An elasto-plastic damage model for reinforced concrete with minimum number of material parameters

An elasto-plastic damage model for reinforced concrete with minimum number of material parameters

Wilfried B.Kr€a tzig a,Rainer P€o lling b,*

a Institute for Statics and Dynamics,Ruhr-University Bochum,44780Bochum,Germany

b Building Contractors of Mesenbrock,Welter Stra?e4,48249D€u lmen,Germany

Received1November2002;accepted4March2004

Available online20April2004

Abstract

Based on a fully3-D elasto-plastic damage theory,the material behavior of all reinforced concrete components––concrete,reinforcement and bond––is,for biaxial loading,realistically modelled,including cyclic action.Thereby emphasis is layed on concrete in tension and compression.The presented model contains a minimum number of material parameters.It further enables to map exact uniaxial stress–strain curves as proposed by modern codes of practise,like the EC2.

All material parameters of the model can be readily interpreted and determined by few standard experiments,or approximated from concrete compression strength.Finally,the concrete model is veri?ed by numerical simulation of experiments.

ó2004Elsevier Ltd.All rights reserved.

Keywords:Concrete;Reinforced concrete;Material damage;Plasticity;Crushing energy;Material model;Softening

1.Introduction

In structural engineering,reinforced concrete is considered as a2-phase composite of uniaxial steel and multi-dimensional concrete components.Hardened concrete itself,predominantly a mixture of aggregates and a cementitious matrix,exhibits a rather complicated deformation behavior,mainly because of initiation and growth of micro-cracking in the matrix.

Constitutive models for concrete,the central concern of this treatise,exist on very di?erent quality levels.Even the most advanced structural codes of practice[1]o?er mainly elastic behavior,at most some monotonic non-linear stress–strain curves without any rheo-mechanical speci?cation.On the other(research)extreme,we?nd highly sophisticated constitutive attempts with few relations to engineering design practice.But for com-puter simulation of advanced structural problems,like failure analyses or life-cycle assessments,constitutive models are required,improved compared to those from codes of practice and more exact,nevertheless of lowest possible complication.Severe de?ciencies of code mod-els are their uniaxial formulation and the lack of infor-mation for cyclic processes.

If one attempts to set up a constitutive model for cyclic behavior of concrete as pictured in Fig.6or7,one observes?rst the residual strains after complete unloading.These generally are interpreted as‘plastic’deformations,although in concrete there will be cer-tainly no plastic slip like in metal plasticity.The second observation is the sti?ness degradation along the cycles which requires a‘damage’component in the model. Obviously in constitutive modeling of concrete,an elasto-plastic component has to be combined with an elasto-damaging one.

Hence the key aim here is a3-dimensional elasto-plastic description of mechanical responses of reinforced

*Corresponding author.Tel.:+49-2548-9193-883;fax:+49-

2548-1085.

E-mail address:rainer.poelling@https://www.360docs.net/doc/2910208676.html,(R.P€o lling).

0045-7949/$-see front matteró2004Elsevier Ltd.All rights reserved.

doi:10.1016/https://www.360docs.net/doc/2910208676.html,pstruc.2004.03.002Computers and Structures82(2004)

1201–1215

https://www.360docs.net/doc/2910208676.html,/locate/compstruc

concrete,supplemented by a damage component.Be-cause of few cyclic experiments for concrete and the problem of reproductivity of results,a minimum number of well-de?ned,experimentally identi?able material parameters shall control the model[2].

Classical elasto-plastic material theories[3,4]are standard in engineering,they need no explanation here. They are‘stress-based’since their yield conditions and yield potentials are formulated in stress-space.‘Strain-based’plasticity theories,with plastic stresses as internal thermodynamic variables and consequently yield con-ditions in strain-space,have their origins in the works [5,6].Remarkably early,Dougill[7]proposed a‘frac-turing theory’for concrete under compression,in our terminology a‘strain-based’damage concept.Opposite to plasticity,its basic idea required that inelastic defor-mations result merely in sti?ness degradations,thus all unloading paths cross through the origin of the stress–strain diagram,and no residual strains remain after unloading.

In a re-work Dougill and Rida[8]de?ned‘micro-fracturing’materials,equipped with sti?ness degrada-tion from micro-crack evolution,linear-elastic behavior during un-/reloading,and without strain/stress residuals after complete unloading.Hence they laid the basis of ‘strain-based’continuum damage theories[9,10],in which the material sti?ness tensor itself acts as internal damage variable.If instead of this the?exibility(com-pliance)tensor is applied,‘stress-based’continuum damage theories can be established[11,12].

In order to properly model the constitutive behavior of concrete in compression or tension,both constitu-ents––plasticity as well as continuum damage theory––have to be combined,both either in‘stress-based’or ‘strain-based’alternatives.Such?rst coupling work was due to Ba z ant and Kim[13].Their‘stress-based’plas-ticity combined with‘strain-based’damage theory led to an‘elasto-plastic fracturing’concept with a total of26 material parameters.Coupling concepts of both con-stituents in stress-space are due to[14],a work speci?ed by[15]to isotropic damage of concrete under com-pression.[16]therein uni?ed both limit surfaces for yielding and fracturing to one.

