1 INTRODUCTION

In the present paper a multidirectional model of bond is considered as a basis for the study of the as-sociated mechanical problem, and for interpretation of the experimental evidence. The model comprises two separate components, namely: (i) A longitudinal component that, for a given input axial bar strain εo at the loaded end, reproduces the attenuation of bar stresses along an anchorage. Phe-nomena occurring along the anchorage length, such as debonding, yield penetration and bond stress redi-stribution with progressive anchorage failure, are explicitly considered. (ii) A transverse component that establishes local bond strength supported by the concrete cover and by passive confinement acting transversely on the anchorage. Cover splitting, fracture energy release and concrete ductility under confinement are the primary behavioral modes considered.

The two components of the model are coupled through frictional and kinematic requirements that relate shear force with transverse pressure on the bar and translation (longitudinal bar slip) with concrete radial displacement respectively. Analytical trends and sensitivities of the model are correlated (such as, the apparent reduction of average bond strength with increasing development length, and the process of yield penetration in yielding anchorages). The fol-lowing sections present derivation of the model components and the relevant coupling relations, as well as corroboration with experimental evidence through correlation of specific tests.

2 LENGTHWISE ATTENUATION OF STRESS

Experimental literature usually reports values for bond strength obtained by averaging the applied load over the contact surface of the bar. An inconsistent result of this simplification is an apparent reduction of bond strength with increasing anchorage length, Lb, whereas constant bond over Lb implies a linear variation of axial bar stress, which may only be ac-ceptable if the bar has not yielded or debonded from the surrounding media. The approximation is nearly valid if the bonded length Lb is short, but it becomes increasingly inaccurate with longer embedded lengths due to rapid attenuation of bar stresses and strains from the loaded end values. For sufficiently large bar strains at the critical section, a part of the bar near the loaded end is detached from concrete, so that no bond stress may develop over that seg-ment. This is the region of debonding for elastic brit-tle bars (FRP bars) or the region of yield penetration for yielding steel bars. Thus, the assumption that constant bond stress develops over the available an-chorage length leads to a conservative estimate of the average bond strength fb

ave as compared with the maximum value fb

max that develops locally. Variable fb

max represents a characteristic property of the inter-face between bar and concrete cover and is indepen-dent of the available anchorage length (Tastani & Pantazopoulou 2006).

The basic equations that describe force transfer in the lengthwise direction from the bar to the sur-rounding concrete through bond are derived from

Modelling reinforcement-to-concrete bond

S.P. Tastani Democritus University of Thrace, Xanthi, Greece

S.J. Pantazopoulou University of Cyprus, Nicosia, Cyprus (on unpaid leave from Democritus University of Thrace, Greece)

ABSTRACT: The distribution of bond on the lateral surface of steel reinforcement embedded in concrete is explored through systematic solution of the governing field equations of the associated mechanical problem. By separating the variables, the state of stress in the concrete surrounding the bar is represented by coupling two independent solutions, each describing attenuation of bond stresses either in the longitudinal or in the radial directions of the cover respectively. Kinematic considerations are used to couple longitudinal slip with the radial translation of the cover, whereas the corresponding stress components developing along the lateral bar surface over the anchorage (radial pressure and bond stress) are related through a frictional relationship. Using the derived solution, various experimentally documented trends are reproduced analytically and inter-preted. These include the processes of debonding and yield-penetration, and the sensitivity of development capacity to important design variables such cover to bar diameter ratio, concrete strength, bar yield strength, bar hardening characteristics and confinement.

force equilibrium applied to an elementary bar seg-ment of length dx and from the compatibility be-tween bar slip, axial bar strain ε and concrete strain εc over dx, namely, (Filippou et al. 1983):

( ) ( ) εεε −≅−−=−= cbb dxdsfDdxdf / ;/4/ (1) where, f the axial stress of the bar, fb the local bond stress and s the relative slip of the bar with respect to the surrounding concrete. Terms in Equation 1 are related through the bond-slip law, fb = fb(s), and the bar material stress-strain relationships, f = f(ε). Solu-tion of Equation 1 is possible through exact integra-tion, resulting in closed form solutions for the state of stress and strain along the anchorage.

2.1 Solution along a straight bar anchorage Solution of Equation 1 requires that the general form of the constitutive relations of the bar and the local bond-slip law are known. A linear elastic – perfectly plastic local bond-slip relationship is adopted (for the case (iv) below regarding an elasto-plastic bar after yielding, the solution requires also a plateau of residual bond strength, fb

res). Figure 1 (a) shows the interrelation between the assumed local bond-slip law, and the corresponding average local bond-slip relationship reduced from test data: fb

max is the cha-racteristic bond strength, fb

ave is the experimental av-erage bond strength (over the available anchorage length), sy is the slip value at the end of the elastic bond region, and su is the slip value at failure of the local bond mechanism. The plateau in the local bond-slip law implies sustained bond strength; to be measured it requires redundancy in the anchorage (availability of longer anchorages to enable force re-distribution before failure). Also the reinforcing bar stress-strain relationship is considered elastic with brittle failure for FRP bars, and elasto-plastic with hardening for steel reinforcement (Fig. 1b).

