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Comput. Methods Appl. Mech. Engrg. 199 (2010) 1371–1385

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Comput. Methods Appl. Mech. Engrg.

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Multi-scale computational model for failure analysis of metal framesthat includes softening and local buckling

Jaka Dujc a,b, Boštjan Brank a,*, Adnan Ibrahimbegovic b

a University of Ljubljana, Faculty of Civil and Geodetic Engineering, Ljubljana, Sloveniab Ecole Normale Supérieure de Cachan, Cachan, France

a r t i c l e i n f o

Article history:Received 11 March 2009Received in revised form 13 May 2009Accepted 3 September 2009Available online 3 October 2009

Keywords:FrameFailure analysisShell-beam modelStrong discontinuityPlasticitySoftening hinge

0045-7825/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.cma.2009.09.003

* Corresponding author.E-mail address: [email protected] (B. Brank).

a b s t r a c t

In this work we present a new modelling paradigm for computing the complete failure of metal frames bycombining the stress-resultant beam model and the shell model. The shell model is used to compute thematerial parameters that are needed by an inelastic stress-resultant beam model; therefore, we considerthe shell model as the meso-scale model and the beam model as the macro-scale model. The shell modeltakes into account elastoplasticity with strain-hardening and strain-softening, as well as geometricalnonlinearity (including local buckling of a part of a beam). By using results of the shell model, thestress-resultant inelastic beam model is derived that takes into account elastoplasticity with hardening,as well as softening effects (of material and geometric type). The beam softening effects are numericallymodelled in a localized failure point by using beam finite element with embedded discontinuity. The ori-ginal feature of the proposed multi-scale (i.e. shell-beam) computational model is its ability to incorpo-rate both material and geometrical instability contributions into the stress-resultant beam modelsoftening response. Several representative numerical simulations are presented to illustrate a very satis-fying performance of the proposed approach.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

The limit load analysis and the complete failure (collapse) anal-ysis of a structural system are important problems in performance-based design procedure. The same is true for structural dynamics.A typical example is the push-over analysis in earthquake engi-neering; a nonlinear static analysis of a building structure, sub-jected to an equivalent static loading that is pushing a structureover its limit capacity (e.g. [1]).

It has been observed from failure modes, produced by seismicactivities and experimental tests, that practical frame structures,composed of columns and beams, fail by exhibiting localized fail-ures in a limited number of critical zones. Those critical zonesare usually described as plastic (inelastic) hinges. A usual approachto compute the limit load of a structural frame, or to compute itscomplete failure, is to model plastic hinges with nonlinear inelasticspring finite elements. Inelastic springs are introduced at prede-fined critical locations in a mesh of conventional elastic beam finiteelements (e.g. [2]), or, alternatively (e.g. [3]), elastic beam elementswith lumped nonlinear spring at both ends are used.

When studying the full collapse of a frame, a softening responseis observed after reaching its limit capacity; the load reduces with

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additional frame deformation. This structural softening responsecan be modelled by an elastoplastic constitutive model with a soft-ening relation between the generalized strain measures and thecorresponding stresses resultants. However, inclusion of strain-softening in the standard finite element approximation results ina physically unrealistic mesh-dependent numerical solutions. Dif-ferent alternatives have been proposed to solve this mesh-depen-dency (e.g. see [4] for a recent review). All of them are related toregularization of ill-posed mathematical problem, which arises asa result of inclusion of strain-softening in the elastoplastic consti-tutive model.

The most frequently used regularization nowadays is so-calledembedded (strong) discontinuity approach; see e.g. Jirasek [5],Armero and Ehrlich [6–8], Ibrahimbegovic et al. [9,10], and Wac-kerfuss [11] for implementation of embedded discontinuity ap-proach for beam and bar finite elements. The key point isintroduction of localized energy dissipation. This is achieved byintroducing strong discontinuity in kinematic fields (e.g. a jumpin rotation of the beam axis), and defining local dissipative mech-anism at that discontinuity in terms of a softening cohesive law(e.g. a softening law between the bending moment and the rota-tion jump). Localized dissipative mechanism eliminates themesh-dependency of numerical solutions. For beams, the intro-duced discontinuity can be naturally regarded as a softening plas-tic hinge.

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1372 J. Dujc et al. / Comput. Methods Appl. Mech. Engrg. 199 (2010) 1371–1385

In the first part of this work, we carry on with the developmentsrelated to numerical treatment of localized failure in beams in or-der to study failure of elastoplastic metal frames. To this end, wederive a planar straight stress-resultant beam finite element withthe following features: (i) Euler–Bernoulli kinematics, (ii) an elas-toplastic stress-resultant constitutive model with isotropic hard-ening, (iii) a localized softening plastic hinge related to thestrong discontinuity in generalized displacements, and (iv) anapproximation of the geometrically nonlinear effects by using thevon Karman strains for the virtual axial deformations.

The derived finite element can be effectively used for the limitload analysis, the push-over analysis and the complete failure anal-ysis of planar metal frames. Localized softening, introduced byembedded discontinuity approach, solves the problem of mesh-dependency. Moreover, the spreading of plasticity over the entireframe and the appearance of the softening plastic hinges in theframe is consistently accounted for in the course of the nonlinearanalysis. With respect to the existing embedded discontinuitybeam finite elements, see Armero et al. [6–8], and Wackerfuss[11], we use more complex material models: stress-resultant elas-toplasticity with hardening to describe beam material behaviorand stress-resultant rigid-plastic softening to describe materialbehavior at the discontinuity.

The second part of this work pertains to a procedure that pro-vides characteristic values of material parameters, used by choseninelastic models. Those values are the yield and the failure (ulti-mate resistance) moments of the beam cross-section, the harden-ing modulus for the stress-resultant beam plasticity, and thesoftening modulus for the softening plastic hinge. Ideally, oneshould for any geometry of beam cross-section, any material typeand any type of beam stress state seek the appropriate experimen-tal results and fit to them the beam model material parameterswith respect to significant quantities (e.g. forces, displacements,energy, dissipation), see e.g. Kucerova et al. [12]. In the absenceof experimental results for metal beams to make any definitiveconclusions, we turn to another approach that belongs under mul-ti-scale label.

The material parameters are obtained by numerical simulationson representative part of a beam by using a refined model, which issuperior to the beam model in a sense that it is able to describe inmore detail the beam response. We focus on rather typical metalframes with thin-walled cross-sections. For this kind of frames,the refined model can be chosen as the nonlinear shell model(e.g. [13,14]). The shell model is superior to the beam model in pro-viding a proper local description of the strain/stress fields and theoverall spread of plasticity. It is also capable of describing localbuckling of the flanges and the web, which is, in bending domi-nated conditions, very often the reason for the localized beam fail-ure. Considering the above, the shell model can be seen as themeso-scale model and the beam model as the macro-scale model.

The outline of the paper is as follows. In Section 2, we derive anelastoplastic Euler–Bernoulli beam finite element with embeddeddiscontinuity. In Section 3, we discuss computation of the beamplasticity parameters and the softening plastic hinge parametersby using the shell model. In Section 4, we present details of thecomputational procedure. Numerical examples are presented inSection 5 and concluding remarks in Section 6.

u1,w1,w1' u2,w2,w2'αu,αθ

xxd

L e

Fig. 1. Beam finite element with embedded discontinuity.

2. Beam element with embedded discontinuity

We consider in this section a planar Euler–Bernoulli beam finiteelement. The element can represent an elastoplastic bending,including the localized softening effects, which are associated withthe strong discontinuity in rotation. The geometrical nonlinearityis approximately taken into account by virtual axial strains of

von Karman type, which allows this element to capture the globalbuckling modes.

