Force-based FE for large displacement inelastic analysis of two-layerTimoshenko beams with interlayer slips

Quang-Huy Nguyen a,n, Mohammed Hjiaj a, Van-Anh Lai b

a Structural Engineering Research Group, INSA de Rennes, Franceb University of Transport Technology, Viet Nam

a r t i c l e i n f o

Article history:Received 4 December 2013Received in revised form21 February 2014Accepted 24 February 2014

Keywords:Two-layer beam/columnPartial interactionShear flexibilityForce-based formulationLarge displacementElastoplastic buckling

a b s t r a c t

This paper presents a novel finite element model for the fully material and geometrical nonlinearanalysis of shear-deformable two-layer composite planar beam/column members with interlayer slips.We adopt the co-rotational approach where the motion of the element is decomposed into two parts:a rigid body motion which defines a local co-ordinate system and a small deformational motion ofthe element relative to this local co-ordinate system. The main advantage of this approach is that thetransformation matrices relating local and global quantities are independent from the choice of thegeometrical linear local element. The effect of transverse shear deformation of the layers is taken intoaccount by assuming that each layer behaves as a Timoshenko beam element. The layers are assumed tobe continuously connected and partial interaction is considered by adopting a continuous relationshipbetween the interface shear flow and the corresponding slip. In order to avoid curvature and the shearlocking phenomena, the local linear element is derived from the force-based formulation. The presentmodel provides an efficient tool for the elastoplastic buckling analysis of two-layer shear deformablebeam/column with arbitrary support and loading conditions. Finally, two numerical applications arepresented in order to assess the performance of the proposed formulation.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

Two-layer composite members are often used in civil engineering.Two typical examples are steel–concrete beams and nailed timbermembers. For these applications, a complete shear layer interactioncannot be obtained and a relative displacement of adjacent layersoccurs. Whereas the transverse separation is often small and can beneglected; the slip tangent to the interface surface influences thebehavior of the composite beam and must be considered.

Several theoretical models, characterized by different levels ofapproximation, have been proposed for the geometrically linearanalysis of elastic composite structures. To the best knowledge ofthe authors, the earliest and most cited work on the partial inter-action of composite beams is due to Newmark et al. in 1951 [1] andit is based on the small deformation elastic analysis consideringEuler–Bernoulli's beam theory for representing the deformation ofbeam layers. Since then, this model was extensively used by manyauthors to formulate analytical models for the static response oflinear elastic [2–7] as well as linear-viscoelastic [8–12] of compositebeams with arbitrary support and loading conditions. In addition,several numerical models based on the same basic assumptions

have been developed to investigate the behavior of compositebeams with partial interaction in the nonlinear range (for materialnonlinearities, see e.g. [13–17], and for geometric nonlinearities, seee.g. [18–20]). The most significant advances in the theory of two-layer beams in partial interaction moved recently toward theintroduction of shear flexibility of both layers according to thewell-known Timoshenko theory (see e.g. [21–35]).

The two-layer members with interlayer slips may develop non-linear geometrical and material behavior, even for small deforma-tions. In contrast with the large body of literature devoted to materialnonlinear and geometrical linear problems of shear deformablelayered beam/columns in partial interaction, only a few numericalmodels which consider both material and geometrical nonlinearities,the interlayer slip and cross-section shear flexibility can be foundin the literature. Recently, Hozjan et al. [36] developed a FE modelfor two-layer beam/column based on the shear-stiff Reissner beamtheory. This model takes into account the exact geometrical (TotalLagrangian approach) and material nonlinearities as well as finite slipbetween the layers. However, the transverse shear deformation isneglected. They developed the fundamental equations of the pro-blem which exactly account for the equilibrium between the contactsurfaces of the layers in the deformed state as well as for thetangential separation of layers at the edges. These equations werethen cast into the discretized weak form by the modified principle ofvirtual work using the unconventional finite element technique.

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Finite Elements in Analysis and Design

http://dx.doi.org/10.1016/j.finel.2014.02.0070168-874X/& 2014 Elsevier B.V. All rights reserved.

n Corresponding author. Tel.: þ33 2 23 23 83 94.E-mail address: qnguyen@insa-rennes.fr (Q.-H. Nguyen).