2.2 Distributions along Lb of a linear elastic bar The solution for elastic bars is valid for the ascend-ing branch of the stress strain law of steel bars (ε≤εsy) and for the entire range of strains to failure for FRP bars (ε≤εfu). (i) For the elastic part of the bond-slip law (s≤sy, Fig. 1a), bar normal strain, slip and bond stress distribu-tions over the available anchorage length (x⊆(0, Lb)) are given by Equation 2, (Fig. 2a):

( ) fusyLxx

Lo b

bee

ex εεεε ωωω

ω or1

)( 22 ≤−

−= −−

− (2a)

( ) yLxx

Lo see

exs b

b≤+

−= −−

−ωωω

ωωε 2

2 )1()( (2b)

( ) maxmax )(/)( bybb fxssfxf ≤⋅= (2c)

where ω=[4fbmax/(E·Db·sy)]0.5. Variable εο is the axial

strain at the loaded end of the anchorage, and E is the modulus of elasticity of the bar. The relation be-tween fb

ave and fbmax is obtained from force equiva-

lence of the two distributions (the assumed uniform and the actual given by Eq. 2c):

max2

0

)(1b

by

oL

bb

aveb f

Lsdxxf

Lf

b

ωε

== ∫ (3)

The bar axial strain at the loaded end, εo = εel(i) is the

limit value beyond which the bond mechanism en-ters the state of plastification (i.e., yielding of bond) over a length lp, while the bar remains elastic. There-fore, variable εel

(i) is directly related with the slip magnitude sy in Figure 1, and may be calculated from Equation 2b after substitution of s(x=0)=sy as:

b

b

L

L

yi

el ees ω

ωωε 2

2)(

11

−

−

+

−= (4)

(ii) If the available bond length, Lb, is sufficient then the bar may sustain a strain value higher than εel

(i). In that case, the maximum bond stress may reach the characteristic strength value, fb

max over a length of bond plastification, lp (Fig. 2b). The complete solu-tion of Equation 1 over Lb comprises two segments: - Distributions over the length lp (for 0≤x≤lp):

( ) xEDf

xb

bo

max4−= εε (5a)

( ) ( ) ( )( ))(5.0 iielpy xxlsxs εε +−+= (5b)

max)( bb fxf = (5c)

where, εel(ii) is the reduced value of bar strain as

compared with the εο at the loaded end, now occur-ring at the end of the bond plastification region, lp: εel

(ii)=εo-4fbmax/(EDb)· lp.

- Distributions over the remaining anchorage length, Lb-lp (for lp≤x≤Lb)). (These are obtained from the elastic solution (Eq. 2) upon substitution of x-lp and Lb-lp in lieu of x and Lb).

( ) ( ))(2

)(2)()()(

1 pb

pbpp

lL

lLlxlxiiel

e

eex

−−

−−−−−

−

−⋅=

ω

ωωωεε (6a)

( ) ( )( ))(2

)(2)()()(

1 pb

pbpp

lL

lLlxlxiiel

e

eexs

−−

−−−−−

−

+⋅= ω

ωωω

ω

ε (6b)

( ) maxmax )(/)( bybb fxssfxf ≤⋅= (6c)

The length of plastification lp may be estimated if continuity of strain and slip are considered at x=lp (by combining Eqs. 5a with 6b and after substitution of s(x=lp)=sy) as,

Figure 1. a) Local (grey line) and measured average local bond - slip law (black line). b) Stress-strain law of reinforcing bar.

b)

fsy

εfu εsu

Es

εsy

Ef

ffu steel

FRP

ε

Esh

a)

s2 ss1

sy su

fbave

fb fbmax

fbres

Figure 2. Elastic bar response while bond-slip law a) remains elastic (case (i)), b) enters in the plastification region (case (ii)) and c) at debonding failure after bond plastification (case (iii)). d) Plastic bar response with bond plastification (case (iv)).

