformulation and a Total Lagrangian formulation that does not useProceeding in a similar way, we use Eqs. (86) and (62) toobtain the discrete geometric stiffness terms as

D2Gint/^,d/^UD/^ZBd/^TGBD/^dx: 90

Therefore, the element geometric stiffness matrix becomes

kG ZBTGBdx: 91

Following the standard steps of the nite element method, theelement and global tangent stiffness matrices are

kT ZBT HTDHGBdx,

KT Xelse 1

kT , 92

where the summation operator is used to represent the niteelement assembly process.

7. Numerical investigations

In this section we show the behavior of the proposed beamelement using different benchmark tests proposed in the literature.Most of existing geometrically exact nite elements cannot dealwith composite materials, so in tests involving composite materialsthe proposed nite element is compared against 3D shell modelsand the formulation presented in [28]. The shell models were builtwith Abaqus S4R elements and contain an average of 50,000 DOF.In order to test the proposed nite elements against other reportedformulations [4,40], we set the material to be isotropic. The resultspresented for the formulations [4,40] were obtained using theresearch software FEAP [41]

7.1. Accuracy assessment 1roll up of a cantilever beam

In the rst test we choose a classical pure bending test; the rollup of a cantilever beam, to test the behavior of the formulationin extreme deformation cases. We use an isotropic material tocompare the formulation against other reported geometricallyexact beam nite element formulations.

The tested specimen is a thin-walled beam with a square crosssection (b0.5, h0.5 and t0.05) and a length of 50. The materialconstants are: E144109 and n0.3. With the Euler formula:directors interpolation should give the same results, except for

7.2. Accuracy assessment 2pure bending of a cantilever beam

We test in this example the behavior of the accuracy of thepresent formulation in a full three dimensional problem where thedeformation is again large. The curved beams reference congura-tion given is a 451 circular segment with radius R100 and lying inthe xy plane (see. Fig. 5), the beam is loaded with a vertical load

7

M.C. Saravia et al. / Thin-Walled Structures 52 (2012) 10211611225

30M1M2the small frame invariance and path independence errors arisingin the Eulerian formulation in [28].

It also important to point out that the present formulationresults to be slower than the non-consistent Eulerian formulation[28], not only because it requires the computation of tangentialmap at the nodes but also because it is necessary to compute thelinearization of the tangential map, which results to be very timeconsuming.

(z direction). The properties of the isotropic material are: E1.010and n0.3. The cross section is a box with b1, h1 and t0.1.

Table 3 shows the results of the bending test for P100. Wehave used an Abaqus 3D shell model as the reference model. As itcan be seen, the present nite element formulation behavesbetter than to the Simo and Vu-Quoc element [4] available inFEAP and the Abaqus B31 beam element. The results obtainedwith the present implementation and the path dependent imple-mentation [28] are essentially the same.

The solution was reached in 5 load steps using an average of8 iterations per step.

Increasing the load to P400 we obtain also very good results(see Table 4). Note that we added to the comparison the Abaqusparabolic beam element B32. The present nite element repre-sents the kinematic behavior of the beam very well.

7.3. Anisotropic casepure bending of a cantilever beam

In this example we present a comparison of the displacementpath of the beam using an anisotropic material, we analyze the451 arc of Fig. 5 laminated with a {45,45,45,45} conguration.The laminas are made of E-Glass bers and an Epoxy matrix [32],

5 0 0 5 10 15 200

5

10

15

20

x

z

Fig. 4. Roll up test.

Table 1Displacements components for M1.

Tip vertical

displacement

Tip horizontal

displacement

Max vertical

displacement

Elements

Simo and Vu-Quoc

(FEAP)

31.673 50.448 31.673 1031.546 50.446 31.546 50

Ibrahimbegovic-Al

Mikad (FEAP)

31.673 50.448 31.673 1031.546 50.446 31.546 50

Analytic 31.831 50.000 31.831

Saravia et al. [28] 31.694 50.405 31.694 1031.567 50.403 31.567 50

Present 31.108 51.258 31.108 1031.554 50.422 31.553 50

Table 2Displacements components for M2.