Finite Elements in Analysis and Design 85 (2014) 1–10

The purpose of this paper is to present a novel finite elementmodel for the fully material and geometrical nonlinear analysis ofshear-deformable two-layer composite planar beams with inter-layer slips. The effect of transverse shear is taken into account usingTimoshenko's beam theory. A co-rotational description is used,which means that the motion of the element is decomposed intotwo parts: a rigid body motion which defines a local co-ordinatesystem and a small deformational motion of the element relative tothis local co-ordinate system. The geometrical nonlinearity inducedby the large rigid-body motion, is incorporated in the transforma-tion matrices relating local and global internal force vectors andtangent stiffness matrices whereas the deformational response,captured at the level of the local co-ordinate system, is assumedto be small and modeled using a geometrical linear element. Themain advantage of the co-rotational approach is that the transfor-mation matrices relating local and global quantities are indepen-dent to the choice of the local linear geometrical element. A secondadvantage of this approach is the separation between geometricaland material nonlinearities. The local formulation is based on theforce-based approach. This choice is motivated by the fact thatshear and curvature locking can be avoided. Furthermore, force-based EF is known for being more effective in dealing with materialnonlinear problems [13]. The present model provides an efficienttool for elastoplastic buckling analysis of two-layer shear deform-able beam with arbitrary support and loading conditions. The maincontribution of the present paper is the incorporation of sheardeformation of the layers which allows for a more general treat-ment of two-layer beams with interlayer slip. This extension addscomplexity to the treatment of large displacement of layered beamswithin a co-rotational formulation. Indeed, the independent shearingof the different layers results in independent cross-section rotation ofthe layers and so in extra degree of freedom which necessarilymodifies the FE formulation itself.

2. Problem definition

Let us consider a planar composite beam element with two layersof possibly different cross-sections and materials and including shearconnectors at the interface which are uniformly distributed along thelongitudinal direction, as shown in Fig. 1. It is assumed that theinterlayer slip can occur at the interface but there is no uplift. In orderto take into account the transverse shear effect, the first-order sheardeformation beam theory of Timoshenko is used.

The co-ordinates of the nodes a1 and a2 in the global co-ordinatesystem ðx; yÞ are ðxa1; ya1Þ and ðxa2; ya2Þ, respectively. For instant,these nodes are chosen to be at the layer interface in order to deriveeasily the kinematic relationships between the global nodal dis-placements and the local ones. The general case of eccentric nodeswill be treated in Section 3.3.

The element has 10 global degrees of freedom in the fixedglobal co-ordinate system ðx; yÞ (cf. Fig. 1). The vectors of globalnodal displacements and forces are defined by

pg ¼ ua1 ub1 va1 θa1 θb1 ua2 ub2 va2 θa2 θb2

h iTð1Þ

Due to the presence of the three rigid body modes in the globalco-ordinate system, the corresponding element stiffness matrix issingular. Consequently, in general there is no flexibility matrixassociated with this global system. For this reason, the proposedforce-based element is formulated in the local system ðxl; ylÞwithout rigid body modes which translates and rotates with theelement as the deformation proceeds. In this local system, theelement has seven degree of freedoms and the vector of localdisplacements is defined as

pl ¼ ua2 ub1 ub2 θa1 θb1 θa2 θb2

h iTð2Þ

where ua2 is the axial displacement of layer a; θm1 and θm2

ðm¼ a; bÞ are the end rotations of layer m. These relative displace-ments correspond to the minimum number of geometric variablesnecessary to describe the deformation modes of the element.

3. Co-rotational formulation

In our work, a co-rotational (CR) approach is adopted to take intoaccount geometric nonlinearity. This approach is a priori based onthe kinematic assumptions: displacements and rotations may bearbitrarily large, but deformations must be small. The main advan-tage of this approach is that the formulation of the element in thelocal basic system is completely independent of the transformation,i.e. in the local system the element can be formulated as geome-trically linear and the geometric nonlinearity can be introduced inthe transformation.

3.1. Co-rotational kinematics for composite beams with partialinteraction

The idea of the co-rotational approach is to decompose themotion of the element into rigid body and pure deformationalparts, through the use of a local basic system ðxl; ylÞ whichcontinuously rotates and translates with the element (see Fig. 1).The origin of the local co-ordinate system is taken at node a1 andthe xl-axis of the local co-ordinate system is defined by the lineconnecting the nodes a1 and a2. The yl-axis is perpendicular to thexl-axis so that the result is a right handed orthogonal co-ordinatesystem. The motion of the element from the original undeformedconfiguration to the actual deformed one can thus be separated intwo parts. The first one, which corresponds to the rigid motion ofthe local frame, is described by the translation of the node a1 andthe rigid rotation of the axes. The deformational part of the motionis always small relative to the local co-ordinate system and ageometrical linear element will be used. The co-rotational methodfor 2D beams is known for many years. However, when consider-ing composite beams with interlayer slips, it is necessary to selectpertinent kinematical local and global variables.

According to the notations defined in Fig. 1, the components ofthe local displacement vector pl can be computed from those ofFig. 1. Degree of freedom in the global and local co-ordinate systems.

