Composite Structures 92 (2010) 2400–2405

Contents lists available at ScienceDirect

Composite Structures

journal homepage: www.elsevier .com/locate /compstruct

Prediction of tensile capacity based on cohesive zone model of bond anchoragefor fiber-reinforce dpolymer tendon

Fei Li *, Qi Lin Zhao, Hao Sen Chen, Jin Quan Wang, Jing Hui DuanEngineering Institute of Engineer Corps, PLA University of Science and Technology, Nanjing 210007, China

a r t i c l e i n f o

Article history:Available online 11 March 2010

Keywords:Bond FRP tendon anchorageCZMTension capacityAnalytical solution

0263-8223/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.compstruct.2010.03.005

* Corresponding author. Tel.: +86 13951035941.E-mail address: lifei609@163.com (F. Li).

a b s t r a c t

In this paper, analytical solutions based on a cohesive zone model (CZM) are developed for the bond fiber-reinforce dpolymer (FRP) tendon anchorage under axial load. With bilinear cohesive laws, the analyticalsolutions of tensile capacity of anchorages are derived. The concept of the minimum relative interfacedisplacement sm is introduced and used as the fundamental variable to express all other parameters, suchas external tensile load. Experimental and analytical results show that the thickness of the anchoringmaterial is main factor affecting tensile capacity. The characteristic bond strength depends mainly onthe properties of the bonding agent-anchoring material, the geometry and surface conditions of the ten-don, and the radial stiffness of the confining medium. A comparison of the calculated and experimentalresults showed good agreement. Formulas based on fracture energy of the tension load capacity derivedin the present work can be directly used in the design of FRP tendon anchorage.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

As a member of the composite material family, increasingimportance is attached to FRP tendon in civil engineering. Com-pared with prestressing steel, the advantages of FRP tendon are:low weight; high tensile strength; high corrosion resistance, etc.Due to the low shear strength and transverse compressivestrength, the establishment of an anchorage system with high per-formance based on the knowledge of the behavior both FRP tendonthemselves and the tendon-anchorage assemblage is essential forthe introduction of FRP tendon into civil engineering. There aremany different kinds of anchorages developed over the world;these anchorages are classified with respect to force transfer mech-anism: clamp anchorage, bond anchorage and wedge-bondanchorage. Bond anchorage can transfer the load by the bond stressand has more reliable anchorage performance than anotheranchorage systems, using this kind of anchor can avoid damageto the tendon [1]. For the same kind of FRP tendon, bond anchoragehas better performance in term of mechanical characteristics thanclamp anchor and wedge-bond anchorage [2]. So the bondinganchorage is more suitable application to civil engineer.

For bond FRP tendon anchorage, different failure modes areencountered depending on the embedment length. when theembedment is small, a pull-out failure can be observed. withembedment increasing, a FRP tendon failure occurs [3]. Differentfailures corresponds to different tensile capacity. In this paper,

ll rights reserved.

the tensile capacity corresponding to pull-out failure is discussed.The tensile capacity can be obtained directly by pull-out experi-ments [4,5]. However, the experimental test of tensile capacityseems to be costly and complicated. It is essential to investigateit by analytical theory. In the field of the analytic theory, The meth-od based cohesive zone model (CZM) have been extensively imple-mented in the finite element analysis for investigating the interfacefracture behavior of structures or specimens under mode I (tensionloading), mode II (in-plane shear loading), and mixed mode I/II.Some analytical solutions for the adhesively bonded pipe joints un-der torsional loading and FRP plate adhesive joint have also beendeveloped [6-8]. However, there are very few studies focusing onthe analytical solution for FRP tendon anchor in the literature. Inthe present study, the cohesive zone model based on analyticalsolutions are developed to solve debonding (fracture) problemfor the FRP tendon anchors under axial load. A typical non-linearmodel–bilinear cohesive law is derived in order to obtain axial loadcapacity as a function of the interface fracture energy. Compared tothe complicated results of most previous analytical solutions, thesimple expressions of axial failure load can be directly used forpractical FRP tendon anchor design.

