Multi-scale Analyses on Seismic Damage and Progressive Failure Of

Finite Elements in Analysis and Design 48 (2012) 1358–1369

Contents lists available at SciVerse ScienceDirect

Finite Elements in Analysis and Design

0168-87

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/finel

Multi-scale analyses on seismic damage and progressive failure ofsteel structures

Z.X. Lin, F.F. Jiang, Y.Q. Tang

Department of Engineering Mechanics, Jiangsu Key Laboratory of Engineering Mechanics, Southeast University, Nanjing 210096, China

a r t i c l e i n f o

Article history:

Received 29 May 2011

Received in revised form

30 July 2011

Accepted 3 August 2011Available online 9 September 2011

Keywords:

Damage evolution

Seismic response

Multi-scale modeling

Progressive failure

Numerical analyses

4X/$ - see front matter & 2011 Elsevier B.V.

016/j.finel.2011.08.002

esponding author. Tel.: þ86 2583790268; fa

ail address: [email protected] (Z.X. Li).

a b s t r a c t

This paper is aimed to develop a numerical procedure for multi-scale analyses on seismic damage and

progressive failure process in steel structures by considering meso-scale damage evolution based on

CDM constitutive model at vulnerable locations. The mechanism of seismic damage or local progressive

failure and their effect on nonlinear dynamic behavior of structures are then explored with the

developed procedure. The multi-scale modeling of structural damage is developed to describe the

phenomena that some joint parts at vulnerable locations with meso-defects in a structure show

nonlinear damage evolution or local failure due to stress concentration while mostly parts of the

structure remain elastic and show linear response. A ductile damage constitutive model is chosen as the

description of meso-scale damage evolution in the numerical analysis on damage and progressive

failure of steel structures under seismic loading, and the developed numerical procedure is imple-

mented with the UMAT subroutine to introduce the damage constitutive equations and incorporate

into multi-scale computations of seismic response performed with the software ABAQUS. Effects of

meso-scale damage evolution on seismic response and progressive failure are analyzed, respectively,

for a dog-bone joint component and a truss structure under ECCS cyclic loading with the developed

numerical procedure. The results show that, the developed numerical procedure is available for

analyses on seismic damage and progressive failure, and is useful for exploring mechanism of seismic

damage and their effect on nonlinear dynamic behavior of structures under seismic excitations with

micro-seismic scale intensities or several earthquake actions. The effect of evolving damage on the

material degradation gives reasonable explanations for the phenomenon of large structural deforma-

tion and progressive failure under seismic loading. The global carrying capacity of the structure

decreases with the increase in damaged area and progressive failure at the welded joint or other

vulnerable areas, which at last leads to the occurrence of structure failure.

& 2011 Elsevier B.V. All rights reserved.

1. Introduction

Damage and crack like defects form during the service ofstructures. The formation of defects is an evolutionary processresulting from the accumulation of damage as a result of workingenvironment. This damage is usually distributed initially in thejoint part or vulnerable locations of structures and only tends tolocalize into discrete cracks just before failure or when thestructure suffers from high loads such as seismic loading. Damageevolution and induced structural deterioration make structuralstrength weakening obvious so that it becomes vulnerable tofailure when these structures suffer from seismic loading. View-ing from the essence of structural failure, each type of failuremode is caused by damage evolution started from defects in

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materials and developed due to long time of service loading orinstantaneous action of disaster loads [1,2]. However, Civil infra-structure such as a long-span bridge is usually in the order ofmagnitude of kilometers while defects in the joint part have to bemodeled in meso-scale. So it is necessary to consider the processof meso-scale damage evolution at vulnerable component ofstructures by developing a numerical procedure for multi-scaleanalyses on seismic damage in order to explore its nonlineardynamic behavior and disaster load-resisting mechanisms, and tounderstand how the resistible capacities and progressive failuremechanisms are affected by various types of meso-scale damageor defects in structures.

Many works have been done on the evaluation of dynamicresponse and damage behavior of structures under seismicexcitations. The structure’s ability to survive an earthquake maybe measured in terms of the expected state of damage of thestructure after the earthquake. Damage is usually quantified usingany of several damage indices defined as functions whose values

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Z.X. Li et al. / Finite Elements in Analysis and Design 48 (2012) 1358–1369 1359

can be related to particular structural damage states. A number ofavailable response-based damage indices are discussed and criti-cally evaluated for their applicability in seismic damage evalua-tion [3]. While Scotta et al. [4] proposed two series of indexes,namely Global Damage Indexes, which are representative of theoverall structure performance, and Section Damage Indexes,which assess the conditions of reinforced concrete beam-columnsections when they performed the seismic risk assessment bydefining compact indexes able to measure the damaging andsafety level of structures, also in view of the economic considera-tions on buildings rehabilitation. Kratzig [5] proposed damageindicators for estimates of seismic vulnerability by extendingstructural damage concepts for aging structures to seismic design,where structures fail by passing the damage evolution over anultimate bound. Shen and Shen [6] established a practical hyster-esis model of steel members, which takes the effects of damageaccumulation and fracture into consideration, and which includesthe calculation of damage index, the effects of damage on theyielding point, modulus of elasticity and hardening coefficient ofsteel, the stress–strain hysteresis relationship and the fracturecriterion. In the work by Dimova and Negro [7], the vulnerabilityof the structure was estimated by fragility analysis based onfitting the numerical models of the structural response in differ-ent seismic intensity levels to the experimental data, in whichwidely used global damage indices, such as Park and Ang overallstructural damage index [8,9] were associated with the condi-tional probability of failure and the damage states of the studiedstructure were expressed in terms of its fragility. Anyway, thedevelopment of seismic design criteria based on performanceindicators and reliability naturally calls for description of theprobable response and behavior of structural damage underseismic excitations with different intensities that may act onthem during their life. Thus that the above methods for estimat-ing the probabilities of failure and the expected damage aredeveloped in terms of global damage indices for a structure.