Han and Chen[17]?rst combined both constituents in complete tensorial manifold in strain space.In order to reduce the number of material constants,they united yield and damage surface,an idea repeated in[10]. Further,[18]o?ered a scalar coupling of plasticity and damage theories introducing again a uni?ed yield/dam-age condition with the key argument that micro-crack-ing in the cementitious matrix is the single source for both inelastic phenomena.

Finally we mention the survey over all basic condi-tions of coupling alternatives in[19].We stress,that there only elasto-plastic-damage concepts are listed with a sound basis in continuum mechanics.Empirical models of reinforced concrete which all combine the mentioned constituents are legion[2];a prominent concept leading to excellent results is that one of Darwin and Pecknold[20].

All mentioned constitutive models call,in principle, for the ability to describe material failure in compression as well as in tension,obviously both strongly localized phenomena[21].To avoid mesh sensitivity of a partic-ular FE-solution,such localized fracture processes re-quire special treatment[22].The intended constitutive law of this work is aimed to the analysis of complete structures,such that the classical‘smeared crack’con-cept shall be maintained[23].Such concept avoids the immense numerical e?ort of modeling hundreds of single cracks,presently to be mastered only by parallel com-puting techniques,in view of the physical irreproduc-ibility of crack-patterns in concrete structures,even under laboratory conditions.

So in order to avoid modelling of localization bands or surfaces as weak or strong discontinuities,or to avoid enhancements by non-local kinematics,the use of crushing respectively fracture energy is chosen here[2]. Advantageously,such regularization requires only the determination of2additional material constants, namely the crushing and the fracture energy.

Consequently,in this paper a‘stress-based’elasto-plastic damage model for reinforced concrete will be derived.It will be demonstrated,that not only the fun-damental ideas of micro-fracturing and damage theories are equal,but that both concepts lead to identical material descriptions[18],if the material sti?ness tensor itself is used as internal thermodynamic variable.It hence is obvious that both phenomena––damage by tension cracks as well as by compressive micro-crack-ing––follow a uni?ed elasto-plastic damage theory.The ?nal constitutive law will possess a minimum number of material parameters,an important aspect for applica-tion.For this reason,the model is adapted without exact description of the response under highly triaxial com-pression.

2.Theoretical basis of the elasto-plastic damage theory

The fundamental framework of the presented con-cept is the‘stress-based’elasto-plastic damage theory. An excellent summary of this theory with the compli-ance tensor as internal variable has been given by Govindjee et al.[12]in complete analogy to plasticity theory.Recently a scalar combination of both damage and plasticity theory has been performed by[16].We start with a brief summary of the underlying concept.

As in[12]our starting point is the assumption of a Helmholtz-free energy W and the de?nition of internal thermo-dynamical variables.To consider both,plastic deformation and sti?ness reduction,we introduce as

1202W.B.Kr€a tzig,R.P€o lling/Computers and Structures82(2004)1201–1215

internal thermo-dynamical variables the plastic strain e pl

as in ordinary plasticity theory and the change D da of the compliance tensor like in [12].Then the Helmholtz-free energy reads:

q 0W ee ;e pl ;D da ;q T?1ee àe pl T:eD 0tD da

|??????{z??????}D

Tà1:ee àe pl

T

tq 0W in

eq T;

e1T

where r and e denote the stress and strain tensors,respectively.D 0abbreviates the initial compliance ten-sor,and q as further internal variable describes the hardening/softening behavior of the material.W in eq Tcontains the free energy associated with progressive degradation and plastic deformations.Finally,with q 0as initial density we identify the thermodynamically associated variables àr ?q 0o W =o e pl ;

àa eq T?q 0o W =o q ;

e2Tà12

r r ?q 0o W =o D da :e3T

Applying the C LAUSIUS –D UHEM inequality q 0_W

6r :_e ,we arrive on one hand at the compliance rela-tion

e ?eD 0tD da T:r te pl !r ?eD 0tD da Tà1:ee àe pl T;

e4T

and on the other hand at the inequality of dissipation P dis ?1

2

r :_D da :r tr :_e pl ta _q P 0:e5T

After introducing the elastic region E in the space of the associated variables (stresses):E :?fer ;a Tj U er r ;a T60g ;

e6T

we deduce from the principle of maximum inelastic dissipation the following normality rules:_e

pl :o r o er r Tt12_D da ?

o U o er r T

á_k ;e7T

_a

?ào U á_

k :e8T

At this point we have to separate the inelastic strains into those of plastic and damaged origin.Because of few available cyclic experiments for later parameter ?tting,the use of just one scalar b ,as proposed in [16]for separation of plastic from damaged parts,is adopted.Thus we assume _D

da ?2_kb o U o er r T

and

_e

pl ?_k e1àb To U o r

:e9T

Obviously with (34),the normality rule (7)is ful?lled.

With the consistency condition _U

?0one ?nally derives the tangential sti?ness relation and the evolution equa-tions for the internal variables as summarized in Table 1.

3.Uni?ed elasto-plastic damage model for reinforced concrete

3.1.Concrete under compression

3.1.1.Yield/damage potential

A yield/damage potential of Drucker–Prager-type has been selected for description of concrete in com-pression.Such type of potential can be handled rela-tively easy,it enables a su?cient modelling quality at least for uniaxial and biaxial proportional loading.This potential reads U c er ;a c T?