The origins of the concept of cohesive zone model (CZM) goback to the work of Barenblatt and Dugdale [9,10]. CZM hasevolved as a preferred method to analyze fracture problems inmonolithic and composite material systems not only because itavoids the singularity but also because it can be easily imple-mented in a numerical method of analysis such as in finite elementmodeling. Therefore, various CZMs have been proposed to investi-gate the fracture process in a number of material systems including

s

τ mτ

resτ

'1s '

2s 1s 2s

1k

2k

s

τ mτ

linear elastic branch linear soften branch

(a) (b)

Fig. 1. (a) Non-linear constitutive law for FRP tendons bond anchorage (b) the bilinear cohesive zone model.

FRP tendon

equivalent lay of steel tube and anchoring material

FRP tendon

anchoring materialsteel tube

Fig. 2. Scheme of the translation of anchoring material and steel tube.

F. Li et al. / Composite Structures 92 (2010) 2400–2405 2401

fiber reinforced polymer composites, metallic materials, ceramicmaterials, concrete materials, and biomaterial systems. All of themstart from the assumption that one or more interfaces can be de-fined, where crack propagation is allowed by the introduction ofa possible discontinuity in the displacement field. Various cohesivezone models (cohesive laws) were proposed [11–14]. The main dif-ference between these models lies in the shape of the traction–dis-placement response, and the parameters used to describe thatshape [8]. The non-linear cohesive zone model for FRP tendonsbond anchorage can be expressed as follows [5], as illustrated inFig. 1a:

ssm¼ 1� exp � s

s01

� �� �2

; s � s01

s ¼ sm �sm � sres

s02 � s01ðs� s01Þ; s01 � s � s02

s ¼ sres; s � s02 ð1Þ

where s01; s02, sm and sres = parameters based on the curve-fitting of

the actual data. Different kinds of FRP tendons and anchoring mate-rials correspond to different parameters. The equivalent bilinearcohesive zone model which is illustrated in Fig. 1b is discussed inthis paper; a typical bilinear cohesive zone model consists of a lin-ear elastic branch and a linear softening branch, as illustrated inFig. 1b. As discussed before, the three-parameter cohesive zonemodel (CZM) may well describe the interface debonding process.The three parameters in the bilinear elastic cohesive law are linearelastic branch interface fracture G1, a linear softening branch inter-face fracture G2 and cohesive shear strength sm. Note that thisequivalent bilinear CZM is based on identical interface fracture en-ergy (the area under the slip–stress curves) and identical cohesiveshear strength sm with the actual non-linear constitutive law, asillustrated in Fig. 1a.

In the linear elastic branch:Z s01

0sm � 1� s

s01� 1

� �2" #

dx ¼ 12

s2m

k1ð1Þ

where k1 is the interface stiffness as illustrated in Fig. 1b; with Eq.(1), we can further obtain:

k1 ¼4sm

3s01ð2Þ

So in the linear elastic branch, the relationship between shear stresss and displacement s can be expressed:

s ¼ k1s ð3Þ

In the linear softening branch:

12ðsm þ srÞ � ðs02 � s01Þ ¼

12

s2m

k2ð4Þ

where k2 is the interface stiffness as illustrated in Fig. 1b, with Eq.(4), we can further obtain:

k2 ¼s2

m

ðsm þ srÞ � ðs02 � s01Þð5Þ

So in the linear softening branch, the relationship between shearstress s and displacement s can be expressed:

s ¼ �k2ðs� s1Þ þ sm ð6Þ

where s1 ¼ smk1

; s2 ¼ smk2þ s1.

Before the derivations, the following assumptions are made inthe current study:

1. The tensile stress of FRP tendon and steel tube are uniformlydistributed in thickness direction.

2. Small deformation is considered.3. The thickness of steel tube is considered, the anchoring material

and steel tube are taken as integer, as shown in Fig. 2, the stiff-ness EA of the integer is calculated.