In the above methods to define and evaluate the behavior ofstructural damage under seismic excitations, damage behaviorstarted from meso-scale defects in materials and evolved atvulnerable area of structures is neglected, which may be properfor the analysis of structural behavior and damage states underseismic excitations with macro-seismic scale intensities. How-ever, for the purpose of exploring mechanism of damage evolvingand progressive failure in a structure under seismic excitations,which may not induce collapse, it is significant to model accu-rately what really happens on the process of meso-scale damageevolution from material degradation to structural failure scale inthe evaluation on seismic damage and possible progressive fail-ure. It is well known that many damage mechanisms, such ascracks or degradation on vulnerable component or welds atstructural connections, occur locally at material points or compo-nent levels, which will in turn influence the entire structuralbehavior, contribute to global structural damage and possiblycause progressive failure. An accurate model for such seismicresponse behavior should incorporate the physical mechanism oflocal damage, treated predominantly at the material point orvulnerable component level, into global structural analysis. Dri-ven by such local deterioration, the structural response behaviorof a structure will change over its service time and under seismicloading, generally, in a nonlinear manner [10,11].

Continuum damage mechanics (CDM) [12] has shown a greatpotential in predicting damage of ductile material and, therefore,can be used to develop reliable tools for the design of componentsundergoing plastic deformation [13]. The theory of continuumdamage mechanics developed in the framework of continuummechanics and irreversible thermodynamics [13,14,15] is consid-ered suitable with the deterioration mechanical behavior in steel

failure description under seismic loading. Many kinds of ductilemodels have been proposed to describe steel damage character-istics [16–18]. However, although these constitutive models havealready been applied in damage analysis for material and compo-nents, more approaches on structural damage analysis do notaddress the physical mechanisms of meso-scale damage initiationand evolution [3,7,19]. The major difficulty now is the lack of aframework for such nonlinear physical-based modeling, espe-cially for infrastructure such as long-span suspension bridges. Thedamage located on vulnerable area should be modeled based onthe theory of continuum damage mechanics for hot-spot stressanalyses coupled with damage evolution and then inserted intothe global model at the structural level for seismic analysis ofdynamic response and progressive failure estimation.

In the past works by the authors’ research group, a methodol-ogy and strategy have been proposed for fatigue damage assess-ment and life prediction of bridge deck sections of existingbridges with online structural health monitoring data [20], inwhich the behavior of the welded joints could not be estimatedaccurately; then an approach was developed for multi-scalemodeling and numerical analysis on dynamic response andstructural deteriorating in long-span bridges under service load-ing [21,22]. However, damage prognosis, as a prediction orestimate of a structutre’s probable course and outcome whenthe structure suffers from high loads such as seismic loading, hasnot to be dealt with.

This paper is aimed to develop a numerical procedure formulti-scale prognosis analyses on seismic damage and progres-sive failure process in steel structures by considering meso-scaledamage evolution based on CDM constitutive model and then toexplore mechanism of damage or local progressive failure andtheir effect on nonlinear dynamic behavior of structures underseismic excitations with micro-seismic scale intensities or severalearthquake actions. A ductile damage constitutive model ischosen as the description of meso-scale damage evolution forthe numerical analysis of progressive failure behavior of steelstructures under seismic loading, and implementation of thenumerical procedure is studied in order to introduce the damageconstitutive equations and incorporate into the multi-scale com-putations of seismic structural response performed with thesoftware ABAQUS.

2. The description on damage in meso- and macro-scale

In order to develop structural damage prognosis and itsarithmetic of analyses for large structures under seismic loading,a multi-scale modeling of structural damage should be firstdeveloped to describe the phenomena that some joint parts of astructure, which contains meso-defects show nonlinear damageor local failure due to stress concentration while mostly parts ofthe structure remain elastic and show linear behavior of responseon initial stage of the structure deteriorating. Fig. 1 shows adescription of the damage in macro- and meso-scale wheredefects exist on a joint part of structures.

2.1. Damage in meso-scale

Suppose that the defects in meso-scale can be described by thedamage variable dL(8xiAVL, where VL is a region with meso-defects on a joint part or vulnerable locations of structure), andthe response in the meso-scale is controlled by the followingequations:

sLij,xjþbL

i ¼ 0 8xiAVL ð1Þ

Fig. 1. Description of the damage in meso- and macro-scale.

Z.X. Li et al. / Finite Elements in Analysis and Design 48 (2012) 1358–13691360

sLij ¼ ð1�DLÞLijkleL

kl ð2Þ

eLij ¼ uL

ði,xjÞð3Þ

and boundary conditions:

uLi ¼ ui, sL

ijnj ¼ _t i ð4Þ

damage evolution law:

_DL¼ f ðDL,eL

kl,eLpÞ if eL

pZep0ð5Þ

where sLij, e

Lkl, uL

i and eLp are, respectively, Cauchy stress, strain,

displacement and equivalent plastic strain in meso-scale; theycan be obtained by solving the Eqs. (1)–(5).

