Effects of Concrete Shrinkage on Tension Stiffening in Cracked Reinforced Concrete Tensioned Members

413

1st International Congress of Serbian Society of Mechanics, 10-13th April, 2007, Kopaonik

EFFECTS OF CONCRETE SHRINKAGE ON TENSION STIFFENING IN CRACKED REINFORCED CONCRETE TENSIONED MEMBERS Borjan D. Popović1 1 Department for Civil Engineering, Faculty of Technical Sciences, The University of Novi Sad, Trg Dositeja Obradovića 6, 21000 Novi Sad, Serbia e-mail: [email protected] Abstract:

The paper presents two basic concepts of tension stiffening behaviour of cracked tensioned reinforced concrete members. The first one is a load sharing approach where the average load carried by the cracked concrete is used to determine the post-cracking stress-strain response of concrete in tension. The second one is tension stiffening strain approach, which evaluates changes in member stiffness to obtain a reduction in member deformation by including the stiffening effect of the tension carried by concrete between cracks.

The concrete shrinkage strain leads to an initial shortening of the tensioned member and this factor must be taken into account in order to evaluate tension stiffening effects properly. The influence of concrete shrinkage strains on cracking, tension stiffening and post-cracking stress-strain response of tensioned reinforced concrete member is considered. The paper presents developed proposal for including concrete shrinkage strains in an analysis of tension stiffening in cracked tensioned reinforced concrete members. The results obtained with analysis based on this proposal are compared with results of previously developed theoretical and experimental study [1]. The illustration of shrinkage effects on tensioned element response under load is presented. Key words: shrinkage, cracking, tension stiffening, reinforced concrete, tensioned members, stress-strain response 1. Introduction

Steel reinforcement in tensioned reinforced concrete elements is used to carry tensile forces across the cracks once the concrete has failed in tension, preventing a premature and brittle failure. After cracking concrete still continues to carry tensile stresses between cracks as a result of bond action, which effectively stiffens the element response and reduces elongation. This phenomenon is called tension stiffening and it plays an important role in assessing serviceability requirements after element cracking.

The estimation of tension stiffening, which represents the contribution from concrete to element stiffness after cracking, is influenced by shrinkage and will lead to an underestimation of this value if the initial element shortening caused by shrinkage is not included in the element response calculations. Additionally, neglecting shrinkage effects causes an apparent reduction in tension stiffening which becomes more predominant as the percentage of reinforcement increases.

B.D.Popović, Effects of Concrete Shrinkage on Tension Stiffening in Cracked Reinforced Concrete Tensioned …

414

2. Tension stiffening effects 2.1 Tension stiffening bond coefficient based on load sharing concept

Axial load force N applied on tensioned reinforced concrete element, after concrete cracking, is shared between concrete and reinforcing steel and for any given element mean strain εm the concrete and steel are carrying average forces cmN and smN , respectively. The average concrete tensile force is based on tensile stresses carried by the concrete between cracks and is transmitted across the cracks by a local increase in the reinforcing steel force. The element axial tensile force capacity is limited by yielding of the reinforcement at the crack locations.

ssyysmcm AfNNNN =≤+= , (1) The average tensile force carried by the steel, along the length of the element, is directly

related to mean steel strain smε that is equal to element mean strain mε :

smsssmssm AεEAεEN == , (2) The quantities sA , sE and syf represent area, modulus of elasticity and yield stress of the

reinforcing steel.

Fig.1. Distribution of axial forces and axial strains along element length for a stabilized crack pattern

The tensioned element cracking will first occur when concrete tensile stress reaches value of the tensile strength of concrete ctf . At this moment axial force applied on element is crN and concrete part of section, which area is cA , carried axial tensile force cr,cN which is equal to:

ccrcccr,cccctcr,c AεEAεEAfN === , (3) Once element cracked, tension force carried by concrete will vary from zero at a crack

location to a maximum value between the cracks which cannot exceed concrete tensile force at cracking cr,cN . The average force carried by concrete Ncm is then expressed as

cr,ccctccmcm NβAfβAσN === , (4) where cmσ is mean concrete stress and β represents a bond coefficient that accounts for the variation of concrete tensile stress between the cracks.