11??3

p àl l I 1?t????J 2p ?àa c eq c T;e10T

with I 1and J 2as ?rst invariant of r and as second

invariant of its deviator s ,respectively.The parameter l herein controls the in?uence of the hydrostatic stress on damage or yield.Further,the term left of the brackets guarantees that during plastic/damage loading a c always corresponds to the negative uniaxial compression stress.Di?erentiation of this potential with respect to r delivers

Table 1

Tangential sti?ness relation and evolution equations of internal variables of the stress-based elasto-plastic damage theory

Elastic loading/unloading _k ?0

Plastic-damaging loading k

>0,U ?0Tangential sti?ness relation:_r ?D à1:_e _r ? D à1àD à1:o U o r o U

o r

:D à1o U :D :o U ào U a ;q o U

!:_e Equations of evolution of internal variables:

_D da ?0_D da ?2b o U á_k or e_D da :r T?b o U _k _e pl ?0_e pl ?e1àb To U o r

á_k _q

?0_q

?o U o a

á_k with _k ?o U

o r

:D à1o U

:

D :o U ào U

a ;q o U

:_e

W.B.Kr €a tzig,R.P €o lling /Computers and Structures 82(2004)1201–12151203

o U c o r ?11??3

p àl l o I 1o r t12????

J 2p o J 2

o r

?11??3

p àl l I ts ts T

4????J 2p :e11T

3.1.2.Modi?cation of the internal variable,yield and

damage rule

We now introduce a new internal variable q ?c ,replacing the original one q c :

_q ?

c ?à2b w er T_q c ?2b w er T_k c with w er T?21??3p àl l

1

t1???2

p

:e12T

Then a c in (10)shall be only a function of q ?c .Due to

Section 3.1.4this transformation allows an analytical solution for the hardening/softening function a c eq ?c Tfrom a given uniaxial stress–strain curve [2],one of our goals.The rate of this variable a c reads

_a c ?d a c ?c _q ?c ?d a c ?c 2b _k c :

e13TFurthermore we assume,that the parameter b ,which separates plastic from damage parts,shall also be a function of q ?c .

The equations of evolution of the plastic strain _e

pl and the strain _e

da ?_D da :r can directly be determined from the potential (10).With respect to (11)and (12)we gain:

_e pl ?e1àb eq ?c TTo U c o r _

k c e14T?w er T

121b eq ?c T à1

11??3

p àl l I t14????J 2p s àts T

á _q ?c e15T

and

_e da ?_D da :r ?b eq ?c

To U c o r _k c e16T?b eq ?c T11

??3

p àl l I t14????J 2p s àts T á

_k c :e17T

Assuming now an isotropic damage evolution due to compression,we ?nd the following compliance evolu-tion law,the correctness of which can be veri?ed by an double scalar contraction with r :

_D

da ;c ?b eq ?

c T1??3

p àl l 1 à16????J 2p I I t14????J 2p eI tI T!á_k c e18T

?w er T1211

??3

p àl l I 1

à1

6????J 2p

I I t1

4????J 2

p eI tI T!

á_q ?c :e19T

The tensor D da ;c describes the compliance evolution and can be represented by two scalars in the isotropic case.Consequently,we replace the fourth-order tensor D da ;c

in the set of internal variables by D da ;c s1and D da ;c

s2as fol-lows:

D da ;c ?D da ;c s1I I tD da ;c s2eI tI T:

e20T

By comparison with the coe?cients in (18)and (20),one

arrives at evolution equations for D da ;c s1and D da ;c

s2:

_D da ;c s1?b eq ?

c T1??3

p àl l 1 à16????J 2p _k c ;e21T_D da ;c s2

?b eq ?c

T1??3

p àl

14????J 2

p

_k c :e22T

3.1.3.Tangential sti?ness relation

Because of all above assumptions the tangential sti?ness relation slightly di?ers from the standard form in Table 1,and one arrives with q ?c at:

_r ?D à1"àD à1:o U c o r o U c o r

:D à1o U c o r :D à1:o U c o r td a c d q ?c

?c w er T

#

:_e e23Tin the case of plastic-damaging loading.

3.1.

4.Concept of determination of the hardening/softening

function

We now consider uniaxial loading with respect to an orthogonal cartesian frame,where the unit base vector i 1coincides with the loading direction.Then the compli-ance relation (4)becomes

e ?1E c

tD da ;c !r te pl with e ?e 11;e pl ?e pl 11;r ?r 11

;1E c

?D 1111;D da ;c ?D da ;c 1111e24T

and the evolution equation (19)for the compliance boils

down to

_D da ;c ?w er T1??3p àl l r "à16??3p r !t14??3p r á2#á_q ?c ?_q ?c :e25TBy integration of this relation one arrives at the

important statement D da ;c ?q ?c ;

e26T

interpreting the internal variable q ?c as the uniaxial change of the compliance due to damage.Furthermore we assume the split of inelastic strains into plastic and damaging parts by the scalar parameter b ,as illustrated in Fig.1.From all previous transformations follows for the plastic strain e pl :

1204W.B.Kr €a tzig,R.P €o lling /Computers and Structures 82(2004)1201–1215

e pl

?b e pl h tD da ;c

á r

ee Ti

!e pl ?b

1àb

D da ;c á r

ee T?

b

1àb q ?

c

á r ee T:e27T

We then obtain by substituting (26)and the latter identity into (24)

e ?

