Advances in Structural Engineering · 2015. 11. 2. · bond-slip model based on the same...

The Tension Stiffening Mechanism in ReinforcedConcrete Prisms

by

Rahimah Muhamad, M.S. Mohamed Ali, Deric John Oehlers and Michael Griffith

Reprinted from

Advances in Structural EngineeringVolume 15 No. 12 2012

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1. INTRODUCTIONThe tension stiffening effect has been widely used inpredicting the tensile behaviour of reinforced concreteprisms (Beeby and Scott 2005; Chan et al. 1992; Goto1971; Gupta and Maestrini 1990; Hegemier et al. 1985;Jiang et al. 1984; Lee and Kim 2008; Marti et al. 1998;Mirza and Houde 1979; Rizkalla and Hwang 1984;Somayaji and Shah 1981; Yankelevsky et al. 2008). Itis not only important in controlling the deflection ofbeams (Bischoff 2005; Gilbert 2007) but also can beutilised for predicting multiple crack spacings andcrack widths (Bischoff et al. 2003; Kong et al. 2007;Wu et al. 2009). The tension stiffening effect allows thestress transfer from the reinforcement to thesurrounding concrete through the interface bond stressslip property (τ−δ ). Hence, the concrete stress isgradually increasing due to this process and also vice

Advances in Structural Engineering Vol. 15 No. 12 2012 2053

The Tension Stiffening Mechanism in Reinforced

Concrete Prisms

Rahimah Muhamad, M.S. Mohamed Ali*, Deric John Oehlers and Michael GriffithSchool of Civil, Environmental and Mining Engineering, University of Adelaide, South Australia 5005, Australia

(Received: 19 August 2011; Received revised form: 5 March 2012; Accepted: 29 March 2012)

Abstract: Tension stiffening is an important phenomenon in reinforced concretebecause it controls not only deflections but also crack spacings, crack widths and theformation of multiple cracks. It is now common practice to study the effects of tensionstiffening in concentrically loaded prisms, which is the subject of this paper, and usethese behaviours as guidance for the effects of tension stiffening in reinforced concretebeams. As tension stiffening is a mechanism for stress transfer between the concreteand reinforcement, the interface bond stress-slip (τ−δ ) properties are of utmostimportance. In this paper, partial interaction theory is used to develop generic closedform solutions for crack spacings and widths, the load to cause primary, secondarycracks and subsequent cracks. Four different types of interface bond characteristics (τ−δ ) are considered: a linear ascending bond slip which is useful at serviceability; alinear descending bond slip which is useful at the ultimate limit state; a nonlinear bondslip characteristic which closely resembles material bond slip behavior at all limits;and the CEB-FIP Model Code 90 (CEB 1992).

Key words: reinforced concrete, tension stiffening, partial interaction, deflections, crack spacings, crack widths,bond stress slip.

versa for stress in the reinforcement. This stress transferprocess continues until the tensile capacity of theconcrete is reached after which cracks occur. Therefore,the tension stress carried by the concrete at any sectionis important in determining the location of cracks.

This tensile concrete stress is directly related to theinterface bond stress slip property which shows that thebond stress is significant in modelling (Cosenza et al.1997; Elighausen et al. 1983) or deriving a governingequation for predicting these behaviours (Mohamed Aliet al. 2008a; Oehlers et al. 2005; Oehlers et al. 2011b).An interface bond stress τ distribution along thereinforcement rather than an interface bond slip materialproperty τ−δ for predicting the crack spacing and crackwidth has been used (Chan et al. 1992; Somayaji andShah 1981). This was a good starting position in theanalysis of tension stiffening. However, as the bond

*Corresponding author. Email address: [email protected]; Fax: +61-8-8303-4359; Tel: +61-8-8303-3968.

stress distribution depends on the bond slip propertiesτ−δ which can vary considerably (Seracino et al. 2007)as well as the size and type of reinforcement (MohamedAli et al. 2008a; Oehlers et al. 2005; Oehlers et al.2011a) this approach will not provided genericsolutions.

A more advanced approach is to use a specific bondslip property τ−δ. The concept of tension stiffeninghas been used by Marti et al. (1998) and Warner et al.(2007) and is referred to as the tension chord model.In their model, the following very simplisticassumptions are made to obtain a solution. A stepped-rigid plastic τ−δ model is proposed to describe the slipbetween the reinforcement and concrete in whichthere is a uniform bond stress slip in the unyieldedregion and a reduced uniform bond stress slip in theyielded region; this change in bond strength suggeststhat the bond properties are not just a materialproperty. Also they made the assumption that thestrain in the concrete between cracks is relativelysmall and can be ignored which limits the accuracy ofthe model as the strain in the concrete is no longerrelated to the formation of cracks. Other researchershave proposed that the bond stress slip is uniform(Choi and Cheung 1996; Gupta and Maestrini 1990;Wu et al. 1991) which is probably better thanassuming it is dependent on yield as it is now amaterial property but still an over simplification.

Extensive experimental investigations into simulatingthe tension stiffening effects of reinforced concreteprisms have been carried out (Bischoff 2003; Jiang et al.1984; Lee and Kim 2008; Mirza and Houde 1979;Rizkala and Hwang 1984; Somayaji and Shah 1981;Tastani and Pantazopoulou 2010; Wu et al. 2008;Yankelevsky et al. 2008). These tests provide a range ofdata that has been used to analyse the crack spacings andcrack widths in order to develop empirical formulaeused in CEB-FIP Model Code 90 (CEB 1992),Eurocode-2 (2004) and by Marti et al. (1998) as shownin Table 1.

The next step in this research is to develop genericclosed form mechanics solutions for crack spacings,crack widths and the loads to cause multiple crackingthat is based on bond-slip τ−δ material characteristicsthat simulate those from tests. This is the subject of thispaper. The governing equations are first presented.Later, these governing equations are solved using thefollowing four wide ranging types of interface bondstress slip τ−δ characteristics to provide closed formsolutions: (1) A linear ascending bond-slip variationwhich is ideally suited for the early stages ofserviceability and which provides nice and simpleclosed form solutions that clearly illustrate the

importance of specific parameters at serviceability; (2) A non-linear bond-slip which gives complexsolutions but closely simulates the shapes of typicalexperimentally determined bond slip shapes; (3) Abond-slip model based on the same exponential shape asthe well accepted CEB-FIB Model Code 90 (CEB 1992)and by Eligehausen et al. (1983) and finally (4) A lineardescending bond-slip variation that is suitable at theultimate limit state such as in the formation of hinges(Haskett et al. 2009a; Mohamed Ali et al. 2008b) andwhich also gives simple closed form solutions andillustrates the parameters that govern at the ultimatelimit state.

The aim of this paper is to develop the fundamentalmechanics that govern the tension stiffening behaviourfor short term loads as it is realised that the short termdeflection of reinforced concrete members is the startingposition for long term deflection. Hence the accuracy ofthe long term deflection depends on the accuracy of theshort term deflection. Hence this paper is on short termloading. However it will be shown in this paper howtime dependent effects can be included.

2. GOVERNING EQUATIONS FOR TENSIONSTIFFENING

The fundamental governing equation for closed formsolutions can be derived from the equilibrium equationsfor any bonded joint such as a reinforced concrete prismunder pure tension as shown in Figure 1(a) in whichshear lag is ignored as is the usual practice. Thederivation of the governing equation for this stresstransfer problem involves four unknown fields whichare: the axial stresses σr = σr (x) in the reinforcement andσc = σc(x) in the concrete; the axial strains εr = εr (x) inthe reinforcement and εc = εc(x) in the concrete; theinterface shear stress across the bonded length τ = τ (x);and the interface slip δ = δ(x) which is the differencebetween the axial displacement ur of the reinforcementand uc of the concrete (Mohamed Ali et al. 2008a;Muhamad et al. 2011a; Wu et al. 2002; Yuan et al.2004).