Lb x

εo

sf

fbe

sο

fb

εo a)

εo

Lb x

sο

fbmax

lp

sf

fbe

εel(ii) εo

b) Lb

lr

x

lp

sdebsy su

sf

fbmaxfb

e

εo,u

εel(iii) εo,u

c)

lr lp

εo ≥ εsy

sdebsy su

sf

fbmaxfb

e

Lb xεo

εel(iv) εsy

fbres

d)

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

+−= )(

)(

ln21

iiely

iiely

bps

sLl

εω

εωω

(7)

At stage of bond plastification Equation 3 for the re-lationship fb

ave and fbmax is re-written as:

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⋅+=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛+= ∫ ∫ 2

)(max

0

max )(1ω

ε

y

iiel

pb

bpl bL

plbb

b

aveb s

lL

fdxxfdxfL

f (8)

The end of bond plastification is followed by the onset of debonding, i.e. the end of interaction of bar-concrete. This phase is marked by excessive slip. In Figures 1 and 2, parameter su corresponds to the end of the plastic branch in the bond - slip law. By substitution of s(x=0) = su in Equation 5b, the criti-cal bar strain εο,u at the loaded end, associated with the onset of debonding, is: εo,u=2(su-sy)/lp-εel

(ii). (iii) Debonding limits the load carried by the bar, i.e. once the debonding process begins, the bar cannot sustain a strain greater than εο,u. While the bar is be-ing pulled out, the strain magnitude εο,u propagates in the anchorage over a length lr, referred to hereon, as debonded length. This process is followed by ex-cessive increase of slip at the loaded end, and by a commensurate reduction of the active anchorage length, to Lb-lr. Using Equations 5 and 6 and by substitution of variables x, Lb and lp with x-lr, Lb-lr and lp-lr respectively, revised expressions for strain and slip distributions are obtained which describe the state of stress in the anchorage after debonding over a length lr (Fig. 2c). Note that slip at the critical section is estimated from integration of strains over Lb as follows: sdeb=su+εo,u·lr.

Regarding the bond stress distribution it is note-worthy that at this stage it tends towards a uniform distribution over the active anchorage length, Lb-lr with a stress-intensity nearly equal to fb

max while failure is anticipated as there are no strength reserves in the anchorage (Fig. 2c).

The expression for the length of bond plastifica-tion is derived as follows: given εo,u at the entrance of the anchorage and assuming values for the sdeb convergence is accomplished when continuity of strain and slip are satisfied at the knee points of the distributions (Fig. 2c). Thus, for the calculation of

the lp, the quadratic equation should be solved:

b

byu

b

buouop ED

fss

EDf

lmaxmax

2,,)2,1(

4)(

8⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−−±= εε (9)

Acceptable value for lp is the result of Equation 9 that also satisfies the requirement of lp+lr≤Lb.

2.3 Distributions along Lb of an elasto-plastic bar Solution of Equation 1 for an elasto-plastic steel bar is explored only after yielding, since the preceding section fully describes the bar elastic behavior. The bar strain at the onset of yielding is denoted by εsy; Esh is the hardening modulus of the stress-strain rela-tionship in the post-yielding regime. (iv) The last case examined concerns yield penetra-tion of the steel bar inside the anchorage with simul-taneous plastification of bond (Fig. 2d). The length of yield penetration is now denoted by lr. In the segment (0, lr) the bond stress is equal to fb

res. Also, the distribution of strains is linear ranging from ε(x=0)=εo at the loaded end, to the value ε(x=lr)=εsy at the end of the yielded region. Slip at each point is obtained from integration of strains from the point considered to the end of the anchorage. A necessary requirement for this case to occur is strain εo exceed-ing the limit value of Equation 4. The strain, slip and bond stress expressions go-verning this problem in the three distinct regions are: - Over the debonded length lr (for 0≤x≤ lr):

( ) xDE

fx

bsh

resb

o4

−= εε (10)

( ) ( ) ( )( ) resbbsyru ffxxlsxs =+−+= ;5.0 εε (11)

- Over the length lp where bond has exceeded the plasticity limit, thus fb(x)=fb

max (for lr≤x≤lr+lp):

( ) ( )rbs

bsy lx

DEf

x −−=max4

εε (12)

( ) ( ) ( )( ))(5.0 ivelpry xxllsxs εε +−++= (13)

- Over the bonded length Lb-lr-lp (for lr+lp≤x≤Lb):

( ) ( ))(2

)(2)()()(

1 rpb

rpbrprp

llL

llLllxllxivel

e

eex

−−−

−−−−−−−−

−

−⋅=

ω

ωωωεε (14)

( ) ( )( ))(2

)(2)()()(

1 rpb

rbrprp

llL

llpLllxllxivel

e

eexs

−−−

−−−−−−−−

−

+⋅= ω

ωωω

ω

ε (15)

Here the relationship between fbave and fb

max is:

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⋅++= 2

)(max1

ω

ε

y

ivel

pbres

brb

aveb

slffl

Lf (16)

In Equations 13-16 εel (iv) is the strain at x= lr+lp

i.e. the point where bond starts being elastic (Fig. 2d); it is calculated from Equation 12. Thus, yield penetration occurs over the lr where strain exceeds εsy; this is accompanied by a sudden increase of slip (Eq. 11) with a commensurate reduction of bond strength to fb

res over the yielded bar length.