Tip vertical

displacement

Tip horizontal

displacement

Max vertical

displacement

Elements

Simo and Vu-Quoc

(FEAP)

0.013 49.545 16.038 100.012 49.554 15.781 50

Ibrahimbegovic-Al

Mikad (FEAP)

0.013 49.545 16.038 100.012 49.554 15.781 50

Analytic 0.000 50.000 15.915 Saravia et al. [28] 0.016 49.494 16.004 10

0.015 49.50 15.752 50

Present 1.263 45.863 14.495 100.024 49.380 15.707 50

zTable 4Maximum displacements in a 451 arc bending test (P400).

Tip y

displacement

Tip x

displacement

Tip z

displacement

Elements

Abaqus Shell 12.201 21.546 50.997 Abaqus B31 12.401 21.311 51.110 50Abaqus B32 12.416 21.310 51.111 50Simo and Vu-Quoc

(FEAP)

12.008 20.692 50.067 50

Saravia et. al. [28] 12.205 21.015 50.880 50Present 12.206 21.019 50.884 50

Table 3Maximum displacements in a 451 arc bending test (P100).

Tip y

displacement

Tip x

displacement

Tip z

displacement

Elements

Abaqus Shell 2.090 3.641 22.611 Abaqus B31 2.574 3.570 22.734 50Simo and Vu-Quoc

(FEAP)

1.986 3.325 22.001 50

Saravia et. al. [28] 2.068 3.495 22.366 50Present 2.069 3.449 22.367 50P

x

y

Fig. 5. Bending of a 451 arc.

the material properties are given in Table 5. The cross section is abox with b1, h1 and t0.1.

To increase the complexity of the stress state in the beam wemodify the applied load to have components Px4.0105, Py4.0105, Pz8.0105. Fig. 6 presents the curves that describethe evolution of the centroidal displacements along the load path(LPF being the Load Proportional Factor) in the tip of the beamand in the middle of the beam (t and m sub indexes, respectively).

It can be seen from Fig. 6 that the correlation of the presentformulation against the Abaqus shell model is excellent. As expected,the present formulation gives the same results than [28]. This is avery good result since in contrast to [28]; the present formulation isframe invariant and path independent (as it will be shown in the nextexamples).

7.4. Anisotropic beam path independence test

We test in this example the path independence property of theproposed formulation. Using the same anisotropic curved beam ofthe previous example we apply a load P(Px,Py,Pz) in six steps andanalyze the resulting displacements at the ending of the load cycle.The loading scheme is shown in Table 6, it must be noted that theload on each step is propagated to the following step. Since theload at the end of the last step is zero in a path independentformulation the resulting displacements must also be zero.

As Table 7 shows, the present nite element is path independent,both the displacements and rotations come back to zero after retiringthe load. Also, it can be observed that this property is independent ofboth the incremental scheme and the number of elements.

xy plane that is rst loaded with a tip force F and then rotatedaround the x, y and z axes. The frame has a leg lying in the x axiswith a length of 10 and a leg parallel to the y axis with a lengthof 5. The cross section is boxed with dimensions h1, b1 and athickness of 0.1; and is made of 4 layers of E-Glass Fiber-Epoxy,laminated in a {45,45,45,45} conguration. The materialproperties are given in Table 5.

The rst load case consist on a tip force of 2107, xed in thez direction; the second load is applied in three different ways:(i) rotation around the z axis, (ii) rotation around the y axis and(iii) rotation around the x axis. For both i, ii, and iii the rotation isimposed in 4000 increments of p/20 rad each, which is equivalentto 100 revolutions.

Fig. 7 shows the evolution of displacements after completingeach revolution; as expected from a frame-indifferent formula-tion, the displacements remain constant along the revolutions.Since the constant displacements are the result of the rst loadcase and we have maintained this load case unaltered, the picturecoincides exactly for both i, ii, and iii.

The following gures (Figs. 810) show the deformed shapesof the frame in the full revolution path. It can be observed that forthe three loading schemes the deformed shapes for the 100revolutions are identical. It may be noted that the displacementsin the beam are really large, this was induced on purpose toemphasize the fact that there is no nontrivial work generated bythe xed force, still if its magnitude is really large.

7.6. Anisotropic beam frame invariance testfollower load

Now, we consider the same elbow presented in the lastexample and analyze the case where the tip load is a follower

10al D

Table 6

M.C. Saravia et al. / Thin-Walled Structures 52 (2012) 102116 1137.5. Anisotropic beam frame invariance test

This example is very similar to that proposed in Criseld andJelenic [13], it is used to show the frame-invariance of the niteelement formulation. It consist on an L-shaped frame lying in the

--20-30-40-50

wt wm

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Vertic

LPF

AbaqusSaravia et. al. [28]Present

Table 5Material properties of E-glass ber-epoxy lamina.