2.2. Damage in macro-scale

On the other hand, the response of the whole structure in themacro-scale is controlled by the following:

sGij,xjþbG

i ¼ 0 8xiAVG�VL ð6Þ

sGij ¼ LijkleG

kl ð7Þ

eGij ¼ uG

ði,xjÞð8Þ

and boundary conditions:

uGi ¼ ui, sG

ij nj ¼ _t i ð9Þ

where sGij , e

Gkl and uG

i are, respectively, Cauchy stress, strain anddisplacement in the region except the joint part with defectsð8xiAVG�VLÞ; they can be obtained by solving the Eqs. (6)–(9).

While the macro-scale damage is expressed as follows:

DG ¼ 1�E0

E8xiAVL; and DG ¼ 0 8xiAVG�VL ð10Þ

The damage in both local details in meso-scale and globalstructure in macro-scale are established. A kind of homogeniza-tion arithmetic specially used for scale transition of damageevolution process in multi-scale is proposed according to thecontinuum damage theory and homogenization method [23], inwhich a key issue is how to define the meso-scale damage modeland to connect conveniently and accurately models of different

scales on the trans-scale boundary, which are explained asfollows.

3. Meso-scale damage model for steel under seismic loading

3.1. The formulation of evolution law for meso-scale damage

In the framework of continuum mechanics and irreversiblethermodynamics, the general expression of damage evolution lawswas first proposed by Lemaitre [24], which relates the growth rateof damage variable, _D (here, damage variable D is actually DL in theabove) to the energy release rate of the damaged material, Y ,through a damage potential function, FD. The evolution law forplastic damage can be written as [2]

_D ¼@FDðY ,p,D,. . .Þ

@Y_pð1�DÞ ð11Þ

where p is the accumulated plastic strain, and Y is the strain energyper volume unit released at constant stress (due to damageincrements in the form of the stiffness loss):

Y ¼ð3=2ÞsD

ijsDij

2Eð1�DÞ2

2

3ð1þnÞþ3ð1�2nÞ sH

seq

� �2" #

¼s2

eq

2Eð1�DÞ2Rv ¼

~s2eq

2ERv

ð12Þ

where seq the Von Mises equivalent stress used in plasticity:seq ¼ ðð3=2ÞsD

ijsDij Þ

1=2; Rv is the triaxiality function: Rv ¼ ð2=3Þð1þvÞþ3ð1�2vÞðsH=seqÞ

2; ~seq ¼ ðseq=1�DÞ is the damage effectivestress of seq.

In the framework of CDM, the final form of damage potentialfunction varies for different researchers when they face differentproblems of damage. Considering the convenience of parameterdetermination and numerical analysis, Lemaitre’s damage evolu-tion law [2,10] is proper for the demand of damage kineticconstitutive equations [24] to describe plastic damage behaviorunder dynamic loading. Other functions for damage potential,such as that proposed by Wang [25,26], or that by Bonora[12,13,16,17], and Chandrakanth and Pandey [27] are too com-plicated to be implemented in seismic damage analyses. Thedamage potential function by Lemaitre can be expressed as

FD ¼Y2

2Sð1�DÞð13Þ

where S is a material constant associated to damage energystrength. Then, the formula of damage evolution law can bederived from the Eqs. (11)–(13).

3.2. Constitutive model for steel with isotropic ductile damage

Von Mises yield function and mixed hardening mode can beintroduced here so that plastic potential function is equal to theyield criterion, which can be written as

Fp ¼ f ¼ seq�sy�R¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3

2ðsD

ij�XDij Þðs

Dij�XD

ij Þ

r�sy�Rr0 ð14Þ

where sDij is the deviatoric component of a stress tensor, XD

ij is thecomponent of back stress tensor, syþR is the subsequent yield stressconsidering isotropic hardening and p is the accumulated plasticstrain.

The sum of the damage potential function given by Eq. (13)and the plastic potential defined by Eq. (14) constitutes thecomplete dissipation potential, which connects the associatedplasticity and associated damage to the plastic damage statecoupling theory. Based on these expressions, the constitutive


model for steel with isotropic ductile damage can be summarizedas follows:

(1)
Damage constitutive equations:� Components of stress and strain tensors:
sij ¼ sDij þ

1

3skkdij, sH ¼

1

3skk ð15Þ

eeij ¼ e

eDij þ

1

3ee

kkdij, eeH ¼

1

3ee

kk, eij ¼ eeijþe

pij ð16Þ

� Elastic damage constitutive equations:

eeij ¼

1þnE

sij

1�D�nE

skk

1�Ddij ð17Þ

� Plastic damage constitutive equations:

depij ¼

3

2

ðsDij�XD

ij ð1�DÞÞ

ðsDij�XD

ij ð1�DÞÞeq

dp¼3

2nij dp ð18Þ

dR¼Hi dpð1�DÞ ð19Þ

dXDij ¼Hknijdpð1�DÞ ð20Þ

(2)

The boundary

Damage evolution laws:

dD¼

s2eq

2ESð1�DÞ2Rvdp pZpD

0 popD

8<: ð21Þ

when D¼Dc and Dc ¼D1cs2

u

ð ~snÞ2r1 and, then meso-crack initiation

ð22Þ

where su is ultimate strength of materials, pD is the thresh-old of accumulated plastic strain and D1c and Dc are thethreshold values of uniaxial and multiaxial damage,respectively.