3 TRANSVERSE ATTENUATION OF STRESS

Based on experiments, bond strength, fbmax, depends

on the concrete cover depth, the bar surface profile, and the presence of confinement along the anchor-age (Tastani & Pantazopoulou 2006, 2010).

To evaluate the deformation and stress state in the concrete surrounding a reinforcing bar the cover is modelled as a thick cylinder with the interior radius equal to that of the bar is used (Tepfers 1979, Panta-zopoulou & Papoulia 2001). An important difference between the classical thick-cylinder model (Tepfers 1979) and the displacement-controlled approach (Pantazopoulou & Papoulia 2001) is that whereas the former solves for the state of stress produced in the cover thickness by a uniform pressure acting on the internal boundary, the latter solves for the state of stress in the cover produced by a fixed radial dis-placement of the internal boundary. The basic form of the model’s result is a characteristic envelope re-lating radial stress and radial strain in the cover. This is a basic measure of the thick cylinder’s capac-ity to accommodate displacement of its inner boun-dary before failure.

3.1 The thick cylinder model The equations that control the boundary value prob-lem of a thick cylinder with internal radius Rb (bar

radius) and external radius Cc, comprising an aniso-tropic non-linear elastic material (i.e., a material having different moduli of elasticity and Poisson’s ratios in the three principal directions (z – r – θ, Fig. Aa) are summarized in the Appendix.

Given a radial displacement ur⎢r=Rb of the inner boundary, Eq. (A.4) is solved numerically to estab-lish the depth of crack penetration (or crack front, Rcr) and the resulting radial pressure (σr⎢r=Rb) applied on the bar lateral surface. The incremental solution uses simple uniaxial stress-strain models for con-crete in compression and tension to model material behavior in the radial and the hoop directions (also being principal stress and strain directions due to the geometric axi-symmetry of the problem). The radial dimension of the thick ring is discretized in Ν+1 equidistant nodes with N segments of constant length h=c/N, where c the clear cover. The first node, i=0 is at the internal boundary which is de-fined by the bar radius, and the last node is on the external boundary of the cover (Fig. Aa).

The radial pressure on the interior boundary σr,o may be estimated through equilibrium across a di-ametric plane of the thick cylinder idealization (de-noted here as σr,ο

eq, Fig. 3a, Eq. 17).

( ) ( )[ ]

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+

−=

++++=

=

−+

−

∑

h

uuE

ru

E

Rh

R

iriririr

i

iri

iriri

NNob

eqor

Ni

eqorb

211

....21

1,1,,,

,,

,,,

,1,1,,,

,,

θθθθ

θ

θθθθ

θ

ννν

σ

σσσσσ

σσ

(17)

Solution is performed incrementally: for the m-th increment in the amount of radial displacement ur,o

(m) the nodal displacements [ur] and the asso-ciated stresses are calculated. For the next incre-ment, m+1, values for the secant moduli Εr and Eθ are estimated from the constitutive relations of con-crete in principal compression and tension (Fig. A), using the state of strain estimated in the m-th step.

Note that hoop stresses and strains (σθ, εθ) are tensile and are responsible for the radial cracking of the cover, whereas the radial stresses and strains (σr, εr) are compressive. Coincident directions of prin-cipal stress and strain and homogenized cracking are assumed. The behavior of concrete in compression is described as a function of compressive strain εc through the familiar parabolic compression stress-strain relationship (Fig. Ab), with initial modulus of Elasticity, Eco=2fck/εo; fck is the uniaxial compressive

σr,o σθ,o

fb fb

F σr,

hr

Bar segment

Figure 3. Equilibrium of a) radial and hoop stresses along a diametric plane of concrete cover and b) radial and bond stressesalong the bar. c) Sliding plane in pulled bar (DTP bond test that was failed by splitting (Tastani & Pantazopoulou 2010).

Sectional stress state of concrete

0 1 σθ,ο

σθ,2 σθ,N

σr,οeq

σr,N

σr,2

2 N

a) transverse crack

hr sr

θ

sliding plane

b) c)

strength and εο the strain at peak stress. The behavior of concrete in tension is assumed linear elastic prior to cracking (εθ≤εcr = 0.00015) with initial elastic modulus Eco and tensile strength obtained either from tests (cylinder splitting) or as fct=0.35~0.5√fck. After cracking the stress versus smeared-strain rela-tionship is defined by the FIP/CEB MC ’90 in terms of the fracture energy Gf and the maximum aggre-gate size dα (Fig. Ac).