2.1. Kinematics

We consider a straight planar frame member, which middle axisoccupies domain X 2 R. Spatial discretization of X leads toNel ðX ¼ ½0; L� ¼ [Nel

e¼1LðeÞÞ finite elements. A typical 2-node finiteelement is presented in Fig. 1. The following notation is used: ui

are nodal axial displacements, wi are nodal transverse displace-ments, w0i are nodal values of the beam axis rotation (derivativeof transverse displacement with respect to the beam axial coordi-nate x 2 ½0; LðeÞ�), and i ¼ 1;2 is node number. In addition to thestandard degrees of freedom at the two nodes, we assume strongdiscontinuity in axial displacement au and beam axis rotation ah

at xd 2 LðeÞ . We also assume that the domain of the discontinuityinfluence corresponds to a single element. The axial displacementis thus defined as:

uhðx; xdÞ ¼ NuðxÞuþMuðx; xdÞau; ð1Þ

where NuðxÞ ¼ f1� x=LðeÞ; x=LðeÞg; u ¼ fu1; u2gT , and Muðx; xdÞ is afunction with zero values at the nodes and a unit jump at xd, i.e.Muð0; xdÞ ¼ MuðLðeÞ; xdÞ ¼ 0 and Muðxþd ; xdÞ ¼ Muðx�d ; xdÞ þ 1. Simi-larly, we can write the transverse displacement as

whðx; xdÞ ¼ NwðxÞwþ Nw0 ðxÞw0 þMhðx; xdÞah; ð2Þ

where

NwðxÞ ¼ 2x

LðeÞ

� �3

� 3x

LðeÞ

� �2

þ 1;�2x

LðeÞ

� �3

þ 3x

LðeÞ

� �2( )

;

w ¼ fw1;w2gT; ð3Þ

Nw0 ðxÞ ¼ LðeÞx

LðeÞ

� �3

� 2x

LðeÞ

� �2

þ x

LðeÞ;

x

LðeÞ

� �3

� x

LðeÞ

� �2( )

;

w0 ¼ fw01;w02gT ð4Þ

and Mhðx; xdÞ is a function with zero values at the nodes and a unitjump of its first derivative at xd, i.e. Mhð0; xdÞ ¼ MhðLðeÞ; xdÞ ¼ 0 andMh0 ðxþd ; xdÞ ¼ Mh0 ðx�d ; xdÞ þ 1.

The beam axial strain can then be written as:

eðx; xdÞ ¼@uh

@x¼ BuðxÞuþ Guðx; xdÞau|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

e

þ dxdau|fflffl{zfflffl}e

; ð5Þ

where BuðxÞ ¼ f�1=LðeÞ;1=LðeÞg; Guðx; xdÞ ¼ @Mu=@x, and dxdis the

Dirac-delta, which appears due to discontinuous nature of axial dis-placement at xd. We further divide the axial strain into a regularpart e and a singular part e. The later can be interpreted as a local-ized plastic axial strain. The beam curvature is computed as:

jðx; xdÞ ¼@2wh

@x2 ¼ BwðxÞwþ Bw0 ðxÞw0 þ Ghðx; xdÞah|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}j

þ dxdah|ffl{zffl}

j

; ð6Þ

J. Dujc et al. / Comput. Methods Appl. Mech. Engrg. 199 (2010) 1371–1385 1373

where

BwðxÞ ¼ � 6

LðeÞ2 1� 2x

LðeÞ

� �;

6

LðeÞ2 1� 2x

LðeÞ

� �� ; ð7Þ

Bw0 ðxÞ ¼ � 2LðeÞ

2� 3x

LðeÞ

� �;� 2

LðeÞ1� 3x

LðeÞ

� �� ð8Þ

and Ghðx; xdÞ ¼ @2Mh=@x2. The curvature j is divided into a regularpart j and a singular part j. The later can be interpreted as a local-ized plastic curvature. The beam strains can be rewritten in a matrixnotation as

� ¼ �þ �; ð9Þ� ¼ Bd|{z}e� þGa; � ¼ dxd

a; ð10Þ

where � ¼ fe;jgT; � ¼ fe;jgT

; e� ¼ fee; ejgT; � ¼ fe;jgT and

B ¼ Bu 0 00 Bw Bw0

� �; d ¼ fuT ;wT ;w0TgT

; ð11Þ

G ¼ DIAG Gu;Ghn o

; a ¼ au;ahf gT : ð12Þ

Kinematic description of the element is concluded by derivation ofG operator. It may be derived indirectly (i.e. without defining Mu

and Mh) through requirement that an element has to be able to de-scribe strain-free mode at some non-zero values of au and ah, seeArmero and Erlich [7]. According to Fig. 2, the generalized nodal dis-placements dhinge ¼ fu1; u2; w1; w2; w01; w

02g

T of such strain-free modeare composed as

dhinge ¼ drigid þ Dhingea; Dhinge ¼0 1 0 0 0 00 0 0 LðeÞ � xd 0 1

� �T

;

ð13Þ

where drigid ¼ fu1; u1; w1; w1 þ w01LðeÞ; w01; w01g

T are generalized nodaldisplacements due to rigid-body motion of a complete beam, andDhingea are generalized nodal displacements due to rigid-body mo-tion of one part of the beam due to imposed strong discontinuitya ¼ fau; ahgT . If we now set strains (9) to zero for dhinge, we have

0 ¼ Bdhinge þ Ga ¼ Bdrigid|fflfflffl{zfflfflffl}¼0

þðGþ BDhingeÞa: ð14Þ

Since the above equation should hold for any a, we get the G oper-ator as

G ¼ �BDhinge; ð15Þ

which leads to

Guðx; xdÞ ¼ �1

LðeÞ; ð16Þ

Ghðx; xdÞ ¼ �1þ 3 1� 2xd

LðeÞ

1� 2x

LðeÞ

LðeÞ

: ð17Þ

The above definition of G matrix concludes kinematic description ofthe geometrically linear element.

Remark 1. By using (1) and (2) to describe strain-free mode ofFig. 2, one can also derive interpolation functions Mu and Mh. By

u1,w1

w1'

u2 u1 u

w2 w1 w1'Le θ Le xd

w2' w1' θ

αuw1' θ

xd

L e

Fig. 2. Strain-free mode of the element.

setting in (1) u1 ¼ u1 ¼ 0; u2 ¼ u2 ¼ au; uh ¼ 0 for x < xd, and uh ¼au for x P xd , one can conclude that Mu¼ Hðx�xdÞ�Nu � f0;1g.Here, Hðx�xdÞ is unit-step function, which is 0 for x< xd and 1for x P xd. Derivation @Mu=@x gives Gu in (16). By using similarprocedure for bending in (2), one can obtain Mh ¼Hðx�xdÞðx�xdÞ�Nw � f0;LðeÞ �xdg�Nw0 � f0;1g. Derivation @2Mu=@x2 givesGh in (17).

In order to account for the geometrically nonlinear effects, andrelated global buckling, we will use the von Karman axial strainwhen computing the virtual axial strain. The real axial strain, usedfor computing the internal forces, will still be assumed as linear, asgiven in Eq. (5). The von Karman axial strain is defined aseVK ¼ @uh

@x þ 12

@wh

@x

2. The corresponding virtual axial strain is thus:

deVK ¼ @duh

@xþ @wh

@x@dwh

@x: ð18Þ

If we choose to interpolate duh; wh and dwh in (18) asduh ¼ NuðxÞduþMuðx; xdÞdau; wh ¼ NwðxÞwþ Nw0 ðxÞw0 anddwh ¼ NwðxÞdwþNw0 ðxÞdw0, where du ¼ fdu1; du2gT is vector of vir-tual nodal axial displacements, dw ¼ fdw1; dw2gT anddw0 ¼ fdw01; dw02g

T are vectors of virtual nodal transverse displace-ments and rotations, and dau is virtual discontinuity in axial dis-placement at xd, the chosen interpolations lead to

deVK ¼ BuðxÞduþ Bu;wðxÞdwþ Bu;w0 ðxÞdw0 þ Guðx; xdÞdau|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}deVK

þ dxddau|fflfflffl{zfflfflffl}de

;

ð19Þ

where

Bu;wðxÞ ¼ CdNw

dx; Bu;w0 ðxÞ ¼ C

dNw0

dx; C

¼ dNw

dx�wþ dNw0

dx�w0

!: ð20Þ

The linear matrix operator B from (11) should be thus replaced withthe nonlinear matrix operator BVK when computing virtual strainsd� ¼ fde; djgT , i.e.

de ¼ deVK

dj

� �¼ Bu Bu;w Bu;w0

0 Bw Bw0

( )|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

BVK

ddþ Gda:ð21Þ

In (21) above, we denote with dd ¼ fduT ; dwT ; dw0TgT the general-ized virtual nodal displacements and with da ¼ fdau; dahgT virtualjumps at xd.

Remark 2.

(a) The tangent stiffness matrix of the beam finite element withvon Karman virtual axial strain has symmetric geometricpart and non-symmetric material part. The matrix can besymmetrized by using B instead of BVK when computing itsmaterial part. Such an approach would lead (for elasticbeams) to the element presented in Wilson [2, Section 11].In this work we use non-symmetric tangent stiffness matrix.

(b) If one uses von Karman definition of axial strains for bothreal and virtual strains, see Reddy [15, Section 4.2], the ele-ment exhibits severe locking.