E11 E22 G12 G23 n12

45.0109 12.0109 5.5109 5.5109 0.3Fig. 6. Bending of an anisotropic cantilever beam0 10 20 30 40

vmum vtut

isplacement

Loading scheme.

Step Px Py Pz

1 0 0 200,000

2 0 100,000 0

3 20,000 0 0

4 0 0 200,0005 20,000 0 06 0 100,000 0displacements vs. load proportional factor.

w y1 y2 y3

1014 0.0 0.0 0.0 6.2810171015 0.0 0.0 0.0 8.291017

1015 0.0 0.0 0.0 1.0110161015 0.0 0.0 0.0 4.911017

1015 0.0 0.0 0.0 2.2310161019 0.0 0.0 0.0 3.451019

M.C. Saravia et al. / Thin-Walled Structures 52 (2012) 102116114Table 7Path dependency test results.

Remaining displacements

Inc. Elements u v

5 50 1.051014 1.8025 9.111015 9.65

10 50 4.491014 1.2525 1.181014 4.04

20 50 5.271014 1.1625 7.031015 5.91force (initially oriented in the z direction) that rotates with theframe around the y axis.

Fig. 11 shows the deformed shapes for the full rotation path of100 revolutions, it can be observed that these deformed shapescoincide for each revolution. From this experiment, we can con-clude that the present formulation is also frame-invariant. We haveonly presented the case where the elbow rotates about the y axis,but the remaining cases give exactly the same conclusion.

Finally we show in Fig. 12 the evolution of displacements forboth the xed force and the follower force.

As it can be seen from Fig. 12, the case with follower force exactlycoincides with the case of non-follower force. It is clear that both u, vand w remain unchanged as the full revolution path evolves.

0 10 20 30 40 50 60 70 80 90 1008

7

6

5

4

3

2

1

0

1

2

Revolutions

Dis

plac

emen

ts

uvw

Fig. 7. Frame invariance test of an anisotropic beamevolution of displacementswith revolutions.

5

0

5

10

5

0

5

6

4

2

0

yx

z

Fig. 8. Deformed anisotropic beam rotating around the z axis.8. Conclusions

A consistent Total Lagrangian geometrically exact nonlinearbeam nite element for composite closed section thin-walledbeams has been presented. The proposed formulation relied on

50

564202

10

10

5

0

5

yx

z

Fig. 9. Deformed anisotropic beam rotating around the y axis.

0

5

10

42

02

8

6

4

2

0

yx

z

Fig. 10. Deformed anisotropic beam rotating around the x axis.

M.C. Saravia et al. / Thin-Walled Structures 52 (2012) 102116 115105

05

10

420

10

5

0

5

10

yx

z

Fig. 11. Deformed anisotropic beam rotating around the y axisfollower force case.the parametrization of the equilibrium equations in terms of thedirector eld and its derivatives, parametrizing the nite rota-tions with the total rotation vector. The weak form of equilibriumwas written in terms of generalized strains, which result from adual transformation of the rectangular GreenLagrange strains.The variables work conjugate to the generalized strains, i.e. thegeneralized beam forces, were deduced from the curvilinear shellstresses before the obtention of the weak form.

The main capability of the proposed formulation is thepossibility of handling composite materials. Since the crosssection properties can be obtained analytically, the proposedapproach is attractive to be used in optimization problems ofcomposite beams with nite deformation such as helicopter rotorblades and wind turbine blades.

Representative numerical experiments showed that the pre-sented thin-walled beam formulation has a very good correlationagainst existing geometrically exact isotropic beam nite ele-ments. For composite materials, the correlation against 3D shellmodels was also very good.

It has been shown that the present implementation maintainsthe path independence and frame invariance properties of thenite element formulation and that interpolated rotations cannotbe fully avoided if it is desired to derive consistent tangentialtensors.

beam structures. International Journal for Numerical Methods in Engineering1979;14:96186.

elasticity. International Journal of Solids and Structures 2008;45:476681.[13] Criseld M, Jelenic G. Objectivity of strain measures in the geometrically

0 10 20 30 40 50 60 70 80 90 10087654321

012

Revolutions

Dis

plac

emen

ts u

v

w

fixed forcefollower force

Fig. 12. Frame invariance of an anisotropic beamfollower force case.exact three-dimensional beam theory and its nite-element implementation.Proceedings of the Royal Society of London. Series A: Mathematical, Physicaland Engineering Sciences 1999;455:112547.