From Figures 1(b) and 1(c), the generic equilibriumequations for a reinforced prism under pure tension canbe written as

(1a)

and

(1b)d

dx

L

Ac p

c

σ τ= −

d

dx

L

Ar p

r

σ τ=

2054 Advances in Structural Engineering Vol. 15 No. 12 2012

The Tension Stiffening Mechanism in Reinforced Concrete Prisms



Ta

ble

1. P

ub

lis

he

d c

rac

k s

pa

cin

gs

an

d w

idth

s

Mo

de

lC

rac

k s

pa

cin

gC

rac

k w

idth

whe

re

whe

re

Eur

ocod

e2S r

,max

=m

axim

um c

rack

spa

cing

wk

=cr

ack

wid

th(2

004)

k 1=

coef

fici

ent o

f th

e bo

nd p

rope

rtie

s of

the

bars

ε sm− ε c

m=

diff

eren

t bet

wee

n m

ean

stra

in o

f st

eel a

nd c

oncr

ete

k 2=

coef

fici

ent o

f di

stri

butio

n of

tens

ile s

tres

sd b

=ba

r di

amet

er

C=

conc

rete

cov

erρ e

ff =

effe

ctiv

e re

info

rcem

ent r

atio

whe

re

whe

re

CE

B-F

IP C

ode

Mod

elσ

s2=

ste

el s

tres

s at

the

crac

kε s

m=

aver

age

stra

in o

f th

e re

info

rcem

ent

(CE

B.1

992)

τ bk

= lo

wer

fra

ctile

val

ue o

f th

e av

erag

e bo

nd s

tres

s =

1.8

f ctm

(t)

(sho

rt te

rm lo

adin

g)f c

tm(t

)=

mea

n va

lue

of th

e te

nsile

str

engt

h of

the

conc

rete

at t

he ti

me

tw

hen

the

crac

k ap

pear

ed 1

+α

eρ s

ef=

1(f

or s

impl

icity

)

Mar

ti et

al.

1998

whe

re

whe

re

S rm

= c

rack

spa

cing

w=

cra

ck w

idth

ρ=

rein

forc

emen

t rat

io o

f te

nsio

n ch

ord

σsr

= m

axim

um s

teel

str

ess

at th

e cr

ack

τ bo

= a

vera

ge o

f bo

nd s

tres

s =

2f c

tσ

sro

= s

teel

str

ess

at c

rack

φ=

dia

met

er o

f re

info

rcem

ent b

arE

s=

ela

stic

mod

ulus

of

the

stee

lf c

t=

tens

ile s

tren

gth

of th

e co

ncre

teλ

= c

oeff

icie

nt o

f cr

ack

spac

ing

wS

Erm

srsr

o

s=

−(

)2

2σλσ

Sf

rmct bo

=λ

φ τρ

2

Sd

rms bk

be

sef

=+

σ τα

ρ2

2

1

1w

Srm

rmsm

=ε

SC

kk

dr

b eff

,max

..

=+

34

042

51

2ρ

wS

kr

smcm

=−

()

,max

εε

and from Figure 1(a), the equation of equilibrium for theprism can be written as

(2)

where Ar and Ac are the cross-sectional areas of thereinforcement and the concrete respectively and Lp isthe circumference of the reinforcement as shown inFigure 1(a). The concrete force Pc = 0 as assumingthat there is an initial crack. The axial tension force Pr inEqn 2 will induce a slip at the interface (δ) between theconcrete and reinforcement

(3)

Differentiating Eqn 3 gives the following slip strain

(4)

which for long term loads can be increased by the

shrinkage strain. As is simply the reinforcement

strain, from the steel and concrete moduli Er and Ec

respectively

(5)

and

(6)

where Er and Ec are the elastic modulus ofreinforcement and concrete respectively where the latter

σ εc c c ccE E

du

dx= =

σ εr r r rrE E

du

dx= =

du

dxr

d

dx

du

dx

du

dxr cδ

= −

d

dx

δ

δ = −u ur c

σ σc c r r r cA A P P+ = +

can be modified to allow for creep under long term loadsif required. Substituting Eqns 5 and 6 into 4 yields

(7)

Differentiating Eqn 7, we get

(8)

and substituting Eqns 1(a) and 1(b) into Eqn 8 yields thefollowing governing equation

(9a)

where

(9b)

The governing Eqn 9(a) can be solved using theinterfacial bond-slip characteristic τ = f(δ) along withthe boundary conditions for this specific tensionstiffening problem that is shown in Figure 2.

β21

=

+

L

A E

A

E Ap

r r

r

c c

d

dx

2

2 2 0δ

β τ− =

d

dx E

d

dx E

d

dxr

r

c

c2

2

1 1δ σ σ=

−

d

dx E Er

r

c

c

δ σ σ= −



r

dx

Steel reinforcement bar

Concrete∆(a) Equilibrium of prism

Concrete

(b) Equilibrium at concrete interface


(c) Equilibrium at reinforcement interface

Pc /2

Pc /2

Pr

c + d c

r + d r

c

r

r Ar

c Ac

τLp/Ar

τLp/Ac

Ar

Ac

Lp

σ

σ

σ σ σ

σ σσ

Figure 1. Free body diagrams for: (a) equilibrium of prism;

(b) equilibrium at concrete interface; and (c) equilibrium at

reinforcement interface

Pc /2

Pr1

Pc /2

Pc /2

Pr1

Pc /2

Pr

Ss = Sp /2

St = Sp /4

Pb = τLpdx

Pr

Pb = τLpdx

PrPr

Secondary crack face

(b) Prism length Sp

(c) Prism length Ss

Primary crack face

(a) Infinitely long prism

Initial crack face

Concreter_p = (0)

Full interaction

δ

= 0δ

Sp

Pc /2Pb = τLbdx

Pr 1

Pc /2


x

Pr

Pc /2

Pc /2

Pr1

dδdx

δ= 0 and = 0∆

r_s = (0)δ∆

r_t = (0)δ∆

Figure 2. Tension stiffening for concrete prism

Figure 2(a) shows the boundary conditions for theformation of primary cracks. Let us assume that there isan initial crack at x = 0 beyond which there are nofurther cracks so it is a question of determining the firstseries of cracks or primary cracks. At x = 0 the strain inthe concrete is zero so that the slip strain is simply thestrain in the reinforcement, hence

(10)

Sp in Figure 2(a) is defined as a position beyondwhich the slip-strain tends to zero that is the start ofwhere the behaviour tends towards full-interaction asshown. Hence

(11)

As full-interaction is first achieved at x = Sp, themaximum stress in the concrete is first achieved at Sp.Hence the primary crack can occur anywhere in the full-interaction region so that Sp is the minimum crackspacing for the primary cracks. As beams are normallysubjected to a moment gradient Sp is also the primarycrack spacing.

Primary cracks will occur at a spacing of Sp along thelength of the prism. Once these primary cracks haveformed, the problem now changes to that shown inFigure 2(b) which is that of a symmetrically loadedprism of length Sp. By symmetry, the boundarycondition at the mid-length of the prism is given by

(12)

where Ss in Eqn 12 and Figure 2(b) is a secondary crackspacing

Once secondary cracks have formed, the prism lengthnow changes to that shown in Figure 2(c). By symmetry,the cracks occurs at a crack spacing of Sp/4 wheneverthe bond is adequately strong, the boundary condition atthis stage can be written as

(13)

where St in Eqn 13 and Figure 2(c) is a tertiary crackspacing

Figures 2(a), (b) and (c) also illustrate the randomnature of cracking. It has already been explained thatthe first primary crack can occur anywhere in the fullinteraction region in Figure 2(a) but will probably

δδ

= ≠ = =0 0 4and atd

dxx S Sp t/

δδ

= ≠ = =0 0 2and atd

dxx S Sp s/

δδ

= = =0 0and atd

dxx Sp

d

dx

P

A Exr

r rr

δδ= =and at = 0∆

occur closer to Sp due to the moment gradient in thebeam. Hence cracks can occur in any position beyondSp from the initial crack and tests on beams wouldsuggest that the crack spacing can be as large as 2Sp.If, for example, the crack spacing is 2Sp, then theprism in Figure 2(b) would be 2Sp long so bysymmetry the next crack would occur at 2Sp/2 and thenext in Figure 2(c) at 2Sp/4. If, as a further example,the crack spacing is 1.4Sp, then the next crack wouldoccur at 1.4Sp/2 and the next at 1.4Sp/4. In thefollowing analyses, it has been assumed that the initialcrack spacing is the minimum crack spacing Sp butthis could be adjusted to a factor of Sp and appliedusing the same boundary condition as in Figure 2(b).It is also worth noting at this stage, that it will beshown that the formation of primary, secondary andtertiary cracks, as in Figures 2(a), (b) and (c), dependson the bond strength. For example the bond may besufficient to form primary cracks in Figure 2(a) butnot sufficient to form secondary cracks as in Figure2(b) as the bonded length in Figure 2(b) is less thanthat in Figure 2(a). This further adds to the randomnature of cracking.