1c t1q ?

c ! r ee T;e28Twhich equation solves with given function r

ee Tfor e as e ?e eq ?

c T,yielding together with the stress–strain instruction ?nally to a c eq ?c T? r ee eq ?c TT:

e29T

Furthermore we need the di?erential quotient d a c =d q ?c

and ?nd

d a c d q ?c ?àd r d

e d e d q ?c ?a c d r d e e1àb T1à1E c tq ?c 1àb d

r d e

h i ;e30T

with d e =d q ?c gained by total di?erentiation of Eq.(28).

Finally we need the explicit form of b eq ?c T.Starting point is the evolution equation (15),which reduces under uniaxial loading conditions to

_e pl

?1?c à1

r á_q ?c ;e31Tfrom which we ?nd directly b eq ?c T?

11t

o e =o q c

c c :e32T

For further transformation we can conclude from (27)

o e pl o q ?c ?b 1àb a c eq ?c T

tq ?

c

d a c d q ?c !;e33Tleading ?nally to b eq ?c T?

1

1tb 1tq ?c

c ?c

d a c

?c

h

i :e34T

3.1.5.Mapping to uniaxial stress–strain curve

For convenience the uniaxial stress–strain curve of concrete,assumed here as given function,shall be sub-divided into three parts as illustrated in Fig.2.

Region 1:Elastic.Below the initial yield/damage stress f c y we assume linear-elastic behavior,de?ned by Young’s modulus E c and Poisson’s ratio m c .The initial yield stress f c y is taken as one third of the failure strength f c ,such that we obtain the initial condition

a c eq ?c ?0T?f c y ?1

3f c :

e35T

Region 2:Hardening.In this region the stress grows until failure strength f c ,consequently the tangential sti?ness decreases from initial sti?ness to zero (hori-zontal tangent).An analytical function of this behavior,which ?ts well with experiments is found in [2].Eq.(2.1-18)therein reads

r

2ee T?E ci e c

te c

2

1à

E ci e c f

c

à2

e e c

f c ;e36T

with f c and e c as failure stress and accompanying strain,respectively.The modulus E ci ;originally denotes the initial modulus of elasticity.But in order to guarantee the stress–strain curve to cross the point eàe c y ;àf c y T,we de?ne E ci ;with respect to (35)as secant modulus

[2]

Fig.1.De?nition of material parameter b

.

Fig.2.Assumed uniaxial stress–strain curve of concrete under

compression.

W.B.Kr €a tzig,R.P €o lling /Computers and Structures 82(2004)1201–12151205

E ci?

1

2E c

f c

e c

2

à

f c

e c

t

3

2

E c:e37T

Region3:Softening.After exceeding the compression strain e c,localization of damage occurs in this softening region.A suitable concept to overcome possible ill-po-sedness there,originally derived for tension softening in smeared crack idealization,is the use of fracture energy: Then the softening branch depends on the fracture en-ergy,a material parameter,and on the characteristic length l eq[24].

Use of fracture energy as material parameter instead of a softening function is generally accepted for tension cracks.The transfer of this concept to softening under compression has been?rst proposed by Feenstra[25] and accepted since that,i.e.[26].To distinguish fracture energy under tension from that one under compression, the latter will be denoted as‘crushing energy’.But one should notice,that because of nonlinear hardening also a di?use crushing energy g cu;exists[27],not included in the presently described localized one.Thus one sets the

volume speci?c localized crushing energy g?

cl (compare

Fig.2)equal to G cl=l eq,where G cl is the material parameter‘crushing energy’and l eq the characteristic length of the respective FE integration point.Clearly,l eq depends on type,quadrature rule and form of the ele-ment[28].

Consequently,the following function has been cho-sen for the softening branch of region3:

reeT?à

1

2tc c f c e c

c

tc c etc c

c

áe2

with c c>0e38T

in which c c is the only free parameter controlling the

area under the stress–strain curve,corresponding to g?

cl .

Note,that this area is?nite.Determination delivers the following relation to the localized crushing energy G cl,as explained in detail in[19]:

c c?

p2f c e c

2G cl

eq à1f c e ce1àbTtb f c

c

h i2:e39TRemark.If the term in square brackets becomes nega-tive,the above equation renders invalid,since this would describe a‘snap-back’-behavior in the considered material point.To avoid this,one should elude elements leading to

l eq6

G cl

f c e ce1àbTtb f c

E c

:e40T

For determination of a ceq?

c T(29)from re-branches,see

Appendix A.3.2.Concrete under tension

For description of concrete behavior in tension we apply the stress-based continuum damage theory again. But for simplicity,all inelastic deformations shall be due to pure damaging,i.e.b?1.Also,opposite to com-pression,we exclude nonlinear hardening before soft-ening.Thus the initial damage surface and the failure surface are identical.

3.2.1.Damage potential

For tension failure,we use the widely accepted damage potential of Rankine type,a criterium in good accordance with biaxial experiments[29].To allow for softening in di?erent directions to model cracks in dif-ferent directions at least a kinematic softening rule is required.Hence introducing the back-stress a t of the stress state r as internal variable,the damage potential reads

U te1Ter;a tT?ne1Tàf ct60;e41Tabbreviating with ne1Tthe?rst eigenvalue of n?ràa t.

The derivatives of this potential with respect to r and a t are

o U t

o r

?Me1T

n

and

o U t

o a t

?àMe1T

n

;e42T

where Me1T

n

denotes the eigenvalue basis of the?rst eigenvalue of n.