Q.-H. Nguyen et al. / Finite Elements in Analysis and Design 85 (2014) 1–102

the global vector pg as

θa1 ¼ θa1�ðβ�β0Þ ð3Þ

θb1 ¼ θb1�ðβ�β0Þ ð4Þ

θa2 ¼ θa2�ðβ�β0Þ ð5Þ

θb2 ¼ θb2�ðβ�β0Þ ð6Þ

ua2 ¼ ln� l0 ð7Þ

ub1 ¼ g1 cos θ1 ð8Þ

ub2 ¼ ua2þg2 cos θ2 ð9Þ

where l0 and ln are, respectively, the undeformed and thedeformed element length defined as

l0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxa2�xa1Þ2þðya2�ya1Þ2

qð10Þ

ln ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxa2þua2�xa1�ua1Þ2þðya2þva2�ya1�va1Þ2

qð11Þ

g1 and g2 denote the global slips at interface which are assumed tobe perpendicular to the average cross-section rotations. Therefore,they are defined by

gi ¼ðubi�uaiÞcos ðθiþβÞ

with θi ¼θaiþθbi

2and i¼ 1;2 ð12Þ

3.2. Element formulation

As can be seen from Eqs. (3)–(12), the local displacement canbe expressed as functions of global ones, i.e.

pl ¼ plðpgÞ ð13Þ

Then pl is used to compute the internal force vector f l and thestiffness matrix Kl in the local co-ordinate system (see Section 4).Note that f l and Kl depend only on the definition of the localstrains and not on the particular form of Eq. (13). The transforma-tion matrix Blg between the local and global displacements isdefined by

δpl ¼ Blg δpg ð14Þ

and is obtained by differentiation of Eq. (13). The global internalforce vector fg and the global tangent stiffness matrix Kg , con-sistent with pg , can be obtained by equating the internal virtualwork in both the global and the local systems, i.e.

fg ¼ BTlgf l; Kg ¼ BT

lgKlBlgþHlg ; Hlg ¼∂ðBTf lÞ∂pg

�����fl

ð15Þ

For the sake of clarity and in order to give explicitly the expressionof transformation matrices, the transformation between the localquantities and the global ones is presented here through twoconsecutive changes of variables only

pl-pa ¼ θa1 θb1 θa2 θb2 ua2 g1 g2h iT

-pg ð16Þ

For the first change of variables between pl and pa, the transfor-mation matrices giving fa and Ka as a function of f l and Kl can be

obtained using Eqs. (8) and (9).

Bla ¼

1 0 0 0 0 0 00 1 0 0 0 0 00 0 1 0 0 0 00 0 0 1 0 0 00 0 0 0 1 0 0

�g12 sin θ1 �g1

2 sin θ1 0 0 0 cos θ1 0

0 0 �g22 sin θ2 �g2

2 sin θ2 1 0 cos θ2

26666666666664

37777777777775

ð17Þand

Hla ¼Hla1f lð2ÞþHla2f lð3Þ ð18Þwhere

Hla1 ¼

�g14 cos θ1 �g1

4 cos θ1 0 0 0 �12 sin θ1 0

�g14 cos θ1 �g1

4 cos θ1 0 0 0 �12 sin θ1 0

0 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0

�12 sin θ1 �1

2 sin θ1 0 0 0 0 00 0 0 0 0 0 0

26666666666664

37777777777775

ð19Þand

Hla2 ¼

0 0 0 0 0 0 00 0 0 0 0 0 00 0 �g2

4 cos θ2 �g24 cos θ2 0 0 �1

2 sin θ2

0 0 �g24 cos θ2 �g2

4 cos θ2 0 0 �12 sin θ2

0 0 0 0 0 0 00 0 0 0 0 0 00 0 �1

2 sin θ2 �12 sin θ2 0 0 0

26666666666664

37777777777775

ð20ÞThe second change of variables from pa to pg is performed usingEqs. (3)–(7) and (12). After some algebraic manipulations, thetransformation matrices giving fg and Kg as a function of fa and Ka

are obtained as

Bag ¼

� sln

0 cln

1 0 sln

0 � cln

0 0� sln

0 cln

0 1 sln

0 � cln

0 0� sln

0 cln

0 0 sln

0 � cln

1 0� sln

0 cln

0 0 sln

0 � cln

0 1

�c 0 �s 0 0 c 0 s 0 0�1c1

�1c1

0 Δu1s12c21

Δu1s12c21

0 0 0 0 0

0 0 0 0 0 �1c2

�1c2

0 Δu2s22c22

Δu2s22c22

2666666666666664

3777777777777775ð21Þ

with

ci ¼ cosθaiþθbi

2þβ0

� �; si ¼ sin

θaiþθbi

2þβ0

� �;

Δui ¼ ubi�uai; i¼ 1; 2 ð22Þ

and

Hag ¼ðrzTþzrTÞ

l2n∑4

i ¼ 1faðiÞþ

zzT

lnfað5ÞþHag1f lð6ÞþHag2f lð7Þ ð23Þ

with

r ¼ �c 0 �s 0 0 c 0 s 0 0� �T ð24Þ

z ¼ s 0 �c 0 0 �s 0 c 0 0� �T ð25Þ

Q.-H. Nguyen et al. / Finite Elements in Analysis and Design 85 (2014) 1–10 3

Hag1 ¼14c31

0 0 0 �2c1s1 �2c1s1 0 0 0 0 00 0 0 �2c1s1 �2c1s1 0 0 0 0 00 0 0 0 0 0 0 0 0 0

�2c1s1 2c1s1 0 Δu1ð2�c21Þ Δu1ð2�c21Þ 0 0 0 0 0�2c1s1 2c1s1 0 Δu1ð2�c21Þ Δu1ð2�c21Þ 0 0 0 0 0

