A two-field mixed variational principle for partially connected composite beams

Finite Elements in Analysis and Design 37 (2001) 929–959www.elsevier.com/locate/!nel

A two-!eld mixed variational principle for partially connectedcomposite beams

Ashraf Ayoub1

Department of Civil and Environmental Engineering, Stanford University, Stanford CA 94305-4020, USA

Abstract

This paper presents a new beam-column element for the nonlinear analysis of partially connectedcomposite beams. The element is based on a two-!eld mixed variational principle, where both variables,forces and deformations, are simultaneously approximated within the element. The model neglects upliftand frictional e+ects. Shear connectors are modeled using a distributed interface element. Two algorithmsfor the proposed model are presented. Stability of both algorithms is discussed. Numerical examples thatclarify the advantages of the proposed model over standard models based on the principle of minimumpotential energy, and discuss the di+erent alternatives of the order of interpolation functions are presented.The studies prove the superiority of the mixed model.? 2001 Elsevier Science B.V. All rights reserved.

Keywords: Composite beams; Mixed formulation; Variational principles

1. Introduction

Composite beams, such as steel–concrete beams, are widely used for 3oor construction inbuildings and bridges. They are considered the preferred structural system for carrying mainlygravity loads. The mechanical bond established by means of shear connectors, forces the concreteslab to act as an integral part of the beam decreasing the weight of the steel beam, while stillincreasing the sti+ness of the 3oor system. The e+ect of composite action in the analysis of suchbeams was studied by several researchers for the linear elastic case. Little attention, however,was given to that e+ect for nonlinear behavior, where most studies assumed perfect interactionbetween steel and concrete.

E-mail address: [email protected] (A. Ayoub).1 Formerly, Research Assistant, Department of Civil and Environmental Engineering, University of California,

Berkeley.

0168-874X/01/$ - see front matter ? 2001 Elsevier Science B.V. All rights reserved.PII: S 0168-874X(01)00076-2

930 A. Ayoub / Finite Elements in Analysis and Design 37 (2001) 929–959

Timoshenko [1] developed a theory for composite beams made of two bonded materials.Bernoulli–Euler beam theory of each part is applied and the constraint of similar transverse dis-placement is enforced. Newmark et al. [2] studied the equilibrium of elastically connected steel–concrete beams. Their work considered slip e+ects but neglected uplift and friction. Adekola[3] extended this work by including uplift and frictional e+ects. He suggested a !nite di+erenceprocedure for solving the di+erential equation of uplift and axial forces.

Ma [4], and Robinson and Naraine [5] further studied the topic of slip and uplift using thesame approach to discuss the e+ect of whether the load is acting on the slab or pulling down onthe steel beam. Their work, however, was limited to elastic cases. Cosenza and Mazzolani [6]proposed a new solution for the same problem for di+erent loading conditions also for elasticcases, as is discussed in Viest et al. [7].

McGarraugh and Baldwin [8] proposed a simpli!ed method for the analysis of steel–concretecomposite beams. They introduced the concept of partial shear connection, where the beamsti+ness was decreased according to its degree of shear connection. The degree of shear con-nection is the ratio of the actual bonded area to the bonded area corresponding to a 100% shearconnection. The latter is de!ned with empirical relations. The relation between the reductionin beam sti+ness and the degree of shear connection is de!ned according to empirical relationsbased on experimental data. Based on those relations, it was proven that the strength interpo-lation between no composite action and full composite action follows a curve higher than astraight line.

Several inelastic models for composite materials have been proposed. Wegmuller and Amer[9] used a layered beam-plate element to study fully bonded composite beams. Hirst and Yeo[10] used quadrilateral isoparametric !nite elements to study partially connected compositebeams. Razaqpur and Nofal [11] used a three-dimensional bar element to model shear con-nectors, a quadrilateral element to model the web, and a 3at shell element to model the 3angesand the slab. Bursi and Ballerini [12] used a smeared crack two-dimensional model to studythe behavior of composite frames.

The use of two- and three-dimensional elements is computationally very demanding and notpractical for large structural systems. For practical reasons, it is better to use one-dimensionalcomposite beam elements. El-Tawil and Deierlein [13] developed a distributed plasticity modelto study composite reinforced concrete–steel (RCS) frames. Their work, however assumed fullcomposite action between steel and concrete.

Arizumi et al. [14] used a displacement-based model with cubic polynomial interpolationfunctions to study partially connected composite beams. Daniel and Crisinel [15] used also adisplacement-based !ber model, where the cross-section is discretized into !bers with nonlinearconstitutive law for the corresponding material. Their work was extended by Amadio and Fra-giacomo [16] to include creep and shrinkage e+ects. Hajjar et al. [17] used the same approachto study the e+ect of interlayer slip in composite concrete-!lled tubes.

Ciampi and Carlesimo [18] proposed a new force-based model for the general analysis ofbeam-type problems. Spacone et al. [19] extended the model in the context of !ber analysis tosimulate the complex hysteretic behavior of reinforced concrete members under cyclic loading.Yassin [20] and Monti et al. [21] used also a force-based model to study bond problems inreinforced concrete structures. Ayoub and Filippou [22] followed their same approach to studycomposite beams with partial interaction. The concrete slab and steel girder were modeled with a

A. Ayoub / Finite Elements in Analysis and Design 37 (2001) 929–959 931

beam element, and the shear connectors were modeled with spring elements. The interface shearstresses were linearly distributed along the element length. Salari et al. [23] extended further thesame idea to study composite beams using a force-based element. Cubic interpolation functionswere used to approximate the interface shear stresses. The slip equation, however, is satis!edonly in a weak sense, and indeed the slip distribution shows a discontinuous behavior acrosselement boundaries.