2.2. Equilibrium equations

The weak form of the equilibrium equations (the principle ofvirtual work) for an element e of a chosen finite element mesh withNel finite elements, can be written as:

Mu

Mu

tM

αθ


dPint;ðeÞ � dPext;ðeÞ ¼ 0: ð22Þ

By denoting the virtual strains as d� ¼ fdeVK ; djgT , where virtualcurvatures dj ¼ djþ dj are of the same form as real curvatures jin (6), we can write a single element contribution to the virtualwork of internal forces as:

dPint;ðeÞ ¼Z LðeÞ

0ðd�ÞTrdx

¼Z LðeÞ

0ddTðBVKÞTrdx|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

standard

þZ LðeÞ

0daTðGT

rþ dxdrÞdx|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

additional

; ð23Þ

The matrices BVK and G are defined in (21) and (15), and

r ¼ fN;MgT ð24Þ

is the vector of beam internal forces that contains axial force N andbending moment M. From the term ‘‘standard” in (23) we obtain thevector of element internal nodal forces

f int;ðeÞ ¼Z LðeÞ

0ðBVKÞTrdx: ð25Þ

From the virtual work of external forces dPext;ðeÞ we can get the vec-tor of element external nodal forces fext;ðeÞ, representing the externalload applied to the element. The finite element assembly of vectorsf int;ðeÞ and fext;ðeÞ, for all elements of the chosen mesh, leads to a set ofglobal (i.e. mesh related) equations

ANele¼1ðf

int;ðeÞ � fext;ðeÞÞ ¼ 0; ð26Þ

where A is the assembling operator.We have only used one part of the right side of Eq. (23) in (22)

when getting the set of global Eq. (26). The other term in (23), de-noted as ‘‘additional” (since it results from additional enrichedkinematics due to embedded discontinuity), will also contributeto the weak form of the equilibrium. However, we will treat thiscontribution locally element by element. Then, in view of (22),the following two equations are obtained for each element of thechosen mesh

hðeÞ ¼ hðeÞN ; hðeÞM

n oT¼Z LðeÞ

0GT

rþ dxdr

dx

¼Z LðeÞ

0GT

rdxþ rjxd|{z}¼t

¼Z LðeÞ

0GT

rdxþ t ¼ 0; 8e2 ½1;Nel�: ð27Þ

We have defined in (27) vector t ¼ rjxd¼ ftN; tMgT with components

tN and tM that represent axial traction and moment (bending) trac-tion at the discontinuity. By using (17) and (24), one can obtain thecomponent form of (27)

hðeÞN ¼Z LðeÞ

0GuNdxþ tN ¼ 0;

hðeÞM ¼Z LðeÞ

0GhMdxþ tM ¼ 0; 8e 2 ½1;Nel�: ð28Þ

The problem of solving a set of global equation (26) together with aset of local (element) equation (27) will be further addressed inSection 4.

2.3. Constitutive relations

We assume that the axial response of the beam material re-mains always elastic, thus discarding the failure by necking, forexample. For the bending behavior of the beam material we choosethe following constitutive models: (i) stress-resultant elastoplasticconstitutive model with linear isotropic hardening, (ii) stress-

resultant rigid-plasticity model with linear softening at the soften-ing plastic hinge. The basic ingredients of the chosen constitutiverelations are built on classical plasticity (e.g. [16]) and can be sum-marized as:

� The regular strains � (10) can be additively decomposed intoelastic part �e and plastic part �p

� ¼ �e þ �p; �e ¼ fee;jegT; �p ¼ fep;jpgT

: ð29Þ

� The axial strain of the beam (5) remains always elastic, thus

e ¼ e ¼ ee; e ¼ 0() ep ¼ 0; au ¼ 0: ð30Þ

� The free energy for the beam material (before localized soften-ing is activated) is assumed to be the sum of the strain energyfunction W and the hardening potential N

Wð�e; nÞ :¼Wð�eÞ þ NðnÞ ¼ 12�eT C�e þ 1

2Khn

2; ð31Þ

where C ¼ DIAGfEA; EIg; E is elastic modulus, A and I are areaand moment of inertia of cross-section, n P 0 is strain-like bend-ing hardening variable, and Kh P 0 is linear bending hardeningmodulus.

� The yield criterion for the beam material is defined in terms ofthe bending moment. The admissible values of the bendingmoment and the stress-like bending hardening variable �qðnÞare governed by the function

/ðM; �qÞ ¼ jMj � ðMy � �qÞ 6 0; ð32Þ

where My > 0 denotes the positive yield moment of the cross-section. Influence of the axial force N on the cross-section yield-ing is taken into account by defining My and �q as functions of N,as shown subsequently.

� The localization (failure) criterion that activates softening at dis-continuity at xd is defined in terms of the bending traction tM

and the stress-like softening bending variable ��qðnÞ (the later isdefined in terms of the bending strain-like softening variable n)

/ðtM; ��qÞ ¼ jtMj � ðMu � ��qÞ 6 0; ð33Þ

where Mu > My > 0 denotes the positive ultimate (failure) mo-ment of the cross-section. Influence of axial force N on thecross-section failure is taken into account by defining Mu and ��qas functions of N, as shown below.

� The bending traction tM at the discontinuity xd is related to therotation jump as shown in Fig. 3

tM ¼ tMðahÞ: ð34Þ

The remaining ingredients of the elastoplasticity with harden-ing can be obtained from the consideration of thermodynamicsof associative plasticity and the principle of maximum plastic dis-sipation (see e.g. [4,17,18]). In the present beam model the elasto-plasticity with hardening happens for a ¼ 0, which leads to � ¼ e�

Fig. 3. Rigid-plastic cohesive law at discontinuity.


and � ¼ 0, see (10). By using (29) and (31) the mechanical dissipa-tion can be written as

0 6 D ¼def :rT _�� _Wð�e; nÞ ¼ r� @W

@�e

� �T

_�e þ rT _�p � @W@n

_n; ð35Þ

where ð _oÞ ¼ @ðoÞ=@t and t 2 ½0; T� is pseudo-time. By assuming thatthe elastic process is non-dissipative (i.e. D ¼ 0), and that the plasticstate variables do not change, we obtain from (35)

r ¼ @W@�e¼ C�e ) N ¼ EAe; M ¼ EIðj� jpÞ: ð36Þ

We can define the hardening variable �q by further considering (35)and (31)

�q ¼ � @W@n¼ � @N

@n¼ �Khn: ð37Þ

By replacing (36) and (37) in (35), the plastic dissipation can be ob-tained as

Dp ¼ rT _�p þ �q _n )ð24Þ;ð29Þ;ð30Þ

Dp ¼ M _jp þ �q _n: ð38Þ

The principle of maximum plastic dissipation states that among allthe variables ðM; �qÞ that satisfy the yield criteria /ðM; �qÞ 6 0, oneshould choose those that maximize plastic dissipation (at frozenrates _jp and _n). This can be written as a constrained optimizationproblem:

nM;�q

max_c

LpðM; �q; _cÞ ¼ �DpðM; �qÞ þ _c/ðM; �qÞh i

; ð39Þ

where _c P 0 plays the role of Lagrange multiplier. By using (38) and(32), the last result can provide the evolution equations for internalvariables

@Lp

@M¼ � _jp þ _c

@/@M¼ 0) _jp ¼ signðMÞ _c; ð40Þ

@Lp

@�q¼ � _nþ _c

@/@�q¼ 0) _n ¼ _c; ð41Þ

along with the Kuhn–Tucker loading/unloading conditions and theconsistency condition

_c P 0; / 6 0; _c/ ¼ 0; _c _/ ¼ 0: ð42Þ

To obtain the remaining ingredients of the rigid-plastic responsedescribing softening at the discontinuity xd, let us isolate the soften-ing plastic hinge. We first define (bending) softening potential atthe discontinuity as N ¼ W ¼ 1

2 K2s n, where W is the strain energy

function due to softening. The softening potential depends on thestrain-like (bending) softening variable n P 0 and the linear (bend-ing) softening modulus Ks 6 0. The dissipation at xd can be thenwritten as:

0 6 D ¼def :

tM _ah �_WðnÞ ¼ tM _ah �

@W

@n

_n; ð43Þ

where tM is the discontinuity bending traction given by (34). Bydefining

��q ¼ � @W@n¼ � @N

@n¼ �Ksn ¼ jKsjn; ð44Þ

the result in (43) can be rewritten as

D ¼ Dp ¼ tM _ah þ ��q_n: ð45Þ

The principle of maximum plastic dissipation at the rigid-plasticdiscontinuity can then be defined as:

mintM ;��q

max_c

LpðtM; ��q;_cÞ ¼ �DpðtM; ��qÞ þ _c/ðtM; ��qÞ

h i; ð46Þ

where _c P 0 is the Lagrange multiplier. By using (45) and (33), weget from (46) above the following evolution equations:

@Lp

@tM¼ � _ah þ _c

@/@tM¼ 0) _ah ¼ signðtMÞ _c; ð47Þ

@Lp

@��q¼ � _

nþ _c@/

@��q¼ 0) _

n ¼ _c: ð48Þ

By observing that signðtMÞ ¼ signðahÞ (see (34) and Fig. 3), it followsfrom (47) that

signðahÞ _ah ¼_n) jahj ¼ n: ð49Þ

The Kuhn–Tucker loading/unloading conditions and the consistencycondition also apply:

_c P 0; / 6 0; _c/ ¼ 0; _c_/ ¼ 0: ð50Þ

With the above results, we are in position to write the total dissipa-tion of the beam finite element when the element is in the softeningregime. Namely, by accounting for the proper definition of strain en-ergy terms for the beam finite element according toW ¼

R LðeÞ

0 WdxþW, the total dissipation in the softening regime canbe written as

DtotLðeÞ¼Z LðeÞ

0rT _�� _Wð�e; nÞ

dxþ tM _ah �_WðnÞ

� �

¼Z LðeÞ

0rT _e� þ rT G _a� rT _�e|{z}

_e�e

þ�q _n

0B@1CAdxþ tM _ah þ ��q

_n

� �

¼Z LðeÞ

0M

_ejp þ �q _n

� �|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}

Dp ;seeð38Þ

dx

þZ LðeÞ

0GhMdxþ tM

!|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} _ah

¼0;seeð28Þ

þ ��q|{z}jKs jn

_n: ð51Þ

It can be seen from (51) that enforcing Eq. (28) will decuple dissipa-tion in the softening plastic hinge from the dissipation in the rest ofthe beam. Therefore, Eq. (28) is further used to compute tM .

We conclude description of constitutive relations by definingplastic work of the beam cross-section in the hardening regime

Wp ¼ M _jp ¼ jMj _n ¼ ðMy þ KhnÞ _n ð52Þ

and plastic work for the beam finite element in the softening regimeas

Wp ¼ tM _ah ¼ jtM j_n ¼ Mu þ Ksn

_n: ð53Þ

3. Computation of beam plasticity material parameters

In the previous section, we have built the framework forstress-resultant plasticity for beam finite element with embed-ded discontinuity. The material parameters that need to beknown for chosen material models are: (i) My and Kh for theplasticity with hardening, and (ii) Mu and Ks for the softeningplastic hinge. In this section we will elaborate on determinationof the above parameters.

The yield moment My can be determined by considering theuniaxial yield stress of the material ry, the bending resistancemodulus of cross-section W, the cross-section area A, and the level


of axial force N. One can associate the yield moment of the cross-section with the yielding of the most-stressed material fiber to get

MyðNÞ ¼Wry 1� jNjAry

� �: ð54Þ

The ultimate bending moment Mu can be derived in a closed formby assuming elastic–perfectly-plastic response of material fibers,e.g. [17]. However, one may try to determine a better estimate forMu, which takes into consideration material hardening, as well aspossibility of local buckling (e.g. buckling of the flanges and/orthe web of the I-beam). This task is addressed in the present workby performing computations with refined finite element modelbased on geometrically and materially nonlinear shell element,which is able to capture local buckling and gradual spreading ofplasticity over the cross-section. The ultimate bending resistanceMu can be obtained by using results of such a shell model computa-tions, as can be moduli Kh and Ks.

To obtain desired results, a part of the frame member with a ref-erence length Lref < Ltot (Ltot is the total length of the frame memberunder consideration) is: (i) modelled with shell finite elements, (ii)subjected to an external axial force bN in the first loading step, and(iii) subjected to a varying external bending moment at the endcross-sections in the second loading step, while keeping bN fixed,see Fig. 4b. It is assumed that such a loading pattern would pro-duce approximately constant internal axial force N ¼ bN duringthe analysis. The computation with shell model takes into accountgeometrical and material nonlinearity that include: plasticity withhardening and strain-softening, strain-softening regularization,and local buckling effects. The results of shell analysis are cast in

M

Κ,α

Mu

My

Κp

αθ

M M

My

Mu

Mu

ξ ξξ ξ

(a) Beam model

Lref

N

MM

N

(c) Moment curvature rotation (beam model)

(e) Moment hardening variable(beam mode)

(f) Moment softening variabl(beam model)

Kh

Ks

11

Fig. 4. Evaluation of beam material paramet

terms of diagrams presented in Fig. 4d and g. One can associatethe ultimate bending moment Mu with the peak point in the dia-gram at Fig. 4d, where applied end moment is plotted versus theend rotation, i.e.

MuðNÞ ¼ Mrefu ðNÞ: ð55Þ

One can also use this point as a border-point between the hardeningregime and the softening regime, where the softening can be due tomaterial and/or geometric effects. To determine the values of thebeam model hardening and softening parameters, we make anassumption that the plastic work at failure should be equal for boththe beam and the shell model. In other words, we want the internalforces of the beam model to produce the same amount of the plasticwork as the internal forces of the shell model, when considering thefull failure of the part of the frame member of length Lref .

Since the plastic work is done in two regimes (hardening andsoftening), we have to assure that the amount of plastic work ineach regime matches for both models, i.e.

EWp ðNÞ ¼ EWp;ref ðNÞ; EWp ðNÞ ¼ EWp;ref ðNÞ: ð56Þ

The plastic work in the hardening regime, EWp;ref, and the plastic

work in the softening regime, EWp;ref, are obtained from the shell

model analysis, Fig. 4g.The plastic work of the beam model in the hardening regime,

EWp, can be determined by observing that each cross-section in

the frame member of length Lref is approximately under the sameforce–moment state during the hardening regime. Integration of(52) allows us to write

N

N

M

M

M

Plastic work

u

u

Muref

EWp,ref

EWp,ref

N const.

θ

(d) Moment rotaion (refined model)

(g) Plastic work (refined model) e

Lref

(b) Refined model

ers by using results of refined analysis.


EWp ¼Z Lref

0

Z t at Mu

0Wpdsdl ¼ Lref My

en þ 12

Khen2

� �: ð57Þ

In (57) above en is the value of hardening variable that correspondsto the bending moment Mu, Fig. 4e. Since we have assumed linearhardening in the beam model (37), we get

en ¼ Mu �My

Kh: ð58Þ

By using (58), (57) and (56), one can obtain an expression for hard-ening modulus as

KhðNÞ ¼M2

uðNÞ �M2yðNÞ

Lref

2EWp;ref ðNÞ: ð59Þ

The plastic work of beam model in the softening regime, EWp, can be

determined by assuming that the softening part of M �u curve inFig. 4d, obtained from the shell model analysis, is produced by avery localized phenomenon (in a single cross-section) related tothe local buckling and/or to the localized strain-softening. By using(53), one can compute the plastic work in the softening regime forthe beam model as

EWp ¼Z t

0Wpds ¼

Z n at tM¼0

0Mu þ Ksn

dn ¼ 12jKsj

en2: ð60Þ

In (60)en is the value of the softening variable that corresponds to

the total cross-section failure, Fig. 4f. Since we have assumed linearsoftening in the beam model, we obtainen ¼ Mu

jKsj: ð61Þ

By using (61), (60) and (56), one can obtain an expression for soft-ening modulus as

jKsðNÞj ¼M2

uðNÞ

2EWp;ref ðNÞ; Ks 6 0: ð62Þ

We note, that the choice of the reference length Lref should havevery small influence on the values of the searched material param-eters. The influence on the value of Kh should be small, since eachcross-section is approximately under the same force–moment stateduring the hardening regime. The influence on the value of Ks

should not be too big neither, since the softening effect is localized.However, one should perform large displacement correction of Mu,if the chosen length of Lref enables large deflections, as shown inExample 5.4.

4. Computational procedure

In this section we will present a procedure for solving the set ofglobal (mesh related) and the set of local (element related) nonlin-ear equations generated by using the stress-resultant plasticitybeam finite element with embedded discontinuity presented inSection 2.

The solution of the set of global nonlinear equation (26), alongwith the set of local nonlinear equation (27) (note that (27) is re-duced to (28) due to assumption (30)), ought to be computed atdiscrete pseudo-time values 0; t1; t2; . . . ; tn�1; tn; tnþ1; . . . ; T bymeans of the incremental-iterative scheme. We will consider thesolution in a typical pseudo-time incremental step from tn totnþ1. Let us assume that all the variables, related to an element eand its integration points ip ¼ 1;2;3 (a 3-point Lobatto integrationscheme is used) are given at tn, i.e.