4. The effect of the circumferential stress and normal stress areignored.

2. Calculation of equivalent stiffness

The previous analytical solution of axial failure load only con-siders the effect of anchoring material and FRP tendon, but the ef-fect of steel tube is ignored [15]. When the thickness of anchoringmaterial is large, the assumption is valid. However when the thick-ness of anchoring material is small, the effect of steel tube cannotbe ignored. In this paper, in order to consider the effect of anchor-ing material thickness to axial load capacity, according to the forcetransfer mechanism of the bond anchorage, the equivalent stiffnessof steel tube and anchoring material can be obtained:

F

s

2402 F. Li et al. / Composite Structures 92 (2010) 2400–2405

TE2A2

¼ TE3A3

þ TE4A4

ð7Þ

E2A2 ¼E4A4 � E3A3

E3A3 þ E4A4ð8Þ

where T is the load on the bond anchorage; E2A2 is the equivalentstiffness; E3 is the elastic modulus of anchoring material; A3 is thesection area of anchoring material; E4 is the elastic modulus of steeltube; A4 is the section area of steel tube.

3. Fundamental equations

A bond anchorage consists of a steel sleeve, anchoring materialand FRP tendons, as shown in Fig. 3. The origin of the coordinate xis situated at the left end of the anchorage. Because of taking theanchoring material and steel tube as integer, the anchorage con-sists of two parts. For a given cross-section, if the displacementsof the two parts are identical, there is no relative displacement be-tween them at that cross-section. If at the given cross-section, thedisplacement of the two parts is different from each other, a rela-tive displacement occurs. Let us introduce the relative interfacedisplacement, which equals to the difference of the individual dis-placement of the two parts at the cross-section. Thus we have:

s ¼ sn � sw; s0 ¼ s0n � s0w; s00 ¼ s00n � s00w ð9Þ

where s is the relative interface displacement; sn is the displace-ment of FRP tendon; se is the displacement of integer of anchoringmaterial and steel tube.

According to the classical mechanical theory, the axial load oftwo parts of anchorage can be expressed as follow, respectively:

s0n � E1A1 ¼ T1; s0w � E2A2 ¼ T2 ð10Þ

where T1 is the axial load of FRP tendon; T2 is the axial load of inte-ger of anchoring material and steel tube.

The equilibrium between external and internal axial load in theanchorage requires:

T2ðxÞ þ T1ðxÞ ¼ T;T2ðxÞ

dxþ T1ðxÞ

dx¼ T

dx¼ 0 ð11Þ

Combine Eqs. (10) and (11), it can be derived that:

s00n � E1A1 þ s00w � E2A2 ¼ 0 ð12Þ

Combine Eqs. (9) and (12), it can be derived that:

s00n ¼E2A2

E2A2 þ E1A1s00; s00w ¼ �

E1A1

E2A2 þ E1A1s00 ð13Þ

The equilibrium of differential element of FRP tendon requires:

2pr1sðxÞ ¼dT1

dxð14Þ

where s(x) is shear stress along the bond length of the FRP tendon;r1 is the radius of FRP tendon

Combine Eqs. (13) and (14), it can be derived that:

E2A2E1A1

E2A2 þ E1A1s00 ¼ 2pr1sðxÞ ð15Þ

FRPtendon

anchoring materialsteel tube

F

x

Fig. 3. Side view of the FRP tendo n anchor.

Substitute Eq. (3) into Eq. (15), the relative displacement s of theelastic zone can be expressed:

E2A2E1A1

E2A2 þ E1A1s00 ¼ 2pr1k1s ð16Þ

In which a1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2A2þE1A1E2A2E1A1

� 2 � p � k1 � r1

q, so Eq. (16) can be written:

s00 ¼ a21s ð17Þ

The general solutions for Eq. (17) can be expressed as follows:

sðxÞ ¼ c1 expð�a1xÞ þ c2 expða1xÞ ð18Þ

where c1, c2 are the unknown coefficients to be determined.Substitute Eq. (6) into Eq. (15), the relative displacement s of

soften zone can be expressed:

E2A2E1A1

E2A2 þ E1A1s00 ¼ 2pr1½�k2ðs� s1Þ þ sm� ð19Þ

The solutions for Eq. (19) can be expressed as follows:

sðx2Þ ¼ d1 cosða2x2Þ þ d2 sinða2x2Þ þ s2 ð20Þ

where d1, d2 are the unknown coefficients to be determined.

a2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2A2þE1A1E2A2E1A1

� 2 � p � k2 � r1

q.

