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VOLUME 12, NO. 2, EDISI XXIX JULI 2004

MEDIA KOMUNIKASI TEKNIK SIPIL 45

IDEALIZED STRESS-STRAIN RELATIONSHIP IN TENSION OF

REINFORCE CONCRETE MEMBER FOR FINITE ELEMENT MODEL

BASED ON HANSWILLES THEORY

Hardi Wibowo 1

ABSTRACT

Untuk penganalisaan kontrol retak (crack control) pada model struktur beton bertulang(reinforced concrete/RC) menggunakan software finite element seperti DIANA diperlukan

pemodelan hubungan tegangan-regangan (stress-strain relationship). Data model hubungantegangan-regangan ini dalam rangka mengakomodasi efek rekatan (bond-slip effect) antaratulangan (reinforcement) dengan beton (concrete) yang melingkupinya. Hubungan tegangan-regangan yang diperlukan adalah nilai rata-rata tegangan-regangan pada tulangan dan beton.Data hubungan tegangan-regangan rata-rata ini bisa diturunkan dengan menggunakan teoriHanswille. Pada tulisan ini akan diuraikan mengenai teori Hanswille untuk menentukanhubungan tegangan-regangan tersebut dan diberikan satu contoh perhitungan dan curvahubungan tegangan-regangan rata-rata dari sebuah batang beton bertulang.

1Staf Pengajar Jurusan Teknik Sipil Fakultas

Teknik Universitas Diponegoro

GENERAL

To model reinforced concrete member forfinite element (FE) analysis, reinforcementsteel bars were modeled as embeddedelements. In this element, the bar elementsdo not have independent degrees offreedom. Instead, the stiffness of the barelements were superposed on that ofmother concrete elements. In FE model,perfect bonding between concrete andembedded reinforcement is assumed.

Bond-slip effect between reinforcement andsurrounding concrete can be taken intoaccount by using an average stress-strain

relationship of reinforced concrete includingtension stiffening effect. The average stress-strain relation derived from the bond-slipdifferential equation proposed by Hanswillewill be explained here.

Fig.1 below shows the schematic figure for

this stress-strain relationship. The state Icorresponds to perfect bonding, while thestate II to perfect cracking. The average

stress of a RC member is expressed interms of average steel stress s and

average concrete stress c ,

n gs a e

D : P o s y i e l d i n g s e

s I i s

, r = s I I

s t a t e Is t a t e I I

11 + c

s

C DA v e r a g e s r a i n

veragesress

A : U n c r a c k e d s a eB : I n i l r a c k i n g s eC : S a b i l z e d c r a c k i

a

(a) Average stress-strain curves

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Idealized Stress-Strain Relationship in Tension of Reinforced Concrete MemberFor Finite Element Model Based on Hanswilles Theory

MEDIA KOMUNIKASI TEKNIK SIPIL46

y

E s

S t e e l s t r a i n s

teelstress s

(b) stress-strain curve for steel

C o n c r e t e s t r a i nc

f c t

E c

(c) stress-strain curve for concrete

Fig.1 Schematic figure for Stress-strain curves of RC member in tension.

scV

dVV

111 .............. (1)

where for a uniaxial stress state,

dV

L

sL

sdV

L

cL

c 0

1,

0

1 ............ (2)

Where :

LtAV is the total volume of the RC

member with the cross-sectional area

sA

cA

tA and length L.

cA

sA : reinforcement ratio

As=Cross-sectional area of reinforcementAc=Cross-sectional area of concrete

Under the assumption of perfectbonding used in the present smeared crackFE analysis, the average concrete strainequals the average steel strain. However,

the average concrete strain c includescontribution of crack opening in concrete.We can separate the average strain into anintact part and a cracking part, as

Lwcm

L

udxdx

du

L

L

dxdx

du

Lc

/

1

0

1

*

......(3)

where cm denotes the average strain over

the intact part *L ; uuuw is thecrack width, and the summation is takenover all cracks in L. In summary, sincenormal stresses are zero in the crackingpart, we have

Lwcmsmsc

smscmc

/

,,

..................(4)

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where the overbar denotes averagedquantities over the total region, while thesubscript m stands for averaged quantitiesover the intact part. In the present smearedFE analysis, we obtain the averagedquantities over the total region as output.