0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

26666666666666666664

37777777777777777775

ð26Þ

Hag2 ¼14c32

0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 �2c2s2 �2c2s20 0 0 0 0 0 0 0 �2c2s2 �2c2s20 0 0 0 0 0 0 0 0 00 0 0 0 0 �2c2s2 2c2s2 0 Δu2ð2�c22Þ Δu2ð2�c22Þ0 0 0 0 0 �2c2s2 2c2s2 0 Δu2ð2�c22Þ Δu2ð2�c22Þ

266666666666666666664

377777777777777777775

ð27Þ

3.3. General finite element with eccentric nodes

The element formulation which is developed just now uses thedisplacements at the interface of the layers as degree of freedom.However, in general the kinematic and static boundary conditionswould be arbitrary. In order to make the present formulation ableto cover any case of prescribed displacements, a change of degreesof freedom at the global level must be performed.

Let us consider the general case where the prescribed displace-ments are applied at the nodes a1, a2, b1 and b2 which are locatedrandomly over the cross-section of the element ends as illustratedin Fig. 2. The new global displacement vector pe is defined as

pe ¼ ua1 ub1 va1 θa1 θb1 ua2 ub2 va2 θa2 θb2

h iTð28Þ

According to the scheme in Fig. 2, the components of pg can beexpressed by the ones of pe as

ub1þxa1 ¼ ub1þxb1þhb1 sin ðθb1þβ0Þua1þxa1 ¼ ua1þxa1�ha1 sin ðθa1þβ0Þva1þya1 ¼ va1þya1þha1 cos ðθa1þβ0Þub2þxa2 ¼ ub2þxb2þhb2 sin ðθb2þβ0Þua2þxa2 ¼ ua2þxa2�ha2 sin ðθa2þβ0Þva2þya2 ¼ va2þya2þha2 cos ðθa2þβ0Þ ð29Þ

The transformation matrices giving fe and Ke as a function of fgand Kg are then obtained

Bge ¼

1 0 0 �ha1ca1 0 0 0 0 0 00 1 0 0 hb1cb1 0 0 0 0 00 0 1 �ha1sa1 0 0 0 0 0 00 0 0 1 0 0 0 0 0 00 0 0 0 1 0 0 0 0 00 0 0 0 0 1 0 0 �ha2ca2 00 0 0 0 0 0 1 0 0 hb2cb20 0 0 0 0 0 0 1 �ha2sa2 00 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 1

26666666666666666664

37777777777777777775

with

ca1 ¼ cos ðθa1þβ0Þsa1 ¼ sin ðθa1þβ0Þcb1 ¼ cos ðθb1þβ0Þca2 ¼ cos ðθa2þβ0Þsa2 ¼ sin ðθa2þβ0Þcb2 ¼ cos ðθb2þβ0Þ

ð30Þ

and the only nonzero terms in the matrix Hge are

Hgeð3;3Þ ¼ ha1sa1fgð1Þ�ha1ca1fgð3ÞHgeð4;4Þ ¼ �hb1sb1fgð2ÞHgeð9;9Þ ¼ ha2sa2fgð6Þ�ha2ca2fgð8ÞHgeð10;10Þ ¼ �hb1cb1fgð7Þ

ð31Þ

4. Local force-based finite element formulation

4.1. Kinematics

Consider a typical straight two-node layered beam element inthe local system ðxl; ylÞ as shown in Fig. 2. The centroidal axis ofthe layer a is taken as the beam reference axis. The layers can slipone on the other but no separation can occur at the interlayer. It isalso assumed that the cross-sections do not distort in their ownplanes. The shear deformation is taken into account by consideringthe first-order shear deformation theory of Timoshenko for eachlayer. Therefore, in the local system, two layers have the sametransversal displacement but different rotations and curvatures. Inthe local system, rotations and displacements are considered to besmall. Based on the above assumptions, the axial, shear andflexural deformations at the layer centroid are related to thedisplacements as follows:

uðx; yÞ ¼uaðx; yÞubðx; yÞvðx; yÞ

264

375¼

uaðxÞ�yθaðxÞubðxÞ�yθbðxÞ

vðxÞ

264

375 ð32Þ

where uaðxÞ and vðxÞ are, respectively, the axial and the transversedisplacement of the reference axis; θiðxÞ is the rotation of thecross-section ið ¼ a or bÞ.Fig. 2. Degrees of freedom of general element with eccentric nodes.