This paper presents a new inelastic one-dimensional composite beam model that combinesthe advantages of both the displacement-based and force-based models, while overcoming mostof their limitations. The model is based on a complete two !eld mixed approach where bothvariables, forces and deformations, are simultaneously approximated within the element. Themodel is an extension of the work by Ayoub and Filippou [24] to model anchored reinforcingbar problems. The model was used by Ayoub [25], and Ayoub and Filippou [26] for seismicanalysis of composite steel–concrete frame structures.

The governing di+erential equations of the strong form of the problem are considered !rst.

2. Governing equations of composite beams

2.1. Equilibrium

The equilibrium of composite beams connected by shear connectors was studied by Adekola[3] for linear elastic behavior. Accordingly, consider an element of length dx of the compositesection shown in Fig. 1. The equations of equilibrium for the top and bottom beam elements

Fig. 1. Composite beam segment.


are:Top beam element: Bottom beam element:Nt; x = sb; Nb; x = − sbVt; x = − (wy + T ); Vb; x =TMt; x =Vt + h1Nt; x ; Mb; x =Vb − h2Nb; x

(1)

where Nt, and Nb are the axial forces in the upper and lower beams, respectively, sb is the shearforce on the connector per unit length, wy is the applied external load, T is the uplift force perunit length, Vt; Mt; Vb; Mb are the shear force and bending moment for the top and bottom beamelements, respectively, h1 and h2 are the distances between the centroid of the top and bottombeam elements, respectively, and the interface level, and a comma denotes derivativation. FromEq. (1)

Mt; xx =Vt; x + h1Nt; xx = − (wy + T ) + h1Nt; xx ;

Mb; ; x =Vb; x − h2Nb; ; x =T − h2Nb; ; x : (2)

The total external moment will be resisted by the sum of moments in top and bottom beamsplus the moment arising from the axial forces

Mt =M + (h1 + h2)N; (3)

where N =Nb = − Nt, and M =Mt + Mb.Using (2), we get

Mt; xx = − wy; M;xx = − wy − (h1 + h2)N;xx : (4)

The axial force can be determined from Eq. (1), and is modi!ed in the region of negativeuplift by adding the frictional contribution

N;x = − sb − �C; (5)

where C is the negative uplift force per unit length, and � is the coeMcient of friction at theinterface level. The frictional resistance in the region of negative uplift acts to reduce the shearforce on the connectors, thus reducing the corresponding slip values.

Eqs. (4) and (5) represent the equilibrium equations of the axial and 3exural forces, respec-tively. Writing the equilibrium equations for partially connected composite beams in matrixform

LTs(x) − LTb sb(x) = w; (6)

where

s(x) = [Nb(x) Nt(x) M (x)]T; L =

d=dx

d=dxd2=dx2

and

Lb = [ − 1 1 H d=dx]

are di+erential operators,

H = h1 + h2 and w = [0 0 wy]T:


2.2. Compatibility

The uplift tension force T arises from the connectors deformation due to di+erential verticaldisplacements between the top and bottom beams. Assuming linear elastic relations

T = kt(vb − vt); (7)

where kt is the uplift tension modulus, vb and vt are the vertical displacements of the bottomand top elements, respectively. For the case of in!nite kt:

vb = vt; v′b = v′t; and v′′b = �b = v′′t = �t; (8)

where �b and �t are the curvatures of the steel and concrete elements, respectively. In this caseT cannot be determined by (7).

For the case of rigid uplift modulus, the compatibility equations are

ub; x = �b; ut; x = �t; (9)

where ub is the bottom beam axial displacement and �b is the bottom beam axial strain, ut isthe top beam axial displacement and �t is the top beam axial strain. And

v;xx = �; (10)

where v is the vertical displacement of the beam, and � is the curvature.Eqs. (9) and (10) represent the compatibility equations of the axial and 3exural responses,

respectively. Writing the compatibility equations for composite beams in matrix form

Lu(x) − e(x) = 0; (11)

where u(x) = [ub(x) ut(x) v(x)]T, and e(x) = [�b(x) �t(x) �(x)]T.The slip eb(x) at the interface between the top and bottom beams is determined as follows:

eb(x) = Lbu(x); (12)

where Lb is as de!ned before in (6).

2.3. Constitutive laws

The section constitutive law is

s(x) =fsec[e(x)]; (13)

where s(x) and e(x) are as de!ned before in (6) and (11), respectively, and fsec is a nonlinearfunction that describes the section force deformation relation. The section force deformationrelation in this study is determined through !ber integration.

The bond constitutive law is

sb(x) =fbond[eb(x)]; (14)

where sb(x) and eb(x) are as de!ned before in (1) and (12), respectively, and fbond is anonlinear function that describes the bond load-slip relation.

In the following discussion a new !nite element model for the composite beam problem ispresented and compared to the classical model based on the principle of minimum potential


energy. Both models assume rigid uplift modulus and neglect frictional e+ects. Robinson andNaraine [5] showed that in general uplift e+ects are small. Both models are implemented in thegeneral purpose !nite element program FEAP developed by Prof. R.L. Taylor, and describedin details in Zienkiewicz and Taylor [27]. The classical formulation based on the principle ofminimum potential energy is presented !rst.

