Treatment of slip locking for displacement-based finite element analysis of composite beam-columns

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2011; 85:805–826Published online 3 August 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.2990

Treatment of slip locking for displacement-based finite elementanalysis of composite beam–columns¶

R. Emre Erkmen1,∗,†,‡ and Mark A. Bradford2,§

1Centre for Infrastructure Engineering and Safety, School of Civil and Environmental Engineering,

The University of New South Wales, UNSW Sydney, NSW 2052, Australia2Faculty of Engineering and Information Technology, University of Technology Sydney,

Broadway, NSW 2007, Australia

SUMMARY

In the conventional displacement-based finite element analysis of composite beam–columns that consistof two Euler–Bernoulli beams juxtaposed with a deformable shear connection, the coupling of the trans-verse and longitudinal displacement fields may cause oscillations in slip field and reduction in optimalconvergence rate, known as slip locking. This locking phenomenon is typical of multi-field problems ofthis type, and is known to produce erroneous results for the displacement-based finite element analysisof composite beam–columns based on cubic transverse and linear longitudinal interpolation fields. Thispaper introduces strategies including the assumed strain method, discrete strain gap method, and kinematicinterpolatory technique to alleviate the oscillations in slip and curvature, and improve the convergenceperformance of the displacement-based finite element analysis of composite beam–columns. A systematicsolution of the differential equations of equilibrium is also provided, and a superconvergent element isdeveloped in this paper. Numerical results presented illustrate the accuracy of the proposed modifications.The solutions based on the superconvergent element provide benchmark results for the performance ofthese proposed formulations. Copyright � 2010 John Wiley & Sons, Ltd.

Received 12 May 2009; Revised 24 August 2009; Accepted 10 June 2010

KEY WORDS: composite beam–columns; slip locking; assumed strain method; discrete strain gap method;kinematic interpolatory strategy; superconvergent element

INTRODUCTION

Composite beams–columns are structural members formed when two beams are connected bymeans of shear connectors to form an interacting unit that is capable of resisting bending momentsand axial forces. The simplest composite beam model with flexible shear connectors was initiallydeveloped by Newmark et al. [1], in which two Euler–Bernoulli beams are connected by assumingthat vertical separation does not occur between the components. Recently, analytical solutionmethods based on Newmark’s model were proposed by Faella et al. [2], Ranzi et al. [3], andRanzi and Bradford [4]. Numerical solution methods by using displacement-based finite elementformulations were developed by Arizumi et al. [5], Daniels and Crisinel [6], and Dall’Asta and

∗Correspondence to: R. Emre Erkmen, Centre for Infrastructure Engineering and Safety, School of Civil andEnvironmental Engineering, The University of New South Wales, UNSW Sydney, NSW 2052, Australia.

†E-mail: [email protected]‡Research Associate.§Professor and Dean.¶This article was publised online on [3 August 2010]. An error was subsequently identified. This notice is included

in the online and print versions to indicate that both have been corrected [14 September 2010].

Copyright � 2010 John Wiley & Sons, Ltd.

806 R. E. ERKMEN AND M. A. BRADFORD

Zona [7]. Typical of multi-field problems; however, displacement-based finite element formulationsof composite beams may suffer from the so-called slip locking phenomenon because of the couplingbetween the displacement fields, and mixed and force-based finite element formulations were alsodeveloped by Salari et al. [8], Ayoub and Filippou [9], Dall’Asta and Zona [10], and Ayoub [11].

On the other hand, in order to improve the performance of the displacement-based finite elementanalysis several strategies have been developed in the past to alleviate shear locking in Timoshenkobeams and Mindlin plates, and membrane locking in curved members. Among those strategiesonly consistent interpolatory strategy, which was initially developed by Tessler and Dong [12] toalleviate shear locking, was employed for the analysis of composite beam–columns by Dall’Astaand Zona [13], and Ranzi and Zona [14]. The strategies that have not yet been employed for theanalysis of composite beam–columns include the assumed strain finite element formulation whichwas initially developed to alleviate membrane locking in arches and thin-shells by Ashwell [15].Simo and Hughes [16] later showed that assumed strain methods can be considered as the specialcases of B-bar procedure which was developed by Hughes and Tezduyar [17] to eliminate shearlocking in Mindlin plates. Tessler and Hughes [18] developed the kinematic interpolatory strategyto produce a locking-free four-node Mindlin plate element, in which the intermediate nodes thatare necessary for consistent interpolations of the primary variables were eliminated by enforcinga continuous Kirchhoff constraint. The discrete strain gap method was initially developed byBletzinger et al. [19] for the locking-free analysis of shear-deformable plates and shells, and laterapplied for the elimination of membrane locking by Koschnik et al. [20].

The aim of this paper is to employ the assumed strain method, kinematic interpolatory strategy,and discrete strain gap method to alleviate locking behavior in displacement-based finite elementanalysis of the composite beam–columns. By changing the conventional primary variables in thecomposite beam–column formulation, a form which allows a more systematic solution of thedifferential equations is also obtained and a superconvergent finite element is developed based onthe homogenous solution of these differential equations. A proof of exactness of this element isalso provided. Numerical examples are presented for comparisons between the performances ofthe developed finite elements and the elements proposed in the literature. Solutions based on thesuperconvergent element provide benchmark results for performance evaluations of the proposedfinite element formulations.

COMPOSITE BEAM–COLUMN ANALYSIS

Displacements and strains

The basic kinematic assumptions adopted in this paper were initially presented by Newmark et al.[1] based on which the composite member is composed of a top and a bottom Euler–Bernoullibeam elements, which are referred to as beams 1 and 2 as shown in Figure 1(a). The compositecross-section is thus represented as A= A1 + A2, where A1 and A2 are the cross-sections of beams 1

Figure 1. Composite beam–column: (a) cross-section; (b) displacements; and (c) strains.