Q.-H. Nguyen et al. / Finite Elements in Analysis and Design 85 (2014) 1–104

The interlayer slip gðxÞ is defined as the relative axial displace-ment at the interface of layer b compared to layer a

gðxÞ ¼ ubðxÞ�uaðxÞ ð33ÞFor the sake of simplicity, ðxÞ is now omitted in all functions of x.As the deformations are assumed to be small compared to unity inthe local co-ordinate, the quadratic part of the Green–Lagrangetensor is negligible compared to the linear part. One obtains

εa;xx ¼ u0a�yθ0

a

γa;xy ¼ v0 �θa

εb;xx ¼ u0b�yθ0

b

γb;xy ¼ v0 �θb ð34Þ

where the prime denotes the differentiation with respect to x. Wedenote e vector of generalized section strains which is related tothe cross-section deformations by the kinematic relations as

eðdÞ ¼

u0a

θ0a

v0 �θa

u0b

θ0b

v0 �θb

ub�ua

2666666666664

3777777777775

ð35Þ

The conjugate internal force (stress resultant) vector D can bedefined as

D¼ Na Ma Ta Nb Mb Tb Dsc� �T ð36Þ

where Ni, Mi and Ti are, respectively, the axial force, the bendingmoment and the shear force of the layer ið ¼ a or bÞ at a givencross-section of co-ordinate x

Ni ¼ZAisi;xx dAi

Mi ¼ �ZAiysi;xx dAi

Ti ¼ZAiτi;xy dAi ð37Þ

and Dsc is the bond force at the interface.

4.2. Equilibrium equations

The equations of equilibrium which are consistent with thekinematic hypothesis stated in Section 4.2, can be obtained fromthe Principle of Virtual Work which is written asZLDTeðδdÞdx�fTl δpl ¼ 0 ð38Þ

where eðδdÞ is the vector of generalized section strains derivedfrom the virtual displacement field δd via the compatibility

Eq. (35); f l ¼ Q1 Q2 Q3 Q4 Q5 Q6 Q7

h iTis the vector of

end forces conjugated to the vector of local displacements pl (seeFig. 1). Note that for simplicity's sake, the element distributedloads (body forces) are omitted in the above expression.

Eq. (38) is rewritten in the expanded form asZL

∑i ¼ a;b

Niδu0iþMi δθ

0iþTiðδv0 �δθiÞ

� þDscðδub�δuaÞ" #

dx�fTl δpl ¼ 0

ð39ÞApplying integration by parts, the above equation is rewritten asZLðN0

aþDscÞδuaþðN0b�DscÞδubþðM0

aþTaÞδθaþðM0bþTbÞδθbþðT 0

aþT 0bÞδv

� �dx

¼ Na δuaþNb δubþMa δθaþMb δθbþðTaþTbÞδv� �L

0�fTl δpl ð40Þ

The above equation must be fulfilled for all kinematically admis-sible variations δui, δv and δθi ið ¼ a or bÞ satisfying the essentialboundary conditions δuað0Þ ¼ δvð0Þ ¼ δvðLÞ resulting in the follow-ing equilibrium equations being obtained:

N0aþDsc ¼ 0

N0b�Dsc ¼ 0

M0aþTa ¼ 0

M0bþTb ¼ 0

T 0aþT 0

b ¼ 0

9>>>>>>=>>>>>>;

in 0; L½ � ð41Þ

with the following natural boundary conditions:

NaðLÞ ¼Q1; Nbð0Þ ¼ �Q2;NbðLÞ ¼Q3

Mað0Þ ¼ �Q4; Mbð0Þ ¼ �Q5;MaðLÞ ¼Q6; MbðLÞ ¼Q7 ð42Þ

4.3. Force interpolation functions

In a force-based FE formulation, the internal forces are expressedin term of end forces by using force interpolation functions. For theregular beamwhich is statically determinate, the force interpolationfunctions, obtained from equilibrium, represent the exact distribu-tion of internal forces along the beam. However, the compositebeam in partial interaction is internally indeterminate. This can beseen from the equilibrium equation (41) where there are only fiveequations for seven unknowns. The exact distribution of internalforces is indeed not available, except for some special cases withlinear elastic behavior [25]. The number of compatibility conditionsis equal to the degree of indeterminacy. However, in our case, thenumber of compatibility conditions is infinity because of continuousproblem. Therefore, some approximations are required to overcomethis indeterminacy. In the present formulation, the axial force Nb

and the bending moment Mb are treated as redundant forces andare linearly interpolated. Moreover, Nb and Mb must satisfy thenatural boundary conditions (42) thus they can be expressed as

Nb ¼xL�1

�Q2þ

xLQ3

Mb ¼xL�1

�Q5þ

xLQ7 ð43Þ

Substituting these expressions into the equilibrium equation (41)and then applying the natural boundary conditions (42), oneobtains a relation between internal forces D and end forces pl

which can be written in matrix form as

D¼ bpl ð44Þwhere

b¼ 1L

L L�x L�x 0 0 0 00 0 0 x�L 0 x 00 0 0 �1 0 �1 00 x�L x

L 0 0 0 00 0 0 0 x�L 0 x

0 0 0 0 �1 0 �10 �1 �1 0 0 0 0

2666666666664

3777777777775

ð45Þ

is the matrix of force interpolation functions. Note that in thisapproach the equilibrium equations are satisfied pointwise (strongform). This is in contrast to the displacement-based formulationwhere the equilibrium equations are satisfied in the average sense(weak form).