In this paper, the primary and secondary crackspacings, crack widths as well as the load to causecracks have been derived for the four different typesof interfacial bond stress slip characteristics τ−δ asshown in Figure 3 and which consist of the following.Firstly a linear ascending bond slips propertyrepresented by O-B in Figure 3 that has a stiffness ke.Secondly a non-linear bond slip property representedby O-B’-E which is characterised by an ascendingnonlinear curve with a peak shear stress of τmax at aslip of δ2 and a descending non-linear curve. Thirdlythe bond slip property of CEB-FIP Model Code 90(CEB 1992) and Eligehausen et al. (1983) for anascending non-linear curve with a peak shear stress ofτmax at a slip of δ1 represented by O-B. And finally thelinear descending bond slip property with a peakshear stress of τmax at a slip of zero and a peak slip ofδmax at a zero shear stress τ represented by O-A-C inFigure 3.



CEB-FIPLinear ascendingNon-linearLinear descending

A B

C E

B'

ke

(N/mm2)

O 2 1 max δδ

maxτ

τ

δ δ (mm)∞

Figure 3. Idealised bond stress slip characteristics

3. SOLUTIONS FOR LINEAR ASCENDINGBOND SLIP CHARACTERISTIC

The bond stress slip for a linear ascending characteristic(Mohamed Ali et al. 2008a; Wu et al. 2002; Yuan et al.2004), as depicted by O-B in Figure 3, can be written asfollows

(14)

where ke is the stiffness of the bond slip property τ−δ.Substituting Eqn 14 into the governing equation of

Eqn 9(a) yields

(15)

Solving the differential equation of Eqn 15 gives theslip variation

(16a)

where

(16b)

Differentiating Eqn 16(a) yields

(17)

Substituting Eqn 16(a) into the linear ascending bondstress slip τ−δ of Eqn 14 results in

(18)

in which the constants c1 and c2 in Eqns 16(a)-18 can besolved through the substitution of boundary conditionsas follows.

3.1. Analysis of Infinitely Long Prism [seeFigure 2(a)]

3.1.1. Linear ascending crack spacings and loadto cause a crack for infinitely long prism

The boundary condition at the initial crack face inFigure 2(a) is given by Eqn 10. Applying this boundarycondition into Eqn 17 yields

(19)cP

A Er

r r2

1=

λ

τ λ λ( ) cosh( ) sinh( )x k c x c xe= +[ ]1 1 2 1

d

dxc x c x

δλ λ λ λ= +1 1 1 1 2 1sinh( ) cosh( )

λ β1 2= ke

δ λ λ( ) cosh( ) sinh( )x c x c x= +1 1 2 1

d

dxke

2

2 2δ

β δ=

τ δ= ke

For the boundary condition in Eqn 11 that is δ = 0and dδ /dx = 0 at x = Sp [refer Figure 2(a)]. Applyingthese boundary conditions and Eqns 19 into 16(a) and17 gives

(20)

(21)

Solving both Eqns 20 and 21 gives the primary crackspacing, Sp as

(22)

where Spλ1 = 2 is based on an assumption that the bondstress is resisted by 97% of the applied load, Pr (Yuanet al. 2004).

Further substitution of Eqns 22 into 21 yields

(23)

The relationship between the force in the bar, Pr atthe initial crack at x = 0 and the forces in reinforcementand concrete, Pr1 and Pc at the initiation of a primarycrack at x = Sp in Figure 2(a), can be written in thefollowing form

(24)

and

(25)

where the definite integral in Eqn 24 yields

the bond force between the concrete and thereinforcement over the length Sp.

Solving Eqns 24 and 25 lead to the relationshipbetween the bond force and concrete force as shownbelow

(26)τL dx f Apx

x S

ct c

p

=

=

∫ =0

τL dxpx

x Sp

=

=

∫0

P P Pr r c= +1

P L dx Pr px

x S

r

p

− ==

=

∫ τ0

1

cP

A Er

r r1

1 2= −

λ tanh

Sp =2

1λ

λ λ λ1 1 1 1 0c SP

A ESp

r

r rpsinh( ) cosh( )+ =

c SP

A ESp

r

r rp1 1

11 0cosh( ) sinh( )λ

λλ+ =



At x = Sp in Figure 2(a), the slip-strain between theconcrete and reinforcement bar is zero which means thatthe strain in the reinforcement, εr1, and strain in theconcrete, εc, are equal from which

(27)

Substituting Eqns 26 and 27 into 24 yields the load tocause a primary crack based on full interaction. Once thetensile stress of the concrete in Eqn 27 reach the tensilestrength capacity of the concrete which σc = fct, the crackwill occur and load to cause the crack can be rewritten as

(28)

The load to cause a primary crack for infinitely longprism based on partial interaction for linear ascendingbond stress slip can be obtained by solving therelationship between the bond force and concrete forceas shown in Eqn 26. Substituting bond stress of Eqns 18into 26 which the constants c1 and c2 as shown in Eqns 23and 19 yield the load to cause a primary crack forinfinitely long prism

(29)

3.1.2. Linear ascending load slip behaviour forinfinitely long prism

The load Pr and slip at the crack face ∆r _ p for infinitelylong prism in Figure 2(a) can be obtained by substitutingEqns 23 and 19 into slip variation of Eqn 16(a), the slipat the initial crack face, x = 0

(30)

As the crack width, wr_ p is twice the slip, ∆ r_ p hencethe crack width for infinitely long prism as

(31)

3.2. Analysis of Prism Length Sp [see Figure 2(b)]3.2.1. Linear ascending crack spacing and load

to cause a crack for prism length SpThe prism length Sp as shown in Figure 2(b) will beanalysed in this section. As the bar of the prism is pulled

wP

A Er pr

r r_ =

2

1λ

∆r pr

r r

P

A E_ =λ1

PA E f A

k Lr cr pr r ct c

e p_ _ =

λ12

Pf

EA E f Ar cr fi

ct

cr r ct c_ _ = +

PE

A Erc

cr r1 =

σ

out, a secondary crack will occur. By symmetry inFigure 2(b), the secondary crack spacing Ss = Sp/2 =1/λ1 can be obtained. Substituting Eqns 19 and 12 into16(a) will leads to the constant c1_s that corresponds tothe prism length Sp as

(32)

Substituting Eqns 32 and 19 into 18 and further intoEqn 26 for boundary limit x = 0 and x = Ss yield the loadto cause a secondary crack for prism length Sp

(33)

3.2.2. Linear ascending behaviour of prismlength Sp

The load Pr and slip at the primary crack face ∆r_s forprism length Sp in Figure 2(b) can be obtained bysubstituting constants c2 and c1_s as shown in Eqns 19and 32 respectively into Eqn 16(a) at x = 0 to give

(34)

As the crack width wr_s = 2 ∆r_s hence the crackwidth for the prism length Sp as

(35)

3.3. Analysis of Prism Length Ss [see Figure 2(c)]3.3.1. Linear ascending crack spacing and load

to cause a crack for prism length SsThe prism length Ss in Figure 2(c) can be analysed in thissection. By symmetry of the prism, a further tertiary crackwill occur at the mid-length of the prism that is St = Sp/4= 1/2λ1. Applying the boundary conditions of Eqn 13 andthe constant c2 of Eqns 19 into 16(a) yields the unknownconstant c1_t that corresponds to the boundary conditionsof Eqn 13 as

(36)cP

A Etr

r r1

10 5_ tanh( . )= −

λ

wP

A Er sr

r r_

tanh( )=

2 1

1λ

∆r sr

r r

P

A E_tanh( )

=1

1λ

PA E f A

k Lr cr sr r ct c

e p_ _ .