Given : dðeÞn ; jp;ipn ; nip

n ; aðeÞh;n; nðeÞn and Mipy ; xðeÞd ; MðeÞ

u : ð63Þ

We have also added in (63): (i) the yield moment at integrationpoint Mip

y (which is only true if hardening plasticity has been acti-vated so far) and (ii) position of the discontinuity xðeÞd and the ulti-mate bending moment MðeÞ

u (which is only true if softening plastichinge has been activated so far). We will then iterate in the pseu-do-time step in order to compute the converged values of the vari-ables at tnþ1, i.e.

find : dðeÞnþ1; jp;ipnþ1; nip

nþ1; aðeÞh;nþ1; nðeÞnþ1 and ðif not given alreadyÞ

Mipy ; xðeÞd ; MðeÞ

u : ð64Þ

The moments Mipy and MðeÞ

u are computed by using (54) and (55).Although they depend on axial force N, we keep them fixed oncedetermined.

The computation of solution (64) is split into two phases:

(a) The global (mesh related) phase computes the current itera-tive values (with ðiÞ as the iteration counter) of nodal gener-alized displacements at tnþ1 while keeping the othervariables fixed, i.e.

global phase : dðeÞ;ðiÞnþ1 ¼ dðeÞ;ði�1Þnþ1 þ DdðeÞ;ði�1Þ

nþ1 : ð65Þ

The computation of iterative update DdðeÞ;ði�1Þnþ1 will be ex-

plained below.
(b) The local (element and integration point related) phase com-
putes the values of variables jp;ipnþ1; nip

n ; aðeÞh;nþ1; nðeÞnþ1 whilekeeping dðeÞ;ðiÞnþ1 fixed. The computation procedure dependson weather the softening plastic hinge has been activatedin the considered element or not. Therefore, the local com-putation procedure on the level of a single element can bebased either on hardening plasticity procedure or on soften-ing plasticity procedure (excluding each other).

In the rest of this section we will first describe the local phase,which will be followed by the description of the global phase. Thehardening plasticity procedure is carried out at each integrationpoint ip (e.g. [19]). We first provide the trial value of the bendingmoment

Mtrial;ipnþ1 ¼ EI j dðeÞ;ðiÞnþ1 ;aðeÞh;n

� jp;ip

n

ð66Þ

and the trial value of the yield function /trial;ip. If the trial yieldcriterion

/trial;ip Mtrial;ipnþ1 ; �q nip

n

� � 6

?

0 ð67Þ

is satisfied, the values of hardening plasticity local variables remainunchanged (the step is elastic)

/trial;ip6 0) jp;ip

nþ1 ¼ jp;ipn ; nip

nþ1 ¼ nipn : ð68Þ

In the case of violation of the trial yield criterion (67), the values oflocal variables are updated by backward Euler integration scheme

jp;ipnþ1 ¼ jp;ip

n þ sign Mtrial;ipnþ1

cip

nþ1; nipnþ1 ¼ nip

n þ cipnþ1; ð69Þ

where cipnþ1 ¼ _cip

nþ1ðtnþ1 � tnÞ. The value of the plastic multiplier cipnþ1

is determined from

/ip Mipnþ1 dðeÞ;ðiÞnþ1 ;jp;ip

nþ1 cipnþ1

; �q nip

nþ1 cipnþ1

¼ /ip cip

nþ1

¼ 0: ð70Þ

For the linear hardening one can determine cipnþ1 explicitly. For a

nonlinear hardening an iteration procedure has to be used. Themain result of the above described hardening plasticity procedureis the new values of the bending moment Mip

nþ1, computed as

εy,σ y

y, y

εu,σu

u, u

ε f ,0f ,0gs

0.15 0.10 0.05 0.05 0.10 0.15ε

40

20

20

40

σ

Fig. 5. Uniaxial stress–strain curve.


Mipnþ1 ¼ EI j dðeÞ;ðiÞnþ1 ;aðeÞh;n

� jp;ip

nþ1

ð71Þ

and the elastoplastic tangent operator @Mipnþ1=@j

ip;ðiÞnþ1 . The updated

value of the rotation jump is aðeÞh;nþ1 ¼ aðeÞh;n.The softening plasticity procedure is carried out at each finite

element e. The discontinuity xðeÞd can only appear at the positionof the integration point with the largest absolute value of the bend-ing moment. We first provide the trial value of the bending tractionat the discontinuity

ttrial;ðeÞM;nþ1 ¼ �

Z LðeÞ

0Gh xðeÞd ; x

M dðeÞ;ðiÞnþ1 ;jp;ipn ;aðeÞh;n

dx ð72Þ

and the trial value of the failure function

/trial;ðeÞ ttrial;ðeÞM;nþ1 ;

��q nðeÞn

6

?

0: ð73Þ

If the trial failure criterion (73) is satisfied, the values of softeningplasticity local variables remain unchanged

/trial;ðeÞ6 0) aðeÞh;nþ1 ¼ aðeÞh;n; nðeÞnþ1 ¼ nðeÞn : ð74Þ

In the case of violation of the trial yield criterion (73), the values oflocal variables are updated by backward Euler integration scheme

aðeÞh;nþ1 ¼ aðeÞh;n þ sign ttrial;ðeÞM;nþ1

cðeÞnþ1; nðeÞnþ1 ¼ nðeÞn þ cðeÞnþ1; ð75Þ

where cðeÞnþ1 ¼_cðeÞnþ1ðtnþ1 � tnÞ. The value of the plastic multiplier cðeÞnþ1

is determined from condition

/ðeÞ tðeÞM;nþ1 aðeÞh;nþ1 cðeÞnþ1

; ��q nðeÞnþ1 cðeÞnþ1

¼ /ðeÞ cðeÞnþ1

¼ 0: ð76Þ

For the linear softening one can determine the plastic multiplierexplicitly, whereas for nonlinear softening an iterative solution pro-cedure is needed. Note, that we compute the bending traction in(76) as

tðeÞM;nþ1 ¼ �Z

XeGh xðeÞd ; x

M dðeÞ;ðiÞnþ1 ;jp;ipn ;aðeÞh;nþ1

dx: ð77Þ

The main result of the above described softening plasticity proce-dure is the new value of softening variable aðeÞh;nþ1, which influencesthe stress state of the whole element by giving the new values ofthe bending moment Mip

nþ1 as

Mipnþ1 ¼ EI j dðeÞ;ðiÞnþ1 ;aðeÞh;nþ1

� jp;ip

n

: ð78Þ

The updated value of the plastic curvature is jp;ipnþ1 ¼ jp;ip

n .Once the local variables are computed, we turn to the global

phase of the iterative loop in order to provide, if so needed, newiterative values of nodal displacements. First, the set of global equi-librium equation (26) is checked with newly computed Mip

nþ1 fromthe local phase

ANele¼1 f int;ðeÞ

nþ1 � fext;ðeÞ;ðiÞnþ1

h i <? tol: ð79Þ

If the convergence criterion (79) is satisfied, we move on to the nextpseudo-time incremental step. If the convergence criterion fails, weperform a new iterative sweep within the present pseudo-timeincremental step. New iterative values of nodal generalized dis-placements of the finite element mesh are computed by accountingfor each element contribution. A single element contribution can bewritten as

KðeÞ Kfa

Khd Kha

" #ðiÞnþ1

DdðeÞ;ðiÞnþ1

DaðeÞ;ðiÞh;nþ1

!¼ fext;ðeÞ

nþ1 � f int;ðeÞ;ðiÞnþ1

0

!; ð80Þ

where the parts of the element stiffness matrix can be formallywritten as

KðeÞ;ðiÞnþ1 ¼@f int;ðeÞ

@dðeÞ

!ðiÞnþ1

; Kfa;ðiÞnþ1 ¼

@f int;ðeÞ

@aðeÞh

!ðiÞnþ1

;

Khd;ðiÞnþ1 ¼

@hðeÞM

@dðeÞ

!ðiÞnþ1

; Kha;ðiÞnþ1 ¼

@hðeÞM

@aðeÞh

!ðiÞnþ1

:

ð81Þ

The static condensation of (80) allows us to form the element stiff-ness matrix bKðeÞ;ðiÞnþ1 that contributes to the assembly

ANele¼1

bKðeÞ;ðiÞnþ1 DdðiÞnþ1

¼ ANel

e¼1 fext;ðeÞnþ1 � f int;ðeÞ;ðiÞ

nþ1

; ð82Þ

where

bKðeÞ;ðiÞnþ1 ¼ KðeÞ;ðiÞnþ1 � Kfa;ðiÞnþ1 Kha;ðiÞ

nþ1

�1Khd;ðiÞ

nþ1 : ð83Þ

Solution of (82) gives the values of iterative update DdðeÞ;ðiÞnþ1 , whichleads us back to (65).

5. Examples

In this section we illustrate performance of the above derivedbeam element when analyzing push-over and collapse of steelframes. We also illustrate the procedure, presented in Section 3,for computing the beam model plasticity material parameters byusing the shell finite element model. The beam model computercode was generated by using symbolic manipulation code AceGenand the examples were computed by using finite element programAceFem, see Korelc [20].