4 LONGITUDINAL TO RADIAL RELATION

The relationship between radial stress σr,o and the imposed radial displacement ur,ο of the interior boundary of the thick cylinder is a characteristic property of the thick cylinder. The radial displace-ment is imposed by the ribs on the inner boundary of the thick concrete ring when the bar begins to slide in the longitudinal direction in response to an ap-plied axial load. The longitudinal and transverse components of bond mechanism are related through: - the frictional model for the stresses (Fig. 3b):

adhorb ff += ,max 2 σ

πμ (18)

- a geometric relationship for the translations: ori us ,⋅= α (19)

where μ is the coefficient of friction. The adhesion component of bond, fadh, is the shear capacity of the interfacial layer (estimated as 1MPa which disinte-grates for limited slip in excess of 0.02mm). Also, s is the bar slip at the loaded end, and ur,o the radial displacement of the cover’s interior boundary (bar surface). Parameter αi in Eq. (19) is an arithmetic coefficient that depends on the rib geometry and the stage of loading (ay for the yield state of bond and au for the end of bond plastification). Seeking a physi-cal significance for ai the kinematics of radial trans-lation of the cover necessary in order to accommo-date longitudinal translation of the lugs is considered in Figure 3c. According to experimental observation (Tastani & Pantazopoulou 2010) the effective angle, θ, of the front of the rib is defined by a plane of slid-ing that forms between the tip of each rib and the base of the preceding one; based on the geometry, the outwards translation needed for this plane to dis-

place forward by an amount of s, is: u=s/tanθ, there-fore, ai=tanθ=sr/hr, where sr the rib spacing and hr the rib height. Evidently, Equation 19 breaks down for s>sr because an infinitesimal increase of Δur,o enables uninhibited slipping of the bar.

Another critical variable of the model is the coef-ficient of friction μ, which is compromised by such parameters as the confining pressure σconf on the ex-ternal boundary of the thick cylinder and slip. The effects of σconf and s on μ have been previously in-vestigated by the authors (Tastani & Pantazopoulou 2010) based on experimental work with steel bars and data from literature (Malvar 1992). The main findings were: (i) there is a limit in the value of the normal pressure on the bar, in the range of 0.4fck, beyond which the coefficient μ degrades from the value of 1.2 to 0.9; here, a linear degradation pattern is assumed up to a normal pressure of fck. (ii) The value of μ decays almost linearly with increasing slip magnitude beyond the limit su, tending to the minimum value as s approximates sr.

Equation 19 is calibrated with the test results of Malvar (1992) for steel bars. Results obtained for the radial displacement ur,o of the interior boundary of the cover and longitudinal bar slip s are plotted in Figure 4 for two limit states of bond (i.e. yield and ultimate). Generally, sustained slip and radial dis-placement of the inner boundary increase with the applied normal pressure. Coefficient 1/αi in Equation 19 is defined by the slope of the average trend, shown in Figure 4 by the solid line, whereas the theoretical limit of sr/hr discussed earlier is shown by a dashed line: for bond at the onset of yielding, αy is chosen as 20 for the analyses of the sectional model below. After bond plastification it appears that slip is independent of radial displacement (ow-ing to the reduced resistance provided by the cover at that stage against sliding of the bar, as compared with the elastic stage of cover behavior); this point is associated with a radial translation approximately equal to the rib height in the case of splitting failure whereas in the case of pullout failure, due to pres-ence of confining pressure, the radial translation is much lower than hr.

Coupling of transverse with the longitudinal com-ponents of the model is possible if the limiting stress and displacement output values (σr,o, fb, ur,o, s) of the transverse model component are taken to correspond to the loaded end of the active anchorage length. Thus, the section of reference recedes towards the end of the anchorage as debonding propagates.

5 CORROBORATION WITH TESTS

In this section the proposed analytical model is cali-brated using data from a detailed parametric experi-mental investigation conducted by the authors on steel reinforcing bars (Tastani & Pantazopoulou 2010) with the objective to assess values for the im-portant parameters characterizing the milestone points of the local bond-slip law (Fig. 1a). An im-portant feature of this experimental program was that it was conducted using tension-pullout speci-

Figure 4. Correlation between radial displacement ur,o withslip s for the first attainment of strength (subscript y) and theend of the plastic branch of bond (subscript u) (Malvar 1992).

1/ αy

1/ (sr /hr) ≈13 σr,ο

hr ≈1mm

σr,ο

mens, a specimen form designed to alleviate any spurious contributions to bond arising from support conditions. Parameters of the experimental program were the related rib area (steel bars with nominal fR ratios 0.065 and 0.15) and the anchorage length (5Db and 12Db, Db=12mm).