While the mean tension force in concrete cmN depends on both the concrete tensile strength


415

and area of concrete affected by the reinforcement, the bond coefficient β is normalized value that is independent of these two variables and can be thought of as a smeared material property. Changes in the reinforcing steel ratio or reinforcement distribution could affect coefficient β if there is corresponding change to the area of concrete affected by the reinforcement.

Generally the coefficient β can vary between 0 for the special case of no bond after concrete cracking and 1 at moment just before cracking. Since the average tension concrete force continues to decrease as more cracks form, there should be a corresponding reduction in value of β until the crack pattern becomes fully developed.

Although tensile stresses can still exist in the concrete between cracks after yielding of reinforcement occurs, the deformed steel bars having a flat yield plateau are unable to transmit a tensile force across the cracks that is greater then yield force of the steel. Hence, tension stiffening is effectively reduced to zero and governed by the stiffness at the crack locations.

2.2 Tension stiffening strain concept

In order to obtain the mean element strain mε for a given axial force value N, the bare steel strain response sbε at a crack location is reduced by a tension stiffening strain value sε∆ , which can be expressed as:

ssbsmm εεεε ∆−== , (5)

Fig.2. Response of a reinforced concrete tension element before and after cracking

The mean steel strain smε is equal to mean member strain mε assuming that there are no thermal effects or prestressing. The mean steel strain smε , using equations (1), (2), (3) and (4), my be expressed as:

⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

−=

−===

sb

max,ssb

cr,c

ssss

cr,c

ss

cm

ss

smsmm ε

εβε

NNβ

AEN

AENβN

AENN

AENεε

∆11 , (6)

where sbε denotes bare steel strain at a crack and maxs ,ε∆ represents the jump in steel strain at crack when concrete first cracks, which is equal:

ss

cct

ss

cr,cmax,s AE

AfAE

Nε ==∆ , (7)


416

Comparing equation (5) with equation (6) indicates that tension stiffening strain value is: max,ss εβε ∆∆ = , (8)

where coefficient β is the same bond coefficient defined earlier and in this case β accounts for the variation of reinforcement strains along the length of the tensioned element.

The concept of tension stiffening strain has been adopted by the CEB-FIP MC 78 [3] and same definition of β coefficient is implemented in earlier version of EUROCODE 2 [5]. The new definition of β coefficient is proposed in CEB-FIP Model Code 90 [4] and this proposal is applied in the last version of EUROCODE 2 [6]. The explanation and comparison of these proposals and proposals given by other researchers is beyond scope of the paper.

3. Theoretical bases of proposed approach for accounting shrinkage effects

Concrete can experience significant amounts of shrinkage before loading and even moist cured concrete will shrink to some extent. Shrinkage of the concrete leads to an initial element shortening and this factor must be taken into account in order to evaluate cracking force, tension stiffening effects and crack widths properly.

3.1 Uncracked element response

The uncracked element total strain after element shrinkage in period of time ( )t,t0 , which is equal element strain change continuously developed during considered period of time ( )t,t0 , applying the algebraic constitutive relations for concrete is obtained as:

( ) ( ) )t,t(εµn

)t,t(εAEA)t,t(E

A)t,t(Et,tεtε shs

*s

shssc

*c

c*c

000

00

11

+=

+=∆= , (9)

where )t,t(E o*c is age-adjusted effective modulus of concrete, )t,t(ε osh is unrestrained (free)

concrete shrinkage strain in period of time ( )t,t0 , *sn is corresponding modulus ratio

)t,t(E/E o*cs and sµ is steel reinforcement ratio cs A/A . Expression (9) is derived substituting

concrete and steel constitutive stress-strain relations in element section equilibrium equation, assuming that strains are uniform over the element cross-section.