5.1. Computation of beam plasticity material parameters

With this example we illustrate computation of beam plasticitymaterial parameters Mu; Kh and Ks as suggested in Section 3. Weconsider a frame member with an I-cross-section with flangewidth bf ¼ 30 cm, flange thickness tf ¼ 1:5 cm, web heightbw ¼ 40 cm and web thickness tw ¼ 0:8 cm. The cross-section areais A ¼ 122 cm2 and the bending resistance modulus isI ¼ 43034:2 cm4. We model a part of the frame member of lengthLref ¼ 2L ¼ 300 cm, which is seven times the height of the section.This length should be sufficient to capture the local softening ef-fects due to local buckling and/or strain softening. The frame mem-ber is made of an elastoplastic material (steel), whose uniaxialresponse is plotted in Fig. 5. Young’s modulus isE ¼ 21000 kN=cm2, yield stress is ry ¼ 24 kN=cm2, ultimate stressis ru ¼ 36 kN=cm2, yield strain is ey ¼ ry=E, strain at ultimatestress is eu ¼ 0:1 and strain at failure is ef ¼ 0:12778.

The example has been computed with the finite element codeABAQUS [21] by using shell finite element S4R with five integrationpoints through the thickness. Only one half of the consideredgeometry was discretized, see Figs. 6 and 7. The symmetry condi-tions uz ¼ ux ¼ uy ¼ 0 were used in the symmetry plane. Themesh consists of equal squared elements. The free-end cross-sec-tion of the model was made rigid by coupling the degrees of free-dom of that cross-section.

L

rigid cross section

MP xz

yN

M

Fig. 6. Boundary conditions for the shell model analysis.

Fig. 7. Failure mode of the representative part of the frame member as computedby the shell model.

0.00 0.05 0.10 0.15 0.20 0.250

100

200

300

400

500

600

MP

MkN

m

00.1 Ny

0.2 Ny

0.3 Ny

0.1 Ny

0.2 Ny

0.3 Ny

Fig. 8. Bending moment versus rotation curves for the end cross-section.

0.00 0.05 0.10 0.15 0.20 0.25

0

20

40

60

80

100

120

140

MP

Plas

ticw

ork

kJ0

0.1 Ny

0.2 Ny

0.3 Ny

0.1 Ny

0.2 Ny

0.3 Ny

Fig. 9. Plastic work versus end cross-section rotation curves.

Table 1Summary of results of the shell model analyses.


The plasticity models with strain softening are mesh-depen-dent. In order to minimize that effect, we have adjusted the postpeak uniaxial stress–strain relation to fit the mesh size, as sug-gested in [4]. According to [4] the linear softening modulus is com-puted as

Kles ¼ �

ler2u

2gs; ð84Þ

where le is a characteristic dimension of the element (in the presentcase le ¼ 5 cm is the side-length of the elements) and gs is the plas-tic work density (plastic work per unit volume) in the softening re-gime of the shell material that corresponds to the gray-colored areain Fig. 5 (in the present case gs ¼ 0:5 kN=cm2). The strain at failure,adjusted to the mesh, is then

elef ¼ eu þ

ru

EKles

EþKles

¼ 0:10727: ð85Þ

The load was applied in two steps. In the first step we applied a de-sired level of axial force N at the mid-point MP of the rigid cross-section, see Fig. 6. In the second step we applied bending momentM at the point MP and performed nonlinear analysis with thepath-following method. Several analyses were carried out with dif-ferent values of the axial force (from N ¼ �0:3Ny to N ¼ 0:3Ny,where Ny ¼ Ary). For each case we monitored the response untilthe bending resistance dropped to zero (or the analysis ran intoconvergence problems).

The results of analyses are presented in Figs. 7–9. Final de-formed configuration of the shell model and distribution of theequivalent plastic strain are presented in Fig. 7 for pure bendingcase ðN ¼ 0Þ. We can see that the considered part of the framemember failed by localized buckling of the bottom flange. We alsonote strong localized yielding of the flange which is concentratedin the neighborhood of the web. Such failure mode was observedalso for all other cases. In Fig. 8, we show the corresponding mo-ment–rotation curves. We can see that the level of axial force has

a significant influence on the peak bending resistance and on theoverall response. In Fig. 9 we present the plastic work versus rota-tion curves. Here, the value of the axial force does not influencemuch the shape of the curve. The relation between rotation andplastic work is almost linear. In Figs. 8 and 9, we marked the pointswith the maximum bending moment. We assume that those pointsseparate hardening regime from softening regime.

The obtained results by the shell model are now used for eval-uation of the beam model material parameters. In Table 1 we sum-marized the following shell model results: the maximum bendingmoment Mref

u , the plastic work in hardening regime EWp;ref, and the

plastic work in softening regime EWp;reffor different values of axial

force N. We can see that Mrefu decreases if N is compressive, whereas

the tensile axial force has only slight effect on Mrefu .

By using Table 1, we determined a bilinear approximation func-tion for Mref

u as


eMrefu ðNÞ ¼

Mref ;0u 1:03þ 0:85 N

Ny

if N < �0:035Ny;

Mref ;0u if N P �0:035Ny;

8<: ð86Þ

where Mref ;0u ¼ Mref

u ðN ¼ 0Þ. We assume that eMrefu ðNÞ can be used to

evaluate the ultimate bending moment of the beam model Mu, i.e.

MuðNÞ ¼ eMrefu ðNÞ; ð87Þ

see Fig. 10. The values for the ultimate resistance obtained with theshell analyses are marked with dots in Fig. 10.

The beam model hardening modulus Kh can be evaluated point-wise by using (59), (54), (87) and third column of Table 1. We getKh ranging from 6:26� 106 kN=cm2 to 1:35� 107 kN=cm2 with anaverage value of 1:06� 107 kN=cm2. Although one could easilyfind a higher-order function that fits these results, we assume forsimplicity that the axial force has no influence on the hardeningmodulus and adopt

KhðNÞ ¼ 1:06� 107 kN=cm2: ð88Þ

The beam model softening modulus Ks can be evaluated point-wiseby using (62), (87) and the last column of Table 1. The gray-coloredfields in Table 1 present unreliable results for EWp;ref

, since for thosecases the shell analysis did not converge. Softening modulus for thefirst three analyses ranges from �2:85� 105 kN=cm2 to�3:97� 105 kN=cm2. We assume that the axial force has no influ-ence on softening modulus and adopt the average value

KsðNÞ ¼ �3:28� 105 kN=cm2: ð89Þ

In Table 2 we make a point-wise comparison between the shellanalysis results Mref

u ; EWp;refand EWp;ref

and the corresponding beammodel results Mu; EWp

and EWp, computed by using approximations

(87)–(89) and expressions (57), (60) and (52). We can see that theerror in ultimate bending moment is small, while the error in dissi-pated plastic work can be quite large.

Table 2Comparison between approximations and shell analyses results.

N=Ny Mu (kN m) Mu�Mref :u

Mref :u

�� (%) EWp(kJ)

0 550 0.00 50�0.1 519 0.26 59�0.2 473 1.54 54�0.3 426 0.22 48

0.1 550 1.94 810.2 550 5.07 1090.3 550 3.72 134

-1000 -500 0 500 1000400

450

500

550

600

N kN

Mu

kNm

Fig. 10. Approximation of the ultimate bending moment of the cross-section.

5.2. Push-over of a symmetric frame

In this example we present a push-over analysis of a sym-metric frame. The geometry is given in Fig. 11, where LB ¼ 500and HC ¼ 250 cm. The material and cross-section properties ofall frame members are equal. They are the same as those pre-sented in Section 5.1. The vertical load is constant and equalsqv ¼ 0:05 kN=cm. The lateral loading is presented in Fig. 11,where F0 ¼ 1 kN is a concentrated force and k is load multiplier.The mesh consists of eight beam finite elements per each framemember.

We performed two analyses, one by using the geometrically lin-ear and the other by using the geometrically nonlinear beam finiteelements. The results are presented in Figs. 12 and 13, where utop ishorizontal displacement of the top right corner of the frame. In theleft part of the Fig. 12 we present the total lateral load versus utop

curves. The points on those curves mark configurations where thesoftening hinge was activated in one of the elements of the mesh.We can see that, even though some parts of the frame are failing,the total resistance of the structure is still growing until the max-imum load is reached at 1527.3 kN for geometrically linear case,and at 1522.3 kN for geometrically nonlinear case.