=λ1

2

0 35

cP

A Esr

r r1

11_ tanh( )= −

λ



Substituting Eqns 36 and 19 into 18 and further intoEqn 26 for boundary limit x = 0 and x = St, will gives theload to cause a tertiary crack for prism length Ss as

(37)

3.3.2. Linear ascending behaviour of prismlength Ss

The load Pr and slip ∆ r_t at the secondary crack face forthe prism length Ss in Figure 2(c) can be obtained bysubstituting constant c2 and c1_t as shown in Eqns 19 and36 respectively into Eqn 16(a) at x = 0 to give

(38)

As the crack width wr_t = 2 ∆ r_t hence the crackwidth for the prism length Ss as

(39)

4. SOLUTIONS FOR NON-LINEAR BONDSLIP CHARACTERISTIC

The non-linear bond slip characteristic shown as O-B’-Ein Figure 3 was proposed by Dai et al. (2006) and can bewritten as

(40a)

where

(40b)

Substituting Eqn 40(a) into the governing equation ofEqn 9(a) leads to

(41)

As

(42a)d

dx

d

dx

d

dx

d

dx

d

d

d

dx

d2

2

δ δ ν νδ

δν=

= =

=ννδd

d

dxe ek k

2

2 24 1δ

τ β δ δ= −− −max ( )

k =0 693

2

.

δ

τ τ δ δ= −− −4 1max ( )e ek k

wP

A Er tr

r r_

tanh( . )=

2 0 5

1λ

∆r tr

r r

P

A E_tanh( . )

=0 5

1λ

PA E f A

k Lr cr tr r ct c

e p_ _ .

=λ1

2

0 113

where

(42b)

using Eqn 42(a) and substituting into Eqn 41 leads to

(43)

Rearranging Eqn 43 and integrating both side of theequation yields

(44a)

where

(44b)

and c3 is a constant of integration.For the boundary conditions of Eqn 11 which

and δ = 0 at x = Sp, unknown constant c3 in

Eqn 44(a) can be determined. Thus, Eqn 44(a) can bewritten as

(45)

Rearranging Eqn 45 yields

(46)

Integrating both side of Eqn 46 will gives thevariation of slip

(47)

where c4 is a constant of integration.Differentiating Eqn 47 will lead to the following

variation of slip strain as

(48)d x

dxA

e

e

k Ax c

k Ax c

δ ( ) ( )

( )=+

+

+

4

4 1

δ ( ) ln ( )xk

ek Ax c= +( )+114

e d

eAdx

k

k

δ

δδ

−=

1

d

dxA e kδ δ= − −( )1

d

dx

δ= 0

A = 2 4 2 2. maxτ δ β

d

dxA e e ck kδ δ δ= − + +− −2 2

3

ννδ

τ β δ δd

de ek k= −− −4 12max ( )

νδ

=d

dx



and variation of bond stress as

(49)

The unknown c4 in Eqns 47 and 48 can be solvedthrough substitution of boundary conditions as infollowing section.


4.1.1. Non-linear primary crack spacing and loadto cause a crack

At the initial crack face of the prism, x = 0 in Figure 2(a),applying the boundary condition of Eqns 10 into 48yields

(50)

At x = Sp in Figure 2(a), applying the boundarycondition of Eqn 11 that is the behaviour tends to fullinteraction (δ = 0 and dδ /dx = 0) and Eqns 50 into 47and 48 gives the primary cracks spacing as

(51)

In solving both Eqns 47 and 48 for boundaryconditions δ = 0 and dδ/dx = 0, eAkSp cannot be zerohence an assumption (eAkSp = 0.135) has to be made toobtain the crack spacing.

Substituting Eqns 50 into 47 and further into Eqn 49yields the bond stress variation. Furthermore,substituting the bond stress variation into therelationship between the bond force and concrete forceof Eqn 26 yields the following load to cause a primarycrack for infinitely long prism

(52a)

where

(52b)c AA Er r5 =

P cc c

L

c

Lr cr pp p

_ _max max

..

.= − −0 5

0 124

6 94

3

55 6 6

τ τ

.. ..max max47 19 85

0 745

6

25

2

6

τ τL c

c

L c

cp p

− + cc5

2

Sp = −1

0 693 22

. maxβτδ

c

PAA E P

k

r

r r r4 =

−

ln

τ τ δ δ( ) max( ) ( )x e ek x k x= −( )− −4 1

(52c)

4.1.2. Non-linear load slip behaviour forinfinitely long prism

The load Pr will induce a slip ∆ r_ p at the initial crackface in Figure 2(a). Slip ∆r_ p can be obtained bysubstituting Eqns 50 into 47 at x = 0 as follows

(53)

As the crack width wr_ p is twice the slip ∆ r_ p hencethe crack width for primary cracks

(54)

4.2. Analysis of Prism Length Sp [see Figure 2(b)]4.2.1. Non-linear crack spacing and load to

cause a crack for prism length SpThe prism length Sp in Figure 2(b) will be considered inthis subsection. As the bar of the prism is further pulledout, secondary cracks will occur. By symmetry, thesecondary crack spacing Ss = Sp/2 in Figure 2(b) wherethe slip is zero is given in Eqn 12. As the slip straindδ /dx is not zero at x = Ss, rearranging Eqn 44(a) yields

(55)

Integrating both sides of Eqn 55 will gives

(56)

Applying the boundary conditions in Eqn 12 to Eqns56 and 44(a) yields the unknown constant c7_s

(57)

and the slip strain

(58)d

dxA c

δ= −3 1

c S Akc

ct t73

13

11_ sinh= − −−

δ ( ) lnsinh

xk

c c Akx c

c=

− −( )( ) +

1 1 13 3 7

3

d

e e cAdx

k k

δδ δ− + +

=− −2 2

3

wk

P

AA E Pr pr

r r r_ ln=

−+

2

1

∆r pr

r r rk

P

AA E P_ ln=−

+

1

1

c f A Akct c6 =



Applying the boundary conditions in Eqns 10 that isδ = ∆r at x = 0 to Eqn 56 yields

(59a)

where

(59b)

Substituting Eqns 59(a) into 44(a) for the boundarycondition of Eqn 10 in which dδ /dx = Pr/ArEr and δ = ∆r

at x = 0 yields

(60)

As slip strain dδ/dx of Eqn 58 is the difference instrain between the strain in the reinforcement εr1 andstrain in the concrete εc, thus Eqn 58 can be rewritten as

(61)

Substituting Eqns 61 into 25 at the stage the concreteis reaches the tensile capacity Pc = fctAc, yields thefollowing load to cause a secondary crack for the prismlength Sp

(62)

Substituting Eqns 62 into 60 gives the value ofconstant c3_s that corresponds to the boundaryconditions of Eqn 12

4.2.2. Non-linear behaviour of prism length SpThe relationship of load-slip P−∆ for the prism length Sp

in Figure 2(b) can be obtained by rearranging Eqn 59(a)

(63)

1 2 2 18 82

3− + − −

+

c c c

f

E A

f A

A E A

s s s

ct

c

ct c

r r

_ _ _

=

+

+

f

E A

f A

A A E E

f A

ct

c

ct c

r r c

ct

2

2

22 cc

r rA E A

2

P A E A cf

EA E f Ar cr s r r

ct

cr r ct c_ _ = − + +3 1

P A E A cf

EA Er r r

ct

cr r1 3 1= − +

P

AA Ec c cr

r rs s

= − +

2

3 8 822 _ _

cc

c c S Ak cs

s8

3

3 3 31_

sinh=

+ − −( )

∆rsk c

=

1 1

8ln

_

and further substituting the resulting equation into Eqn60 to give

(64)

Thus the crack width for prism length Sp can beobtained by wr_s = 2 ∆r_s.