(�G, �G, DG)Transition zone

(�L, �L, dL) ∀xi ∈VL)

Fig. 3. Connection between meso- and macro-scale FE modeling.

4. Implementation of multi-scale numerical analyses onseismic damage

A new computational technique should be developed formulti-scale numerical calculation on seismic damage and struc-tural failure with continuum damage evolution on the criticallocations of structures, and the technique should be based on oneof commercial software for the purpose of seismic analyses onlarge infrastructures. The commercial software package ABAQUS

FE model of the studied structure

To determine vulnerable locations

of the structure

Constitudamaged m

ev

Computational tfor dynamic recoupled with d

Streda

Technique of Connection between meso- and

macro-scale modeling

Multi-scale FE model of the studied structure

Fig. 2. Main flow chart of seismic response analyses with

is chosen since it is a good tool for nonlinear finite elementanalyses and has the capability of user-defined material featurereferred to as UMAT subroutine. The computation on seismicresponse of steel structures with damage evolution on the criticallocations should be implemented by specific supplemental soft-ware developed on the basis of ABAQUS, by which the main flowchart of computational procedure for seismic response andprogressive failure with meso-scale damage evolution on thecritical locations is shown in Fig. 2, in which two of the keyissues in the computation are (1) to connect conveniently andaccurately models of different scales on the trans-scale boundaryand (2) introduce the meso-scale damage model explained in theSection 3 at the critical locations of the structure where the meso-scale model is developed.

4.1. Connection between meso- and macro-scale modeling

In order to show how models of different scales could beconveniently connected on the trans-scale boundary, let’s see asimple case of trans-scale boundary in multi-scale modeling of acomponent with some type of defects, for example, a very smallhole, as shown as Fig. 3. The part with defects is modeled withsolid elements in meso-scale while the other part of the compo-nent is modeled with beam elements in macro-scale. In order toconnect conveniently and accurately models of different scales onthe trans-scale boundary, two kinds of connecting methods havebeen adopted by introducing displacement consistency and stresscontinuous conditions in the computation with multi-scale mod-els. Comparing the results of displacements and stresses calcu-lated by the trans-scale models with two kinds of methods for the

Seismic excitations and boundary conditions

Criteria of local failure and/or crack

initiation

Analyses on seismic response and damage

or failure process

tive equation of aterial and damage

olution law

echnique sponse amage

ss, strain and mage fields

meso-scale damage evolution on vulnerable locations.


trans-scale connections with the singe-scale models, it was foundthat the results by multi-scale model fit well with those by thesingle model. However, in order to accord with the requirementof the trans-scale connection methods and assuring accuracy onthe concerned part, the trans-scale boundary must be far awaythe damaged zone in local details, that is to say, a transition zoneas shown in Fig. 3 is needed.

4.2. Implementation of the meso-scale damage model via UMAT

The another key issue of the developed procedure is toimplement the material models and damage evolution equationsdescribed in the previous sections with programming codescompiled via User-defined Material Mechanical Behavior (UMAT)in ABAQUS, in which material plasticity, damage evolutionbehavior and damage threshold are defined and coupled intocalculation of stresses and strains through the algorithm for stressupdating and consistent tangent modulus solving. Structuralelasto-plastic behavior is related to loading and deformationhistory so that incremental equations of plastic damage theoryshould be implemented in the form of programming to convertnonlinear problems into linear form.

The flow chart of implementing damage constitutive equationand coupled calculation of stresses with damage provided in thesubroutine UMAT is given in Fig. 4, in which three steps areincluded: the description of plastic damage material model, stressand damage increment solution and the consistent tangent

Mises yield criterion

Start

Trial stress estimates for judgment of elastic-plastic

Calculate total strain and input threshold of plastic strain

No

Yes

Jacobian matrix computation and state varia

End of the current i

D>0Yes No

Elastic stiffness matrix

Co

u

0(1 )E E D= −

0(1 )y y Dσ σ= −

0E E=

0y yσ σ=

Fig. 4. Flow chart of introducing meso-damage equation and coupled cal

modulus matrix definition. In the first step, elastic constitutiverelations are applied to compute the predictive value of stress andstress increment. If the predictive value of stress is greater thanthe yield strength, material plastic and damage state can beacquired on the basis of subsequent yield condition, kinematichardening rule and damage evolution law. Subsequently, stressand damage increment is capable of being solved in accordancewith existing strain increments and constitutive expressions. Theradial return mapping algorithm based on the backward Eulermethod (integration algorithm) and the Newton–Raphson itera-tion (solution algorithm) is selected to guarantee calculationaccuracy and convergence speed. Finally, the consistent tangentmodulus matrix (Jacobian matrix) is assembled and updated toreplace continuum elastic-plastic modulus matrix in order toavoid the occurrence of pseudo loading and pseudo unloadingphenomena.