For the general bond anchorage, the E2A2 > E1A, sR > sL (sR is slipdisplacement of end which is loaded, sL is slip displacement of theend which is unloaded), the relative slip displacement along thebond length is in illustrated in Fig. 4.

For the sake of convenience, we reassign a coordinate systemfor the bilinear cohesive zone model as illustrated in Fig. 4 andtwo coordinates x1, x2 are introduced for the elastic zone and soft-ening zone, respectively. For the bilinear cohesive zone model, theorigin x1 is always set at the cross-section where minimum relativeinterface displacement sm is located. The origin of x2 is situated atthe cross-section(s = s1), which separates the elastic zone and thesoftening zone.

The general solutions for the elastic zone and softening zonecan be written as follows, respectively:

When s < s1,

sðx1Þ ¼ c1 expð�a1x1Þ þ c2 expða1x1Þ

When s > s1,

sðx2Þ ¼ d1 cosða2x2Þ þ d2 sinða2x2Þ þ s2 ð21Þ

When s < s1, the entire bond length consists with elastic zonelength: L = d0R + d0L; when s > s1, the entire bond length is separatedby four parts: d0R, d0L, d1R and d1L as illustrated in Fig. 4. in which d0R

represents the elastic zone length on the right side of the cross-sec-tion where sm is located; d0L represents the elastic zone length on

softening zoneat rigth end

elastic zone

softeningzone at left

end

Ls

R

1x2x2x

ms

1Ld1Rd

0 0L Rd d+

Fig. 4. Relative interface displacement s with coordinates for the bilinear cohesivezone model.

FRP tendon

anchoring materialsteel tube

F

plastic tube

10mm

46mm

40mm

26mm

20mm

10mm

50mm

Fig. 5. Geometry of bond anchorage models.

Table 1Physical and mechanical properties of FRP tendons in this study.

Material d (mm) E (GPa) mxy

FRP tendons 10 59 0.26Anchoring material – 22.6 0.11Steel tube – 210 0.3

F. Li et al. / Composite Structures 92 (2010) 2400–2405 2403

the left side of the cross-section where sm is located; d1R representsthe softening zone length on the right side of the cross-sectionwhere s1 is located; d1L represents the softening zone length onthe left side of the cross-section where s1 is located; Obviously,the entire bond length can be written as: L = d0R + d0L + d1R + d1L

s0ðx1Þjx1¼0 ¼ 0 sðx1Þjx1¼0 ¼ sm ð22Þ

From Eq. (22), we can get: c1 ¼ c2 ¼ sm2

According to the definition of elastic zone length s(x1 = d0R) = s1,so

sðx1Þjx1¼d0R¼ sm

2ðexpð�a1d0RÞ þ expða1d0RÞÞ ð23Þ

With Eq. (23), we can get: d0R ¼ 1a1

arccos h s1sm

� �.