Since the elastic perfectly plastic model isassumed for stress-strain relation ofreinforcement steel as shown in Fig.1(b),the stress-strain relation for concrete asshown in Fig.1(c) was derived from theaverage stress-strain relation fromHanswilles. The derived stress-strain

relation of concrete is modeled as a multi-linear curve in FE analysis.

CONSITUTIVE MODEL AND BOND-SLIPDIFFERENTIAL EQUATION

Consider a differential length dx ofreinforcement embedded in concrete asshown in Fig.2, then the force due to bondbetween steel reinforcement and thesurrounding concrete must be same as thechange of axial force on steel or theconcrete cross-section.

v

dxc+dc

s

c

s+ ds

vc

dx

s

Fig.2 Stresses in differential element of RC member.

Considering equilibrium of forces in thelongitudinal direction

-dc(x)Ac = ds(x)As = v(x)Usdx ............ (5)

Where :v = bond stressc = Stress in concretes = Stress in concreteUs = Perimeter of cross-section of

reinforcing bar =dsds = Diameter of steel barx = Longitudinal coordinate of the

memberDividing Eq.5 by Asdx

)(4

)()(

xvsdsA

sUxvxd

xsd

...............(6)

Assuming that the cross-section remains

constant then the slip v or relativedisplacement between steel and concrete isequal to the difference of strain betweensteel and concrete. From Fig.2

v = s-c- ........................................(7)

or 0 csdx

dv..........................(8)

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where :v = relative displacement between steel

and concrete

c = displacement in concretes = displacement in steel = displacement due to shrinkage0 = Shrinkage strainc = Strain in concretes = Strain in steelIn Eq.8, s and c can be replaced by steelstress and concrete stress respectively and

n=modular ratio=cE

sE

Es = modulus of elasticity of steel

Ec = modulus of elasticity of concrete

0)()(1

xcnxssEdx

dv................ (9)

Hanswille used the following constitutivemodel defined in exponential form for bondstress and relative displacement or slip

v(x) = AfcwvN(x) ................................ (10)

where A and N are constants and areobtained empirically, and fcw=compressive

strength of cube of concrete. Using thevalue ofv(x), we get

)(14

2

2xNv

sE

n

sd

cwAf

dx

vd

.............. (11)

Now the solution of v=0 and 0dx

dv , will

give the length of slip region v(x) :

N

N

x

N

sd

cwAf

sE

n

N

N

Ndx

dv

1

1

1

1

)1(2

1

21

1

2

..........................................................................(12)The boundary condition described abovemeans that at a location where there is noslip between steel reinforcement and theconcrete is taken as the origin of coordinatesystem.

FIRST CRACK

In a RC member subjected to axial force,when the stress attains the tensile strengthof concrete, the first crack appears and thenthe stress of concrete and steel at thecracking region changes and a relativedisplacement (slip v) between steel andconcrete is produced. The stress conditionas shown in Fig.3 is produced. The length ofthe region where crack produces relativedisplacement is denoted here by LER and is

called introduction length or transmissionlength.

s , r

L E R

s, r s , r

s

s , r

s I

0

s ( x )

c x )

x

fc

Fig.3 Stress of concrete and steel reinforcement after first crackin a RC member subjected to axial tensile force

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To describe the stress c(x) and s(x) ofconcrete and steel in the region of lengthLER, these quantities are represented as afunction of slip. The coordinate systemchosen is shown in Fig.3. At the origin ofthis coordinate system, no slip is producedbetween steel and concrete, i.e. at x = 0,

0dx

dv

From Eq.9, for x=0, we have

00)0()0(1

cnssEdx

dv......... (13)

s(x) at x=LER is s,r and s(x) at x=0 is s,I.

When at x=LER, s(x) is s,r then c(0)=fct.Equating the total force at the crack and atthe section x=0s,rAs = Acc + Ass or s,r = c + s

At x = 0, s,r = c(0) + s(0)And then we find

0

1,

sE

nctfrs

.................... (14)

rs

sErsn

n

sErs

rs

,

0,1

1

0,,

.... (15)

dx

dv

n

sErs

1, ............................. (16)

INITIAL CRACKING STATE

In a reinforced concrete member subjectedto axial tensile force, when the stressattains the tensile strength of concrete, thefirst crack appears and the relative slipbetween steel reinforcement andsurrounding concrete is produced. In this

method for crack width evaluation, thesame constitutive relation between the bondstress v and the slip v was used as used byHanswilles theory, given in Eq.10. In this

equation, N is not a non-dimensionalparameter but dimensional one. If the unitof length is cm, then Hanswille reported that

A=0.58 and N=0.3 are standard values for adeformed bar.