The total element steel resultant compressive force )t(Ns at the end of observed period of time ( )t,t0 , which is opposite to the total element concrete resultant tensile force )t(Nc , is finally obtained as:

( ) ( ) ( ) ( )s

*s

shssccssµn

)t,t(εAEt,tNtNt,tNtN+

=∆−=−=∆=1

1000 , (10)

The total resultant steel and concrete forces at moment t, )t(Ns and )t(Nc , are equal to corresponding steel and concrete resultant force changes, )t,t(Ns 0∆ and )t,t(Nc 0∆ , which are continuously developed during period of time ( )t,t0 .

3.2 “Decompression” state of concrete part of element cross-section

The fictitious element instantaneous elastic strain change at moment t, which is necessary to bring previously tensioned concrete part of the cross-section in state of zero concrete stress, is equal to:

( )s

*s

sssh

s*s

s*s

shc

*c

cc

cd

µnµn)t,t(ε

µnµn)t,t(ε

)t(E)t,t(E

A)t(E)t(Ntε

+=

+=−=∆

1100

0 , (11)


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where ( )tEc is concrete modulus of elasticity at moment t, and sn is corresponding steel to concrete modulus ratio ( )tE/E cs . This fictitious section state is usually called fully decompression state of section concrete part, because previously developed stresses in concrete part of element cross section, that need to be unstressed, are usually compressive stresses.

The fictitious concrete and steel resultant force changes, )t(N d,c∆ and )t(N d,s∆ , which corresponds to considered fictitious elastic strain change )t(εd∆ , are equal to:

( ) )t(NtN cd,c −=∆ , (12)

( ) )t(εAEtN dssd,s ∆=∆ , (13) The fictitious concrete resultant force change )t(N d,c∆ and the fictitious steel resultant force

change )t(N d,s∆ are in treated case compressive forces. The applied on element fictitious force change )t(Nd∆ , which correspond to fictitious elastic

strain change )t(εd∆ is:

( ) ( ) ( )s

*s

ssshssd,sd,cd

µnµn)t,t(εAEtNtNtN

+

+=∆+∆=∆

11

0 , (14)

The total strain in element cross-section at moment t, after fictive elastic unstressing of concrete part of section, is equal to sum of previously developed strain ε(t) during time period of element shrinkage and fictitious elastic strain change )t(εd∆ :

( ) ( ) ( )s

*s

ssshdd

µnµn)t,t(εtεtεtε

+

+=∆+=

11

0 , (15)

This total strain is ussally called fully decompression strain. At this fictitious section strain state the corresponding resultant force of total stresses in

concrete part of section is equal to zero.

( ) ( ) ( ) 0=∆+= tNtNtN d,ccd,c , (16)

At fully decompression section state the corresponding resultant force of total stresses in steel part of section is equal to sum of previously developed steel resultant force )t(Ns∆ during time period of element shrinkage and fictitious elastic steel resultant force change )t(N d,s∆ :

( ) ( ) ( )s

*s

ssshssd,ssd,s


+

+=∆+=

11

0 , (17)

The total force applied on the section at fully decompression section state is equal to total force carried by steel at this state:

( ) ( ) ( )s

*s

ssshssd,sd,cd


+

+=+=

11

0 , (18)

In considered case the total force )t(Nd applied on the section at fully decompression state is equal to fictitious element force change )t(Nd∆ because element is not externally loaded during precedent period of time.