Copyright � 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2011; 85:805–826DOI: 10.1002/nme

TREATMENT OF SLIP LOCKING FOR COMPOSITE BEAM–COLUMNS 807

and 2, respectively. The displacement field adopted in the formulation consists of the verticaldisplacement v of the selected reference axis, and the longitudinal displacements w1 and w2 ofthe centroids of beams 1 and 2, respectively as shown in Figure 1(b). The slip at the interfacebetween the two components of the composite beam is �, as shown in Figure 1(b), which is dueto the difference in longitudinal displacements w1 and w2, and the rotation of both cross-sectionsv′, i.e.

�=w2 −w1 +hv′, (1)

where h is the distance between the centroids of the beams, and ()′=d()/dz. Strain expression foreach component can be determined by using the Euler–Bernoulli beam kinematics, hence in termsof the extensions w′

1 and w′2 and the curvature v′′ due to bending deformations as

ε1 = w′1 −(y−h1)v′′, (2)

ε2 = w′2 −(y−h2)v′′, (3)

where h1 and h2 are the coordinates of the centroids of beams 1 and 2 with respect to the referenceaxis, respectively as shown in Figure 1(c).

FINITE ELEMENT FORMULATIONS

Displacement-based finite element formulation

A displacement-based finite element formulation can be developed by employing the total potentialenergy functional, i.e.

�= 1

2

∫L

∫A1

E1ε21 dA dz+ 1

2

∫L

∫A2

E2ε22 dA dz+ 1

2

∫L

∫b��2 dxdz−�ext, (4)

where the first and second integrals are the elastic bending energies of the two Euler–Bernoullibeam components, the third integral is due to the elastic deformations of the shear connection inwhich � is its elastic stiffness (force/length3) which is the shear stress in longitudinal direction perunit slip, b is the width of the effective interface surface between the two beam components, and�ext is the work done by the external forces. In a displacement-based finite element formulation,the longitudinal displacement fields w1 and w2 and the derivative of the vertical displacement fieldv′ can be expressed in terms of the selected interpolation functions as⎧⎪⎪⎨

⎪⎪⎩w1

w2

v′

⎫⎪⎪⎬⎪⎪⎭=

⎡⎢⎢⎣

M(z) 0 0

0 M(z) 0

0 0 N′(z)

⎤⎥⎥⎦⎧⎪⎨⎪⎩

wN1

wN2

vN

⎫⎪⎬⎪⎭=B(z)U, (5)

where M(z) and N(z) are the vectors of the interpolation functions for the longitudinal and thevertical displacement fields, respectively. In Equation (5), B(z) is the discrete slip matrix and U isthe vector of nodal displacements composed of the vectors of nodal longitudinal displacements atthe centroids of both beams wN1, wN2 and the vertical displacement vN , i.e.

UT =〈wTN1 wT

N2 vTN 〉. (6)

By using Equations (1)–(3) and (5) in Equation (4), the total potential energy functional can bewritten as

�= 1

2UT

(∫L

BTd (z)DBd (z)dz

)U+ 1

2UT

(∫L

BT(z)D�B(z)dz

)U−FU (7)



in which Bd (z)=B′(z), F is the energy equivalent nodal external load vector,

D=

⎡⎢⎣

E1 A1 0 0

0 E2 A2 0

0 0 E1 I1 + E2 I2

⎤⎥⎦ (8)

and

Dq=�b

⎡⎢⎢⎣

1 −1 −h

−1 1 h

−h h h2

⎤⎥⎥⎦ , (9)

where I1 and I2 are the second moments of area of the beams with respect to their horizontalprincipal axes passing through the centroids of each cross-section. From the first variation of thetotal potential energy functional, the weak form of the equilibrium equations is obtained as

(Kb +Ks)U=F, (10)

where Kb is the stiffness matrix associated with the bending and the axial deformations of thebeam components and Ks is associated with the slip energy i.e.

Kb =∫

LBT

d (z)DBd (z)dz, (11)

Ks =∫

LBT(z)D�B(z)dz. (12)

The simplest element that satisfies the compatibility conditions can be developed by using linearinterpolations for the longitudinal displacements and cubic interpolation for the vertical displace-ment, i.e.

M(z) = 〈(1−z/L) z/L〉, (13)

N(z) =⟨(

1− 3z2

L2+ 2z3

L3

) (z− 2z2

L+ z3

L2

) (3z2

L2− 2z3

L3

) (− z2

L+ z3

L2

)⟩(14)

and the vector of nodal displacement U can be written as

UT =〈w1(0) w1(L) w2(0) w2(L) v(0) v′(0) v(L) v′(L)〉 (15)

This element will be referred as the basic element (BE) herein. It has been reported that for stiffshear connection, i.e. for the limit case of �→∞, element BE suffers from slip locking (e.g.Dall’Asta and Zona [7]). The locking behavior was alleviated by Dall’Asta and Zona [13] byusing consistent interpolations strategy i.e. by selecting the polynomial order of the interpolationfunctions for the vertical displacement to be one order higher than those of the longitudinaldisplacements. The element derived based on the consistent interpolations strategy will be usedherein for comparison purposes and referred as the consistent interpolation element (CE). Thecause of locking in BE can be seen by substituting Equations (13) to (15) into Equation (1) toobtain the discretized form of the slip � as

�=�(0)(

1− z

L

)+�(L)

z

L+c

6h

L2

(z− z2

L

), (16)

where �(0) and �(L) are nodal slip values, i.e.

�(0)=w2(0)−w1(0)+hv′(0) (17)



and

�(L)=w2(L)−w1(L)+hv′(L). (18)

In (16), constant c can be written as

c=v(L)−v(0)− L

2[v′(L)+v′(0)], (19)

which is non-zero unless the vertical displacement field v is parabolic rather than cubic. Thecondition �=0 for the limit case of �→∞ causes Equation (19) to vanish which enforces aconstant value for the curvature v′′, thus for stiffer connection stiffness � the convergence rate ofelement BE reduces. On the other hand, c is not zero in general; therefore; constant slip cannotbe produced by using Equation (16) and that causes oscillations in the slip field. Since shearlocking in Timoshenko beams has been widely investigated in the literature, it is of interest to notethe similarities and differences between the finite element analyses of these two beam models.Substituting Equations (1) to (3) into Equation (4) produces

�= 1

2(E1 I1 + E2 I2)

∫Lv′′2 dz+ 1

2kG A

∫L

[v′− 1

h(w1 −w2)]2 dz−�ext (20)

in which

kG A=h∫

b�dx,

have been used. It should be noted that in obtaining Equation (20), it has been assumed that noaxial load is carried and thus the strain energy due to axial deformations vanishes, i.e.