For the linear elastic branch, when

x2 ¼ 0 : sðx2Þjx2¼0 ¼ d1 þ s2 ¼ s1 ð24Þ

From Eq. (24), the coefficients d1 can be expressed as: d1 = s1–s2

The continuous boundary condition requires:

a2

a1d2 ¼

sm

2ð� expð�a1d0RÞ þ expða1d0RÞÞ ð25Þ

with Eq. (25), the coefficient d2 can be written as: d2 ¼ � a1a2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

s21 � s2

m

q, Note that for the right side of the anchor(x2 > 0), the value

of d2 is positive; for the left side of the anchor(x2 < 0), the value of d2

is negative. So Eq. (21) can be expressed:

when s < s1; sðx1Þ ¼sm

2expð�a1x1Þ

þ sm

2expða1x1Þsðx2Þ ¼ ðs1 � s2Þ cosða2x2Þ�

when s > s1;a1

a2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2

1 � s2m

qsinða2x2Þ þ s2

ð26Þ

According to boundary condition:

E2A2 � s0jx2¼�d1L¼ �T E1A1 � s0jx1¼d1R

¼ T

We can get:

E2A2 � ½a2ðs2 � s1Þ sinða2ðL� 2 � d0R � d1RÞÞ

þ a1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2

1 � s2m

qcosðL� 2 � d0R � d1RÞ� ¼ T

E1A1 � ½a2ðs2 � s1Þ sinða2d1RÞ þ a1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2

1 � s2m

qcosða2d1RÞ� ¼ T ð27Þ

For a given minimum relative interface sm, there is a correspondingsoftening zone length d1R, according to Eq. (27), with the deter-mined value of d1R, one can readily obtain the tension load T by uti-lizing the second term in Eq. (27) Through seeking the maximum ofEq. (27), the tensile capacity of anchorage can be obtained.

4. Example and verification

In order to verify the effectiveness of the proposed model, pull-out tests were to investigated. The test program involved one typeof FRP tendons provided by feng hui corporation; one type ofanchoring material; two bonded lengths of 100 and 150 mm andtwo anchoring material thickness of 5 and 15 mm, as illustratedin Fig. 5. The FRP tendon is in a circular form with a diameter of10 mm. The surface of the tendons is sand coated to improve bond.The physical and mechanical properties of the FRP rods, anchoringmaterial, are summarized in Table 1.

The anchorage sleeve was a cylindrical steel tube with an outerdiameter of 46, and 26 mm and a wall thickness of 3 mm. The FRPtendon was inserted into the tube and bonded with the anchoringmaterial. The bond-free length of 50 mm above the bonded lengthwas obtained by covering the rod with thin plastic tube (Fig. 5).The anchorages of the specimen were tested using a Universal

Testing Machine at 28 days after grouting. The rate of loadingwas fixed at 22 kN/min. During the tensile tests; the applied loadwas monitored with a load cell. The FRP rod displacements weremeasured using two LVDTs on the surface of the rod at a predeter-mined distance from the surface of the anchoring material. Thereadings from the load cell and LVDTs were collected using a dataacquisition system (Fig. 6).

The pull-out behavior of anchoring material grouted anchorageswere expressed by load–slip curves, characterized by maximumand residual loads. The slip values were calculated as in the pull-out tests, the load and displacement values were used to calculatethe corresponding nominal bond stress and slip. The average bondstress was calculated from the applied tensile force T as follows [5]:

s ¼ Tpdla

ð28Þ

Where T is applied pull-out load; d is tendon diameter; and la isbond length of the tendon.

The slip values were calculated as:

s ¼ smea � sbar ð29Þ

In which s is tendon slip along the bonded length; smea is measuredtotal displacement; sbar is tendon elastic elongation.

The values of smea were measured continuously with at increas-ing loads and at a distance from the surface of the anchoring mate-rial by the two LVDTs as indicated in Fig. 6. The tendon elongationwas calculated as follows:

Table 4Theoretical and experimental results with 10 mm bonded thickness anchoragespecimens.

Bond length(mm)

Specimen Teult

(kN)Tt

ult

(kN)jTe

ult � Ttult j

(N)

Teult�Tt

ultTe

ult� 100%

100 L1 24.80 22.28 2520 10.2L2 21.67 19.59 2080 9.6L3 22.92 20.65 2270 9.9L4 25.75 22.58 3170 12.3

150 L1 33.44 31.46 1980 5.9L2 39.56 36.63 2930 7.4

Table 5Theoretical and experimental results with 5 mm bonded thickness anchoragespecimens.