4.3. Analysis of Prism Length Ss [see Figure 2(c)]4.3.1. Non-linear behaviour with crack spacing

Ss and load to cause a crackConsider the prism of length Ss in Figure 2(c). Bysymmetry, the tertiary crack spacing is St = Sp /4 inFigure 2(c) where the slip is zero as given in Eqn 13.The boundary condition of Eqn 13 in which δ = 0 atx = St = Sp /4 when applied to Eqn 56 will give theunknown constant c7_t that corresponds to theboundary condition of Eqn 13

(65)

Substituting Eqns 65 into 56 for the boundarycondition δ(x) = ∆r at x = 0 yields

(66a)

where

(66b)

Substituting Eqn 66(a) into Eqn 44(a) for theboundary condition of Eqn 10 in which dδ/dx = Pr /ArEr

and δ = ∆r at x = 0 yields

(67)

Substituting Eqn 62 into Eqn 67 gives the value ofconstant c3_t that corresponds to the boundary conditionsof Eqn 13

(68)

1 2 2 18 82

3− + − − +

c c c

f

E A

f A

A E At t tct

c

ct c

r r_ _ _

=

+

+

f

E A

f A

A A E E

f Act

c

ct c

r r c

ct2 2

22 cc

r rA E A

2

P

AA Ec c cr

r rt t

= − +

2

3 8 822 _ _

cc

c c S Ak ct

t8

3

3 3 31_

sinh=

+ − −( )

∆rtk c

=

1 1

8ln

_

c S Akc

ct t73

13

11_ sinh= − −−

e eP

AA Eck k r

r rs

r r− −− =

−2

2

32∆ ∆_



The load to cause a tertiary crack for prism length Ss

is given by Eqn 62 with c3 = c3_t from Eqn 68.

4.3.2. Non-linear behaviour of prism length SsThe relationship of load-slip P−∆ for the prism length Ss

in Figure 2(c) is given in Eqn 64 where c3_t iscorresponding to the boundary conditions of Eqn 13 asshown in Eqn 68. Thus the crack width for prism lengthSs can be obtained by wr_t = 2 ∆r_t .

5. SOLUTIONS FOR NONLINEAR CEB-FIPBOND SLIP CHARACTERISTIC

The ascending non-linear curve of bond stress for theCEB-FIP Code Model 90 (CEB 1992) and Eligehausenet al. (1983) shown as O-B in Figure 3 can be written as

(69)

Substituting bond stress of Eqn 69 into governingequation of Eqn 9(a) leads to

(70)

Using Eqn 42(a), Eqn 70 can be rewritten as

(71)

Rearranging Eqn 71 and integrating both sides of theequation yields

(72a)

where

(72b)

and c9 is a constant of integration.Rearranging Eqn 72(a) and further integrating the

equation yields

(73a)δ

α αλ δ α

Hyp Fc2 1

21

9

1

2

1

11

1

1, , ,

++

+−

+

== +( )x c c10 92

λβ τ

δ α22

1

= max

d

dxc

δ λ δα

α

=+

++2

122

1

9

ννδ

β τδδ

αd

d=

21

max

d

dx

2

2 21

δβ τ

δδ

α

=

max

τ τδδ

α

=

max1

where Hyp2F1 represents a 2F1 hypergeometric functionand it is a series of slip function. Solving that slip seriesfunctions in mathlab, an assumption has been made thatthis series is approximately 1.0 hence the slip variationin Eqn 73(a) can be rewritten as

(73b)


5.1.1. CEB model primary crack spacing and loadto cause for infinitely long prism

For the boundary conditions of Eqn 11, in which δ = 0at x = Sp, the constant c10 in Eqn 73(b) can be obtainedas

(74)

Substituting Eqn 74 into the slip variation of Eqn73(b) and further into the bond force and concrete forcerelationship of Eqn 26 gives the primary crack spacing

(75)

Substituting the boundary conditions of Eqn 10 intothe slip strain variation of Eqn 72(a) and slip variationof Eqn 73(b), will lead to the constant c9_ p

(76)

where Pr in Eqn 76 is the load to cause the primarycrack spacing Sp that is Pr = Pr_cr Substituting the loadto cause a primary crack Pr_cr from Eqns 28 into 76 willgive the constant c9_ p

(77)

5.1.2. CEB model load slip behavior for infinitelylong prism

The load Pr and the slip ∆r at the initial crack face inFigure 2(a) can be obtained by applying the boundary

cc f A

L

f

E

f A

Ap

p ct c

p

ct

c

ct c9

9 2 120 5_

_

max.− = − +

λ δ

τ

α

rr rE

2

cc f A

L

P

A Epp ct c

p

r

r r9

9 2 120 5_

_

max.− = −

λ δ

τ

α 22

Sf A

L cp

ct c

p

=+( )

( )

+1

2

1

9

11

α δ

τ

α

α

α

max

c Sp10 = −

δ = +( )x c c10 92



conditions of Eqn 10 in which dδ/dx = Pr /ArEr and δ = ∆r at x = 0 into the slip strain of Eqn 72(a)

(78)

Thus the crack width for prism length Sp can beobtained by wr_ p = 2 ∆r_ p.

5.2. Analysis of Prism Length Sp [see Figure2(b)]

5.2.1. CEB-FIP model crack spacing and load tocause a crack for prism length Sp

The prism length Sp in Figure 2(b) will be analysed. Asthe bar of the prism is further pulled out, secondarycracks will occur. By symmetry in Figure 2(b), thesecondary crack spacing Ss = Sp /2 and applying theboundary condition of Eqns 12 into 73(b) will leads tothe constant c10_s corresponding to the prism length Sp

(79)

Substituting Eqn 79 into the slip variation of Eqn73(b) and further into the bond force and concrete forcerelationship of Eqn 26 gives the constant c9_s thatcorresponds to the prism of length Sp

(80)

Substituting the constant c9_ s of Eqn 80 and also Eqn 79into the slip variation of Eqn 73(b) at x = 0 yields thefollowing slip at the occurrence of the secondary crackfor the prism of length Sp

(81)

Applying the boundary conditions of Eqn 10 and alsoEqn 81 into Eqn 72(a) yields the following load to causea secondary crack for the prism length Sp

(82)P A E

Sf A

Lr cr s r r

sct c

_ _

max=

+( ) +( )+2

1

12 1 1λα

α δ

τ

αα

pp s

ct c

p s

S

f A

L S

( )

++( )

+

+

1

1

11

α

αα

αα δ

τmax (( )

+1

2

α

α

∆r cr s sc ct

p s

SA f

L S_ _

max

( )=

+

( )

+

δ α

τ

α

α1

1

1

1α

cf A

L Ss

ct c

p s9

11

2

0 51

_max

.=+( )

( )

+

α δ

τ

α

α

αα

c Ss s10 _ = −

λα

α2

1 2 2

10 5 0 5

∆ ∆r r

p

r

r rS

P

A E

+

++

=

. .

5.2.2. CEB model load slip behavior of prismlength Sp

The load Pr and the slip ∆r_s at the primary crack facesin Figure 2(b) can be obtained by substituting theboundary conditions of Eqn 10 in which dδ /dx =Pr /ArEr and δ = ∆r at x = 0 into the slip strain of Eqn72(a). This gives the load-slip relationship as well ascrack width as shown in Eqn 78 with replace Sp = Ss.

5.3. Analysis of Prism Length Ss [see Figure 2(c)]5.3.1. CEB model crack spacing and load to

cause a crack for prism length SsThe prism length Ss in Figure 2(c) will be used in thissubsection. By symmetry of the prism, the crack willoccur at the mid-length of the prism that is St = Sp/4.These boundary conditions of Eqn 13 in which δ = 0 atx = Sp/4 can be applied and give the load to cause atertiary crack for prism length Ss as shown in Eqn 82with replace Ss = St.