1

2

∫L

∫A1

E1w′21 dA dz+ 1

2

∫L

∫A2

E2w′22 dA dz =0 (21)

In this case Newmark’s model can be identified as the Timoshenko beam model by replacingthe derivative of the vertical displacement field v′ of Newmark’s model with the rotation of thecross-section � of the Timoshenko beam model, and the slip � of Newmark’s model with the shearstrain � of the Timoshenko beam model, and thus Equation (20) becomes

�= 1

2E I

∫L�′2 dz+ 1

2kG A

∫L

(�−u′)2 dz−�ext (22)

in which u′ denotes the rotation of the beam axis in Timoshenko beam model, i.e.�=�−u′. Basedon the variational form in Equation (22), a C0 finite element formulation can be developed byusing linear interpolations for both fields � and u, and thus the discretized form of the shear straincan be written as

�=�(0)+ u(L)−u(0)

L+[�(L)−�(0)]

z

L, (23)

based on which, zero shear condition, i.e. �=0 for the limit case of kG A→∞, causes zero bendingenergy [21]. That is because the discretized form of the curvature term in Equation (22) based onlinear interpolations is

�′ = [�(L)−�(0)]1

L=0 (24)

since each polynomial term in Equation (23) vanishes. It should be noted that unlike Equation (22),the variational form in Equation (20) does not allow a C0 finite element formulation becauseBE requires C1 continuity for the vertical displacement field v. It should also be noted that inBE formulation, the longitudinal displacements w1 and w2 have linear expressions as oppose toconstant u′ used in C0 Timoshenko beam finite element formulation based on Equation (22).Therefore, the interpolation space of BE is richer than that of the C0 Timoshenko beam finite



element formulation and thus slip locking does not cause as severe degradations in the convergencerate as in the C0 Timoshenko beam finite element analysis. Nevertheless, the treatment techniquespreviously developed for Timoshenko beam analysis can be adopted to improve the convergencerate and alleviate oscillations in slip in composite beam–column analysis.

Assumed strain formulation

Assumed strain formulation can be used to provide some freedom in the selection of the interpo-lation functions for the displacement fields w1, w2 and v while preserving a locking-free behavior.In order to obtain an assumed strain finite element formulation, partial Hu–Washizu principlecan be used to derive a variationally consistent modified functional (e.g. Simo and Hughes [16]),in which the slip as well as the displacements are contained as independent variables, i.e.

�= 12

∫L

∫A1

E1ε21 dA dz+ 1

2

∫L

∫A2

E2ε22 dA dz+∫

L �(�−�̄)dz+ 12

∫L

∫b ��̄

2dx dz−�ext (25)

Owing to the third integrand in Equation (25), the independent slip field �̄ satisfies Equation (1)in the average integral sense. On the other hand, from the Euler–Lagrange conditions of themodified functional the Lagrange multiplier � can be identified as the shear flow between thebeam components caused by the slip field �̄ when the constitutive relation for the shear studs aresatisfied a-priori (e.g. Stolarski Belytschko [22]), i.e. �=∫

b ��̄dx . By employing the displacementinterpolations used in the previous section and selecting an independent interpolation field H(z)for the slip, the slip can be written as

�̄=H(z)ws, (26)

where ws is the vector of nodal slip values. The modified functional in Equation (25) can bewritten in terms of nodal slip and displacement values as

� = 1

2UT

(∫L

BTd (z)DBd (z)dz

)U+wT

s

(∫L

H(z)TC�B(z)dz

)U

−1

2wT

s

(∫L

H(z)T�bH(z)dz

)ws −FU (27)

in which Cq=∫

b �〈−1 +1 h〉dx have been used. From the first variation of the functional inEquation (27) and condensing out the nodal strain vector ws in the element level [23], the equi-librium equations are obtained as

(K̄b +K̄s)U=F, (28)

where K̄b is identical to the stiffness matrix associated with the bending and the axial deformationsin Equation (11) and the stiffness matrix associated with the slip energy K̄s can be written as

K̄s =∫

LBT(z)CT

�H(z)dz ·[∫

LH(z)T�bH(z)dz

]−1

·∫

LH(z)TC�B(z)dz. (29)

However, due to the limitation principle of de Veubeke [24], when the slip field derived from thedisplacements is contained within the assumed strain field, nothing is gained with the assumedstrain formulation and in that case the stiffness matrices in Equations (12) and (29) are identical(e.g. Stolarski Belytschko [22]). The slip field in Equation (1) based on the interpolation functionsin Equations (13) and (14) is parabolic, thus linear interpolation is used in Equation (26), i.e. H(z)=M(z), and this element will be referred as the assumed strain element (AE).



Figure 2. Euler–Bernoulli mode of deformation.