3.3 Cracking state of element cross section

The fictitious element instantaneous elastic strain change at moment t, which is necessary to bring previously unstressed concrete part of the cross-section in state immediately before concrete cracking, i.e. in state when concrete stress is equal his tensile strength )t(fct , is equal:


418

( ) ( ) ( )t,ε)t(E)t(ft,εtε cr

ds

c

ctcr

dc

dcr ∆==∆=∆ , (19)

The fictitious force change )t(N dcr∆ applied to element, which correspond to elastic element

strain change )t(εdcr∆ is equal:

( ) [ ] ( ) )µn(A)t(ftεAEA)t(EtN sscctdcrsscc

dcr +=∆+=∆ 1 , (20)

The fictitious concrete and steel resultant force changes, )t(N dcr,c∆ and )t(N d

cr,s∆ , which are

correspond to considered elastic strain change )t(εdcr∆ , are equal to:

( ) ( ) cctdcrcc

dcr,c A)t(ftεA)t(EtN =∆=∆ , (21)

( ) ( ) cctssdcrss

dcr,s A)t(fµntεAEtN =∆=∆ , (22)

The total strain in element cross-section at moment t, immediately before concrete cracking is equal to sum of previously developed decompression strain )t(εd and fictitious elastic strain

change )t(εdcr∆ :

( ) ( ) ( ) ( )( ) ⎥

⎥⎦

⎤

⎢⎢⎣

⎡

∆+

++∆=+∆=

tε)t,t(ε

µnµntεtεtεtε d

cr

sh

s*s

ssdcrd

dcrcr

0

111 , (23)

The total force, which is equal to total force change at moment t, applied on the section immediately before cracking is equal:

( ) ( ) ( ) ( ) ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡

∆+++=∆+∆=∆=

)t(ε)t,t(ε

µnµnµnA)t(ftNtNtNtN d

cr

sh

s*s

sssscctd

dcrcrcr

0

111 , (24)

The concrete and steel resultant force changes, which are correspond to elastic strain change )t(εcr∆ that causes concrete cracking, are equal to:

( ) ( ) ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡

∆++=∆+∆=∆

)t(ε)t,t(ε

µnµnA)t(ftNtNtN d

cr

sh

s*s

sscctd,c

dcr,ccr,c

0

11 , (25)

( ) ( ) ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡

∆++=∆+∆=∆

)t(ε)t,t(ε

µnµnµnA)t(ftNtNtN d

cr

sh

s*s

sssscctd,s

dcr,scr,s

0

11 , (26)

The total concrete and steel resultant forces, which are correspond to concrete cracking at moment t, are equal to:

( ) ( ) ( ) cctccr,ccr,c A)t(ftNtNtN =+∆= , (27)

( ) ( ) ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡

∆+

++=+∆=

)t(ε)t,t(ε

µnµnµnA)t(ftNtNtN d

cr

sh

s*s

sssscctscr,scr,s

0

111 , (28)

The effects of an initial element shortening are shown on Figure 3, indicating that element cracks at a lower external load, since the concrete is under an initial tensile stress:


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( ) ( )s

*s

s*s

sh*c

c

cc

µnµn)t,t(ε)t,t(E

At,tNtσ

+−=

∆=

100

0 , (29)

The reduced external cracking force at moment t, which is defined with equation (24), can be expressed as part of element cracking force for the case when concrete shrinkage is ignored:

( ) ( ) ( ) ⎥⎦

⎤⎢⎣

⎡−∆=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∆++∆=∆

)t(f)t(σtN

)t(ε)t,t(ε

µnµntNtN

ct

cdcrd

cr

sh

s*s

ssdcrcr 1

11 0 , (30)

The elastic strain change )t(εcr∆ that causes concrete cracking at moment t is also reduced related to element strain change for the case when concrete shrinkage is ignored and is equal:

( ) ( )( )

( )( )

( ) )t(εtεtε)t(εtε

tε)t,t(ε

µnµntεtε d

dcrd

cr

ddcrd

cr

sh

s*s

ssdcrcr ∆+∆=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∆

∆+∆=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∆++∆=∆ 1

11 0 , (31)

3.4 Post cracking state of tensioned element

The point that has coordinates ( ) ( )( )tN,tε dd is origin of the coordinate system with abscissa

)t(εd∆ and ordinate )t(N d∆ in which member response under subsequent loading can be idealized as instantaneous member response at moment t.