Of the total 50 specimens tested, only those spe-cimen pairs that were designed to illustrate the effect of each parameter on the response data are consi-dered for model calibration in the following sec-tions. The primary response indices of specimens se-lected for further study are summarized in Table 1.

5.1 Model longitudinal component: correlation Two characteristic example cases of steel anchorag-es are considered for model calibration (Table 1). Figure 5 plots distributions of bar strain, slip and bond stress along the anchorage. Two stages are identified: the first corresponds to the plastic re-sponse of the bond-slip law (thin black line) whereas the second corresponds to the bar response just be-fore failure and after debonding has taken place (bold black line). The dashed grey line represents the experimental average bond strength, fb

ave, estimated over the nominal anchorage length, Lb.

Subgroups h(1.1)MnpA - h(1.1)LnpA (Fig. 5) had

identical parameters (rib height hr=1.1mm, no con-finement, same concrete A) apart from the anchor-age length (Lb

M=60mm and LbL=144mm). The model

converged to a characteristic bond stress-slip law with milestone values: fb

max=10MPa, sy=0.2mm; the ultimate slip magnitude su is affected by the longer Lb, thus the model solution results in su=0.23mm and su=0.33mm respectively (Table 1). Note here that the experimental value of s2 (Fig. 1a and Table 1) refers basically to the debonding slip, sdeb, and not to the termination of the plateau of the theoretical bond stress - slip law, su. With a longer anchorage length in case of subgroup h(1.1)LnpA as compared with h(1.1)ΜnpA, bond is plastified over a longer segment of the bar: 122mm for h(1.1)LnpA ascom-pared to 54mm for h(1.1)ΜnpA measured from the loaded end of the bar. The region with constant strain (Fig. 5) is associated with debonding failure rather than bar yielding. This is also confirmed by the maximum load sustained in both specimen groups considered which was lower than the yield force of the bar (εsy=0.26%), so that the bar re-mained in the elastic range of the stress-strain curve.

From the analysis cases presented thus far it is clear that: (i) In the limit case of bond plastification the bond stress distribution is non-uniform even in the cases of very small anchorage lengths (e.g. 5Db). The implication of this finding is that as the anchor-age length is increased, the difference between the bond stress values at the start and at the end of the available anchorage also increases (e.g. in the case of h(1.1)MnpA the difference is 1.3MPa while in the case of h(1.1)LnpA the difference is 3MPa for an in-crease in the anchorage by 7Db). (ii) In the limit case of debonding this difference is usually observed in the neighbourhood of the loaded end whereas bond plastification prevails in the remainder of the an-chorage length (i.e. fb(x=Lb-lr)=fb

max). Particular reference is made to those cases where

bar yielding was observed. This concerns two of the three specimens of subgroup h(0.5)LnpΑ. The analy-sis was conducted using as an average experimental bond strength the value of fb

ave =11.5MPa whereas

the bar material stress-strain relationship is assumed elastic – plastic with a mild hardening slope (Fig. 1b) with material properties: fsy=520MPa, fsu=580MPa, εsy=0.0026, εsu=0.01 and Esh=(fsu-fsy)/(εsu-εsy)=8.1GPa. The residual bond strength is assumed fb

res=20%fbmax, a value that is consistent

Table 1. Experimental and calibrated bond – slip limit valuesfor steel bars (units: mm, MPa. Specs. geometry: Lb/Db=5 for the S and M specs., Lb/Db=12 for the L specs., c/Db=2 for the S specs. and c/Db=3.7 for the M and L specs).

Specs. Exp. results Model results

fbave s1 s2 fb

max sy - su lp - lr

h(1.1)SnpA 6.7 0.1 0.2

7.5 0.14

- 0.15

53.6 -

5.6 7 0.2 0.28

6.6 0.12 0.21

h(1.1)MnpA 9.6 0.2 0.27

10 0.2 -

0.23

54.2 -

4.8 8.8 0.2 0.9 8 0.45 0.45

h(0.5)LnpA 12.5 0.5 4.26

13 0.4 -

0.56

120 -

24 10.6 0.5 0.95 8.75 0.26 0.71

h(1.1)LnpA 8.9 0.75 0.9

10 0.2 -

0.33

122 -

18.7 8.7 0.6 0.6 8.6 --- 0.29

Figure 5. Distributions of bar strain, slip and bond stress for specimen subgroups h(1.1)MnpA - h(1.1)LnpA.

fbave(Lb=5Db)

fbave(Lb=12Db)

L b=6

0mm

L b=1

44m

m

debonding (case (iii))

plastification (case (ii))

with the results of the analyses of the thick cylinder model for the cross-section of the anchorage. The anchorage response at two limit cases is plotted in Figure 6: onset of yielding at the loaded end with simultaneous bond plastification (thin black lines) and attainment of the strain at observed anchorage failure, εο,u

max=0.0052 where yield penetration has occurred over a length lr=24mm (=2Db) and bond plastification over the remaining active anchorage length (≈120mm, bold solid black lines). In the latter limit case the calculated slip magnitude at the loaded end is sdeb=0.65mm, i.e. a value similar to the expe-rimentally estimated s2 values (Table 1).