3. Minimum potential energy formulation (displacement model)

In the minimum potential energy formulation, the displacements serve as primary variables.The governing equations of the problem are cast by assuming prede!ned continuous displace-ment !elds along the beam length:

u(x) = a(x)v; (15)

where u(x) = [ub(x) ut(x) v(x)]T is the vector of displacements at a given point x; ub(x) andut(x) are the bottom and top axial displacements, respectively, v(x) is the vertical displacement,a(x) is a matrix of 3×nd displacement shape functions, nd depends on the order of displacementshape functions and equals the total number of displacement degrees of freedom, and v is thevector of element end displacements. Similarly

e(x) = B(x)v; (16)

eb(x) = Bb(x)v; (17)

where B(x) = La(x) and Bb(x) = Lba(x); L and Lb are as de!ned before in (6).The !nite element formulation is derived by considering the following variational equation:

Iv =∫" uT(x)LTs(x) d" −

∫"sb(x)Lb u(x) d"= 0; "= [0; L]: (18)

For simplicity of calculations, it is easier to replace L and Lb in the above equation by OL = IL,and OLb = ILb, respectively, where

I =

−1

−11

:

Integrating by parts the !rst term and taking the transpose of the second term

Iv = uT(x)S1OLTs(x)|L0 −

∫"(L1 u(x))TS1OL

Ts(x) d"

−∫" uT(x)OL

Tb sb(x) d"= 0; "= [0; L]; (19)


where the integral operator

S1 =

∫dx ∫

dx ∫dx

and the di+erential operator

L1 =

d=dx

d=dxd=dx

:

It is important to note that S1LT1 = I, the identity matrix, and

S1OLT

=

−1

−1d=dx

= L

T:

Integrating by parts again the last two terms and substituting for LT:

Iv = uT(x)LTs(x)|L0 − (L1 u(x))TS2L

Ts(x)|L0 − uT(x)S2OL

Tb sb(x)|L0

+∫"(L2L1 u(x))TS2L

Ts(x) d" +

∫"(L2 u(x))TS2OL

Tb sb(x) d"= 0; "= [0; L];

(20)

where the integral operator

S2 =

− s

− s ∫dx

;

s is a function that is equal zero at the boundaries and one inside the domain, and the di+erentialoperator

L2 =

1

1d=dx

= IL:

It is important to note that S2LT2 = LT

2S2 = I inside the domain, S2LT

= I the identity matrix,and L2L1 = L.

Using the previous relations yields

Iv =∫" u(x)TLTs(x) d" +

∫" uT(x)LT

b sb(x) d" − Boundary Terms (BT) = 0;

"= [0; L]; (21)

where BT = − [ uT(x)LTs(x)|L0 − ( uT(x)LT

1 s(x)|L0 − uT(x)S2OLTb sb(x)|L0].


An incremental Newton–Raphson iterative technique is used to solve the nonlinear problem.Accordingly

I iv = I i−1v +

@@v

I i−1v dvi = 0; (22)

where i is the iteration number.Substituting (21) into (22)

I iv =∫" u(x)TLTsi(x) d" +

∫" uT(x)LT

b sib(x) d" − Boundary Terms (BT) = 0;

"= [0; L]: (23)

Substituting the prede!ned displacement shape functions into (23), we get

I iv = vT[∫

"B(x)Tsi(x) d" +

∫"

Bb(x)Tsib(x) d" − Boundary Terms]

= 0;

"= [0; L]; (24)

where B(x), and Bb(x) are as de!ned in (16) and (17), respectively.Eq. (24) represents the weak form of the minimum potential energy formulation.The second term of (22) is used to derive the tangent sti+ness matrix and is equal to

@@v

I i−1v dvi

=

[ vT

(∫"

B(x)T @s(x)@e(x)

@e(x)@v

∣∣∣∣i−1

d" +∫"

Bb(x)T @sb(x)@eb(x)

@eb(x)@v

∣∣∣∣i−1

d"

)]dvi;

=[ vT

(∫"

B(x)Tf′i−1sec (x)B(x) d" +

∫"

Bb(x)Tf′i−1bond(x)Bb(x) d"

)]dvi;

"= [0; L]: (25)

The !rst term of (22) is used to derive the resisting load vector and is equal to

I i−1v = vT

[∫"

B(x)Tsi−1(x) d" +∫ L

0Bb(x)Tsi−1

b (x) d" − Boundary Terms]; "= [0; L]:

(26)

Substituting (25) and (26) in (22), and from the arbitrariness of v, we get

(Ki−1 + Ki−1b ) dvi = P − qi−1 − qi−1

rb ; (27)

where

Ki−1 =∫"

BT(x)ki−1(x)B(x) d"; "= [0; L]

is the section element sti+ness matrix, where k(x) =f′sec(x),

Ki−1b =

∫"

BTb (x)ki−1

b (x)Bb(x) d"; "= [0; L]


is the bond element sti+ness matrix, where kb(x) =f′bond(x),

qi−1 =∫"

BTsi−1(x) d"; "= [0; L]

is the section element resisting load vector,

qi−1rb =

∫"

BTb (x)si−1

b (x) d"; "= [0; L]

is the bond element resisting load vector and P is the vector of applied external loads.Because (27) include !rst derivatives of the axial and second derivatives of the vertical

displacement interpolation functions a(x), these functions have to be C0 continuous for theaxial part, and C1 continuous for the vertical part. Thus, this formulation does not enforce axialstrain, and curvature continuity at element boundaries.