Bondlength(mm)

Specimen Teult

(KN)Tt

ult

(KN)jTe

ult � Ttult j

(N)

Teult�Tt

ultTe

ult� 100%

100 S1 19.4 16.13 3270 11.4S2 18.6 17.65 950 7.8S3 19.9 19.00 900 8.3S4 20.6 17.94 2660 14.7

150 S5 23.9 21.74 2160 19S6 25.0 28.90 3270 5.6

Note: Teuit and Te

ult are experimental and theoretical ultimate load bearing capacityof bond-type anchorage, respectively.

1

2

2

3

3

1.anchorage specimen

2.LDVT

3.testing machine

Fig. 6. Test system.

2404 F. Li et al. / Composite Structures 92 (2010) 2400–2405

sbar ¼TL

EpApð30Þ

where T is applied load; and L is distance from the top of the bondedlength to the measuring point of the LVDT. Tables 2 and 3 show thepull-out test results of the specimen.

The test results of anchorage specimens with a bonded thick-ness of 5, and 15 mm are summarized in Tables 2 and 3 and usedto define the characteristic bond strength and corresponding slip ofthe bond anchorage in this paper. It is observed in Tables 2 and 3that the thickness of anchoring material has an insignificant effect

Table 2Pull-out test results of bond anchorages with 15 mm bonded thickness.

Bond length (mm) Specimen sm (MPa) sres (MPa) Slip at loaded end

S01 (mm) S02 (mm)

100 L1 7.9 2.1 2.15 7.32L2 6.9 1.7 2.4 6.62L3 7.3 2.4 1.82 7.52L4 8.2 2.7 2.42 6.48

150 L5 7.1 3.1 3.21 7.56L6 8.4 3.6 2.95 7.3

Table 3Pull-out test results of bond anchorages with 5 mm bonded thickness.

Bond length (mm) Specimen sm (MPa) sres (MPa) Slip at loaded end

S01 (mm) S02 (mm)

100 S1 5.8 1.4 1.29 3.71S2 6.1 1.6 1.05 5.1S3 6.6 2.5 0.68 5.71S4 6.7 1.6 0.64 3.9

150 S5 5.7 2.3 1.7 5.3S6 6.5 2.2 1.6 5.5

Table 6Theoretical and experimental results of bond anchorages for arapree tendon.

Bond length (mm) sm (MPa) sres (MPa) S01 (mm) S02 (mm

40 14 5.8 1.15 2.980 14.85 3.93 1.22 3.25

Note: experimental results obtained by Burong Zhang, Brahim Benmokrane (2002).

on the bond characteristics. The specimen with a thickness of5 mm shows poor bond strength probably because the thicknessof anchoring material is larger; the degree of stress concentrationis lower.

The theoretical tensile capacity of the anchorage is determinedfrom the established Eq. (27). Through seeking the maximum of Eq.(27), the tensile capacity of anchorage can be obtained. Tables 4and 5 shows the theoretical tensile capacities of 5, and 15 mmthick bond anchorage specimens for different bond length. In com-parison with the experimental data, the theoretical model provideslower magnitudes of tensile capacity with a difference of <15% ofthe test values. Except one with 19%. The larger discrepancy maybe caused by the lower experimental result.

In Table 6, the test data are obtained from the work of BurongZhang and Brahim Benmokrane [5]. The anchorage specimens weremade from a single Arapree tendon with cement mortar inside asteel pipe. The Arapree tendon has a diameter of 7.5 mm. The mod-ulus of elasticity and Poisson ration are 62.5 and 0.38 GPa, respec-tively. The theoretical versus experimental results show that mostof them have good agreement with a difference <10%.