Discrete strain gap method for composite beam–columns

By selecting the assumed slip field ¯̄� to be orthogonal to the difference of the slip field derivedfrom the displacement field and the assumed slip field, i.e.∫

L

¯̄�(�− ¯̄�)dz =0 (30)

the stiffness matrix associated with the slip energy ¯̄Ks can be obtained directly without condensation.In that case, the modified functional in Equation (25) becomes

�= 1

2

∫L

∫A1

E1ε21 dA dz+ 1

2

∫L

∫A2

E2ε22 dA dz+ 1

2

∫L

∫b� ¯̄�2

dx dz−�ext. (31)

An assumed slip field ¯̄� satisfying the orthogonality condition in Equation (30) can be obtainedby adopting the so-called discrete strain gap method. Initially, Bletzinger et al. [19] applied themethod for the elimination of shear locking and used the name discrete shear gap method for theirprocedure. Later, the method was generalized for membrane locking problems by Koschnik et al.[20] and a more general name discrete strain gap was adopted, which is also used herein. In thismethod, the total deformation is decomposed into a slip deformation mode and a pure Euler–Bernoulli beam–column deformation mode in which the slip deformations vanish as in Figure 2.In the Euler–Bernoulli mode, the increment of vertical displacement between coordinates z0 and zcan be expressed in terms of the axial displacements of the centroids of the two composite beamcomponents as (Figure 2).

v=∫ z

z0

�dz =∫ z

z0

(w1 −w2)/h dz. (32)

On the other hand, from the integral of Equation (1), the gap in the relationships between thevertical displacement increment and the axial displacements can be obtained, i.e.∫ z

z0

¯̄�dz =�sp(z)=∫ z

z0

(w2 −w1 +hv′)dz, (33)

where �sp(z) occurs due to slip and called slip gap herein. As the gap caused by the slip inbetween coordinates z0 and z is of interest, in Equation (33) �sp(z0) is assumed zero. In thediscrete strain gap method, the idea is to discretize the gap by using the interpolation functions.Thus, the discretized slip gap can be determined by interpolating the nodal slip gaps as

�sp(z)=G(z)Ds, (34)

where Ds is the vector of nodal slip gaps, i.e. DTs =〈Ds(z0) Ds(z1) . . .Ds(zN )〉, and z0, z1, . . . , zN

correspond to the nodal coordinates of the finite element of N +1 nodes. The relationships betweenthe nodal values of the slip gap Ds(zi ) and the nodal displacements can be determined by usingEquation (33), i.e.

Ds(zi )=∫ zi

z0

M(z)dz(wN2 −wN1 )+h[N(zi )−N(z0)]vN . (35)



The BE can be modified by assuming linear interpolation for the discrete slip gap �sp(z) in

Equation (34) which produces a constant slip field ¯̄�, i.e.

¯̄�=〈−1 +1 h〉 ¯̄BU, (36)

where

¯̄B=

⎡⎢⎢⎣

12

12 0 0 0 0 0 0

0 0 12

12 0 0 0 0

0 0 0 0 −1L 0 1

L 0

⎤⎥⎥⎦ . (37)

By using Equations (5) and (36), and the relation �=〈−1 +1 h〉BU, it can be shown that∫

L¯̄�(�−

¯̄�)dz is satisfied, hence formulation is variationally consistent when the modified functional inEquation (31) is used [16]. From the first variation of the modified functional in Equation (31), and

by using Equation (36), the stiffness matrix ¯̄Ks associated with the slip energy can be obtained as

¯̄Ks =∫

L

¯̄BTD�

¯̄Bdz. (38)

The stiffness matrix associated with the bending and the axial deformations ¯̄Kb is identical to Kbin Equation (11). This element will be referred as the discrete strain gap element (DE). It shouldbe noted that assuming quadratic interpolation for the discrete slip gap in Equation (34), i.e. linear

interpolation for the slip field ¯̄� produces a stiffness matrix ¯̄Ks identical to the stiffness matrix K̄sof element AE which is based on linear assumed strain field.

Kinematic interpolatory strategy for composite beam–columns

In this method, the constraint condition of zero slip, i.e. �=w2 −w1 +hv′ =0 for the limit case of�→∞ is enforced to be satisfied in the pointwise sense. Thus, in Equation (4) the third integralidentically vanishes and the total potential energy functional becomes

�= 1

2

∫L

⎛⎜⎜⎝〈w′

1 w′2 v′′〉E

⎡⎢⎣

E1 A1 0 0

0 E2 A2 0

0 0 E1 I1 + E2 I2

⎤⎥⎦⎧⎪⎪⎨⎪⎪⎩

w′1

w′2

v′′

⎫⎪⎪⎬⎪⎪⎭

⎞⎟⎟⎠dz−�ext. (39)

By using the constraint condition, i.e. w1 =w2 +hv′ to eliminate w1 and from the first variationof Equation (39), the weak form of the equilibrium equations is obtained as

∫L

(〈�w2 �v〉

[(E1 A1+E2 A2)D2 E1 A1hD3

E1 A1hD3 (E1 A1h2+E1 I1+E2 I2)D4

]{w2

v

})dz−��ext−��Boun

∣∣∣∣∣L

0

=0 (40)

in which D=d/dz is the differential operator with respect to the beam axis z. Integrations byparts have been used in obtaining Equation (40). The boundary conditions are satisfied at z =0 andz = L , i.e. ��Boun|L

0 =0, thus the homogenous form of the field equations can be obtained fromthe first integral in Equation (40) from which the solution for v and w2 can be obtained as

v(z) = N(z)vN , (41)

w2(z) = M(z)wN2 + E1 A1h

E1 A1 + E2 A2[N′(z)−M∗(z)]vN , (42)

where vTN =〈v(0) v′(0) v(L) v′(L)〉, wT

N2 =〈w2(0) w2(L)〉, M(z) and N(z) are as in Equa-tions (13) and (14), respectively, and

M∗(z)=〈0 (1−z/L) 0 z/L〉. (43)



From Equations (41) and (42), by employing the constraint condition again, i.e. w1 =w2 +hv′ thelongitudinal displacement field w1 can be written as

w1(z)=M(z)wN1 +(

E2 A2h

E1 A1 + E2 A2

)[N′(z)−M∗(z)]vN , (44)

where wTN1 =〈w1(0) w1(L)〉, and w1(0)=w2(0)+hv′(0) and w1(L)=w2(L)+hv′(L) have been

used. A displacement-based finite element formulation can be derived by substituting Equa-tions (41), (42) and (44) into Equation (4). This element will be referred as the kinematic interpo-latory element (KE).