4. Mixed variational principle formulation

In the mixed formulation, the di+erential equations are solved based on both a displacement!eld and a force !eld. Several forms of the mixed formulation are available: The u − s !eld,the u − e − s !eld and the u − e − sb !eld. For the composite beam problem, it is knownthat the curvature !eld shows a steep distribution especially in the plastic zone, while the slip!eld shows a much smoother distribution along the beam length. Consequently, it is moreadvantageous from a numerical standpoint to approximate the displacement !eld rather than thecurvature !eld. Therefore, the u − s !eld is adopted in the present study. The u − s model isbased on assuming both prede!ned continuous displacement !elds and prede!ned force !eldsalong the beam length. Accordingly

u(x) = a(x)v; (28)

where u(x) = [ub(x) ut(x) v(x)]T is the vector of displacements at a given point x; ub(x) andut(x) are the bottom and top axial displacements, respectively, v(x) is the vertical displacement,a(x) is a matrix of 3 × nd shape functions, nd depends on the order of displacement shapefunctions and is equal the total number of displacement degrees of freedom, and v is the vectorof element end displacements. Similarly

e(x) = B(x) v; (29)

eb(x) = Bb(x) v: (30)

Also

s(x) = b(x)q; (31)

where s(x) = [Nb(x) Nt(x) M (x)]T is the vector of forces at a given point x, Nb(x) and Nt(x)are the bottom and top axial forces, respectively, M (x) is the sum of top and bottom beammoments, b(x) is a vector of 3 × ns force interpolation functions, ns depends on the order offorce interpolation functions and is equal the total number of force degrees of freedom, and qis the vector of element end forces. It is important to note that nd and ns do not have to beequal.


The !nite element formulation is derived by considering the following two variational equa-tions:

Iq = −∫" sT(x)e(x) d" +

∫"uTLT s(x) d"= 0; "= [0; L]; (32)

Iv =∫" uT(x)LTs(x) d" −

∫"sb(x)Lb u(x) d"= 0; "= [0; L]: (33)

Taking the transpose of the second term in (32), it could be written as

Iq = −∫" sT(x)e(x) d" +

∫" sTLu(x) d"= 0; "= [0; L]: (34)

Using an incremental Newton–Raphson iterative technique to solve the nonlinear problem

I iq = I i−1q +

@@q

I i−1q dqi = 0: (35)

Substituting (34) into (35)

I iq = −∫" sT(x)ei(x) d" +

∫" sTLui(x) d"= 0; "= [0; L]: (36)

Eq. (36) represents the !rst weak form equation of the mixed formulation.Substituting the prede!ned force interpolation functions into (36)

I iq = qT[−∫"

bT(x)ei(x) d" +∫"

bTLui(x) d"]

= 0; "= [0; L]: (37)

The second term in (35) used to determine the tangent sti+ness matrix is equal to

@@q

I i−1q dqi =

[ qT

(−∫"

b(x)T @e(x)@s(x)

@s(x)@q

∣∣∣∣i−1

d"

+∫"

b(x)TL@u(x)@v

@v@q

∣∣∣∣i−1

d"

)]dqi; "= [0; L]:

=[ qT

(−∫"

b(x)T(f′−1sec )i−1(x)b(x) d"

+∫"

b(x)TLa(x)@v@q

∣∣∣∣i−1

d"

)]dqi; "= [0; L]: (38)

Eq. (38) could be written as

@@q

I i−1q dqi =− qT

(∫"

b(x)T(f′−1sec )i−1(x)b(x) d"

)dqi

+ qT(∫

"b(x)TB(x) d"

)dvi; "= [0; L]: (39)


The !rst term in (35) used to determine the resisting load vector is equal to

I i−1q = qT

[−∫"

bT(x)ei−1(x) d" +∫"

bTLui−1(x) d"];

= qT[−∫"

bT(x)ei−1(x) d" +∫"

bTLa(x) d"vi−1];

"= [0; L]: (40)

Substituting (39) and (40) into (35) and using the arbitrariness of q yields

T dvi − Fi−1 dqi − vi−1r = 0; (41)

where

T =∫"

bT(x)B(x) d"; "= [0; L]; (42)

Fi−1 =∫"

bT(x)f i−1(x)b(x) d"; f (x) =f′−1sec (x) = k−1(x); "= [0; L]; (43)

vi−1r =

∫"

bT(x)ei−1(x) d" − Tvi−1; "= [0; L]: (44)

F is the element 3exibility matrix, and vr is the element residual deformation vector.Integrating by parts the !rst term of (33) and taking the transpose of the second term, while

using OL and OLb de!ned in (18) instead of L and Lb:

Iv = uT(x)S1OLTs(x)|L0 −

∫"

(L1 u(x))TS1OLTs(x) d" −

∫" uT(x)OL

Tb sb(x) d"= 0;

"= [0; L]; (45)

where the integral operator S1, and the di+erential operator L1 are as de!ned before in (19). Itis important to note that S1LT

1 = I, the identity matrix, and S1OLT

= LT.

Integrating by parts again the last two terms and substituting for LT:

Iv = uT(x)LTs(x)|L0 − (L1 u(x))TS2L

Ts(x)|L0 − uT(x)S2OL

Tb sb(x)|L0

+∫"

(L2L1 u(x))TS2LTs(x) d"

+∫"

(L2 u(x))TS2OLTb sb(x) d"= 0; "= [0; L]; (46)

where the integral operator S2, and the di+erential operator L2 = IL are as de!ned before in(20). It is important to note that S2LT

2 = LT2S2 = I inside the domain, S2L

T= I the identity

matrix, and L2L1 = L.


Using the previous relations yields

Iv =∫" uT(x)LTs(x) d" +

∫" uT(x)LT

b sb(x) d" − BT = 0; "= [0; L]; (47)

where

BT = − [ uT(x)LTs(x)|L0 − uT(x)LT

1 s(x)|L0 − uT(x)S2OLTb sb(x)|L0]:

Using an incremental Newton–Raphson iterative technique to solve the nonlinear problem

I iv = I i−1v +

@@v

I i−1v dvi = 0: (48)

Substituting (47) into (48)

I iv =∫" uT(x)LTsi(x) d" +

∫" uT(x)LT

b sib(x) d" − BT = 0; "= [0; L]: (49)

Eq. (49) represents the second weak form equation of the mixed formulation.Substituting the prede!ned displacement shape functions and force interpolation functions into

(49)

I iv = vT[∫

"B(x)Tsi(x) d" +

∫"

Bb(x)Tsib(x) d" − BT]