3.2.2.Modi?cation of internal variable,kinematic soften-ing and damage rule

If one follows strictly the thermodynamic concept of the continuum damage theory,one has to introduce a

second-order tensor q

t

,thermodynamically conjugate to a t.Furthermore one has to?nd an evolution law as normality condition for this tensor and a tensor-valued function a teq tTas softening rule.The evolution law then reads_a t?o a t=o q t:_q t.Instead of such complexity determination,we replace the fourth-order tensor

o a t=o q t by a scalar function Zea1

t n

T,with a1

t n

?a t:Me1T

n

as normal component of the back-stress in crack-direction.

Introduction of an explicit conjugate variable q1

t n

can be omitted,because no hardening regime has been as-

sumed,and the function q1

t n

ea1

t n

Ttherefore is bijective. These simpli?cations lead to the softening rule,illus-trated in Fig.3:

_a t?Zea1

t n

T

o U t

o a t

_k

te1T

?à_k te1TZea1

t n

TMe1T

n

e43T

corresponding to Prager’s hardening law.(43)guaran-tees crack formation in each principal stress direction, independent of other existing cracks.

Now the normality rule can be derived from the potential U te1T(41)reading

1206W.B.Kr€a tzig,R.P€o lling/Computers and Structures82(2004)1201–1215

_e da ;t :?_D da ;t :r ?_k t e1To U t e1T?_k t e1TM e1Tn :e44T

The anisotropic damage rule:_D

da ;t ?1n

_k t e1TM e1Tn M e1T

n

with

r 1n

?

M e1T

n

:r e45T

has been taken over from [12].This rule guarantees the accomplishment of (44),and preserves symmetry of the compliance tensor.

3.2.3.Consideration of further cracks

As well known,a second crack in 2D-plane and a third one in 3D-space can appear,each orthogonal to the existing ones.Therefore two more damage potentials are introduced analogously to the original one (41):U t e2Ter ;a t T?n e2Tàf ct ;60;e46TU t e3Ter ;a t T?n e3Tàf ct ;60:

e47T

The complete theory thus develops to a non-smooth

multi-surface continuum damage theory.According to Koiter’s rule [30],we then obtain _a

t ?àX 3i ?1

_k t ei TZ ea i t n

TM ei Tn ;e48T

_D

da ;t ?X 3i ?1

1r i n _k t ei TM ei Tn M ei T

n with r i n ?M ei T

n :r :

e49T

Loading conditions have to been introduced for each crack direction separately:_k

t ei TP 0;U t ei T60;

_k

t ei TáU t ei T?0:e50T

3.2.

4.Closure of cracks

Crack closures results in instantaneous local re-sti?-ening of the concrete,where the cracks remain as ‘pas-sive’ones.Following an idea of Ortiz [11]we thus

disregard in the compliance tensor D the constituent D da ;t ,which represents all cracks,but maintain D da ;ta of all ‘active’cracks:D ?D 0tD da ;c tD da ;ta :

e51T

Kraj c inovi c [31]obtained the following relation between D da ;ta and D da ;t ,assembling only those active cracks in principal directions,which belong to positive principal stresses:

D da ;ta ?P t:D da ;t :P t;

e52T

where P tdenotes a fourth-order projection tensor.According to [31],one possible form of P twith H eTas Heaviside-function is:

P t

?X 3i ?1

H r ei TàáM ei T

r M ei T

r :e53T

3.2.5.Tangential sti?ness relation

With all these assumptions the tangential sti?ness relation will expectedly slightly di?er from that one in Section 2.For its evaluation,we determine the consis-tency parameters by use of the consistency condition _U

t ei T?0,the damage rule (49)and the kinematic soft-ening rule (48),?nally arriving at:

_r ?D à1"

àX 3i ?1H _k t ei T D à1:M ei Tn M ei Tn :D à1M ei Tn :D à1

:M ei Tn

àZ a i t n àá#:_e :e54T

3.2.6.Assumed uniaxial stress–strain curve of concrete

under tension

In order to relate our theory to experiments,the softening function Z a i

t n àáwill be determined directly from a given uniaxial curve analogously to the proce-dure in Section 3.1.5.Due to Fig.4we hence distinguish:Region 1:Elastic region.Below the tension strength,the initial damage stress f ct ,linear-elastic behavior is

assumed.

Fig.3.Illustration of the assumed kinematic softening rule.

W.B.Kr €a tzig,R.P €o lling /Computers and Structures 82(2004)1201–12151207

Region 2:Softening.In the literature softening curves for concrete in the post-peak region generally are linear or exponential [32].We will base our derivation on the exponential function r ee T?f ct áe ee t àe T=c t

e55T

in which c t controls the area under the stress–strain curve and thus the volume-speci?c fracture energy g ?t .In analogy to the compression regime,the post-peak curve here depends on the material parameter G f ,the fracture energy,and on the equivalent length l eq in the quadra-ture point,relating:c t ?

G f l eq f ct à12f ct

E c

:e56T

Again we emphasize,that for large element sizes a ‘snap-back’-phenomenon may occur.The critical length scale

is for tension l eq ?2G f E c =f 2

ct ,however Ba z ant and Oh [24]recommend,not to violate the condition

l eq 6

G f E c

f 2ct

:e57T

For too small element sizes,Ba z ant and Oh [24]rec-ommend to reduce the failure stress f ct ‘arti?cially’,such

that the condition (57)is ful?lled.The determination of the softening function Z a i

t n àáfrom a given re -curve is detailed in Appendix B.