Superconvergent finite element formulation for composite beam–columns

The homogenous solution of composite beam–column differential equations can be obtained ina systematic fashion by changing the primary variables to the vertical and axial displacementsdefined on the arbitrary reference axis v and w, respectively and the slip �, thus the strain fieldscan be written as

ε1 = w′− yv′′−�′, (45)

ε2 = w′− yv′′. (46)

From Equations (45) and (46) by substituting into the first variation of Equation (4), performingintegration by parts and assuming that the boundary conditions are satisfied, the homogenous formof the differential equilibrium equations can be obtained as⎡

⎢⎢⎣E ID4 −E SD3 --

-

E1 A1h1D3

−E SD3 E AD2 ---

−E1 A1D2

−−−−−−−−−−−−−−−−−−−E1 A1h1D

3 −E1 A1D2 --

-

E1 A1D2 −�b

⎤⎥⎥⎦⎧⎪⎨⎪⎩

v

w

�

⎫⎪⎬⎪⎭=

⎧⎪⎨⎪⎩

0

0

0

⎫⎪⎬⎪⎭ (47)

where E I = E1(I1 + A1h21)+ E2(I2 + A2h2

2), E S = E1 A1h1 + E2 A2h2 and E A= E1 A1 + E2 A2. Thevertical and axial displacements do not contribute to the slip energy directly and this choiceof primary variables allows the slip field � to be expressed in terms of the vertical and axialdisplacements by using the first row and the derivative of the second row of the differentialequilibrium equations in Equation (47) as{

D4v

D3w

}=−

[E I −E S

−E S E A

]−1{E1 A1h1

−E1 A1

}D3�=

{C1

C2

}D3�. (48)

The clear advantage of working with these primary variables is that the relationships between theslip field � and the displacement fields w and v that satisfy the equilibrium equations as wellas the kinematic relations can be easily obtained from Equation (48) which can be used in thederivative of last row of Equation (47) to obtain a differential equation in terms of the slip field only,so that

RD3�−�bD�=0, (49)

where

R = E1 A1 −〈E1 A1h1 − E1 A1〉[

E I −E S

−E S E A

]−1{E1 A1h1

−E1 A1

}. (50)

The solution of Equation (49) can be used in the second row of Equation (48) to determine w

and then the solutions of both fields � and w can be used in the first row of Equation (47) todetermine v. This procedure produces ten integration constants, two of which can be shown to be



dependent by substituting the solutions into the last two rows of Equation (47), i.e. the displacementfields can be written in terms of eight unknown constants as

u=LC. (51)

where uT =〈v w �〉, CT =〈a1 a2 a3 a4 a5 a6 a7 a8〉 and

L=

⎡⎢⎢⎣

C1 sinh�z�

C1 cosh�z� 0 0 z3 z2 z 1

C2 cosh�z C2 sinh�z z 1 3E SE A z2 0 0 0

cosh�z sinh�z 0 0 − 6E1 A1 E2 A2h�bE A 0 0 0

⎤⎥⎥⎦ (52)

in which �=√�b/R. The vector of unknown constants C can be determined in terms of the

end displacements at z =0 and z = L , and the superconvergent finite element formulation can beobtained based on the exact homogenous solution i.e.

u=LG−1MU, (53)

where

M =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 1

0 0 1 0 0 −h2 0 0

0 0 0 1 0 0 0 −h2

−1 0 1 0 0 h 0 0

0 −1 0 1 0 0 0 h

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, (54)

G =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0C1

�0 0 0 0 0 1

C1 0 0 0 0 0 1 0

C1 sinh�L�

C1 cosh�L� 0 0 L3 L2 L 1

C1 cosh�L C1 sinh�L 0 0 3L2 2L 1 0

C2 0 0 1 0 0 0 0

C2 cosh�L C2 sinh�L L 1 3L2 E SE A 0 0 0

1 0 0 0 − 6E1 A1 E2 A2h�bE A 0 0 0

cosh�L sinh�L 0 0 − 6E1 A1 E2 A2h�bE A 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(55)

and U is as in Equation (15), i.e. the finite element nodal unknowns are the nodal displacements w1,w2, v and v′ at z =0 and z = L . By using Equations (45) and (46), and substituting the displacementfunctions v, w and � and their derivatives into the potential energy functional in Equation (4),the stiffness matrices associated with the bending and the axial deformations, and the slip energy,respectively can be obtained as

Kbe =∫

LMT(G−1)TBT

e (z)DeBe(z)G−1Mdz, (56)

Kse =∫

LMT(G−1)TLT

�(z)�bL�(z)G−1Mdz, (57)



where

De =

⎡⎢⎣

E I −E S E1 A1h1

−E S E A −E1 A1

E1 A1h1 −E1 A1 E1 A1

⎤⎥⎦ , (58)

Figure 3. Analysis results for the simply supported beam: (a) vertical deflection (�=0.2N/mm3);(b) vertical deflection (�=200N/mm3); (c) axial deflection w2 (�=0.2N/mm3); (d) axialdeflection w2 (�=200N/mm3); (e) slip (�=0.2N/mm3); (f) slip (�=200N/mm3); (g) curvature

(�=0.2N/mm3); and (h) curvature (�=200N/mm3).



Be =

⎡⎢⎢⎣

C1�sinh�z C1�cosh�z 0 0 6z 2 0 0

C2�sinh�z C2�cosh�z 1 0 6E SE A z 0 0 0

�sinh�z �cosh�z 0 0 0 0 0 0

⎤⎥⎥⎦ , (59)

and

L�(z)=⟨cosh�z sinh�z 0 0 − 6E1 A1 E2 A2h

�bE A 0 0 0⟩. (60)

This element provides exact nodal values and referred as the exact element (EE) herein. Proof ofexactness of the nodal values for general loading is given in Appendix A.

APPLICATIONS

The evidence of slip locking and the performances of the proposed elements are discussed on acomposite beam–column example subject to different boundary and loading conditions. The totalspan of the horizontal member is L =4m in all cases. The cross-sectional and material properties

(a) (b)

(c) (d)

(e)

Figure 4. Accuracy and convergence rate for the simply supported beam: (a) energy error (�=0.2N/mm3);(b) energy error (�=200N/mm3); (c) convergence rate (�=0.2N/mm3); (d) convergence rate

(�=200N/mm3); and (e) energy error versus connection stiffness based on four elements.