= 0; "= [0; L]: (50)

The second term of (48) is used to derive the tangent sti+ness matrix and is equal to

@@v

I i−1v dvi =

[ vT

(∫"

B(x)T @s(x)@q

@q@v

∣∣∣∣i−1

d"

+∫"

Bb(x)T @sb(x)@eb(x)

@eb(x)@v

∣∣∣∣i−1

d"

)]dvi; " = [0; L]

=

[ vT

(∫"

B(x)Tb(x)@q@v

∣∣∣∣i−1

d"

+∫"


]dvi; "= [0; L]: (51)

Eq. (51) could be written as

@@v I i−1

v dvi = vT(∫

"B(x)Tb(x) d"

)dqi + vT

(∫"


)dvi;

"= [0; L]: (52)


The !rst term of (48) is used to derive the resisting load vector and is equal to

I i−1v = vT

[ ∫"

B(x)Tsi−1(x) d" +∫"

BTb (x)si−1

b (x) d"

−Boundary Terms]; "= [0; L]:

= vT[ ∫

"B(x)Tb(x) d"vi−1 +

∫"

Bb(x)Tsi−1b (x) d"

−Boundary Terms]; "= [0; L]: (53)

Substituting (52) and (53) into (48), and using the arbitrariness of v:

TT dqi + Ki−1b dvi = P − TTqi−1 − qi−1

rb ; (54)

where

Ki−1b =

∫"

BTb (x)ki−1

b (x)Bb(x) d"; "= [0; L];

is the bond element sti+ness matrix, where kb(x) =f′bond(x),

qi−1rb =

∫"

BTb (x)si−1

b (x) d"; "= [0; L];

is the bond element resisting load vector, P is the vector of applied external loads, and T is asde!ned before in (42).

Writing Eqs. (41) and (54) in matrix form[−Fi−1 TTT Ki−1

b

] [dqi

dvi

]=[

vi−1r

P − TTqi−1 − qi−1rb

]: (55)

Eq. (55) represents the matrix form of the mixed variational approach.It is important to note that at convergence, the residual deformation vector vr reduces to zero

inside each element satisfying compatibility.Since the matrix form given above includes !rst derivatives of the axial displacement shape

function, and second derivatives of the vertical shape functions, the former has to be C0 con-tinuous and the latter has to be C1 continuous, while the stress interpolation function b(x) neednot be continuous.

4.1. State determination of mixed variational principle

Two algorithms for the mixed variational principle are present. The !rst retains the forcedegrees of freedom at the structural level, while in the second the force degrees of freedom arecondensed out at the element level and only the displacement degrees of freedom are retainedat the structural level. The two algorithms are discussed next.


4.1.1. Algorithm 1In the !rst algorithm, the system of equations in (55) is solved for globally with the dis-

placements and forces as degrees of freedom. The only term on the right-hand side of the!rst equation is the residual deformation vector −vr, since no external loads are applied tothe force degrees of freedom. At convergence, the residual deformation vector reduces to zeroindependently inside each element. Therefore, only an external global iteration, typically ofNewton–Raphson type, is needed to solve for the incremental global problem, while no internalelement iteration is needed.

This algorithm enforces force continuity at element boundaries by solving for the forces asindependent global degrees of freedom.

4.1.2. Algorithm 2In the second algorithm, the force degrees of freedom are condensed out at the element level

resulting in a generalized displacement sti+ness matrix. Accordingly

From the !rst of equations (55): dqi = [Fi−1]−1[T dvi − vi−1r ]: (56)

Substituting (56) in the second of equations (55), we get

TT[Fi−1]−1[T dvi − vi−1r ] + Ki−1

b dvi = P − TTqi−1 − qi−1rb : (57)

During each global iteration, an element state determination is required. Two alternatives ofthis state determination are possible:

(a) A direct state determination following the proposal of Neuenhofer and Filippou [28]. Inthis case, the element state determination works in tandem with the global iteration according to(57) and no internal element iteration is required. At each global iteration, element forces satisfyconstitutive laws, but do not satisfy equilibrium, and thus are not compatible with the assumedforce !eld. Consequently, unbalanced forces arise. By correcting these unbalanced forces in thenext global iteration, the element iteration becomes unnecessary. Upon convergence, the residualdeformation vector reduces to zero, and thus compatibility is satis!ed. The algorithm, referredto as a the non-iterative algorithm, is as follows:

4.1.2.1. Non-iterative algorithm. With dvi−1 known at the current global iteration, we enterthe element.

(1) Determine the element force increments from Eq. (56)

dqi−1 = [Fi−2]−1[T dvi−1]: (58)

(2) Update the element forces

qi−1 = qi−2 + dqi−1: (59)

(3) Determine force increments at the integration points.The force increments at the integration points are calculated using the prede!ned force inter-

polation functions and the unbalanced section forces at the previous iteration

dsi−1(x) = b(x) dqi−1 + si−2(x); (60)

where si−2(x) is the vector of section unbalanced forces at the end of the previous iteration.