Note that for an infinitesimal increase of the strain at the loaded end, over the value of 0.0052, the an-chorage is led to failure with a sudden loss of load owing to a collapse of the bond mechanism since there are no strength reserves or possibility for redi-stribution of bond stress. Yield penetration over lengths exceeding Lb=12Db has been also reported by Bonacci (1994) and Bonacci & Marquez (1994) who noted that a small change in the anchorage length from 11Db to 12Db could convert the failure mode from combined slipping/pullout without yield-ing to yield penetration followed by pullout.

5.2 Model transverse component: correlation For the assessment of the cross-sectional model, the specimen subgroups h(1.1)MnpA are evaluated. Fig-ure 7 plots the radial pressure at the bar-cover inter-face resulting from applied radial displacement for each specimen group separately. From comparison of the displacement histories it is concluded that the two estimations of σr,o obtained using Equation 17

and the algorithm by Pantazopoulou & Papoulia (2001) –denoted here as eq. A- are not the same, the difference being attributed to the thick cover (i.e. c=3.7Db).

In estimating the bond stress through Equation 18 the more conservative estimation of σr,o is used. Us-ing Equations 18 and 19 the analytical relationship between bond stress and slip is obtained for the spe-cimens (in Fig. 7b are also given for comparison the relevant experimental average bond – slip curves). Based on the maximum pressure developed in the cover (Fig. 7a) the coefficient of friction was set to μ(h(1.1)MnpA)=1.18. The values of bond stress and slip at the onset of failure obtained from the trans-verse bond model coincide with sufficient accuracy both with the experimental measurements as well as with the analytical values obtained using the longi-tudinal bond model. For example, the average expe-rimental bond strength for the subgroup h(1.1)MnpA was fb

ave=9.2MPa with an analytical characteristic value (longitudinal component) fb

max=10MPa, whe-reas the sectional thick-cylinder model yielded a value of 9.8MPa (parameter αy was previously de-fined as 20). Similarly, experimental value of slip sustained at attainment of average bond strength was s1=0.2mm (Table 1), the analytical value at the initi-ation of bond plastification was sy=0.2mm (Fig. 5, Table 2), whereas the value obtained from the sec-tional model (Fig. 7) was sy=0.18mm.

6 CONCLUSIONS

A detailed analytical model was developed to interp-ret the various aspects of bond behavior. The model comprises two independent but mutually comple-mentary components: a “sectional” and a “longitu-dinal” model. The sectional model idealizes the state of stress in the concrete cover surrounding the bar using the thick-ring analogy and displacement con-trol. Longitudinal bond stress is related to the radial pressure on the inner boundary of the ring using a frictional relationship. The state of stress lengthwise along the anchorage is resolved by the “longitudinal model”, which represents the exact solution to the differential equation of bond, obtained through the assumption of a bilinear local bond stress-slip law; properties for this law are obtained through the “sec-tional model”. Apart from the frictional model to re-

Figure 7. a) Radial pressure – radial displacement plot (σr,o –ur,o) at the bar-cover interface and b) bond stress –slip curve (fb– s) for specimen h(1.1)MnpA.

model

b)

h1.1MnpA-1

a)

17

)

Figure 6. Distributions of strain, slip and bond stress for spec. h(0.5)LnpΑ where yield penetration has been observed before feature.

Plastic strain – yield penetration

εsy plastification (case (ii))

Yield penetration withdebonding (case (iv))

late stresses, the two models are also related by the kinematics imposed by the bar surface deformations as they slide through the cover.

The solution in the longitudinal direction distin-guishes several phases such as plastification of bond partial debonding along the anchorage (for elastic bars) or yield penetration (for steel bars) as failure propagates from the loaded to the free end. Bond stress is generally non-uniform along the anchorage, rendering the local bond strength, fb

max, greater than the average bond strength, fb

ave, which is reported from tests as the ratio of applied load to bar contact surface. It is relevant to note that the local bond stress-slip law described by the CEB/FIP Model Code ’90, had been obtained from pullout tests on short length steel bar anchorages.) It is concluded that: (i) the experimental value of fb

ave, obtained upon the assumption of uniform distribution general-ly does not coincide with the real situation, as represented by the exact solution of the governing equation in the longitudinal direction, whereby the bond stress values at the opposite ends of the an-chorage are at significant discrepancy. (ii) The expe-rimental average bond strength fb