3.3.Summary of the concrete model

Both previously presented models for concrete in compression and in tension can be united in a multi-surface material theory.The elastic region then is limited by 4damage surfaces U c ?0,namely (10)and U t ei T?0(41),(46)and (47).The loading conditions,the tangen-tial sti?ness relation and the evolution equations of all internal variables are summarized in Table 2.

The complete material model contains nine material parameters,all to be determined by standard experi-ments.If just Young’s modulus E c and the compression strength f c is known for a particular concrete,all remaining parameters can be approximated from for-mulae in [2].These 9parameters are:

?Compression strength f c :The mean value of the uni-axial strength under compression,determined by a cylinder compression test [2].

?Initial Young’s modulus E c and Poisson’s ratio m c :These elastic constants are due to standard

tests.

Fig.4.Assumed uniaxial stress–strain curve of concrete under tension.

Table 2

Loading conditions,tangential sti?ness relation and evolution equations of internal variables of ?nal concrete model Loading conditions:_k

c P 0,U c 60,_k c U c ?0_k

t ei TP 0,U t ei T60,_k t ei TU t ei T?0Compliance tensor:D ?D 0tD da ;c tD da ;ta

with D da ;ta ?P t:D da ;t :P t

Tangential sti?ness relation:

_r ?D à1àD à1:o U c o r o U c

o r :D à1

o U

c

:D à1

:o U c td a c ?c

2b eq ?c TáH e_k c TàáááàX 3i ?1D à1:M ei Tn M ei Tn :D à1M ei Tn :D à1:M ei Tn àZ a i t n àáH _k t ei T "

#:_e Evolution equations of

internal variables:

_D da ;c s1?b eq ?

c T1?3

p àl l

1

à1???2

p

_k c _D da ;c s2?b eq ?c T1?3

p àl 1???2p _k c _e

pl ?e1àb eq ?c TTo U c o r

_k c q ?c ?

2b eq ?c Tw er T_k c

with _k c ?o U c

o r

:D à1o U

c

o r

:

D à1:o U c

o r td

a c d q c 2

b eq ?

c Tw er T

:_e

_D da ;t ?P 3

i ?11i t n

M ei Tn M ei Tn _k t ei T

_a t ?P 3i ?1Z a i t n àáM ei Tn _k t ei T

with _k t ei T?M ei T

n :D à1

M ei T

n :D à1:M ei T

n àZ a i

t n eT

:_e

1208W.B.Kr €a tzig,R.P €o lling /Computers and Structures 82(2004)1201–1215

?Crushing strain e c:Strain of concrete,if compression strength is reached.For medium strength concrete this strain can be approximated as2.2&[2].?Tension strength f ct.

?Fracture energy G f:In reinforced specimen,G f can be falsi?ed by bond e?ects(see Section3.5).Thus we employ the energy G fc;related to a single crack,the determination of which is not trivial.However, approximations generally su?cient for structural analysis can be taken from[2].

?Localized crushing energy G cl:Up to now,only Vonk

[27]o?ers experimental values for the complete

(localized and di?use)crushing energy,values be-tween10and25kN/m for medium strength concrete with aggregate sizes64–8mm.Roughly speaking, this value is200–500times the fracture energy.Since more accurate values are rare in literature,the local-ized crushing energy has been approximated by ?tting the analytical softening branch with experi-mental data,see Section4.1.

?The damage parameter b,illustrated in Fig.1,can be determined by curve-?tting to cyclic uniaxial com-pression tests,delivering values around0.5[19].?l:This parameter controls the in?uence of the hydro-static stress on the yield/damage potential.Numerical tests have demonstrated[19],that a value of0.05 adopts the presented material model well to experi-mentally recorded biaxial failure curves[29].

3.4.Steel

Reinforcement steel typically is used as bars in rein-forced concrete structures,thus only uniaxial steel behavior has to be modelled here.For failure analyses simulation at least elasto-plastic behavior is required. For more sophisticated models,mapping hardening and the Bauschinger-e?ect under cycling loading to any de-sired degree of accuracy,the reader is referred to [19,33,34].

3.5.Bond

In design and dimensioning of reinforced concrete structures,generally rigid bond is assumed.For ad-vanced treatments special attention is spend to softening and sti?ening e?ects for crack development and growth. Out of several modelling alternatives we adopt the concept by Feenstra[25]as follows:The complete bond e?ect is virtually separated into a‘tension-softening’-part,considered on concrete side,and a‘tension-sti?-ening’-part,usually allocated to steel.For simplicity,the latter will be neglected in our examples.

The basic postulate of the following concept con-cerning‘tension-softening’in reinforced concrete is,that the complete fracture energy G f inside a?nite element with the characteristic length l eq shall be equal to the sum of the fracture energies of all primary and second-ary cracks inside the equivalent length.Thus before starting any computer simulation we estimate the ex-pected mean crack distance l s,depending on the per-centage of reinforcement q s,the reinforcement diameter d s and the concrete cover c,all in the framework of[2]. Consequently we set for the fracture energy

G f?max G fc;

l eq

s

G fc

&'

;e58T

with G fc;from[2],as explained before.For the mean crack distance l s we?nd in[2]:

l s?