[14] Jelenic G, Criseld MA. Geometrically exact 3D beam theory: implementationof a strain-invariant nite element for statics and dynamics. ComputerMethods in Applied Mechanics and Engineering 1999;171:14171.

[15] Ibrahimbegovic A, Taylor R. On the role of frame-invariance in structuralmechanics models at nite rotations. Computer Methods in Applied[3] Simo JC. A nite strain beam formulation. The three-dimensional dynamicproblem. Part I. Computer Methods in Applied Mechanics and Engineering1985;49:5570.

[4] Simo JC, Vu-Quoc L. A three-dimensional nite-strain rod model. Part II:computational aspects. Computer Methods in Applied Mechanics and Engi-neering 1986;58:79116.

[5] Simo JC, Vu-Quoc L. On the dynamics in space of rods undergoing largemotionsA geometrically exact approach. Computer Methods in AppliedMechanics and Engineering 1988;66:12561.

[6] Cardona A, Geradin M. A beam nite element non-linear theory with niterotations. International Journal for Numerical Methods in Engineering1988;26:240338.

[7] Simo JC, Vu-Quoc L. A geometrically-exact rod model incorporating shear andtorsion-warping deformation. International Journal of Solids and Structures1991;27:37193.

[8] Ibrahimbegovic A. On nite element implementation of geometrically non-linear Reissners beam theory: three-dimensional curved beam elements.Computer Methods in Applied Mechanics and Engineering 1995;122:1126.

[9] Ibrahimbegovic A. On the choice of nite rotation parameters. ComputerMethods in Applied Mechanics and Engineering 1997;149:4971.

[10] Gruttmann F, Sauer R, Wagner W. A geometrical nonlinear eccentric 3D-beam element with arbitrary cross-sections. Computer Methods in AppliedMechanics and Engineering 1998;160:383400.

[11] Gruttmann F, Sauer R, Wagner W. Theory and numerics of three-dimensionalbeams with elastoplastic material behaviour. International Journal forNumerical Methods in Engineering 2000;48:1675702.

[12] Auricchio F, Carotenuto P, Reali A. On the geometrically exact beam model: aconsistent, effective and simple derivation from three-dimensional nite-Acknowledgments

The authors wish to acknowledge the supports from Secretarade Ciencia y Tecnologa of Universidad Tecnologica Nacional andCONICET.

Appendix A

A.1. Beam forces

The explicit expression of the beam forces vector gives

S

N

M2

M3

Q2Q3T

P2

P3

P23

266666666666666664

377777777777777775ZS

Nxx

Mxxx02Nxxx3

Mxxx03Nxxx2

Nxsx02Nxnx

03

Nxnx02Nxsx

03

Mxsx022 x

023 Nxsx

03x2x

02x3Nxnx

02x2x

03x3

Mxxx03x2 12Nxxx

2

2

Mxxx02x3 12Nxxx

2

3

Nxxx2x3Mxxx02x2x

03x3

0BBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCA

ds,

A1where N is the axial beam force, M2 and M3are the beam exuralmoments, Q2 and Q3 are beam shear forces, T is the beam torsionmoment and P2, P3 and P23 are high order exural moments.

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M.C. Saravia et al. / Thin-Walled Structures 52 (2012) 102116116

A consistent total Lagrangian finite element for composite closed section thin walled beamsIntroductionKinematicsBeam mechanicsStrain fieldConstitutive relationsBeam forces

Variational formulationFinite rotations and director variationsVirtual generalized strainsInternal virtual workExternal virtual workWeak form of equilibrium

Linearization of the weak formFinite element formulationInterpolations and directors updateObjectivity of the generalized strain measuresDiscrete virtual directorsDiscrete virtual strainsTangent stiffness matrix

Numerical investigationsAccuracy assessment 1--roll up of a cantilever beamAccuracy assessment 2--pure bending of a cantilever beamAnisotropic case--pure bending of a cantilever beamAnisotropic beam path independence testAnisotropic beam frame invariance testAnisotropic beam frame invariance test--follower load

ConclusionsAcknowledgmentsAppendix ABeam forces

References