4.4. Section constitutive relations

The relation between internal forces D and generalized strainse depends on the material properties and the cross-sectiongeometry of the beam. For two-layer beam in partial interaction

Q.-H. Nguyen et al. / Finite Elements in Analysis and Design 85 (2014) 1–10 5

with nonlinear material behavior, this relation can be expressed ingeneral form as

D¼ΩðeÞ ð46Þ

where Ω represents a general function that permits the computa-tion of internal forces for given generalized strains. The lineariza-tion of Eq. (46) is obtained using the tangent section stiffnessmatrix which is given as

kt ¼ka;t 0 00 kb;t 00 0 ksc;t

264

375 ð47Þ

where ksc;t is the tangent shear bond stiffness; ki;t denotes thetangent section stiffness of layer ið ¼ a or bÞ, given as

ki;t ¼

RAiEi;t dA �R

AiEi;ty dA 0

�RAiEi;ty dA

RAiEi;ty

2 dA 0

0 0RAik

si Ei;t dA

2664

3775 ð48Þ

with

Ei;t ¼∂si;xx

∂εi;xx

Gi;t ¼∂τi;xy∂γi;xy

and ksi is the shear factor that depends on the cross-section shapeof layer i.

The section tangent flexibility matrix ft , necessary in the force-based formulation, is obtained by inverting the tangent sectionstiffness matrix kt . Finally, the linearized force–deformation rela-tion for two-layer beam in partial interaction can be expressed as

ejCej�1þΔej ¼ ej�1þf j�1t ðDj�D

j�1Þ ð49Þ

where j denotes the element current Newton–Raphson state; Dj�1

denotes the section resisting forces obtained through the straindriven constitutive equations at the state j�1. In the presentformulation, to evaluate the integrals of Eq. (48) for cross-sectionswith arbitrary geometry, they are subdivided into regions of regularshapes, over which the Gauss–Lobatto quadrature integration ruleis employed. Note that this method is more accurate that the fibermethod which usually uses the midpoint integration rule [37].

4.5. Weak form of compatibility equations

The weak form of the compatibility between the deformationsderived from the implicit element displacements equation (35)and the corresponding deformations derived from the internalforces via the constitutive law equation (46) may be expressed as

ZL

∑i ¼ a;b

δNiu0iþδMiθ

0iþδTiðv0 �θiÞ

� �þδDscðub�uaÞ�δDTe

( )dx¼ 0

ð50Þ

where δD¼ ½ δNa δMa δTa δNb δMb δTb δDsc �T are theweighting functions that satisfy the differential equations ofequilibrium (41) and they are chosen as δD¼ b δpl with b definedin Eq. (45).

Applying integration by parts and considering the kinematicboundary conditions, the above equation is rewritten as

δfTl pl ¼ZLδDTe dxþ

ZL

δN0aþδDsc

δN0b�δDsc

δM0aþδTa

δM0bþδTb

δT 0aþδT 0

b

26666664

37777775

Tua

ub

θa

θb

v

26666664

37777775dx ð51Þ

This equation is satisfied for all statically admissible variations δD.Therefore, the second term on the right hand side of Eq. (51) isequal to zero. Furthermore, substituting from Eq. (44) into Eq. (51),the following expression is obtained:

δfTl pl ¼ δfTl

ZLbTe dx ð52Þ

Since δf l is a vector of arbitrary virtual forces, Eq. (52) must bevalid for all values of δf l. Therefore, this equation may besimplified in the following form:

pl ¼ZLbTe dx ð53Þ

This equation, which represents element compatibility equations,allows for the determination of the element end displacements interm of section deformations along the element.

4.6. Linearization of the element compatibility equation

Using the linearized force–deformation relation (49) andEq. (44), Eq. (53) can be expanded about the current elementstate j as follows:

Fj�1t Δf jl ¼Δpj

lþ ~p j�1l ð54Þ

where

Fj�1t ¼

ZLbTf j�1

t b dx ð55Þ

is the element tangent flexibility matrix; and

~pj�1l ¼ pj�1

l �ZLbTðej�1þf j�1

t ðDj�1�Dj�1ÞÞ dx ð56Þ

represents the element nodal displacements due to the lack ofcompatibility at the element level.

In order to use the present formulation in the general co-rotational framework, the flexibility matrix Fj�1

t must be invertedto obtain the element stiffness matrix Kj�1

t at the end of the lastiteration. As the present local formulation is derived for theelement with rigid body modes therefore Fj�1

t can be directlyinverted and Eq. (54) can be rewritten as

Kj�1t Δpj

l ¼Δf jl� ~fj�1l ð57Þ

where

~fj�1l ¼ Fj�1

t

h i�1~pj�1l ¼Kj�1

t ~pj�1l ð58Þ

4.7. Nonlinear state determination algorithm

In a standard displacement-based formulation, the state determi-nation is a strain-driven process, i.e., the stresses are obtained from thestrains which are computed from the element displacements throughthe deformation shape functions. This is a sharp contrast with force-based formulation where there are no interpolation functions to relatethe section deformations to the deformation field inside the elementto the end node displacements. Therefore, the state determinationprocedure for a force-based element is not straightforward and morecomplicated. In the present model, the state determination procedure

Q.-H. Nguyen et al. / Finite Elements in Analysis and Design 85 (2014) 1–106

developed for regular beam/column developed by Spacone et al. [37]is employed and extended in large displacement. The nonlinearsystem of equations is iteratively solved by Newton–Raphson'smethod using three imbricated loops at different levels (structurallevel, element level and cross-section level).