(4) Determine deformation increments at the integration points.The deformation increments at the integration points are calculated using the section 3exibility

at the end of the previous iteration

dei−1(x) = f i−2(x) dsi−1(x): (61)

(5) Update the total deformations at the integration points

ei−1(x) = ei−2(x) + dei−1(x): (62)

(6) Determine the section sti+ness and resisting load at the integration points.The total deformations at the integration points along with the deformation increments are

used to determine the section sti+ness and resisting load from the section constitutive law

ei−1r (x) =fsec(ei−1(x); dei−1(x)); ki−1(x) =f′

sec(ei−1(x); dei−1(x)); (63)

where er is the section resisting load, k is the section sti+ness, and fsec and f′sec are the functions

de!ning the section constitutive law.(7) Determine the residual section deformations

ri−1(x) = ki−1−1(x)(si−2(x) + dsi−1(x) − si−1

r (x)): (64)

(8) Determine the new element 3exibility matrix and residual vector.The section 3exibility matrix is obtained by inverting the section sti+ness matrix. The element

3exibility matrix Fi−1 and residual vector vi−1r are calculated as in (43) and (44), respectively

Fi−1 =∫"

bT(x)ki−1−1(x)b(x) d"; "= [0; L]; (65)

vi−1r =

∫"

bT(x)(ei−1(x) + ri−1(x)) d" − Tvi−1; "= [0; L]: (66)

(9) Update the element resisting forces

qi−1 = qi−2 + dqi−1 − Fi−1−1vi−1

r : (67)

(10) Update the section forces and the unbalanced forces

si−1(x) = si−1r (x); (68)

si−1(x) = b(x)qi−1 − si−1(x): (69)

(11) Calculate the total element sti+ness and resisting load

Ki−1total = {TT[Fi−1]−1T + Ki−1

b }; (70)

qi−1total = TTqi−1 + qi−1

rb : (71)

(b) An iterative element state determination in which the deformation vector is adjusted untilcompatibility is satis!ed in the element and the residual deformation vector vr is reduced tozero, before returning to the global iteration. Two nested iterations are thus needed: an internalelement iteration, and a global iteration, typically of Newton–Raphson type. Let j denotes theiteration counter for the internal element iteration. The algorithm, referred to as the iterativealgorithm, is as follows:


4.1.2.2. Iterative algorithm.(a) 1st iteration (internal): with vi−1 known at the current global iteration, we enter the ele-

ment iteration.

(1) For j= 1 set qi−1; j−1 = qi−2; Fi−1; j−1 = Fi−2; (72)

vi−1; j−1r = vi−2

r = 0, since the residual deformations vanish at the end of the state determinationin every element.

(2) Determine the element force increments from Eq. (56)

dqi−1; j = [Fi−1; j−1]−1[T dvi−1 − vi−1; j−1r ] (73)

when j= 1; T dvi−1 is applied to advance the stresses to the state of vi−1; when j¿ 1; T dvi−1

= 0.(3) Update the element forces

qi−1; j = qi−1; j−1 + dqi−1; j : (74)

(4) Determine force increments at the integration points.The force increments at the integration points are calculated using the prede!ned force inter-

polation functions

dsi−1; j(x) = b(x) dqi−1; j : (75)

(5) Determine deformation increments at the integration points.The deformation increments at the integration points are calculated using the section 3exibility

and the residual deformation at the end of the previous iteration counter j.

dei−1; j(x) = f i−1; j−1(x) dsi−1; j(x) + ei−1; j−1(x): (76)

(6) Update the total deformations at the integration points

ei−1; j(x) = ei−1; j−1(x) + dei−1; j(x): (77)

(7) Determine the section sti+ness and resisting load at the integration points.The total deformations at the integration points along with the deformation increments are

used to determine the section sti+ness and resisting load from the section constitutive lawthrough !ber integration.

si−1; jr (x) =fsec(ei−1; j(x); dei−1; j(x)); ki−1; j(x) =f′

sec(ei−1; j(x); dei−1; j(x)); (78)

where sr is the section resisting load, k is the section sti+ness, and fsec and f′sec are the functions

de!ning the section constitutive law.(8) Determine the residual section deformations

ri−1; j(x) = ki−1; j−1(x)(si−1; j(x) − si−1; j

r (x)): (79)

(9) Determine the new element 3exibility matrix and residual vector.


The section 3exibility matrix is obtained by inverting the section sti+ness matrix. The element3exibility matrix Fi−1; j and residual vector vi−1; j

r are calculated as in (43) and (44), respectively

Fi−1; j =∫"

bT(x)ki−1; j−1(x)b(x) d"; "= [0; L]; (80)

vi−1; jr =

∫"

bT(x)(ei−1; j(x) + ri−1; j(x)) d" − Tvi−1; "= [0; L]; (81)

(10) Set j= j+1, and repeat steps (2)–(9) until vi−1; jr vanish to within a speci!ed tolerance.

(11) At convergence, set qi−1 = qi−1; j, and Fi−1 = Fi−1; j.(b) 2nd iteration (global): once the internal iteration converges, the residual deformation vec-

tor reduces to zero leading to

TT[Fi−1]−1T dvi + Ki−1b dvi = P − TTqi−1 − qi−1

rb ; (82)

where TTFi−1T is the element global section sti+ness, Ki−1b is the element global bond sti+ness,

TTqi−1 is the element global section resisting load, and qi−1rb is the element global bond resisting

load. The previous equation could be written as follows:

Ki−1total dvi = P − qi−1

total; (83)

where

Ki−1total = {TT[Fi−1]−1T + Ki−1

b }; qi−1total = TTqi−1 + qi−1

rb :

Ktotal is the element total generalized global sti+ness matrix, and qtotal is the element totalgeneralized global resisting load vector.

Since forces are condensed out inside each element, this algorithm does not guarantee forcecontinuity at element boundaries. However, with the proper choice of interpolation functions,the jump in force distributions at element boundaries could be minimized as described in thenext section.

4.2. Stability of mixed formulation

It is known in general that the imposition of excessive continuity in a !nite element solutionmight yield unstable behavior. Furthermore, as in the case of algorithm 1, the imposition offorce continuity might produce erroneous results. Consider the case of a composite beam withconcentrated loads at intermediate nodes. While the exact solution shows a jump in the forcedistribution, using algorithm 1 of the mixed formulation will yield a continuous force !eld,since it solves for the forces as nodal degrees of freedom. In that case the imposition of forcecontinuity is incorrect. In general, using algorithm 1 will produce highly oscillating displacementdistributions. As discussed by Zienkiewicz and Taylor [27], it is recommended that the forcecontinuity be relaxed locally. That can be accomplished by using algorithm 2, where the forcesare condensed out at the element level.