5.3.2. CEB model load slip behaviour of prismlength Ss

The load Pr and the slip ∆r_t at the secondary crack faceas well as crack widths in Figure 2(c) can be obtained asshown in Eqn 78 with replace Sp = St.

6. SOLUTIONS FOR LINEAR DESCENDINGBOND SLIP CHARACTERISTIC

The bond stress for a linear descending bond stress slipcharacteristic (Haskett et al. 2009b) O-A-C in Figure 3can be written as

(83)

Substituting Eqn 83 into the governing equation ofEqn 9(a) leads to

(84a)

where

(84b)

Solving the differential equation of Eqn 84(a) givesthe slip variation

(85)δ λ λ δ( ) sin cos maxx c x c x= ( ) + ( ) +11 3 12 3

λτδ

β3 2= max

max

d

dx

2

2 32δ

λ δ δ= −( )max

ττδ

δ δ= −( )max

maxmax



Differentiating Eqn 85 yields the slip strain variation

(86)

Further substituting Eqn 85 into the bond stress slipτ−δ of the linear descending bond-slip in Eqn 83 resultin the bond stress variation

(87)

where the constants c11 and c12 in Eqns 85 to 87 can besolved through substitution of the boundary conditionsas follows.


6.1.1. Linear descending crack spacing and loadto cause a crack for infinitely long prism

The boundary condition at x = Sp in Figure 2(a) is givenin Eqn 11. Substituting these boundary condition intoEqns 85 and 86 yields

(88)

and

(89)

Substituting both Eqns 88 and 89 into Eqn 86 andfurther substituting into Eqn 10 will lead to

(90)

where Pr in Eqn 90 is the load to cause the primarycrack spacing Sp that is Pr = Pr_cr. Substituting bothconstants of Eqns 88 and 89 into the relationship of thebond force and concrete force as given in Eqn 26 resultsthe load to cause a primary crack

(91)

Substituting Eqns 91 into 90 gives the primary crackspacing

(92)S

f AL

p

ct c

p=

arcsin

max

λτ

λ

3

3

PA E f A

Lr cr pr r ct c

p_ _

max

max=

λ δτ

32

S

PA E

p

r

r r=

arcsin

maxλ δ

λ3

3

c Sp12 3= − ( )δ λmax cos

c Sp11 3= − ( )δ λmax sin

ττδ

λ λ( ) sin cosmax

maxx c x c x= − ( ) + ( ) 11 3 12 3

d x

dxc x c x

δλ λ λ λ

( )cos sin= ( )− ( )3 11 3 3 12 3

6.1.2. Linear descending load slip behavior forinfinitely long prism

The load Pr and slip ∆r_ p can be obtained bysubstituting the constants c11 and c12 in Eqns 88 and 89into 85 at x = 0 as follows

(93)

Substituting Eqn 93 into wr = 2 ∆r yields the crackwidth for infinitely long prism as

(94)

6.2. Analysis of Prism Length Sp [see Figure 2(b)]6.2.1. Linear descending crack spacing and load

to cause a crack for prism length SpAs the bar of the prism length Sp in Figure 2(b) is pulledout, secondary cracks will occurs. By symmetry, thesecondary crack spacing Ss = Sp /2. Substituting theboundary conditions of Eqns 10 and 12 into Eqns 86 and85 respectively gives the constants c11_s and c12_s thatcorresponds to a prism of length Sp

(95)

and

(96)

Substituting both constants of Eqns 95 and 96 into 87and further substituting into Eqn 26 gives the load tocause a secondary crack

(97)

6.2.2. Linear descending load slip behavior ofprism length Sp

The load Pr and slip ∆r_s relationship for a prism lengthSp can be obtained by substituting Eqns 95 and 96 into85 at x = 0

(98)

As the crack width wr_s is twice the slip ∆r_s hence thecrack width for a prism length Sp as

(99)wS

P

A ESr s

s

r

r r_ max cos

tan= −( )

−2 1

1

3 33δ

λ λλ ss( )

∆r ss

r

r rsS

P

A ES_ max cos

tan= −( )

−δ

λ λλ1

1

3 33(( )

Pf A A E S A E

r cr sct c r r s r r

_ _max macos

=( )−δ λ λ λ τ3

23 3 xx max

max

sin

cos

δ λ

τ λ

L S

L Sp s

p s

3

31

( )− ( )( )

cA E P S

S A Esr r r s

s r r12

3 3

3_

max sin

cos=− − ( )

( )δ λ λ

λ λ33

cP

A Esr

r r11

3_ =

λ

wP

A Er pr

r r_ max

maxcos arcsin= −

2

3δ

λ δ +

δmax

∆r pr

r r

P

A E_ maxmax

cos arcsin= −

δ

λ δ3++δmax



6.3. Analysis of Prism Length Ss [see Figure 2(c)]6.3.1. Linear descending crack spacing and load

to cause a crack for prism length Ss

By symmetry of the prism of length Ss, the tertiary crackwill occur at the mid-length of the prism that is St = Sp /4.Substituting the boundary condition of Eqns 13 into 85gives the constant c12_t that corresponds to a prismlength Ss as shown in Eqn 96 with replace Ss = St.

Substituting Eqn 95 and the constant c12_t into bondstress variation of Eqn 87 and further into Eqn 26 for theboundary limit from x = 0 to x = St yields the load tocause a tertiary crack for the prism length Ss as shown inEqn 97 with replace Ss = St.

6.3.2. Linear descending load slip behavior ofprism length Ss

The load Pr and slip ∆r_t relationship for the prism oflength Ss can be obtained by substituting Eqn 95 and theconstant c12_t into the slip variation of Eqn 85 at x = 0 asshown in Eqn 98 with replace Ss = St. As the crack widthwr is twice the slip ∆r hence the crack width for the prismlength Ss is given in Eqn 99 with replace Ss = St.

7. SUMMARY AND COMPARISONSThe results of this mechanics based analysis of tension-stiffening have been used to derive the short termdeflection of steel reinforced beams (Muhamad et al.2011b), FRP reinforced beams (Oehlers et al. 2011b)and the behaviour of hinges (Visintin et al. 2012) andgive good correlation with test results. However, it isfelt that a strength of this mechanics based approach isto isolate the parameters that affect tension-stiffeningand this will be studied in this section.

7.1. Parametric ComparisonTable 1 lists published crack spacings Srm which, ingeneral, have been derived empirically. It can be seenthat the main empirically derived parameters thatcontrol crack spacing are: bond properties such as k1;bar diameter db; ratio of reinforcement area to that ofconcrete ρ; steel stress at crack σs2; and the tensilestrength of concrete fct. These empirically identifiedparameters are also those identified in the mechanicsmodels that have been developed in this paper. Take forexample the analysis based on the linear ascending bondcharacteristic in Figure 3 where crack spacing is givenby Eqn 22 which can be written as follows

(100)S

kL

A EA

E A

p

ep

r r

r

c c

=

+

2

1

Comparing the parameters in the mechanics modelof Eqn 100 with the empirical parameters listedabove: ke in the mechanics model is the bond stiffnessand equivalent to the bond property k1 in theempirical model, Lp is the bar circumference used inthe mechanics model as opposed to the bar diameterdb in the empirical model, and Ar /Ac is thereinforcement ratio ρ in the empirical model. Thecrack spacing of Eqn 100 is the crack spacing ofprimary cracks which is simply twice the spacingafter secondary cracks occur. As the reinforcementbar load to cause primary cracks is given by Eqn 29and that to cause secondary cracks by Eqn 33, it canbe seen that the crack spacing is also dependent onthe stress in the bar at the crack that is the parameterσs2 in the empirical model. The crack spacing of Eqn100 for the linear ascending bond slip properties aswell as that for the non-linear bond slip properties ofEqn 51 is not dependent on the tensile strength of theconcrete fct. In contrast, that for the CEB-FIP ModelCode 90 (CEB 1992), Eqn 75, and that for the lineardescending bond properties, Eqn 92, is dependent onfct which explains why some of the empirical modelsin Table 1 show a dependence on fct (Marti et al.1998) and others (Eurocode 2 2004; CEB-FIP ModelCode 90 1992) do not.