Two sets of relations are provided by the UMAT subroutineafter being called by ABAQUS/Standard, namely, the trial stressestimates and the trial tangent constitutive coefficients, for the ithiteration of the current kþ1th solution increment. For a linearelastic brittle material, the trial local tangent stiffness coefficientsare independent of the current strain values and are defined usingthe local secant stiffness approach. For the damaged material thatexhibits a nonlinear relationship in terms of strains, specialconsideration must be taken in defining these terms. Thereforethe UMAT subroutine actually defines a trial stress state for theend of the increment and as such, damage can evolve as the

Flow rule

Iterative algorithm for stress updating with

damage

Yes

Damagecriterion

No

Not needed

bles storage (stress, strain and damage)

ncrement step

Crack initial criterion

Material failure

mpensate value of stresses

pdating values of strain and stress

Damage evolution law

Yes

No

Yes

culation of stresses with damage evolution in the subroutine UMAT.

1st Step of analyses

Convergence?

1st iteration

Call UMAT at every integration point to form initial stiffness matrix

Call UMAT at every integration point to compute constitutive equations

2nd iteration

ith iteration

2nd Increment

1st Step completed

2nd Step completed

Jobcompleted


Convergence?


Yes

No

Convergence?

Yes

No

Yes

1st Increment

kth Increment Repeat R1Repeat R1

Repeat R2

R1

R2

Fig. 5. Flow chart of calling the subroutine UMAT in the calculation with ABAQUS/Standard.


nonlinear solution procedure iterates to find the solution for thecurrent increment.

The flow chart of the subroutine UMAT incorporated into thecomputations with ABAQUS/Standard for a particular iterationand solution increment is given in Fig. 5. As all of us know, in thenonlinear iteration process, the next solution goes from a con-verged solution increment k to the solution for the ith iteration ofsolution increment kþ1. The UMAT subroutine is called byABAQUS/Standard at every material integration point, so thatupdated variables such as the trial stress estimates and the trialconstitutive coefficients are returned to ABAQUS/Standard fromthe UMAT subroutine at each material point and incorporatedinto the computations that generate the internal force vector forthe residual vector (or force imbalance vector) and the tangentstiffness matrix for a particular iteration and solution increment.

5. Effects of meso-damage evolution on seismic behavior ofjoint components

In order to demonstrate the proposed procedure for calculat-ing structural seismic response with meso-scale damage evolu-tion and progressive failure process, a dog-bone joint is firstanalyzed as an example to show the effect of meso-scale damageevolution on the seismic behavior of joint components. The dog-

bone joint is a main form of steel joints designed to improveductility and bending resistance of components under seismicloading. The sectional geometry and its size of a dog-bone framejoint are given in Fig. 6 and Table 1, in which the value of a, b andc are determined according to the seismic design recommendedby Federal Emergency Management Agency (FEMA) [28] asfollows:

a¼ ð0:220:25Þbf ; b¼ ð0:520:7Þbf ; c¼ ð0:6520:85Þdb ð23Þ

The typical mechanical properties of the steel Q235 and therelative parameters for damage evolution at room temperatureare summarized in Table 2, where Hk is the modulus of kinematichardening and h is mixed hardening factor. The relative para-meters for the damage evolution of the steel Q235 given inTable 2 are determined from the experiment results by Song [29].

The traditional and multi-scale FE model of the dog-bone framejoint including boundary conditions of displacement and loadingare shown in Fig. 7, in which two types of loading, ‘‘Load I’’ and‘‘Load II’’, are used, respectively. ‘‘Load I’’ is a cyclic reversed loadingin horizontal direction, while ‘‘Load II’’ is in vertical direction andaccording to the standard ECCS (the European Convention forConstructional Steelwork) loading procedure [30] (cyclic testingprocedure recommended by ECCS in order to simulate severeseismic actions) for traditional seismic performance in which the

Table 1Size of dog-bone frame joint (mm).

Component Height,

db

Width,

bf

Flange thickness,

tf

Web thickness,

tw

Length,

L

Beam 600 200 17 11 3600

Column 400 400 21 14 3600

Table 2Parameters of Q235 steel mechanical and damage properties.

Parameters of material property

E u sy h Hk

200 GPa 0.3 235 MPa 0.2 1400 MPa

Parameters for damage evolution

pD S D1c

0.01 0.281 MPa 0.18

b c

tf

db

b c

a

R = (c2+4a2)/8

bf

Fig. 6. Geometry and size of the dog-bone connection. (a) Front and (b) top views.

Vulnerable area

Load II

Load I

Fig. 7. The finite element model of the dog-bone frame joint with boundary conditions

of displacement and loading. (a) Traditional and (b) Multi-scale models.

-0.02

400

320

240

160

80

0

-80

-160

-240

-320

Eff

ectiv

e st

ress

es (

MPa

)

Strain, �-0.01 0.00 0.01 0.02 0.03 0.04 0.05

With damage evolution

Without damage

Fig. 8. Comparison of the stress–strain hysteresis loop curves under cyclic loading

calculated with and without considering meso-damage evolution.


full mode of the displacement loading condition is as follows:

ð1Þ Dþy =2�D�y =2 1 cycle ð2Þ Dþy �D�y 1 cycle

ð3Þ 2Dþy � 2D�y 2 cycles ð4Þ 3Dþy � 3D�y 2 cycles

ð5Þ 4Dþy � 4D�y 2 cycles ð6Þ � � � � � � � � � � � �

ð24Þ

where Dy is the value of displacement at the beam end of thecomponent under monotone loading when the flange section ofminimal width in vulnerable area goes into plasticity.