Table I. Average slopes from the log–log error curves for the simply supported beam.

Element

BE AE DE KE CE(1) (2) (3) (4) (5) (6)

�=0.2N/mm3 2.69 2.72 1.98 1.69 3.95�=200N/mm3 0.93 1.04 2.44 4.00 4.00

Table II. Percentages of the bending and slip energies based on four-element solutionfor the simply supported beam.

Element

EE BE AE DE KE CE(1) (2) (3) (4) (5) (6) (7) (8)

�=0.2N/mm3 Bending (%) 82.0 81.6 81.6 82.0 82.8 82.0Slip (%) 18.0 18.4 18.4 18.0 17.2 18.0

�=200N/mm3 Bending (%) 99.8 99.5 99.3 99.8 99.8 99.8Slip (%) 0.2 0.5 0.7 0.2 0.2 0.2

Table III. Slip deflections at the mid-points for the simply supported beam for�=200N/mm3 based on four-element solution.

Element

L (mm) AE×10−4 (mm) DE×10−4 (mm) EE×10−4 (mm)(1) (2) (3) (4)

500 7.37 5.80 5.221500 2.35 1.45 1.742500 −2.35 −1.45 −1.743500 −7.37 −5.80 −5.22

for beam 1 and beam 2 are E1 =200×103 MPa, A1 =6000mm2 and I1 =1122×102 mm4, E2 =26×103 MPa, A2 =7104mm2 and I2 =124×106 mm4. The axis is selected to pass through thecentroid of beam 2 and the distance between the centroids of the beams is 163.5mm. Analyseswere undertaken for two different slip stiffness parameters �=0.2N/mm3 (�L =2.64) and �=200N/mm3 (�L =83.38). The effective intersection surface width between the two components istaken as b=150mm.

Simply supported beam under vertical uniform distributed load

For the simply supported case (i.e. v(0)=w2(0)=v(L)=0) the beam is subjected to 3 kN/mvertical uniform distributed load. Analyses were made for one-, two- and four-element models.Figure 3 shows the vertical and longitudinal displacements, slip, and curvature based on four-element solutions, respectively. The normalized errors in total strain energy ‖e‖en are shown inFigures 4(a) and (b), and the convergence performances of the elements are shown in log–logscale in Figures 4(c) and (d). The effect of slip stiffness on the total strain energy is shown inFigure 4(e). The rate of convergence (i.e. p in ‖e‖en =Chp where C is an arbitrary constant andh is the element size) can be determined from the slope of the log–log error from Figures 4(c)and (d), and given in Table I. Bending and slip energies and their ratios to the total strain energyare shown in Table II. Table III shows the slip at the mid-points of the elements based on AE, DEand EE solutions.



Figure 5. Analysis results for the cantilever beam–column: (a) vertical deflection (�=0.2N/mm3);(b) vertical deflection (�=200N/mm3); (c) axial deflection w2 (�=0.2N/mm3); (d) axial deflection w2(�=200N/mm3); (e) slip (�=0.2N/mm3); (f) slip (�=200N/mm3); (g) curvature (�=0.2N/mm3); and

(h) curvature (�=200N/mm3).

From Figures 3(a), (c) and (e) it can be verified that for the flexible shear connection (�=0.2N/mm3) deflections based on all finite elements are close to the exact solution for which theslip energy is around 18% of the total strain energy (Table II). On the other hand for stiff shearconnection (�=200N/mm3), the slip strain energy is 0.2% of the total strain energy (Table II).In that case, the vertical and axial deflection curves are still satisfactory for all finite elementsolutions as shown in Figures 3(b) and (d), respectively; however, the nodal values of slip based



(a) (b)

(c) (d)

(e)

Figure 6. Accuracy and convergence rate for the cantilever beam–column: (a) energy error (�=0.2N/mm3);(b) energy error (�=200N/mm3); (c) convergence rate (�=0.2N/mm3); (d) convergence rate

(�=200N/mm3); and (e) energy error versus connection stiffness based on four elements.

Table IV. Average slopes from the log–log error curvesfor the cantilever beam–column.

Element

BE AE DE KE CE(1) (2) (3) (4) (5) (6)

�=0.2N/mm3 2.15 1.92 3.59 2.01 3.22�=200N/mm3 2.06 2.38 2.12 0.28 0.27

on element BE are totally erroneous and severe oscillations occur in the slip field as shownin Figures 3(f). These results are consistent with those reported in Reference [13]. In addition,Figure 3(h) shows that the curvature tends to become a constant value in between nodes ofelement BE. This reduces the accuracy of the analysis. By comparing Figures 4(a) and (b), it canbe verified that the relative error in the total strain energy increases significantly for element BEwhen the connection becomes stiffer. Figures 4(c) and (d) show that the convergence rate reducesi.e. slope changes from 2.69 to 0.93 as given in Table I. Thus, locking in element BE is evident



Table V. Percentages of the bending and slip energies based on four-elementsolution for the cantilever beam–column.

Element

EE BE AE DE KE CE(1) (2) (3) (4) (5) (6) (7) (8)

�=0.2N/mm3 Bending (%) 89.8 89.7 89.7 89.7 89.9 89.8Slip (%) 10.2 10.3 10.3 10.3 10.1 10.2

�=200N/mm3 Bending (%) 100 99.9 99.9 100 100 100Slip (%) 0 0.1 0.1 0 0 0

Table VI. Slip at the mid-points for the cantilever beam–column for �=200N/mm3

based on four-element solutions.