ave is a more con-servative measure as compared with the characteristic value fb

max since it is estimated without considering that near failure, the anchorage is par-tially debonded, a fact that reduces the active devel-opment length. (iii) The experimental slip value, s2, at the end of the plateau in the local bond-slip law (Fig. 1a), as shown through the analytical investiga-tion, corresponds to the measured slip at the onset of debonding, sdeb. The value of sdeb comprises partly the slip of the bonded length, su, and partly the bar extension that occurs over the debonded part of the anchorage. The proposed model mitigates some of the empiricism attached to experimentally derived bond stress–slip laws. It is also valuable for estima-tion of member’s rotation capacity through consid-eration of yield penetration effects.

7 REFERENCES

Bonacci J. 1994. Bar yield penetration In monotonically loaded anchorages, ASCE J. of Str. Eng. 120(3): 965-986.

Bonacci J., Marquez J. 1994. Tests of yielding anchorages un-der monotonic loadings, ASCE Str. Eng. 120(3): 987-997.

CEB-FIP 1993. Model code for structural concrete (FIP/CEB MC’90). Thomas Telford Pubs., London.

Filippou F., Popov E., Bertero V. 1983. Modeling of R/C Joins under Cyclic Excitations, ASCE Str. Eng. 109(11):2666-84.

Malvar J. 1992. Bond of reinforcement under controlled con-finement, ACI Materials Journal 89(6): 593-601.

Malvar L. J., Cox J. V., Cochran B. K. 2003. Bond between Carbon Fiber Reinforced Polymer Bars and Concrete. I: Experimental Study, ASCE Comp. for Constr.7(2):154-163.

Pantazopoulou S.J. and Papoulia K.D. 2001. Modeling cover-cracking due to reinforcement corrosion in r.c structures, ASCE J. of Engineering Mechanics 127(4): 342 -351.

Tastani S.P., Pantazopoulou S.J. 2006. Bond οf G-FRP Bars in Concrete: Experimental Study and Analytical Interpreta-tion, ASCE J. of Comp. for Construction 10(5): 381-391.

Tastani S.P., Pantazopoulou S.J. 2010. Direct Tension Pullout Bond Test: Exp. Results, ASCE Str. Eng. 136(6):731-743.

Tepfers R. 1979. Cracking of concrete cover along anchored deformed reinf. bars, Mag. Concr. Research 31(106): 3-12.

8 APPENDIX: THE THICK CYLINDER MODEL OF THE COVER CROSS SECTION

Field Equations in polar coordinates:

0=−+ θσσ

σ rdr

d rr

(A.1)

ru

drdu rr

r == θεε , (A.2)

For the anisotropic elastic material these are resolved in terms of σz as follows:

( )

( )⎪⎪⎩

⎪⎪⎨

⎧

−+

++−

=

−+

++−

=

zrr

zrzrrrr

rr

zrr

rzzrrrr

rrr

EE

EE

σνννννενε

ννσ

σνννννενε

ννσ

θθ

θθθθθ

θθθ

θθ

θθθθθ

θθ

111

111

(A.3)

Upon substitution of Eqs. (A.3) and (A.2) in Eq. (A.1) the basic differential equation for the radial displacement of the problem, ur, is obtained:

( )

( )( ) ( )[ ]

( ) 01

11

1

2

2

2

=−−

+++−−+

+−

zrrzr

zrzrzzrzrrr

rrzz

Er

u

EEdr

dur

dr

udE

νν

ννννννν

νν

θ

θθθθθθ

θθ

(A.4)

This is solved numerically after consideration of bound-ary conditions at r=Rb and r=Cc.

Solution is obtained for plane stress whereas the Pois-son’s effect associated with stress or strain in the z-direction are neglected. Thus, σz=0, νθz=νrz=0, νzθ = νθz Ez / Eθ =0 and νzr=νrz Ez / Er =0, where Er, Eθ and Ez are the secant moduli of concrete in compression (r) and in ten-sion (θ, z). The non-zero Poisson ratios νrθ and νθr are related as νθr = νrθ Eθ /Er. Solution follows the numerical algorithm presented by Pantazopoulou & Papoulia (2001).

Figure A. a) Discretization of the thick cylinder in the radial di-rection. Constitutive laws for concrete: b) compression and c)tension (Pantazopoulou & Papoulia 2001).

Ecο f

εo

Er σr

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−=

2

co

2

co

2ckc ε

εεε2fσ

εr

b)

Εcο

εu

fct

0.15fct

εk εcr

σθ

c)

a)

Rb ri

c

0

h

1 2 N N-1