Despite the relaxation of shape functions continuity by using algorithm 2, a limitation in thechoice of individual shape functions exists. Babuska and Brezzi [29,30] discussed the reasonsfor that diMculty. The Babuska–Brezzi (B–B) condition of stability for the composite beamproblem could be easily derived as follows. By considering Eq. (57), for the limiting case


where Ki−1b = 0, and assuming linear elastic behavior where vi−1

r = 0, it is necessary to ensurethat the term TT[Fi−1]−1T is non-singular for the system without rigid body modes. Singularityof that term will happen if its rank is smaller than the number of unknowns in the vector vafter excluding rigid body modes, namely nd′ . Since the rank of that term cannot be greaterthan the rank of [Fi−1]−1 which is equal ns, the number of unknowns in the vector q, the B–Bcondition of stability becomes

ns¿ nd′ : (84)

The previous condition is necessary but not suMcient for a convergent element.While algorithm 2 seems to be superior to algorithm 1, certain care should be exercised when

applying it, concerning mainly the choice of each independent interpolation function. These haveto satisfy condition (84). Moreover, according to De Veubeke’s principle of limitation [31], theorder of a stress !eld cannot be greater than that of the corresponding strain !eld that satis!esthe strain–displacement compatibility condition. Accordingly, the equality of condition (84), i.e.ns = nd′ is the most eMcient choice from a computational standpoint for linear elastic behavior.Furthermore, in the nonlinear range, it minimizes the jump in force distributions across elementboundaries, which results from condensation of the force degrees of freedom.

To clarify the previous condition, consider a linear elastic composite beam. For Eq. (36)to be satis!ed, the integral of the arbitrary s(x) !eld with the deformation !eld e(x), shouldbe equal the integral of the arbitrary s(x) !eld with the deformation !eld that satis!es thestrain–displacement compatibility condition Lu(x). From the arbitrariness of s(x), the order ofthe Lu(x) !eld will be the same of that of the e(x) !eld. For linear elastic behavior the orderof the force !eld s(x) also equals that of the deformation !eld e(x), verifying the principle oflimitation.

If the inequality of Eq. (84) is used, i.e. ns¿nd′ , the higher order force interpolation functionswill be discarded by the element, and no additional accuracy is gained in the linear elastic rangeover the case of ns = nd′ . Moreover, in the nonlinear range, the force distributions will showa big jump at element boundaries and an abrupt redistribution in the yielding portion of thebeam.

In this study, algorithm 2 of the mixed model is used with linear axial force, and quadraticaxial displacement shape functions. As for the 3exural behavior, since di+erential vertical dis-placements between top and bottom beams are neglected in the model (i.e. uplift modulus isassumed in!nite), conventional hermitian polynomial shape functions are used to approximatethe vertical displacements, and linear force interpolation functions are used to approximate theexternal bending moments, in case of no application of external distributed loads. It is knownthat the hermitian polynomial shape functions are C1 continuous. These choices of interpolationfunctions are in agreement with De Veubeke’s principle of limitation.

5. Evaluation of model by numerical studies

In order to evaluate the eMciency of the new model, a comparison with the standard modelbased on the principle of minimum potential energy is conducted on a sample composite beamproblem. The beam is under midspan loading. The top and bottom beam sections are 60×30 mm,


Fig. 2. Global response (minimum potential energy principle).

and the beam span is 4000 mm. The beam section properties are derived by !ber discretization.The beam material is assumed to be elastic perfectly plastic with its yield stress fy = 300 MPa,and its initial sti+ness E = 2 × 105 MPa. The connector shear load slip relation is also assumedto be elastic perfectly plastic, with its initial sti+ness kb = 500 MPa and its yield force per unitlength +y = 150 N=mm. The global load–displacement response is shown in Figs. 2 and 3 for theminimum potential energy model (displacement model) and mixed variational model, respec-tively. Convergence was achieved with 16 elements per half span for the minimum potentialenergy model, while it was achieved with only one single element for the mixed model. Thereason can be explained by examining the distributions of the di+erent parameters of the prob-lem along the beam length. The curvature distributions at three di+erent load stages, for boththe minimum potential energy and mixed models are shown in Figs. 4 and 5, respectively. Thethree load stages A, B, and C are shown in Figs. 2 and 3.

A !ne mesh that represents the numerically converged solution is shown for comparison.The !ne mesh consists of 16 elements using the minimum potential energy formulation. Thecurvature of the !ne mesh shows a steep distribution in the plastic zone in the region nearmidspan. Since the minimum potential energy model assumes a linear curvature !eld, it failsto represent the curvature in the plastic zone with few elements as shown in Fig. 4. The mixedmodel, however, successfully represents the curvature distribution as shown in Fig. 5 with justtwo elements since it assumes a prede!ned force !eld. Figs. 6 and 7 show the bending momentdistributions for both models. The exact total resisting moment is linear, in the absence of anydistributed loads, irrespective of the presence of shear forces at the interface level. The mixedmodel predicts the exact linear moment distribution as shown in Fig. 7, while the minimumpotential energy model predicts a nonlinear moment distribution, and exhibits big jumps atelement boundaries. The superiority of the mixed model is even more con!rmed by examiningthe axial force distributions shown in Figs. 8 and 9. The mixed model clearly better predicts


Fig. 3. Global response (mixed variational principle).