There is a remarkably good agreement on theparameters that control the crack width in the empiricalrules in Table 1 where it can be seen that in all threeempirical rules the crack width depends on the crackspacing Srm and the reinforcement strain εsm. The crackwidth for the linear ascending bond-slip properties isgiven by Eqn 31 which can be written as

(101)

This mechanics model also depends on the crackspacing Sp and reinforcement strain Pr/ArEr and isvirtually the same equation as that in Eurocode 2 (2004)in Table 1.

7.2. Analysis of Concentrically Loaded PrismThe mechanics based solutions are now used to analysea concentrically loaded prism, such as that shownin Figure 1, with the steel reinforcement properties ofAr = 1385 mm2, Er = 200 GPa and Lp = 132 mm, concreteprism properties of Ac = 2215 mm2, Ec = 25 GPa fc = 30MPa and fct = 2.74 MPa, and bond-slip properties of τmax = 6.85 MPa, δ1 = 1.5 mm and δ2 = 2.59 mm asshown in Figure 4. The CEB-FIP Model Code 90(CEB.1992) bond-slip property O-B in Figure 4 is Eqn 69

wS P

A Erp r

r r=



with the code recommended value for the exponent α inEqn 69 of 0.4. As the exponent α increases from 0.4 tounity, the CEB-FIB Model Code 90 (CEB 1992)variation approaches that of the linear ascending Eqn 14with the stiffness ke as shown in Figure 4. The bondstiffness ke used in Figure 4 is therefore a lower bound tothe CEB-FIB Model Code 90 (CEB 1992) stiffness’s.

The results of the analysis depicted in Figure 2(a) todetermine the crack spacing Sp is shown in Figure 5 for thelinear ascending bond characteristics in Figure 4. It can beseen in Figure 5 that the concrete stress builds up along thelength of the bar and peaks at a distance of 554 mm fromthe crack face. It can also be seen that the shape of thisdistribution remains unchanged, that is, it peaks at Sp = 554 mm which is independent of the applied load Pr.Hence the crack spacing is independent of fct as Eqn 100suggests. In contrast for the CEB-FIB Model Code 90(CEB 1992) bond in Figure 6, the distance from the crackface at which the stress in the concrete peaks is a functionof the reinforcement force so that the crack spacing is nowa function of the tensile strength of the concrete. Hence itcan be seen that the shape of the bond-slip propertydetermines the dependence of the crack spacing on thetensile strength of the concrete which explains why some

empirical models in Table 1 include the dependence on fct

and others do not.The results of analysing a concentrically loaded

prism using both the mechanics models developed inthis paper and the empirical models in Table 1 are listedin Table 2. The linear ascending results are given in



Table 2. Results from structures mechanics models

Bond model Primary crack Primary crack load Secondaryspacing* (mm) (kN) crack load (kN)

(1) Linear ascending 554 36 103(2) Non-linear 438 42 104(3) CEB-FIP (α = 0.4) 72 [36]** 447(4) CEB-FIP (α = 0.5) 106 [36]** 323(5) CEB-FIP (α = 0.6) 151 [36]** 246(6) CEB-FIP (α = 0.99) 596 [36]** 90(7) Full interaction — [36]** —(8) Eurocode 2 (2004) 54 — —(9) CEB-FIP Model

Code 90 (CEB.1992) 93 — —

(10) Marti et al. 1998 27 — —

* If secondary cracks occur, they will have half this crack spacing** [ ] values from full interaction value

CEB-FIPLinear ascendingNon-linear

A B B'

Cke

6.85

01.5 2.59

(N/mm2)τ

δ (mm)∞

Figure 4. Different value of bond stress material properties

Pr = 10 kN

Sp = 554

Pr = 25 kN

Pr = Pr–cr = 36 kNƒct = 2.74 MPa

3

2.5

2

1

1.5

0.5

00 100 200 300 400 500 600

Distance from crack face (mm)

Con

cret

e st

ress

(M

Pa)

Figure 5. Tensile concrete stress along prism length for linear

ascending τ−δ

Distance from crack face (mm)

Con

cret

e st

ress

(M

Pa)

ƒct = 2.74 MPa3

B2.5

2

1

1.5

0.5

00 10 20 30 40 50 70 8060

Pr = Pr–cr = 36 kN

Pr = 13.8 kN

Sp = 7255

Figure 6. Tensile concrete stress along prism for CEB-FIP τ−δ

Row 1. The primary crack spacing is 554 mm and theload to cause this crack is 36 kN which is virtually thesame load as that from the full interaction analysis Eqn28 given in Row 7. The reinforcement load has toincrease substantially from 36 kN to 103 kN to formsecondary cracks. The non-linear results in Row 2 aresimilar to the linear ascending results in Row 1. TheCEB-FIP Model Code 90 (CEB 1992) bond-slip modelrecommends at value of α = 0.4 as plotted in Figure 4.The results are given in Row 3 where the primarycrack spacing is 72 mm, the primary crack load is 36 kN(as this analysis uses the full-interaction results in Eqn28 and the secondary crack load is 447 kN. The value ofα is gradually increased in Rows 4 and 5 where it can beseen that this reduction in bond stiffness causes anincrease in the primary crack spacing but a reduction inthe secondary crack load. When α → 1 in Row 6, thebond-slip properties tend to that of the linear ascendingin Figure 4 so the results in Row 6 in Table 2 tend to thelinear ascending results in Row 1. The empirical crackspacing in Rows 8 and 9 are similar to that in Row 3which uses the recommended CEB-FIB Model Code 90(CEB.1992) bond model. These results emphasise theimportance of the bond properties on crack spacing andwidths.

8. CONCLUSIONSGeneric mechanics based models have been developedfor various idealised bond characteristics to predictcrack spacings, crack widths and the load to causeprimary cracks, secondary cracks and subsequent cracksfor short term loads. A comparison between thecontrolling parameters from the mechanics models andthose from empirical models shows that the empiricalresearch has identified the major parameters that affecttension stiffening but that the mechanics equations aretoo complex to be derived empirically. It is suggestedthat this research provides an in-depth understanding oftension stiffening and the random nature of cracking andprovides the fundamental mechanics parameters thatcould be calibrated experimentally to develop moreaccurate design rules. The research also shows how theformation and behavior of primary cracks andsubsequent cracks are different due to differentboundary conditions.

ACKNOWLEDGEMENTSThis research was supported by the Australian

Research Council Discovery grant DP0985828 “Aunified reinforced concrete model for flexure andshear”. The first author also thanks the UniversitiTeknologi Malaysia and the Ministry of HigherEducation of Malaysia for financial support.

REFERENCESBeeby, A.W. and Scott, R.H. (2005). “Cracking and deformation of

axially reinforced members subjected to pure tension”, Magazine

of Concrete Research, Vol. 57, No. 10, pp. 611−621.

Bischoff, P. (2003). “Tension stiffening and cracking of steel fiber-

reinforced concrete”, Journal of Materials in Civil Engineering,

ASCE, Vol. 15, No. 2, pp. 174–182.

Bischoff, P. (2005). “Reevaluation of deflection prediction for

concrete beams reinforced with steel and fiber reinforced polymer

bars”, Journal of Structural Engineering, ASCE, Vol. 131, No. 5,

pp. 752–767.

CEB (1992). CEB-FIP Model Code 90, Thomas Telford, London, UK.

Eligehausen, R., Popov, E.P. and Bertero, V.V. (1983). Local Bond

Stress-Slip Relationship of Deformed Bars Under Generalized

Excitations, Report no. UCB/EERC-83/23, Earthquake

Engineering Research Center, University of California, Berkeley,

CA, USA.