)t(ε)t(ε)t(ε)t(ε)t(ε ddd ∆−∆=−=∆ , (32)

)t(N)t(N)t(N)t(N)t(N ddd ∆−∆=−=∆ , (33)

Fig.3. Effects of shrinkage on member response before cracking

The member initial strain ( )tε i,m at moment t, which is equal to initial concrete strain ( )tε i,c , is equal previously developed uncracked member strain ε(t) caused by shrinkage in previous time period ( )t,t0 . The point that has coordinates ( )( )0,tε i,c is origin of coordinate system with abscissa

)t(εexp and ordinate )t(Nexp , in which is usually represented element test response under loading. The adjusted member origin accounts the developed concrete shrinkage strain, but in this case bare steel response is necessary to shift right to correctly account shrinkage effects on tension stiffening, which causes corresponding axial force offset )t(N off,m (see Fig.4).


420

Fig.4. Effects of shrinkage on member response after cracking

Tension stiffening is measured by subtracting the bare steel response from the measured member response, and it is usually assumed that both of these responses begin at same origin. This leads to an apparent bond coefficient βexp which is incorrect when shrinkage effects are neglected, since the bare steel response needs to be offset by the initial element shortening to obtain the true bond β coefficient.

dcr

i,mss

dcr

i,mss

i,mssctc

i,mssctc

off,mexp,cr,c

off,mexp,cm

exp,cr,c

exp,cmexp

ε

εµn

ε

εµnβ

εAEfAεAEfAβ

NNNN

NN

β

∆+

∆+

=++

=−

−==

1, (34)

The correct tension stiffening bond coefficient β is obtained rearranging equation (34) as:

( ) ( ) dcr

dexpexpd

cr

i,mssexpexp

εεββ

ε

εµnβββ

∆

∆−−=

∆−−= 11 , (35)

The correct value of the average tensile force carried by the concrete is:

cexp,cms

*s

shssexp,cmi,mssexp,cmcm NN

µnεAENεAENN +=+

−=−=1

, (36)

4. Comparison with previous research proposal

The results obtained with described proposal are compared with results of previously developed theoretical and experimental study [1]. In that study shrinkage effects on tensioned member response are taken into account by an elastic analysis, which is significant simplification.

The uncracked element total strain ε(t), total element steel resultant compressive force )t(Ns and total element concrete resultant tensile force )t(Nc , after element shrinkage in period

of time ( )t,t0 , are obtained in that study as:

( ) ( ) )t,t(εµn

)t,t(εAEAE

AEt,tεtε shss

shsscc

cc000 1

1+

=+

=∆= , (37)

( ) ( )ss

shsscs µn)t,t(εAEtNtN+

=−=1

10 , (38)

The fictitious element decompression strain change )t(εd∆ and corresponding fictitious force change )t(Nd∆ at moment t are obtained in study [1] as:


421

( )ss

sssh

cc

cd µn

µn)t,t(εA)t(E)t(Ntε

+=−=∆

10 , (39)

( ) )t,t(εAEtN shssd 0=∆ , (40)

The total decompression strain )t(εd in element cross-section and corresponding element decompression force )t(Nd at moment t, are obtained in study [1] as:

( ) ( ) ( ) )t,t(εtεtεtε shdd 0=∆+= , (41)

( ) )t,t(εAEtN shssd 0= , (42)

The total strain in element cross-section immediately before concrete cracking )t(εcr and corresponding cracking force )t(Ncr , at moment t, are obtained in study [1] as:

( ) ( ) ( ) ( )( ) ⎥

⎥⎦

⎤

⎢⎢⎣

⎡

∆+∆=+∆=

tε)t,t(εtεtεtεtε d

cr

shdcrd

dcrcr

01 , (43)

( ) ( ) ( ) ( ) ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡

∆+++=∆+∆=∆=

)t(ε)t,t(ε

µnµnµnA)t(ftNtNtNtN d

cr

sh

ss

sssscctd

dcrcrcr

01

11 , (44)

The reduced external cracking force at moment t, expressed as part of element cracking force for the case when concrete shrinkage is ignored, is obtained in study [1] as:

( ) ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡

∆++∆=∆

)t(ε)t,t(ε

µnµntNtN d

cr

sh

ss

ssdcrcr

01

1 , (45)

The correct tension stiffening bond coefficient β is obtained in study [1] as:

( ) ( ) dcr

sh

ss

ssexpexpd

cr

i,mssexpexp

ε)t,t(ε

µnµnββ

ε

εµnβββ

∆+−−=

∆−−= 0

111 , (46)

The member initial strain ( )tε i,m at moment t, which represents position of the adjusted origin of member response under load, and corresponding axial force offset )t(N off,m determined on basis of presented proposal have greater values. Thus, obtained reduction of element cracking force at moment t is greater, as well as correct tension stiffening bond coefficient β value.

5. Illustration of shrinkage effects on tensioned element response under load

The values of concrete creep coefficient and shrinkage strain for discrete time moments t0 and time periods t-t0 are taken from tables in national codes [2] for value of relative humidity RH=40% and mean element depth dm=20cm. The age-adjusted effective modulus of concrete is calculated with constant aging coefficient equal to value χ=0.8. The change during time of concrete modulus of elasticity and tensile strength are calculated using relations presented in [2], whose are based on CEB-FIP MC 90. The steel modulus of elasticity has value Es=200GPa.

The calculated values of ratio of external axial cracking force at moment t, after concrete shrinkage period t-t0, to element axial cracking force at moment t when concrete shrinkage is ignored are presented on Fig. 5 for usually reinforcement coefficient µs values. The reduction of external cracking force is significant for longer time periods t-t0 and for greater reinforcement coefficient µs values.


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Fig.5. The influence of shrinkage in period of time (t0, t) on tensioned element cracking force at moment t

6. Conclusions

The level of load that causes cracking of tensioned reinforced concrete element and post cracking tension stiffening element response are affected significantly by shrinkage in previous period of time. The element behavior depends on both the amount of time dependent developed concrete shrinkage and creep strain and reinforcement percentage. Measured values of bond coefficient, obtained by test, must be corrected for shrinkage effects using presented expressions.

The presented proposal, which is based on application of well known algebraic constitutive relations for concrete is insignificantly complex related to previously proposal based on simplified linear elastic analysis. It gives more accurate results and is almost equally appropriate for practical application. After described corrections, whose are necessary to take into account shrinkage effects, tension stiffening effects in post cracking tensioned element response my be analyzed applying some of relevant code proposals [4] and [6] or some of actual research proposals. References [5] Bischoff, P., Effects of Shrinkage on Tension Stiffening and Cracking in Reinforced Concrete,

Canadian Journal of Civil Engineering, Vol. 28, No.3, 2001., pp. 363-374. [6] Concrete and Reinforced Concrete According National Code BAB 87, Vol.1 Guide & Vol.2

Appendices (in Serbian), Građevinska knjiga, Beograd, 1991., pp. 780. & pp.702. [7] Comite Euro-International du Beton and Federation International de la Precontraint: CEB-FIP Model

Code for Concrete Structures, CEB, Paris, 1978., pp. 174. [8] Comite Euro-International du Beton: CEB-FIP MODEL CODE 1990, DESIGN CODE, Thomas

Telford Services Ltd., London, 1993., pp. 360 [9] Eurocode 2: Design of Concrete Structures, Part 1: General Rules and Rules for Buildings, ENV

1992-1-1, December 1991., pp. 202. [10] Eurocode 2: Design of Concrete Structures, Part 1: General Rules and Rules for Buildings, Final draft,

October 2001., pp. 230.

0.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

1.000

0 1 2 3 4

µs (%)

∆Ncr(t)

/∆N

d cr(t)

t-t0=7dt-t0=14d

t-t0=28d

t-t0=90d

t-t0=365d

t0=7d RH=40% dm=20cm

Effects of Concrete Shrinkage on Tension Stiffening in Cracked Reinforced Concrete Tensioned Members

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