Fig. 4. Curvature distribution (minimum potential energy principle).

the axial force distribution than the minimum potential energy model which also exhibits bigjumps at element boundaries. The minimum potential energy model, however, represents theslip distribution fairly well as shown in Fig. 10. The mixed model predicts the slip distributioneven better as shown in Fig. 11. Both models succeed to represent the spread of yielding of theinterface shear as shown in Figs. 12 and 13. The above-mentioned results verify the superiorityof mixed models compared to standard models based on the minimum potential energy principle.


Fig. 5. Curvature distribution (mixed variational principle).

Fig. 6. Bending moment distribution (minimum potential energy principle).

To clarify the e+ect of choice of interpolation functions on the response of mixed models,consider a sample composite steel–concrete simply supported beam under three point bending.The beam was tested by McGarraugh and Baldwin [8] as a part of a series of tests to investigatethe e+ect of lightweight concrete on the behavior of composite beams. The beam consists of


Fig. 7. Bending moment distribution (mixed variational principle).

Fig. 8. Axial force distribution (minimum potential energy principle).

a W 14 × 30 steel section connected to a reinforced concrete slab by shear connectors. Theslab depth is 4.5 in and its width is 36 in. Five 3=4′′ × 3′′ shear studs per half-span were used.The steel yield stress is 36:6 ksi and the concrete compressive strength is 5:56 ksi. The concretematerial model used in the analytical study follows the model by Scott et al. [32], while the


Fig. 9. Axial force distribution (mixed variational principle).

Fig. 10. Slip distribution (minimum potential energy principle).

steel material model used is a simple bilinear one. A softening shear force slip relation for theshear connectors is used in the analytical study with a yield force equal 17.6 kips and an initialsti+ness equals 440 kips=in. The dimensions, loading arrangements, and cross-section dimensionsof the specimen are shown in Fig. 14.


Fig. 11. Slip distribution (mixed variational principle).

Fig. 12. Interface shear force distribution (minimum potential energy principle).

The !nite element model used for the analysis consists of a number of composite elements.Five integration points for the beam elements and three for the bond element were used. Thecross-section of the steel beam is made up of 12 !bers while the cross-section of the concretebeam is made up of 4 !bers.


Fig. 13. Interface shear force distribution (mixed variational principle).

Fig. 14. McGarraugh and Baldwin test beam.


Fig. 15. Global response for softening beam.

A comparison of di+erent interpolation functions of the parameters for the mixed model ispresented. The cases considered are: (a) a linear axial force and a linear bending moment witha quadratic axial displacement, (b) a parabolic axial force and a linear bending moment with aquadratic axial displacement, and (c) a parabolic axial force and a parabolic bending momentwith a quadratic axial displacement.

The global load–displacement response for the three di+erent cases of interpolation functionsfor the mixed model is shown in Fig. 15. Four elements were used for all three cases. All threecases give essentially identical results in the elastic range, but di+erent results in the post-yieldrange. The case of linear axial force and linear bending moment represents the converged so-lution. The axial force distribution along the beam length at load point A of Fig. 15 is shownin Fig. 16. The linear axial case shows a smooth variation while both parabolic axial casesshow a big jump at element boundaries. These results verify De Veubeke’s principle of limi-tation: The increase in the order of the axial force interpolation function does not produce anyadditional accuracy in the elastic range while it produces big jumps at element boundaries inthe inelastic range. Fig. 17 shows the bending moment distributions for all three cases at loadpoint A. The assumption of linear moment is exact and produces correct results. The higherorder parabolic moment distribution produces jumps at element boundaries verifying again theprinciple of limitation. The curvature distributions are shown in Fig. 18. The case of linearaxial force produces the best results since it succeeds to describe the forces accurately. Fig. 19shows the slip distributions. All three cases give essentially the same results since they assumea quadratic axial displacement along the beam length. Using higher order interpolation func-tions is only possible if the equality of condition (84) is satis!ed. Accordingly, quadratic axial


Fig. 16. Axial force distribution for softening beam.

Fig. 17. Bending moment distribution for softening beam.


Fig. 18. Curvature distribution for softening beam.

Fig. 19. Slip distribution for softening beam.


Fig. 20. Axial force distribution (parabolic force and cubic displacement).

Fig. 21. Slip distribution (parabolic force and cubic displacement).


force interpolation functions could only be used if the order of axial displacement interpolationfunctions is raised to a cubic shape. The resulting distributions of axial force and slip, for thatcase, are shown in Figs. 20 and 21 respectively. The results show that the jump in axial forcevalues is small at element boundaries verifying the preceding conclusion.

6. Conclusion

A new model for composite beams with partial interaction is presented. The model is basedon a two-!eld mixed variational principle, where both variables, forces and deformations, aresimultaneously approximated within the element. The model neglects uplift and frictional ef-fects. Two algorithms could be used for the proposed model: an algorithm that enforces forcecontinuity at element boundaries, and a preferred algorithm where force continuity is relaxedlocally at the element level. Stability of both algorithms is discussed. Numerical examples thatclarify the advantages of the proposed model versus the standard model based on the principleof minimum potential energy are presented. The numerical examples proved the accuracy ofthe proposed model in describing local forces and deformations with very few !nite elements,which results in a considerable reduction of the computational cost. A discussion of the di+er-ent choices of interpolation functions used for the mixed model is also presented. The studiesveri!ed the principle of limitation as the main criteria in selecting the appropriate choice ofinterpolation functions.

Acknowledgements

This work was conducted as part of the Ph.D. dissertation of the author at the Universityof California, Berkeley under the supervision of Prof. Filip C. Filippou. The author greatlyappreciates the continuous advice of Professor Filippou regarding the formulation and imple-mentation of the mixed variational principle. The author also appreciates the constant assistantof Professor Robert L. Taylor regarding the use of the !nite element program FEAP.

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A two-field mixed variational principle for partially connected composite beams