5. Conclusion

In the current study, the cohesive zone model based the theo-retical models are successfully developed for the FRP tendonanchorage. With bilinear cohesive laws, the analytical solutionsof tensile capacity of anchorages are derived. The study involvedthe experimental verification tests and theoretical calculation on

) Teult (KN) Tt

ult (KN) jTeult � Tt

ult j(N) Teult�Tt

ultTe

ult� 100%

13188 12500 738 5.627977.4 26000 198 7

F. Li et al. / Composite Structures 92 (2010) 2400–2405 2405

the tensile capacity for bond anchorages. A comparison of theoret-ical versus experimental results has been reported. It should benoted that many parameters affect the bond behavior of groutedFRP tendons, including dimensions and properties of the steel cyl-inder; geometry, properties, and constituent of the anchoringmaterial; and properties of the tendon. Based on the study, the fol-lowing conclusions are drawn:

(1) With bilinear cohesive laws, the analytical solutions of ten-sile capacity can be obtained. The models can be extendedto other types of cohesive law, such as multi-linear cohesivelaws with similar methodology by applying continuityconditions.

(2) The characteristic bond strength and the corresponding slipof a bond anchorage with a steel tube depend mainly on thesurface conditions of the tendon and the thickness of thegrout, having no relation with the bonded length and loadlevel. The transfer length and the slope of the bond stressesfrom peak bond stress to residual bond stress due to deb-onding are constant for a given anchorage.

(3) Based on the pull-out test results of 5 and 15 mm thickanchorage specimens, the comparison with the calculatedand experimental results shows a good agreement. The dif-ference between the theoretical and experimental resultsis <20% for 100 mm long Anchorage specimens and 10% for150 mm long anchorage specimens. The proposed analyticalmodel and equations can be used to predict the tensilecapacity of bond anchorages.

References

[1] Mei Kui-hua. Analysis of mechanical behavior of CFRP cable bonding anchors.Bridge Constr 2007;3:80–3 [in China].

[2] Holte LE, Dolan CW, Schmid TRJ. Epoxy socketed anchors for nonmetallicprestressed tendons. In: Proceeding of international symposium on fiber-reinforced-plastic reinforcement for concrete structure. Publication ACI, SP;1993. p. 381–400.

[3] Pecce M, Manfredi G, Realfonzo R, Cosenza E. Experimental and analyticalevaluation of bond properties of GFRP bars. J Mater Civ Eng 2001:282–90.

[4] Cosenza E, Manfredi G, Realfonzo R. Development length of FRP straight rebars.Composites: Part B 2002;33:493–504.

[5] Zhang Burong, Benmokrane Brahim. Pullout bond properties of fiber-reinforced polymer tendons to grout. J Mater Civ Eng 2002;14:399–408.

[6] Francesco Ascione. Mechanical behavior of FRP adhesive joints: a theoreticalmodel. Composites: Parts B 2009(40):116–24.

[7] Francesco Ascione. Ultimate behavior of adhesively bonded FRP lap joints.Composites: Parts B 2009(40):107–15.

[8] Zhenyu Ouyang, Guo qiang Li. Cohesive zone model based analytical solutionsfor adhesively bonded pipe joints under torsional loading. Solids Struct2009:1205–17.

[9] Barenblatt GI. The formation of equilibrium cracks during brittle fractureGeneral idea sand hypothesis. Axisymmetrical Cracks, PMM 1959;3:434–44.

[10] Dugdale DS. Yielding of steel sheets containing slits. J Mech Phys Solids1960;8:100–4.

[11] Hilleborg A, Modeer M, Petersson PE. Analysis of crack formation and crackgrowth in concrete by means of fracture mechanics and finite elements. CemConcr Res 1976(6):773–82.

[12] Xu XP, Needleman A. Void nucleation by inclusion debonding in a crystalmatrix. Model Simulat Mater Sci Eng 1993(1):111–32.

[13] Rose JH, Smith JR, Ferrante J. Universal features of bonding in metals. Phys RevB 1983(28):1835–45.

[14] Namas Chandra. Evaluation of interfacial fracture toughness using cohesivezone model. Composites: Parts A 2002(33):143–1447.

[15] Zhang Burong, Benmokrane Brahim. Prediction of tension capacity of bondanchorages for FRP tendons. Compos Constr 2000;4:34–9.