In the right part of Fig. 12 we present the plastic work versusutop displacement curves. The results of the geometrically linearand the geometrically nonlinear elements are completely the same.At the beginning of the analysis there is no energy dissipation sincethe material response is elastic. The non-dissipative period is fol-lowed by a short period with dissipation due to material hardeningonly, which ends with the first activation of softening plastic hingein one of the beam finite elements. For a while we have a combinedhardening and softening energy dissipation, which is finally fol-lowed by a period when the structure is dissipating energy onlydue to softening. On the left part of Fig. 13 we present the final de-formed configuration of the frame. In the right part of Fig. 13 wepresent locations where the softening plastic hinges appeared dur-ing the analysis.

EWp�EWp;ref :

EWp;ref :

�� (%) EWp(kJ) EWp�EWp;ref :

EWp;ref :

�� (%)

42 46 1220 41 817 34 1912 28 ?14 46 ?27 46 ?

6 46 ?

λF0

λ2F0

λ3F0

λ4F0

λ2F0

λ4F0

λ6F0

λ8F0

λ2F0

λ4F0

λ6F0

λ8F0

λF0

λ2F0

λ3F0

λ4F0

LB LB LB

HC

HC

HC

HC

utop

qv

qv

qv

qv

qv

qv

qv

qv

qv

qv

qv

qv

Fig. 11. Symmetric frame: geometry and loading.

0 10 20 30 40 50 600

200

400

600

800

1000

1200

1400

utop cm

Lat

eral

load

kNLinear

Nonlinear

0 10 20 30 40 50 600

100

200

300

400

500

600

utop cm

Dis

sipa

ted

ener

gykJ Hardening: Lin., Nonlin.

Softening: Lin., Nonlin.Total: Lin., Nonlin.

Fig. 12. Load versus displacement and dissipated energy versus displacement curves.

Fig. 13. Deformed shape and locations of softening plastic hinges at utop � 60 cm.


5.3. Push-over of an asymmetric frame

In this example we analyse an asymmetric frame presented inFig. 14, where LB1 ¼ 600 cm; LB2 ¼ 500 cm, LB3 ¼ 400 cm andHC ¼ 250 cm. All the other geometrical, material and loadingparameters are the same as in the previous example.

The results are presented in Figs. 15, 16. The total lateral loadversus utop curves, where utop is horizontal displacement at thetop-left corner of the frame, are presented on the left part ofFig. 15. The results of geometrically linear and geometrically non-linear analyses are nearly the same before the ultimate resistanceis reached at 1581.8 kN for geometrically linear case and at1578.4 kN for geometrically nonlinear case. After that point the dif-ference between those two analyses is bigger. The final computedequilibrium configuration for the geometrically nonlinear case is atutop ¼ 26:52 cm. In the next load step one additional softeningplastic hinge is activated, which results in the global failure mech-anism. Since our path-following algorithm is only governed by the

λF0

λ2F0

λ3F0

λ4F0

λ2F0

λ4F0

λ6F0

λ4F0

λ2F0

λ4F0

λ3F0

λF0

λ2F0

LB1 LB2 LB3

HC

HC

HC

HC

utop

qv

qv

qv

qv

qv

qv

qv

qv

qv

Fig. 14. Asymmetric frame: geometry and loading.

increase of utop, we are unable to capture the remaining part of theload–displacement curve.

In the right part of Fig. 15 we present the dissipated energy ver-sus utop curves. The shapes of the curves are very similar to thosefrom the symmetric frame case. Namely, first there is the elasticnon-dissipative phase, followed by the pure hardening dissipationphase, followed by the combined hardening and softening dissipa-tion phase and finally the pure softening dissipation phase. In thegeometrically nonlinear case we do not have the final pure soften-ing dissipation phase due to activation of global failure mecha-nism. On the left part of Fig. 16 we present the deformedconfiguration of the frame at utop ¼ 26:52 cm. Locations, wheresoftening plastic hinges were activated at utop � 26:52 cm, are pre-sented in the middle part of Fig. 16 for the geometrically linear caseand in the right part of the same figure for the geometrically non-linear case.

5.4. Bending of beam under constant axial force

In this example we compare results of the beam model with re-sults obtained by using the shell finite element model from ABA-QUS. For the comparison we choose the problem of the bendingof the beam of length Lref under a constant axial force, presentedin Section 5.1. For that reason, the geometric and material proper-ties are the same as those in the Section 5.1. The beam model anal-yses are performed with two sets of material parameters, wherethe first set (SET1) is given by (87)–(89). In the second set (SET2) we replace the expression (87) with

M�u ¼ Mð87Þ

u � NjDuyj; ð90Þ

where Mu is the maximum concentrated moment applied at the endcross-section (point MP, see Fig. 6), N is the applied axial force (po-sitive when producing tension) and Duy is the relative displacementin the y direction between the point MP and the position of the localfailure. The difference between the applied concentrated moment atthe point MP (see Fig. 6) and M�

u thus arises due to large displace-

0 5 10 15 20 25 30 350

500

1000

1500

utop cm

Hor

izon

tall

oad

kN

Linear

Nonlinear

0 5 10 15 20 25 30 350

100

200

300

400

utop cm

Dis

sipa

ted

ener

gykJ

Hardening Lin.Softening Lin.Total Lin.Hardening Nonlin.Softening Nonlin.Total Nonlin.

Fig. 15. Load versus displacement and dissipated energy versus displacement curves.

Fig. 16. Deformed shape and locations of softening plastic hinges.


ments correction. When the yielding and local buckling of the beamare significant and the displacements in the y direction are no long-er negligible, the contribution of the axial force N to the bendingmoment must be taken into account. In this particular case we haveN ¼ �0:1Ny; Mu ¼ 521 kN m and Duy ¼ 0:15 m; which leads toM�

u ¼ 565 kN m.Five beam finite elements are used to model one-half of the

beam under consideration, since the symmetry is taken into ac-count. The symmetry conditions at the symmetry plane areu ¼ w ¼ w0 ¼ 0. The load was applied in two steps. In the first stepthe beam was loaded with compressive axial force N ¼ �0:1Ny. Inthe second step the moment M was applied at the free-end of thebeam. In order to ensure the proper activation of softening in thegeometrically linear analyses the ultimate bending moment ofthe finite element near the symmetry plane was slightly weakened.

In Fig. 17 the results for geometrically linear and geometricallynonlinear cases are compared with the results of the shell modelfrom ABAQUS. On the left part of Fig. 17 we present curves relatingapplied moment to free-end rotation. The ultimate bending mo-ments of the shell model, the geometrically linear SET1 beam mod-el and the geometrically nonlinear SET2 beam model are veryclose, whereas the geometrically linear SET2 beam model givesslightly bigger and the geometrically nonlinear SET1 beam modelgives slightly lower value for ultimate bending moment.

On the right part of Fig. 17 we present the plastic work versusrotation of the end cross-section curves. There is hardly any differ-

0.00 0.05 0.10 0.15 0.20 0.25 0.300

100

200

300

400

500

MkN

m

ABAQUSLinear SET1

Nonlinear SET1Linear SET2

Nonlinear SET2

Fig. 17. Comparison of results for the bending o

ence between the beam model results when u is smaller than 0.15.After that point the difference becomes bigger. The prediction ofthe beam model for plastic work in hardening regime is in the caseof geometrically linear analysis with SET1 material parameters 80%of the shell model prediction, and the prediction for plastic work insoftening regime is 92% of the shell model prediction. This is inagreement with the results of Table 2. The predictions for plasticwork of other beam analyses give bigger differences compared tothe shell model. We note that one could get better agreements inplastic work by using better approximations for hardening andsoftening moduli in place of simplifications (88) and (89).

5.5. Collapse of a simple frame

In this example we compare results of the nonlinear beam mod-el with the results of the shell model. We consider a simple framepresented in Fig. 18. The geometry of the beam model (middle axesof the beam model correspond to the middle axes of the shell mod-el) is presented on the left part of Fig. 18. The geometry and the fi-nite element mesh of the shell model is presented on the right sideof Fig. 18. The cross-section and the material properties of the shellmodel are the same as those in the Section 5.1. In the shell modelwe made connections between the beam and the columns rigid bycoupling the degrees of freedom of the corresponding end cross-sections. The beam model mesh consists of eight finite elementsper each frame member. Two different sets of material parameters

0.00 0.05 0.10 0.15 0.20 0.25 0.300

20

40

60

80

100

120

140

Dis

sipa

ted

ener

gykJ

ABAQUSLinear SET1

Nonlinear SET1Linear SET2

Nonlinear SET2

f the beam under compression axial force.