5.1. Effects of evolving meso-damage on material hysteresis

behavior

Damage evolving in material directly leads to material stiff-ness degradation expressed by the decrease in material constitu-tive parameters such as elastic modulus, plastic hardeningmodulus and the area of stress–strain hysteresis loop. In orderto investigate the effect, the calculation of the variation of thesematerial constitutive characteristics on the vulnerable area of thecomponent under ‘‘Load I’’, that is cyclic reversed loading, iscarried out, in which the component is loaded in each cycle fromthe level 0.85 Fmax increasing to Fmax (Fmax is the peak value ofloading, which leads to material failure in vulnerable area of thecomponent). In the seismic response analysis, the component isconsidered losing loading capacity if the value of the accumulatedplastic strain at the whole vulnerable area achieves 0.15.

Fig. 8 shows the comparison of the calculated results of thestress–strain hysteresis loop under the same cyclic loading with

and without considering damage evolution at vulnerable area, inwhich the curve by real line is the result with damage evolutionin calculation while that by the dashed line is obtained by thecalculation without damage. It can be seen that significantdecrease is observed in the value of the elastic modulus and theyield stress caused by material damage accelerate plastic flow anddamage evolution, so that the degree of plastic deformation has

Fig. 9. Damage distribution on vulnerable area of the component under cyclic reversed loading. (a) By traditional and (b) by multi-scale models.

0

0.02

0.04

0.06

0.08

0.1

0.12

-0.25

Coordinates for the beam web location, x (m)

Dam

age

Val

ue

t = 13st = 12st = 11st = 10st = 9st = 8st = 7st = 6st = 5s

b c tf

db

Front view

x

-0.15 -0.05 0.05 0.15 0.25

Fig. 10. Damage distribution on the beam web and its evolution at different time

of loading.

b c

abf

Top view

x

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.05Coordinates for beam flange location, x (m)

Dam

age

Val

ue

t = 13st = 12st = 11st = 10st = 9st = 8st = 7st = 6st = 5s

0.1 0.15 0.2 0.25 0.3 0.35 0.4

Fig. 11. Damage distribution on the beam flange and its evolution at different

time of loading.


accumulated to a higher level compared with results showed bythe dashed line. When considering meso-damage, the slopechange of constitutive curve represents elasto-plastic stiffnessdegradation, so that it can be concluded that strain-damagecoupled constitutive equations can reflect material stiffnessdegradation considering decrease effects of the elastic modulus,the yield stress and the hardening modulus caused by damage.The effect of evolving damage on the hysteresis behavior givesreasonable explanations for the phenomenon of components’large deformation and progressive failure under seismic loading.

5.2. Distribution of damage and its evolution process

In order to investigate the damage behavior and seismicresponse of the joint component under seismic loading, thecalculation is carried out for the component under the ‘‘Load II’’,that is the cyclic load according to ECCS loading procedure (seeFig. 7 and Eq. (24)). The result of damage distribution onvulnerable area of the component calculated by the traditionaland multi-scale model, respectively, is shown in Fig. 9, in whichthe applied loading reaches 4Dy. It can be seen that, damage ismainly occurred in the vulnerable area and its amplitude is

proportional to the degree of beam section reduction, and similarresults are obtained using two different models. It is suggestedthat the multi-scale model not only can take place of the refinedmodel in analyses on nonlinear response of structure and thelocal damage, but also can greatly improve the efficiency of thecalculation.

Further descriptions of damage distribution on the beam weband flange at different times of loading are shown in Figs. 10 and 11,respectively. In these two figures, x-coordinate gives the calculatedpoints at location of beam web (for Fig. 10) and flange (for Fig. 11)and the location of x-axis on the joint component is shown by thesmall chart in Figs. 10 and 11. It can be seen from Figs. 10 and 11that the moment of damage initiation becomes earlier and thespeed of damage evolution accelerates with the reduction of thecomponent section. The local damage at the maximum reducedlocation of the vulnerable area has accumulated to a high value,which means where local failure and progressive rupture mayoccur, the component stiffness, seismic resistible and carryingcapacity will reduce here if the component continues carrying load.The rate of damage evolution in the beam web (Fig. 10) and flange(Fig. 11) has increased 85% and 75%, respectively, under the sameloading condition, which suggests that severe damage can greatly


accelerate the process of deterioration of the component and thenprogressive failure possibly.

5.3. Effects of meso-damage evolution on seismic response

Based on the above descriptions on local damage distribution,the following analysis is further carried out to investigate theeffect of the above distributed local damage and its evolutionprocess on seismic behavior and loading capability, in which thedistribution of dynamic stresses and seismic performance of thecomponent under the ‘‘Load II’’ according to ECCS loading proce-dure are studied through comparison between the relative resultscalculated by considering local damage evolution and thosewithout evolving damage in the computation.