After first cracking of concrete, furtherincrease of the axial force increases thenumber of cracks and accordinglydeformation due to cracking, and thusspacing between adjacent cracks reduces.Hence, depending upon the spacingbetween the cracks, two cracking states canbe defined, one is initial cracking state andthe second one is stabilized cracking state.

In the initial cracking state as shown inFig.4(a), the transmission length of twoadjacent cracks do not overlap each otherand there is a state I region between twoadjacent cracks. Since there is no relativeslip in the state-I region, there is nointeraction between two adjacent cracks,i.e. the opening of one crack does not affectthe width of around cracks.

Further increase of axial force causes thegeneration of more cracks till the spacingbetween cracks become so small so that thetransmission lengths of cracks overlap eachother. This is called stabilized cracking stateas shown in Fig.4(b). In this state, theopening of one crack affects the widths ofaround cracks. Crack width expression inboth cracking states is given by w=2v at thecrack position.

The crack width wR for the first crack isderived as given by Eq.18 for isIs ,with rs, and rs, are given in Eq.14

and

ctf

rs , respectively.

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s s

concrete

reinforcing steel

state ILER

a=LER (

s

cfct

sII

,c IIsmsI

LER

(a) Initial cracking stage

s s

a=LER (

s

c

s,c

sI Ism

cm

(b)Stabilized cracking stageFig.4: Stress distributions in cracked steel reinforced concrete member.

is=Average strain at the boundary betweeninitial cracking state and stabilized crackingstate. is is obtained from the average steelstrain s,m in the initial cracking state. Wehave

IsIIsN

ms ,,1

,

................. (17)

where is defined as

N

N

sErs

sEIIs

1

1

0,

0,

....................... (18)

and

I,sII,sII,s At the boundary of initial and stabilizedcracking states, =1thus s,II=s,r and s,II=s,r=s,II-s,I

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is a non-dimensional factor defining crackspacing in terms of transmission length.

Hanswille has considered the possibility of

stress drop due to further cracking in theinitial cracking state and thus he has alsoproposed expression for crack width otherthan first crack, thus giving crack width lessthan first crack. As long as initial crackingstate prevails, the crack width expressionfor first crack gives the maximum crackwidth. This procedure is intended toevaluate the maximum crack width. Thus, inour proposed method, we assume aconstant stress state in the initial crackingstate as shown in Fig.1.

STABILIZED CRACKING STAGE

In the stabilized cracking stage, we have:

N

N

sEsr

sEsII

1

1

0

0

........................ (19)

and =a/LER is the non dimensional crackspacing. Hanswille proposes the followingexpression as a mean value of m on thebasis of the statistical experimental data.

)**(1.1

21.1/max m ............ (20)

where

2

2

1

1

10

0

2

2

1

1

0

0

N

N

N

sEsr

sEsII

N

N

N

sEsr

sEsII

......... (21)

When evaluating the crack width, it isneeded to know sII from FE analysis.

sII is the stresses in steel in state II as

defined in Fig. 4(b). The relationshipbetween sII and the average stresses in

FE analysis can be derived from theequilibrium condition.

Nc=Axial force of RC slabAt uncracked location, the force will beshared by concrete and steel reinforcement.

ssAccAcNt

Ac

N

......................... (22)

At crack, the force is taken by onlyreinforcing steel, and the resulting stresseson the reinforcing bars is sII.

sA

cNsII ........................................ (23)

Putting the value of Nc from Eq.2.40 intoEq.2.41 we get the relation

scsA

cAsII .............................. (24)

Or scsII

1

...................... (25)

Thus the Eq.24 can be used for getting thevalue of sII from c and s . The c

and s are obtained from out put of FEM

analysis. The average strain derived fromthe bond-slip differential is given byHanswille, as follows:

Nm

m

N

sII

sII

sE

sIIsm

1

2

2

111

111

......................................................... (26)

From this average strain expression, we canspecify the average stress-strain of a RCmember, which is used in the FE analysis.