5. Validation and numerical examples

5.1. A simply-supported composite beam

The main objective of this numerical example is to analyze theeffect of geometrical nonlinearity on the beam deflection in theelastic range as well as in plastic range. To do so, we consider asimply-supported steel–concrete beam. The beam has a spanlength of 2800 mm loaded by a single concentrated force at mid-span. The steel section of the beam is IPE 330. The slab is 800 mmwide and 100 mm thick, longitudinally reinforced by 5 steel barsof 14 mm diameter at the mid-depth. The geometric character-istics and the material properties of the beam are shown in Fig. 3.

The von Mises plasticity model with combined isotropic andkinematic hardening is adopted for steel. An extensive descriptionfor formulation constitutive rate equations of this model can befound in [38,39]. As to the constitutive law of concrete, for the sakeof simplicity it is assumed that the shear and tension/compressionbehaviors are uncoupled and therefore the 1D constitutive law isused. The elastic linear shear behavior is adopted while the 1Delasto-plastic model developed in [40] is used for concrete intension/compression. The connection is modeled by an elastic-perfectly plastic model. Table 1 presents the constitutive modelparameters which are used for the computer analysis. Note that, inthis table, all symbols are defined in the corresponding citedreferences.

The numerical integrations over the layer cross-section areperformed using 5 Gauss–Lobato points. The same number of theGauss–Lobato point is used for the numerical integrations over theelement length.

In order to assess the performance of the present force-basedformulation, the results will be compared with the ones obtained withthe classical displacement-based formulation (see [39]). Figs. 4 and 5show the comparisons between the global load/deflection responsesobtained by linear and nonlinear geometric analyses. For the sake ofclarity in these figures, the following notations are used for thelegends of the curves: E means element; D means the displacement-based model; F means the force-based model; ML means materiallinearity; MN means material nonlinearity; GL means geometric

Fig. 3. Geometrical characteristics of studied simply-supported beam.

Table 1Input values of the constitutive models utilized for computer analysis.

Steel model [38]

f y (MPa) E (MPa) υ b c(MPa) k1 (MPa) k2 (MPa)

300 200,000 0.3 0.26 2000 17,000 21

Concrete model [40]

f c(MPa) Ec(MPa) εc(%) υ Gcl(kN/m) leq;c(mm) f t(MPa) Gf t(N/m) leq;t (mm)

34.7 31,200 2 0.2 30 100 3 60 35

Connector model

Dy (N/m) Esc(MPa)

200 300

Fig. 4. Elastic load/deflection curves.

Q.-H. Nguyen et al. / Finite Elements in Analysis and Design 85 (2014) 1–10 7

linearity and GN means geometric nonlinearity. It can be observedfrom Fig. 4 that, the same number of element force-based anddisplacement-based models gives almost the same elastic load/deflec-tion curves. This is to confirm the well-known conclusion that inelastic range the force and displacement approaches are equivalentbecause the force field is linearly related to displacement field.Regarding the geometrical nonlinearity, it starts to affect significantlythe beam deflection as the deflection exceeds 400 mm (about L/7).For instance, we can observe that the deflection increases by about100 mm (20%) at a load level of 50,000 kN. However, it can be seenfrom Fig. 5 that such beam behaves elastically up to a load equalto 500 kN. Therefore, the elastic deflection of the composite steel–concrete beam can be computed by neglecting the effect of geome-trical nonlinearity.

In the case where all materials have an elasto-plastic behavior,load/deflection curves for different numbers of displacement-based and force-based elements are shown in Fig. 5. It is shownthat a mesh of 4 force-based elements gives almost the same curveof the one of 10 force-based. It is to say that only 4 force-basedelements are required to obtain the converged nonlinear responseof the beam. However, 10 displacement-based elements do notgive yet the converged solution. This is to say that when thematerial nonlinearity is considered, the force-based model pro-duces a better global load/deflection response. The improvedaccuracy of the force-based solution is related to the fact thatthe equilibrium is satisfied in strict sense even in the plastic range.Furthermore, once again we can conclude from Fig. 5 that in thisexample the beam can be considered in large displacement whenthe deflection exceeds 14% beam length.

5.2. Buckling of a two-layer column

The buckling behavior of the two-layer column depicted inFig. 6 is investigated. The column is clamped at one end andsubjected to a compression force at the other end. For simplicitypurposes, the layers are made from the same material and theyhave the same cross-section dimension as indicated in Fig. 6. Anelastic perfectly plastic behavior is assumed for each layer.