5.3.1. The effect on the distribution of dynamic stresses

Figs. 12 and 13 give, respectively, the comparison of results ofstress distribution on the beam web and flange of the componentat the end of loading, in which the two curves are calculated byconsidering local damage evolution and those without evolvingdamage in the computation. In these two figures, x-coordinategives the calculated points at location of beam web (for Fig. 12)and flange (for Fig. 13) and the location of x-axis on the jointcomponent is shown by the small chart in Figs. 12 and 13. It canbe seen from the numerical results showed in Fig. 12 that the

-320

-240

-160

-80

0

80

160

240

320

-0.3

Coordinates for the beam web location, x (m)

Eff

ectiv

e st

ress

(M

Pa)

With damage evolutionWithout damage

b c tf

db

Front view

x

-0.2 -0.1 0 0.1 0.2 0.3

Fig. 12. Comparison of stress distribution on the beam web at the end of loading.

230

245

260

275

290

305

320

0.05Coordinates for beam flange location, x (m)

Eff

ectiv

e st

ress

(M

Pa)

-

With damage evolution

Without damage

b c

abf

Top view

x

0.1 0.15 0.2 0.25 0.3 0.35 0.4

Fig. 13. Comparison of stress distribution on beam flange at the end of loading.

effective stress calculated by considering local damage evolutionon the beam web has increased about 7.5% compared with thecalculated results neglecting local damage. It can be seen from thenumerical results showed in Fig. 13 that effective stresses in beamflange calculated by considering and neglecting evolving damageare almost the same in the edge of the vulnerable area; howeverlocal damage leads to the increment of effective stresses in thecentral part of the beam flange, and the varying range isapproximately 4%.

5.3.2. The effect on hysteretic behavior of loading versus

displacement

Load–displacement curve is generally used in seismic analysisto describe the hysteretic behavior of components or structures.Here, in order to evaluate the energy dissipation capacity of thecomponent, the load–displacement curves of the component aregiven in the form of the resultant force applied on the beam endin the vertical direction versus the displacement of the beam end.Fig. 14 shows the comparison of the calculated results of theload–displacement hysteresis loop calculated with and withoutconsidering damage evolution at vulnerable area, in which thecurve by real line is the result with damage evolution in calcula-tion while that by the dashed line is obtained by the calculationwithout damage. It can be seen from the hysteretic behaviorsshown in Fig. 14 that, the local damage evolution leads to a littledecrease in the applied force at the beam end under the same

-8.00Displacement (cm)

With damage

Without damage

40

30

20

10

0

-10

-20

-30

-40

App

lied

load

P (

kN)

-6.00 -4.00 -2.00 0.00 2.00 4.00 6.00 8.00

Fig. 14. Load–displacement curves of the joint under cyclic loading calculated

with and without considering evolving damage.

0.96

0.97

0.98

0.99

1.00

1.01

0Loading time (s)

Nor

mal

ized

for

ce, P

D/P

0

2 4 6 8 10 12 14

Fig. 15. Curve of deterioration of the supporting capacity on the loading process.


displacement condition, which suggests deterioration of thecomponent’s carrying capacity with the increase in local damageat vulnerable area.

Furthermore, Fig. 15 shows the curve of the carrying capacityon the loading process, in which the normalized force at the beamend, PD/P0, is defined as the ratio of the supported force calculatedwith and without considering damage evolution at vulnerablearea. The calculation results show that when the accumulatedplastic strain at vulnerable area of the connection reaches theultimate value, the value of local damage there will accumulate toabout 0.12, and the carrying capacity of the component decreases

Suspender

The longitudinaltruss

Fig. 16. Longitudinal truss available typically as a stiffening frame in long-span

bridge decks: (a) the stiffening frame in bridge deck and (b) the multi-scale model

of the truss.

Fig. 17. Damage distribution in the

almost 4% compared to that calculated without considering localdamage.

It should be noticed that, compared to the stress–strainhysteresis loops shown in Fig. 8, one can easily find that theeffect of local damage evolution on structural hysteresis behaviorgiven by Fig. 14 is less than that on local stress–strain hysteresisbehavior showed in Fig. 8. The finding suggests that indeed theeffect can be neglected for the seismic analysis of structuressuffered seismic loading with macro-seismic scale intensities,however, it may not be for structures suffered multiple seismicloading with micro-seismic scale intensities since the effect maybe accumulated and lead to wrong analytical results if localdamage evolution is neglected.

6. Effects of evolving damage on progressive failure of trussstructures

In this section, a longitudinal truss in steel bridge decks isselected as a case to study how the proposed procedure forcalculation of seismic response with local damage evolution tobe applied to analyses on progressive failure of truss structures.The longitudinal truss is typically available as a stiffening frame inlong-span bridge decks as shown as Fig. 16(a). The multi-scalemodel of the truss is shown as Fig. 16(b). The cyclic loading, P,which is same to the ‘‘Load II’’ in the above section, i.e. loadingaccording to ECCS loading procedure, is applied to the FE model.

The multi-scale FE model is then used to calculate seismicresponse and progressive failure of the truss under the load, P2,according to ECCS loading procedure. Firstly, the value of dis-placement at the load point, Dy, on the truss under monotoneloading when the vulnerable area goes into plasticity is calculatedin order to determine the amplitude of the displacement con-trolled load. The detailed loading procedure has been given byEq. (24).

Damage distribution in the truss under ECCS cyclic loading isshown in Fig. 17. It can be seen that considering the symmetry ofthe truss model and its boundary conditions with respect tovertical member of middle web under the loading, distribution ofstructural plastic zone and damage zone is symmetric. The majormode of failure will be ductile failure induced by meso-scaledamage accumulation while the damage and progressive failureappear firstly on the area of the joints, so that the attentionshould be paid to these locations for the analyses on progressivefailure process.

truss under ECCS cyclic loading.