Element


500 8.58 7.27 5.801500 6.88 4.80 5.802500 5.85 7.30 5.803500 4.87 2.31 5.80

since for stiffer connection parasitic slip as well as reductions in the accuracy and convergencerate occur. A comparison of accuracy performances between elements BE and CE and the effect ofincreasing connection stiffness on the accuracy was initially made in Reference [13]. Figure 4(e)is consistent with that provided in Reference [13] which also shows the performances of elementsAE, DE and KE based on four-element solutions. From Figures 4(b), (d) and (e) it can be verifiedthat the improvement gained by element AE is very little. Nodal slip values based on element AEstill show severe oscillations however AE provide good results for the slip at element mid-pointsas shown in Table III. On the other hand, the slip based on element DE are step functions asshown in Figure 3(e) and (f) and these functions follow the exact curves and agree well with theexact solutions at the mid-point of the elements as shown in Table III. Thus, the oscillations canbe prevented with element DE. It should also be noted that element DE can provide upper boundsto the exact total strain energy while the other finite element solutions provide lower bounds. InFigure 3 the deflection curves for stiff connection based on elements KE and CE agree perfectlywell with the exact solution. Figure 4 shows that the improvement in accuracy and convergence ratebased on element KE (Figure 4(d) and (e)) is significant for stiffer connection and its performanceis as good as element CE which has the best performance in all cases. On the other hand, theperformance of element KE reduces for flexible connections as can be verified from Figure 4(e).However, the deflection fields are still represented well with element KE and the piecewise linearcurvature field agrees with the exact solution at element mid-points.

Cantilever beam–column subject to vertical and axial loads at the tip

The cantilever beam–column (i.e. v(0)=v′(0)=w1(0)=w2(0)=0) is subjected to 5 kN verticaland −20kN longitudinal loads both acting at the free end. Finite element analyses were made forone-, two- and four-element models. Figure 5 shows the vertical and longitudinal deflections, slip,and curvatures based on four-element solutions. The convergence and accuracy performances ofthe elements with different connection stiffnesses based on the energy norm are given in Figure 6.Table IV gives the rate of convergence based on the slope of the log–log error given in Figures 6(c)



Figure 7. Analysis results for the propped cantilever: (a) vertical deflection (�=0.2N/mm3);(b) vertical deflection (�=200N/mm3); (c) axial deflection w2 (�=0.2N/mm3); (d) axial deflection w2(�=200N/mm3); (e) slip (�=0.2N/mm3); (f) slip (�=200N/mm3); (g) curvature (�=0.2N/mm3); and

(h) curvature (�=200N/mm3).

and (d). Bending, slip, and the total strain energies are shown in Table V. Table VI shows the slipat the mid-points of the elements based on AE, DE, and EE solutions.

Similar to the simply supported case all elements provide sufficiently accurate results for theflexible connection; however, for stiff shear connection element BE shows severe oscillations,and the convergence rate and accuracy diminish due to locking. Figure 5(f) shows that slip dueto element AE also oscillate for the stiff connection; however, the mid-point values are accurate(Table VI). On the other hand, the accuracy of element DE is better than element BE for bothflexible and stiff connections and oscillations in slip are prevented for stiff connection. Element KE



(a) (b)

(c) (d)

(e)

Figure 8. Accuracy and convergence rate for the propped cantilever: (a) energy error (�=0.2N/mm3);(b) energy error (�=200N/mm3); (c) convergence rate (�=0.2N/mm3); (d) convergence rate

(�=200N/mm3); (e) energy error versus connection stiffness based on four elements.

Table VII. Average slopes from the log–log error curves for the propped cantilever.

Element

BE AE DE KE CE(1) (2) (3) (4) (5) (6)

�=0.2N/mm3 2.37 2 0.45 2.11 3.91�=200N/mm3 2.28 2.43 2.24 1.84 2.43

has a very good performance for stiff connection and its accuracy is much better than elements BE,AE, and DE. For stiff connection, the accuracy and convergence rates of element KE (Figure 6(b),(d) and (e)) is almost as good as element CE which has the best performance in all cases.

Propped cantilever under vertical uniform distributed load and vertical tip load

The cantilever beam–column is propped at the mid-span (i.e. v(0)=v′(0)=w1(0)=w2(0)=v(L/2)=0) and subjected to 3 kN/m vertical uniformly distributed load and 5 kN vertical point



Table VIII. Percentages of the bending and slip energies based on four-elementsolutions for the propped cantilever.

Element

EE BE AE DE KE CE(1) (2) (3) (4) (5) (6) (7) (8)

�=0.2N/mm3 Bending (%) 85.9 85.9 85.9 86.0 86.2 85.9Slip (%) 14.1 14.1 14.1 14.0 13.8 14.1

�=200N/mm3 Bending (%) 99.6 99.1 99.2 99.5 99.6 99.6Slip (%) 0.4 0.9 0.8 0.5 0.4 0.4

Table IX. Slip at the mid-points for the propped cantilever for�=200N/mm3 based on four-element solutions.

Element


250 −1.33 −1.19 −1.04750 −1.30 −1.20 −1.211250 −1.50 −1.34 −1.391750 −2.00 −1.79 −1.552250 1.60 1.41 1.182750 1.10 0.93 1.013250 0.84 0.88 0.843750 0.62 0.63 0.67

load at the free end. Finite element analyses were made for two-, four- and eight-element models.Deflections, slips and curvatures are reported in Figure 7. Figure 8 shows the convergence andaccuracy performances of each element based on different connection stiffness. Table VII givesthe rate of convergence. Bending and slip energies and their ratios to the total strain energy aregiven in Table VIII, and slip at the mid-points of elements AE, DE and EE are given in Table IX.The results show that oscillations that occur in stiff connection due to locking in element BE canbe circumvented with element DE. In order to further improve the accuracy of the analysis forstiff connections, element KE can be used which provides a better accuracy than elements DE,AE, and DE. However, Figure 8(b) and (d) shows that element CE has a slightly better accuracyin the energy norm than element KE.