Due to the geometric and material nonlinearities, the finiteelement problem is numerically solved in an incremental way.A specific technique is implemented within this numerical proce-dure, following [41], in order to detect the bifurcation points alongthe fundamental equilibrium path. At the end of each increment, itmust be checked whether one has gone across one or several criticalpoints. The detection of critical points is based on the singularity ofthe tangent stiffness matrix, which may be factorized following the

Crout formula Kg ¼ LdLT, where L is a lower triangular matrix withunit diagonal elements and d is a diagonal matrix. Since the numberof negative eigenvalues of Kg is equal to the number of negativediagonal elements (pivots) of d, the critical points are determined bycounting the negative pivot number and comparing its valuebetween the successive increments.

5.2.1. Elastic bucklingThe elastic buckling force of the two-layer column is now

investigated. The material parameters used for the calculation are:E¼ 8000 MPa and G¼ 3200 MPa. For this numerical example, thecolumn length of 1 m is considered. In order to access the accuracyof the present FE model, the numerical buckling load, computedwith 10 and 100 elements will be compared with the analyticalsolution developed by Le Grocgnec et al. [30] for three values ofconnection stiffness: ksc ¼ 1 MPa (no bond); ksc ¼ 1000 MPa;ksc ¼ 1;000;000 MPa (perfect bond). The results are given inTable 2. It can be seen that with 10 elements the relative errorin buckling load is about 2.68% in the case of no bond and goesdown to 1.95% in the case of perfect bond. These relativedifferences are significantly improved with a mesh of 100 ele-ments. The proposed model is thus accurate in predicting thebuckling load of composite column in partial interaction takinginto account the shear flexibility of the layers.

5.2.2. Elastoplastic bucklingThe influence of the material nonlinearity on the critical buckling

load of the two-layer column depicted in Fig. 6 is now investigated.For the sake of simplicity it is assumed that the shear behavior islinear elastic given by τ¼ G γ while the tension/compressionbehavior is elastic perfectly plastic as shown in Fig. 6. The constitu-tive law of connection is taken to be linear. Fig. 7 displays theevolution of the ratio elastoplastic buckling load and plastic loadversus the slenderness of the column for different values ofconnection stiffness. The plastic load is defined as Ppl ¼ ðAaþAbÞsy

where sy ¼ 70 MPa being the yield strength. The slenderness of the

column is defined as λ¼ LffiffiffiffiffiffiffiA=I

pwhere A¼ AaþAb is the total area

and I ¼ Iaþ Ib is the total inertia moment of the cross-section withrespect to its centroid. It can be observed that for the slendernesssmaller than 30 and for any value of connection stiffness the columnbucks after reaching the yield strength of the cross-section. As anelastic perfectly plastic model was adopted for the material law, thebuckling load is thus equal to the plastic load. Furthermore, it clearlyappears that for the slenderness greater than 30 buckling load thebuckling load hardly depends on the connection stiffness. Forinstant, with a slenderness of 60 the buckling load decreases about68% from no bond to perfect bond.

Fig. 5. Elasto-plastic load/deflection curves.

Fig. 6. Geometrical, material and loading data of the two-layer column.

Q.-H. Nguyen et al. / Finite Elements in Analysis and Design 85 (2014) 1–108

6. Conclusion

In this paper, novel geometric nonlinear FE formulation for theanalysis of two-layer beams/columns with interlayer slips has beenderived. A co-rotational description has been adopted and theelement is obtained using rather simple transformation matricesbetween local and global quantities where the local quantities arederived using a geometrical linear formulation. The main advantageof the present approach is that the transformation matricesbetween local and global quantities are independent to the choiceon the local linear element. This means that, using the present co-rotational framework, different geometrical linear elements can beeasily transformed into nonlinear ones. The local element has beenderived using force-based formulation. The effect of transverseshear of cross-sections was taken into account using Timoshenko'sbeam theory. Two numerical examples have been carried out inorder to validate and investigate the performance of the proposedFE model. It has been shown that our results are in good agreementwith the ones obtained with the existing displacement-based FEmodel in terms of the load/deflection curve of composite beam andwith the analytical model in terms of elastic buckling load ofcomposite column. The numerical results showed that in the caseof full material and geometrical linearities the force-based elementperformed better the load/deflection response than the classicaldisplacement-based element. Furthermore, it has been seen thatthe connection stiffness plays a very important role on the elasto-plastic buckling load of slender columns.

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Table 2Comparison of elastic buckling load between the present numerical model andanalytical solution.

ksc[MPa] Panalyticalcr [kN] Pnumerical

cr [kN] (relative error [%])

10 elements 100 elements

1 3307 3396 (2.68%) 3338 (0.94%)1000 9831 10,040 (2.13%) 9901 (0.71%)1,000,000 12,619 12,865 (1.95%) 12,698 (0.63%)

Fig. 7. Elastoplastic buckling load versus slenderness for different connectionstiffnesses.

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