Eurocode-2 (2004). Eurocode 2: Design of Concrete Structures-Part

1: General Rules and Rules for Buildings, European Committee

for Standardization, Brussels, Belgium.

Chan, H.C., Cheung, Y.K. and Huang, Y.P. (1992). “Crack analysis

of reinforced concrete tension members”, Journal of Structural

Engineering, ASCE, Vol. 118, No. 8, pp. 2118–2132.

Choi, C.K. and Cheung, S.H. (1996). “Tension stiffening model for

planar reinforced concrete members”, Computers & Structures,

Vol. 59, No. 1, pp. 179–190.

Cosenza, E., Manfredi, G. and Realfonzo, R. (1997). “Behaviour and

modeling of bond of FRP rebars to concrete”, Journal of

Composites for Construction, ASCE, Vol. 1, No. 2, pp. 40–51.

Dai, J., Ueda, T. and Sato, Y. (2006). “Unified analytical approaches

for determining shear bond characteristics of FRP-concrete

interfaces through pull out test”, Journal of Advanced Concrete

Technology, Vol. 4, No. 1, pp. 133–145.

Gilbert, R.I. (2007). “Tension stiffening in lightly reinforced

concrete slabs”, Journal of Structural Engineering, ASCE,

Vol. 133, No. 6, pp. 899–903.

Goto, Y. (1971). “Cracks formed in concrete around deformed

tension bars”, ACI Journal Proceedings, Vol. 68, No. 4,

pp. 244–251.

Gupta, A.K. and Maestrini, S.R. (1990). “Tension stiffening model

for reinforced concrete bars”, Journal of Structural Engineering,

ASCE, Vol. 116, No. 3, pp. 769–790.

Haskett, M., Oehlers, D.J., Mohamed Ali, M.S. and Wu, C. (2009a).

“Rigid body moment-rotation mechanism for reinforced

concrete beam hinges”, Engineering Structures, Vol. 31, No. 5,

pp. 1032–1041.

Haskett, M., Oehlers, D.J., Mohamed Ali, M.S. and Wu, C.

(2009b). “Yield penetration hinge rotation in reinforced

concrete beams”, Journal of Structural Engineering, ASCE,

Vol. 135, No. 2, pp. 130–138.

Hegemier, G.A., Murakami, H. and Hageman, L.J. (1985). “On

tension stiffening in reinforced concrete”, Mechanical Materials,

Vol. 4, pp. 161–179.





Jiang, D.H., Shah, S.P. and Andonian, A.T. (1984). “Study of the

transfer of tensile forces by bond”, ACI Journal Proceedings,

Vol. 81, No. 3, pp. 251–259.

Kong, K.L., Beeby, A.W., Forth, J.P. and Scott, R.H. (2007).

“Cracking and tension zone behaviour in reinforced concrete

flexural members”, Proceedings of ICE: Structures and

Buildings, Vol. 160, No. 3, pp. 165–172.

Lee, G.Y. and Kim, W. (2008). “Cracking and tension stiffening

behaviour of high strength concrete tension members subjected

to axial load”, Advances in Structural Engineering, Vol. 11,

No. 5, pp. 127–137.

Marti, P., Alvarez, M., Kaufmann, W. and Sigrist, V. (1998).

“Tension chord model for structural concrete”, Structural

Engineering International, Vol. 8, No. 4, pp. 287–298.

Mirza, S.M. and Houde, J. (1979). “Study of bond stress-slip

relationships in reinforced concrete”, ACI Journal Proceedings,

Vol. 76, No. 1, pp. 19–46.

Mohamed Ali, M.S., Oehlers, D.J., Griffith, M.C. and Seracino, R.

(2008a). “Interfacial stress transfer of near surface-mounted

FRP-to-concrete joints”, Engineering Structures, Vol. 30, No. 7,

pp. 1861–1868.

Mohamed Ali, M.S., Oehlers, D.J. and Griffith, M.C. (2008b).

“Simulation of plastic hinges in FRP plated RC beams”, Journal of

Composites for Construction, ASCE, Vol. 12, No. 6, pp. 617–625.

Muhamad, R., Mohamed Ali, M.S., Oehlers, D.J. and Sheikh, A.H.

(2011a). “Load-slip relationship of tension reinforcement in

reinforced concrete members”, Engineering Structures, Vol. 33,

No. 4, pp. 1098–1106.

Muhamad, R., Oehlers, D.J. and Mohamed Ali, M.S. (2011b)

“Discrete rotation deflection of RC beams at serviceability”,

Proceedings of ICE: Structures and Buildings. (in press)

Oehlers, D.J., Liu, I.S.T. and Seracino, R. (2005). “The gradual

formation of hinges throughout reinforced concrete beams”,

Mechanics Based Design of Structures and Machines, Vol. 33,

No. 3–4, pp. 373–398.

Oehlers, D.J., Mohamed Ali, M.S., Haskett, M., Lucas, W.,

Muhamad, R. and Visintin, P. (2011a). “FRP reinforced concrete

beams – a unified approach based on IC theory”, Journal of

Composites for Construction, ASCE, Vol. 15, No. 3,

pp. 293–303.

Oehlers, D.J., Muhamad, R. and Mohamed Ali, M.S. (2011b).

“Serviceability flexural ductility of FRP and steel RC beams: a

discrete rotation approach”, Construction and Building

Materials. (submitted)

Rizkalla, S.H. and Hwang, L.S. (1984). “Crack prediction for

members in uniaxial tension”, ACI Journal Proceedings, Vol. 81,

No. 6, pp. 572–579.

Somayaji, S. and Shah, S.P. (1981). “Bond stress versus slip

relationship and cracking response of tension members”, ACI

Journal Proceedings, Vol. 78, No. 3, pp. 217–225.

Seracino, R., Raizal Saifulnaz, M.R. and Oehlers, D.J. (2007).

“Generic debonding resistance of EB and NSM plate-to-concrete

joints”, Journal of Composites for Construction, ASCE, Vol. 11,

No. 1, pp. 62–70.

Tastani, S.P. and Pantazopoulou, S.J. (2010). “Direct tension pullout

bond test: experimental test”, Journal of Structural Engineering,

ASCE, Vol. 136, No. 6, pp. 731–743.

Visintin, P., Oehlers, D.J., Wu, C. and Haskett, M. (2012). “A

mechanics solution for hinges in RC beams with multiple

cracks”, Engineering Structures, Vol. 36, No. 3, pp. 61–69.

Warner, R.F., Foster, S.J. and Kilpatrick, A.E. (2007). Reinforced

Concrete Basics: Analysis and Design of Reinforced Concrete

Structures, Pearson Education, Australia.

Wu, H.Q. and Gilbert, R.I. (2008). An Experimental Study of

Tension Stiffening in Reinforced Concrete Members under Short

Term and Long Term Loads, Report No. R-449, University of

New South Wales, Sydney, Australia.

Wu, H.Q. and Gilbert, R.I. (2009). “Modelling short-term tension

stiffening in reinforced concrete prisms using a continuum-based

finite element model”, Engineering Structures, Vol. 31, No. 10,

pp. 2380–2391.

Wu, Z., Yoshikawa, H. and Tanabe, T. (1991). “Tension stiffness

model for cracked reinforced concrete”, Journal of Structural

Engineering, ASCE, Vol. 117. No. 3, pp. 715–732.

Wu, Z., Yuan, H. and Niu, H. (2002). “Stress transfer and fracture

in different kinds of adhesive joints”, Journal of Engineering

Mechanics, ASCE, Vol. 128, No. 5, pp. 562–573.

Yankelevsky, D.Z., Jabareen, M. and Abutbul, A.D. (2008).

“One-dimensional analysis of tension stiffening in reinforced

concrete with discrete cracks”, Engineering Structures, Vol. 30,

No. 1, pp. 206–217.

Yuan, H., Teng, J.G., Seracino, R. and Wu, Z.S. (2004). “Full range

behaviour of FRP-to-concrete bonded joints”, Engineering

Structures, Vol. 26, No. 5, pp. 543–691.

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