2

3

l s;max

with l s;max?

d s

s;eff

and q s;eff?

A s

c;eff

:e59T

Above A s denotes the area of reinforcement and A c;eff the e?ective tension area of concrete.For surface-like structures,reinforced orthogonally in x-and y-direction, [2]o?ers for the maximum crack-distance

l s;max?

j cos u c j

l s x;max

t

j sin u c j

l s y;max

à1

e60T

in which u c denotes the angle between the expected crack-line and the x-direction.l s x;max as well as l s y;max denote the expected maximum crack-distances in x-and y-direction in accordance with Eq.(59).

4.Examples

https://www.360docs.net/doc/2910208676.html,puter simulation of uniaxial stress–strain curve of concrete

As demonstrated in this example,the parameter G cl can be determined from uniaxial compression tests performed in de?ection-controlled manner up to the softening region.For such re-calculation,the compres-sion tests of van Mier et al.[35]have been selected.All required material parameters are given in Fig.5.

First we varied G cl and found the best agreement of the stress–strain curve for specimen with height of100 mm using a value of G cl?20kN/m.In fact,this value is larger than in[27],what can be explained with larger aggregate sizes(16mm)in the experiments.For all varied l eq,the gained results are also plotted in Fig.5in comparison with the tests.In all simulations,the height of the specimen has been chosen as characteristic length l eq.

The excellent agreement of the calculated re-curves with the experiments demonstrates the correctness of the ‘crushing energy’concept and the high quality of the derived analytical function(38)for the softening region.

W.B.Kr€a tzig,R.P€o lling/Computers and Structures82(2004)1201–12151209

4.2.Model capability for cyclic processes

One of the key features of the new concrete model is

its capability for cyclic loading processes.Fig.6dem-onstrates this in comparison to the cyclic compression tests of [36].l eq therein has been selected equal to the specimen height of 15.0cm,and the crushing energy G cl

is chosen such that the monotonic post-strength path is modeled su?ciently exact.

Fig.7depicts the simulation of a cyclic tension-compression test in [37].The equivalent length l eq ?35mm has been chosen identically to observations in the experiment;all other material parameters are due to the publication.

Both Figures demonstrate strong and weak points of the derived model.Generally,the correspondence with the tests is excellent.Small deviations are due to the struggle for greatest possible simplicity:the de-and re-loading paths lack any hysteresis,and plastic defor-mations in tension are neglected (Fig.7).All these deviations can be healed:for the price of additional material parameters.4.3.Reinforced concrete beam

The following response up to failure of a reinforced concrete test beam has been simulated by use of the FE-code FEMAS 2000[38].These nonlinear numerical simulations have been performed in a deformation controlled technique using Newton–Raphson-iterations.The test beam with high reinforcement ratio

was

https://www.360docs.net/doc/2910208676.html,parison of experimental compression tests [21]with presented material

model.

Fig.6.Simulation of cyclic compression test

[36].

Fig.7.Simulation of cyclic compression–tension test [37].

1210W.B.Kr €a tzig,R.P €o lling /Computers and Structures 82(2004)1201–1215

experimentally investigated by Bresler and Scordelis [39].Geometry,reinforcement,loading and material parameters are depicted in Fig.8.

In the FE-analysis layered beam elements were used. For symmetry reasons,only one half of the structure with14elements was discretized.20concrete layers serve as internal re?nement,and the reinforcement bars are modelled as two extra layers.

The calculated load–displacement curves for di?erent crushing energies G cl are shown in Fig.9,compared with the experiments and the analysis of Zahlten[40].The accordance of calculated and experimental response curves seems excellent,especially,since no?tting parameters are used.All material properties stem from [39]or from approximative formulae in[2].

Note,that the failure load depends on the localized crushing energy G cl.Obviously,the ratio of both failure loads calculated with G cl?1000N/m and G cl?10N/m, is1.14.This fact demonstrates,that the limit load––for highly reinforced specimen and failure triggered by crushing of concrete––is dominated by a size e?ect.In this example the failure load seems to be predicted best with the choice G cl?10kN/m,however,this value overestimates the experimental one of353.6kN by approximately10%.

4.4.Reinforced concrete plate

To demonstrate the applicability of the presented model to structural problems,a plate-experiment of McNeice[41].is re-calculated.The quadratic plate was orthogonally reinforced and corner supported.Geo-metric properties,reinforcement and material parame-ters are shown in Fig.10.

The load–de?ection curve,calculated with3·3shell-elements of10layers each under use of symmetry conditions,is depicted in Fig.11,together with the experimental results of McNeice.The correspondence between both curves is convincing.However in the re-gion just beyond crack-initiating the response of

the Fig.8.Geometry,structure and material parameters of beam according to

[39].

Fig.9.Discretisation and load–deformation curve of beam according to[39].

W.B.Kr€a tzig,R.P€o lling/Computers and Structures82(2004)1201–12151211

numerical system seems to be a bit sti?.Better results are

gained,if the tension strength is reduced from 2.96to 2.30MPa.5.Conclusions

In this present work we have derived a 3-D material model for concrete,based on a uni?ed thermodynami-cally based elasto-plastic damage theory.This model for concrete is able to represent exactly uniaxial stress–strain curves from codes of practise [2].An important aspect is its low number of 9material parameters.All of them can be determined from standard experiments,or auxiliary estimated from the concrete compression strength f c [2].Thus this model is considered as optimal for use in advanced structural analysis problems of concrete structures.Acknowledgements

The present work originated from projects of the Special Research Center SFB 398‘Life-time oriented

structural design’at the Ruhr-University Bochum.Financial support of the DFG,the National German Science Foundation,is gratefully acknowledged.