AN EXAMPLE

Finally, we have conclusion that stress-strain relationship to model RC member forFEM has three stages i.e.: un-cracked stage,initial cracking stage, and stabilized crackingstage (see fig.1a). In the un-cracked stage

(line 0-a), range of strain is 0 sI and

average stress is

1

1 nctf . In the

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initial cracking state (line a-b) range of

average strain is sI is where is isaverage strain at the boundary between

initial cracking state and stabilized crackingstate. In this point =1 and =2, then.

02

1

nN

sE

ctfis and average

stress is

1

,rs. And in the Stabilized

cracking state range of average strain is is = sm and equations below arehold:

Nm

m

N

sII

sII

sE

sIIsm

1

2

2

111111

In this condition 0C therefore

1

,IIsand IIsS , , where

IIsrs ,, Yield strength of

reinforcement.

Following is an example of calculation usingdata as below: Modulus of elasticity of steelE = 210000 N/mm2, Compressive strengthof concrete = cy = 40 N/mm2 , Tensilestrength of concrete fct = 2.69 N/mm

2,Reinforcement ratio = 0.0191, Modularratio no = 7, Bond slip constant A = 0.291(in mm), Bond slip constant N = 0.3,

Shrinkage strain of concrete o = 0. Thecalculation result can be presented as atable and figures as shown below:

Ave.strain

Ave.RCstress(Mpa)

Ave.reinf.stress(Mpa) Note

0,00000 0,00000 0,00000 Point 0

0,00009 2,99250 0,35291 Point A

0,00032 2,99250 1,27677 Point B

0,00035 2,99872 1,39261

0,00042 3,18614 1,65887

0,00048 3,37356 1,87031

0,00053 3,56099 2,07046

0,00058 3,74841 2,26533 Stabilized

0,00062 3,93583 2,45719 cracking

0,00067 4,12325 2,64715 state

0,00072 4,31067 2,83583

0,00077 4,49809 3,02361

0,00082 4,68551 3,210750,00086 4,87293 3,39741

0,00091 5,06035 3,58372

0,00096 5,24777 3,76975

0,00101 5,43519 3,95558

0,00101 5,45393 3,97416

0,00102 5,51015 4,02987

Table 1.: Calculation result

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0

1

2

3

4

5

6

0 200 400 600 800 1000 1200

Average Strain ( )

AverageStress(MPa)

RC Member

Reinforcement

a b

(a). Stress-strain relationship for RC member and reinforcement

0

0.5

1

1.5

2

2.5

3

0 200 400 600 800 1000 1200

Strain ( )

Stress(MPa)

(b). Stress-strain relationship for concrete

0

0.5

1

1.5

2

2.5

3

0 200 400 600 800 1000 1200

Strain ( )

S

tress(MPa)

(c). Multi-linear tension softening curve for DIANA input data

Fig. 5. Result of calculation of stress-strain relationship andMulti-linear tension softening curve

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REFERENCES

AASHTO, 1998, LRFD Bridge Specifications,

SI units, Second edition, AmericanAssociation of State Highway andTransportation Officials, 444 North CapitolStreet, N.W., Suite 249 Washington, D.C.20001

Clark, A. P., 1956, Cracking in reinforcedconcrete flexural members, ACI Journal,Proceedings, Vol.52, No.8.

Crisfield, M. A. and Wills, J.,1989, Analysisof R/C panels using different concretemodels, Journal of Engineering Mechanics,

ASCE, Vol. 115, No. 3

Hanswille, G., 1986, ZurRibreitenbeschrnkung bei

Verbundtrgern, Technisch- Weissenschaftliche Mitteilungen, Institut frKonstruktiven Ingenieurbau Ruhr-

Universitt Bochum, Mittelilung Nr. 86-1.

Holmberg, A. and Lindgren, S., 1970, Crackspacing and crack width due to normal forceor bending moment, Document D2, 1970,National Swedish building research.

JSCE, 1998, Committee of Concrete,Standard specifications for concretestructures[Design], (in Japanese).

azaqpur, A. G. and Nofal, M., 1990,Analytical modelling of non-linear behaviour

of composite bridges, Journal of StructuralEngineering, ASCE, Vol. 116, No. 6.

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