Q QλF0 λF0

utop

520 cm

320

cm

500 cm

300

cm

Q QλF0λF0

utop

rigid

rigid

Fig. 18. Simple frame: the beam and the shell model.


are used for the beam model analysis. The first set (SET1) is givenby 54 and (87)–(89) and the second set (SET2) by

My ¼ 1:2Mð54Þy ; Mu ¼ 1:2Mð87Þ

u ; Kh ¼ 0:6Kð88Þh ; Ks

¼ Kð89Þs : ð91Þ

Support conditions are (u ¼ w ¼ w0 ¼ 0 for the beam model andux ¼ uy ¼ uz ¼ ux ¼ uy ¼ uz ¼ 0 for the shell model).

The load applied to the frame is presented in Fig. 18. The verti-cal load is constant and equal to Q ¼ 500 kN, while the horizontalload multiplier kðF0 ¼ 1 kNÞ is controlled with the path-followingmethod.

The results are presented in Figs. 19–21. The total lateral loadversus utop curves (utop is horizontal displacement of the top-rightcorner of the frame) are presented on the left part of Fig. 19. Thedissipated energy versus utop curves are presented on the right partof Fig. 19. On the left part of Fig. 20 we present the equivalent plas-tic deformation on the deformed configuration of the shell model.The deformed configuration of the beam model, along with posi-tions where the softening response was activated, is presentedon the right part of Fig. 20. Note, that in both models the localizedfailure appears at the ends of the columns. In Fig. 21 we present theinternal forces at the right support of the shell model. Note that theaxial force is not constant at the beginning of the loading, but oncethe response becomes nonlinear it hardly changes.

The lateral load versus utop curve (left part of Fig. 19) of thebeam model with SET1 material parameters has a similar shapeas the shell model curve, but the prediction of the maximum resis-tance of the beam model is around 84% of the shell model’s resis-tance. We have a similar situation as in the previous example,where the resistance of the cross-section was greater than theone obtained by analysis in Section 5.1. On the bottom-right partof Fig. 21 one can see that the axial force that corresponds to themaximum bending moment Mmax ¼ 523 kN m is aroundNðMmaxÞ ¼ �720 kN � �0:25Ny. If we compare Mmax toMuð�0:2NyÞ ¼ 473 kN m from Table 2, we can see that we havemore than 10% bigger bending resistance. One must also considerthat plasticity (hardening and softening) in the beam model is trig-

0 5 10 15 200

100

200

300

400

500

600

700

utop cm

Lat

eral

load

kN

ABAQUSNonlinear SET1Nonlinear SET2

Fig. 19. Comparison of results

gered at positions where we have rigid connections in the shellmodel, which also decreases the resistance of the beam modelcompared to the shell model.

The dissipated energy versus utop curve (right part of Fig. 19) ofthe beam model with SET1 material parameters has a similar shapeto the shell model’s curve. The prediction of the SET1 beam modelfor the value of the dissipated energy that corresponds toutop ¼ 20 cm is around 71% of the shell model prediction.

These results are significantly improved, and we obtain muchbetter fit to the shell model, when SET2 beam parameters are used;see Fig. 19. We recall that the latter accounts for the large displace-ment correction of ultimate resistance.

5.6. Darvall–Mendis frame

We consider the clamped portal frame under vertical loadingfirst studied by Darvall and Mendis [22] and later examined byArmero and Ehrlich [7] and Wackerfuss [11]. The geometry of theframe is presented in Fig. 22. The length of the columns and thebeam is L ¼ 3:048 m, cross-section area of all the members isA ¼ 0:103 m2 and their moment of inertia is I ¼ 0:001 m4. Theelastic material response is defined by the Young’s modulusE ¼ 2:068� 107 kN=m2. The inelastic response is defined by theultimate bending moment Mu;C ¼ 158:18 kN m for the columns,the ultimate bending moment Mu;B ¼ 169:48 kN m for the beam,and the softening modulus Ks ¼ aEI

10L, where the values a ¼ 0,�0.04, �0.06, �0.0718 are considered. Note, that in this examplethe inelastic response does not include any material hardening.We also consider that the axial force has no influence on the ulti-mate bending resistance of the frame members. The mesh consistsof eight geometrically linear beam finite elements, see Fig. 22. Theframe is loaded with a vertical load kF0 ðF0 ¼ 1 kNÞ applied at thenode 5, see Fig. 22. In the numerical simulations we control theload multiplier k and the vertical displacement uv at the node 5by the path-following method.

The vertical load versus vertical deflection curves are presentedin Fig. 23, where the points on the curves mark configurationswhere the softening plastic hinge was activated in one of the ele-

0 5 10 15 200

20

40

60

80

100

120

140

utop cm

Dis

sipa

ted

ener

gykJ

ABAQUSNonlinear SET1Nonlinear SET2

for simple frame example.

Fig. 20. Deformed shapes of the simple frame.

0 5 10 15 20

-750

-700

-650

-600

-550

-500

-450

utop cm

Axi

alfo

rce

kN

0 5 10 15 200

100

200

300

400

500

utop cm

Ben

ding

mom

ent

kNm

0 5 10 15 200

50

100

150

200

250

300

utop cm

Shea

rfo

rce

kN

-750 -700 -650 -600 -550 -500 -4500

100

200

300

400

500

Axial force kN

Ben

ding

mom

ent

kNm

Fig. 21. Internal forces at the right support of the shell model.

λF0, uv

L

0.55 L

L

1

1

2

2

33

44

55

66

7

7

8

8

9

Fig. 22. Geometry and loading of the portal frame.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40

100

200

300

400

uv cm

λF0

kN

a 0

a 0.04

a 0.06

a 0.0718

Fig. 23. Vertical load versus vertical deflection curves.


ments of the mesh. In all cases hinges form at the same locations.The first hinge forms in elements 4 and 5 at node 5, the second

hinge forms in element 7 at node 7 and the third hinge forms inelement 2 at node 3. This example shows the significant influenceof the softening modulus on the ultimate load of the structure. Theultimate load in the case of perfectly plastic response of the hingesða ¼ 0Þ is 434 kN when the third hinge forms. In cases a ¼ �0:04and a ¼ �0:06 the structure fails when the second hinge formswhen the vertical load reaches 383 kN and 350 kN, respectively.In the case of a ¼ �0:0718 the structure fails when the first hingeforms at 336 kN. In Table 3 we compare our results with the results

Table 3Comparison of the presented formulation with the literature.

a Hinge Darvall and Mendis [22] Armero and Ehrlich [7] Wackerfu [11] Present

uv ðcmÞ kF0 ðkNÞ uv ðcmÞ kF0 ðkNÞ uv ðcmÞ kF0 ðkNÞ uv ðcmÞ kF0 ðkNÞ

0 1 0.50 336 0.50 337 0.53 342 0.50 3362 1.14 427 1.14 428 1.13 435 1.13 4273 1.34 433 1.34 434 1.33 440 1.34 434

�0.04 1 0.50 336 0.50 337 0.53 349 0.50 3362 1.14 387 1.18 388 1.16 401 1.19 383

�0.06 1 0.50 336 0.50 337 0.52 348 0.50 3362 1.19 336 1.29 337 1.23 349 1.23 350

�0.0718 1 0.50 336 0.50 337 0.50 336


obtained by Darvall and Mendis [22], Armero and Ehrlich [7] andWackerfuss [11].

6. Conclusion

The presented multi-scale model for computing the limit load ofplanar metal frames under the push-over and the full collapseanalysis combines the best of two worlds: on one side the effec-tiveness and robustness of the macro-scale beam model for the en-tire structure, and on another side, a refined representation oflocalized instability effects (both geometric and material) bymeso-scale effects based upon the geometrically nonlinear elasto-plastic shell formulation. The latter is captured and stored withinthe macro-scale beam model in the manner which is compatiblewith enhanced beam kinematics with embedded discontinuity.The most appropriate choice of the meso-scale shell model canbe further guided by the error-controlled adaptive finite elementmethod for shell structures (by using model error estimation, seee.g. [23]), which could automatically find the most appropriatemodel for representing a particular local phenomena underconsideration.

The multiscale procedure proposed in this paper belongs tothe class of weak coupling methods, where we carry out thesequential computations. The results of the shell model compu-tations, accounting for material and geometric localized instabil-ity, are stored to be used within the beam model softeningresponse. As presented by numerical simulations, performanceof the proposed multi-scale computational approach is very sat-isfying. One of its main features is that detection and develop-ment of the softening plastic hinges in the frame is fullyautomatic, and spreads gradually in accordance with stressredistribution in the course of the nonlinear analysis. This is incontrast with many standard computational approaches to thelimit load, under the push-over and the full collapse analysisof frames, which rely on predefined locations of plastic hingesand the corresponding inelastic deformations.

Acknowledgement

The authors gratefully acknowledge financial support of Solve-nian Research Agency.

References

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[2] E.L. Wilson, Three dimensional static and dynamic analysis of structures, CSi,2002.

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http://www.fgg.uni-lj.si/Symech

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