Appendix A.Final determination of function a c (q ?c )The principle of determination of a c eq ?c T,explained in Section 3.1.4,will now be applied for the stress–strain law of Section 3.1.5.First of all and depending on q ?c ,we have to decide,if the material is in the hardening or

softening region.Keeping in mind the identity q ?c ?D da ;c

and Fig.2,we ?nd the limit b e c f c à1E c

P q ?c :region 2;

c :region 3:&eA :1T

Now we determine the function a c eq ?c Tfor the regions 2

and 3separately:

Region 2:The function (36)has been substituted into (28),and this equation has been subsequently solved for e ,denoted then as e 2

:

Fig.10.Geometry,structure and material parameters of RC-plate according to

[41].

Fig.11.Discretisation and load–deformation curve of plate according to [41].

1212W.B.Kr €a tzig,R.P €o lling /Computers and Structures 82(2004)1201–1215

e 2eq ?c T?E ci á1

c

tq ?c

à1h i f c e 2c E ci ;e 2c à2f c e c t1E c

tq ?

c

1àb f 2c :eA :2T

Thereafter we write with help of (29)for the hardening

region

a c eq ?c T?à r 2e e 2eq ?c TT:

eA :3T

To determine the derivative d a c =d q ?c ,we furthermore need d r

2=d e as d r 2d e ?e e 2te c TE ci e c ee c à e 2Tt2 e 2f c e c eE ci e 2e c àe2

e 2te c T

f c Tf 2c :eA :4T

Region 3:By substituting (38)in Eq.(28),we gain the

cubic equation

2tc c f c e c c e tc c e 2tc c c e 3?à

1c tq ?c

!

;eA :5Twhich can be solved by standard methods,e.g.[42,p.132].For solution,we consider the reduced form of (A.5),namely ^e 3tp 1^e tp 2?0

with :^e ?e t23

e c

p 1?2e c c c f c à13e

2

c

p 2?16e 2c 27à4e c t2c c f c e 3c 3c c f c

t1E c tq ?c

1àb !

2e c c c ;eA :6T

the discriminant D p ?ep 1=3T3tep 2=2T2and the auxil-iary variable Y p ?sign ep 2T????????????j p 1j =3p .The solution of (A.6)reads then:

ea Tp 1?0:^

e ??????

p 23p eA :7Teb Tp 1>0:

^e ??????????????????????????

àp 22t??????D p p 3

r àp 1?????????????????????????????àp 2=2t??????D p

p 3q eA :8T

ec Tp 1<0_D p >0:

^e ?à2Y p cosh

v 3with v ?arcosh p 2

2Y 3p eA :9T

ed Tp 1<0_D p 60:^e ?à2Y p cos

v 3with v ?arccos p 2

2Y 3p

:eA :10T

In case (d)three real-valued solutions exist in fact,from

which we need only the smallest one given in (A.10).Finally we determine e 3?^e à23e c and substitute this expression into (29):a c eq ?c T?à r 3e e 3eq ?c TT:

eA :11T

Again we need for the derivative d a c =d q ?c the term

d r

3=d e ,reading here d r 3d e ?c c tc c

c e 3

2tc c f c e c c

tc c e 3tc c c á e 2

3h i 2:eA :12T

An overview over the computations of a c ,d a c =d q ?c

and b eq ?c Tfrom q ?

c is given in the ?ow chart depicte

d in Fig.12.

Appendix B.Determination of softening function Z (a i t n )Opposite to [16],which searches for an appropriate softening law by trial and re-checking then the obtained uniaxial curve,we deduce the softening law directly from a given uniaxial stress–strain curve.Our starting point is the compliance relation (4)in rate form

_e ?D 0?tD da ;t ?

:_r t_D da ;t ár :eB :1TThe parameter _k

t e1Tas function of _r is determined by use of the consistency condition as

_k t e1T?à1Z a 1t n

àáM e1Tn

:_r :eB :2

T

Fig.12.Flow chart to compute a c eq ?c T,d a c =d q ?c and b eq ?

c T.

W.B.Kr €a tzig,R.P €o lling /Computers and Structures 82(2004)1201–12151213

Substituting this expression into the damage rule (45)and the result further into relation (B.1),one obtains

_e ?D 0"tD da ;t à1Z a 1t n àáM e1Tn M e1T

n

#

:_r ;eB :3Ta constitutive statement,which reduces under uniaxial conditions to

_e ?1c "

tD da ;t à1Z a I t n àá#_r

:eB :4TIn this case the compliance 1=E c tD da ;t changes to e =r obtaining

_e ?e r "

à1Z a 1t n àá#_r !d e d r ?e r à1Z a 1t n àá:

eB :5TThe assumed stress–strain law (55)can be solved for e and further di?erentiated with respect to r .Introducing these expressions into (B.5)then leads directly to the desired function Z a 1t n àá

,if one keeps r ?f ct ta 1t n in mind:Z a 1t n àá?

f ct ta 1

t n e t tc t 1àln 1ta t n =f ct

àá?

?:

eB :6T

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