Fig. 18. Progressive failure process at the vulnerable area. (a) 1st cycle at D¼ 3Dy , (b) 1st cycle at D¼�3Dy , (c) 2nd cycle at D¼ 3Dy , (d) 2nd cycle at D¼�3Dy and (e) 1st

cycle at D¼ 4Dy

-800

-600

-400

-200

0

200

400

600

800

-2Δy 0 2Δy 4Δy-4ΔyCyclic loading

Car

ryin

g ca

paci

ty (

kN)

Fig. 19. Variation of the carrying capacity of the truss under ECCS cyclic loading.


With the continuation and development of the cyclic loadingprocess, the damage value and extension of damage area willenlarge, which leads to local failure at vulnerable area of the truss,and then loading capacity of the structure is lost when theaccumulated plastic strain reaches its limit and/or damagereaches its threshold for failure. In order to describe the progres-sive failure process, the computation on the seismic response ofthe truss under ECCS cyclic loading is carried out by consideringlocal damage evolution, in which the vulnerable area is deter-mined by the distribution of damage as well as the maximumstresses, and the computation is done with the continuation anddevelopment of the cyclic loading process until the truss is failedin supporting the load. Fig. 18 shows the progressive failureprocess at the vulnerable area, one of the welded joints, in whichthe amplitude of damage value and area are given when the trussunder the ECCS loading procedure of (a) the 1st cycle at D¼ 3Dy,(b) the 1st cycle at D¼�3Dy, (c) the 2nd cycle at D¼ 3Dy, (d) the2nd cycle at D¼�3Dy and (e) the 1st cycle at D¼ 4Dy. It can beseen that the damage value as well as the damaged area areenlarged with the increase in the cycle times as well as theamplitude of the cyclic loading once the damage starts its

evolution in vulnerable area, as a result, which will induceprogressive failure of the welded joint.

Fig. 19 shows variation of the carrying capacity on the processof local damage evolution in the truss under ECCS cyclic loading.It can be seen that the global carrying capacity no longer increasein linear after the cycle at 2Dy when progressive failure processstarts at the vulnerable area of the welded joint of the truss, andthen carrying capacity is expected to decrease with the increasein deformation and damaged area. Local failure happens first onthe vulnerable area then redistribution of stresses and internalforces will appear in the truss due to local loss of supportingcapacity, and progressive failure process from the first vulnerablearea to the second, and other locations of the truss will lead to theoccurrence of structure failure.

7. Conclusions

A numerical procedure has been developed for multi-scaleanalyses on seismic damage and progressive failure of steelstructures with meso-damage evolution on vulnerable areas,and the procedure has been implemented by introducing themeso-damage constitutive equations and incorporating into thecomputations of seismic structural response performed with thesoftware ABAQUS. Effects of meso-damage evolution on seismicresponse and progressive failure were analyzed for the dog-bonejoint component and a truss structure under ECCS cyclic loadingwith the developed numerical procedure. Based on the results forthis study, the following conclusions can be made:

(1)
The developed numerical procedure is available for analyseson seismic response of steel structures and their progressivefailure process in which meso-scale damage evolution isconsidered based on CDM constitutive model. It is useful forexploring mechanism of seismic damage or progressivefailure and their effect on nonlinear dynamic behavior ofstructures under seismic excitations with micro-seismicscale intensities or several earthquake actions.
(2)
The calculated results of seismic response coupled with meso-scale damage reflect material degradation on the process ofseismic loading, in which parameters of material propertieshave significant variation compared with those without


consideration of damage evolution. The effect of evolvingdamage on the material degradation gives reasonable expla-nations for the phenomenon of large structural deformationand progressive failure under seismic loading.

(3)
The effect of meso-scale damage evolution on structuralhysteresis behavior is less than that on local stress–strainhysteresis behavior, which suggests that, indeed the effectcan be neglected for the seismic analysis of structuressuffered seismic loading with macro-seismic scale intensi-ties. However, if meso-damage evolution is neglected for astructure suffered multiple seismic loading with micro-seismic scale intensities, wrong analytical results could begenerated since the significant effect of evolving damage onlocal hysteresis behavior may be accumulated and lead towrong description on progressive failure process of thestructure.
(4)
With the increase in amplitude of seismic loading, thedamage value and extension of damage area enlarge moreand more, which leads to local failure at vulnerable area ofthe structure, and then loading capacity of the structure islost when the accumulated plastic strain reaches its limitand/or damage reaches its threshold for failure. The globalcarrying capacity of the structure decreases with theincrease in damaged area and progressive failure at thewelded joint or other vulnerable area, which at last will leadto the occurrence of structure failure.
(5)
The calculated results show that the multi-scale model cannot only take place of the refined model in analyses on thedamage evolution and the failure process, but also cangreatly improve the efficiency of the calculation. It isspecially significant for the large structure, which couldnot be modeled by the refined model in single scale; thedeveloped procedure for multi-scale analyses will be theonly way to explore the process of damage evolution andprogressive failure of the structure under high loads.
Acknowledgments

The work described in this paper was substantially supportedby the grant from the National Natural Science Foundation ofChina (No. 11072060), which are gratefully acknowledged.

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