CONCLUSIONS

The displacement-based finite element formulation based on linear interpolation of longitudinaldisplacements and cubic interpolation of vertical displacements suffers from locking when usedfor the analysis of composite beam–columns that consist of two Euler–Bernoulli beams juxtaposedwith deformable shear connections. The main problem with the displacement-based formulation isthat for stiff connections the nodal values of slip may be totally erroneous and severe oscillationsoccur in the slip field. In this study, the assumed strain method, kinematic interpolatory techniqueand discrete strain gap method were employed first time to alleviate locking behavior in thedisplacement-based finite element analysis of composite beam–columns. The results show that theperformance of the displacement-based finite element analysis can be improved by using thesemethods. A systematic solution of the differential equations of equilibrium was also provided, anda superconvergent element was developed in this paper. Comparisons of the finite element resultswith those based on the exact solution show that the elements developed based on the assumed



strain and discrete strain gap methods slightly improve the accuracy in the energy norm and provideaccurate slip at element mid-points. The element based on the kinematic interpolatory techniqueimproves the accuracy of the results significantly for stiff connections and totally eliminates sliposcillations. Thus, the finite elements developed in this study are suitable for the analysis ofcomposite beam–columns with stiff shear connections.

APPENDIX A

The exact displacement field uT =〈v w �〉 for a member subject to end displacements andmember loads can be considered to consist of two components u=uh +up corresponding to thehomogenous solution uh and the particular solution up respectively, where uT

h =〈vh wh �h〉 anduT

p =〈vp wp �p〉. By expressing the homogeneous displacement vector uh in terms of nodaldisplacements one obtains

uh =LG−1MUh (A1)

in which Uh is the vector of nodal displacements based on the superconvergent finite elementsolution. The particular solution vector up can be expressed as the sum of two components

up =l+LG−1MUp. (A2)

In Equation (A2), the components of the displacement vector lT =〈�v �w ��〉 consists of anydisplacement functions satisfying the equilibrium equations in the presence of member load terms.The second displacement field vector consists of any linear combination of the homogeneoussolution. Thus, if vector l is a particular solution for the differential equations of equilibrium,vector up must also be a particular solution irrespective of the magnitude of the arbitrary constantsin vector Up. Conventionally, once a particular solution is determined, the entries vector Up areselected to vanish for simplicity. In contrast, we deviate from this convention in this proof byselecting the entries Up in order for the particular solution vanish at the boundaries, i.e.

vp(0)=v′p(0)=wp(0)=�p(0)=vp(L)=v′

p(L)=wp(L)=�p(L)=0. (A3)

The vector of displacement derivatives w can be written in terms of the homogeneous and particularparts as

w=b(z)+BeG−1M(Uh +Up) (A4)

in which w=〈v′′ w′ �′〉T and bT =〈�′′v �′

w �′�〉. It should be noted that the variation of

the prescribed components of the displacement vector in Equation (A4)‖ vanishes, i.e.dw=BeG−1MdUh . Thus, for the general non-homogeneous loading case, the virtual work expressioncan be written as

��=�UTh (Kbe +Kse)Uh −�UT

h F+��e (A5)

in which

��e =∫

L〈�v′′

h �w′h ��′

h〉

⎡⎢⎣

E I −E S E1 A1h1

−E S E A −E1 A1

E1 A1h1 −E1 A1 E1 A1

⎤⎥⎦⎧⎪⎪⎨⎪⎪⎩

v′′p

w′p

�′p

⎫⎪⎪⎬⎪⎪⎭ dz+�b

∫L��h�p dz. (A6)

‖Correction made here after initial online publication.



In the following, it will be shown that the extra term in virtual work expression ��e vanishes.This signifies that the stiffness matrix and the energy equivalent nodal force vector for thenon-homogenous case is identical to that of the homogenous case. By integrating the termsincluding w′

p and �′p once by parts, and those including the term v′′

p twice by parts, ��e can bewritten as

��e =∫

L〈vp wp �p〉

⎧⎪⎪⎨⎪⎪⎩

E I�vivh − E S�w′′′

h + E1 A1h1��′′′h

−E S�v′′′h + E A�w′′

h − E1 A1��′′h

−E1 A1h1v′′′h + E1 A1�w′′

h − E1 A1��′′h +�b��h

⎫⎪⎪⎬⎪⎪⎭ dz

+(E I�v′′h − E S�w′

h + E1 A1h1��′h)v′

p |L0 −(E I�v′′′

h − E S�w′′h + E1 A1h1��′′

h)vp|L0

−(E S�v′′′h − E A�w′′

h + E1 A1��′′h)wp|L

0

+(E1 A1h1�v′′′h − E1 A1�w′′

h + E1 A1��′′h)�p|L

0 . (A7)

From Equation (A7) it can be shown that the extra term in virtual work expression ��e vanishesbecause the displacement fields vh,wh and �h satisfy the homogenous form of the equilibriumequations and vp, v′

p, wp and �p satisfy the boundary conditions in Equation (A3)‖. Exactintermediate values for the general loading can be obtained by post processing. For this purpose anintermediate point j between the element ends can be selected. The discretized exact equilibriumequations between the element node I and any intermediate point j can be written as

[Ka Kb−−−−−KT

b Kc----

--

]{UI−−U j

}={

FI−−F j

}, (A8)

where the stiffness matrix, nodal displacements and the energy equivalent nodal load vectors arewritten in partitioned form. The span in calculating the stiffness matrices Kbe and Kse, and thusthe sub-matrices Ka , Kb and Kc should be calculated from node I to the intermediate point j .Since at node I the displacement vector UI and the energy equivalent force vector FI are known,by using Equation (A8) the displacements and internal forces at the intermediate point j can becalculated as

U j = K−1b (Fa −KaUI ), (A9)

F j = KaUI +KTb U j . (A10)

These expressions can be adopted along with numerical integration to calculate the exact strainenergies of the deformed beams.

ACKNOWLEDGEMENTS

The work in this paper was supported by the Australian Research Council through a Discovery Projectawarded to the second author. An additional support to the first author was provided by the Faculty ofEngineering at the University of New South Wales through an Early Career Research Grant. Their supportis gratefully acknowledged.

‖Correction made here after initial online publication.



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Treatment of slip locking for displacement-based finite element analysis of composite beam-columns

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