BOND STRESS-SLIP RELATIONSHIP IN REINFORCED CONCRETE: NEW RELATIONSHIP AND COMPARATIVE STUDY

Sung-Nam Hong*, Sungkyunkwan University, Korea Jun-Myoung Park, Sungkyunkwan University, Korea

Tae-Wan Kim, Sungkyunkwan University, Korea Kyoung-Bong Han, Sungkyunkwan University, Korea

Sun-Kyu Park, Sungkyunkwan University, Korea Won-Jun Ko, Induk Institute of Technology, Korea

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33rd Conference on OUR WORLD IN CONCRETE & STRUCTURES: 25 – 27 August 2008, Singapore

BOND STRESS-SLIP RELATIONSHIP IN REINFORCED CONCRETE: NEW RELATIONSHIP AND COMPARATIVE STUDY

Sung-Nam Hong*, Sungkyunkwan University, Korea Jun-Myoung Park, Sungkyunkwan University, Korea Tae-Wan Kim, Sungkyunkwan University, Korea

Kyoung-Bong Han, Sungkyunkwan University, Korea Sun-Kyu Park, Sungkyunkwan University, Korea Won-Jun Ko, Induk Institute of Technology, Korea

Abstract

This paper deals with the relationship between bond stress and slip based on experimental data that were obtained from axial tension force on both side specimens. The relationship proposed in this paper utilizes conventional bond-slip theory as well as the characteristics of deformed bar and the cross-sectional area of concrete. An analytical equation for the estimation of the bond stress is formulated as the function of dimensionless factors (e.g. bond stress, relative slip, etc.). The analytical results presented in this paper indicate that the proposed relationship can effectively estimate the bond-slip behavior of axial tension members.

Keywords: bond stress, slip, axial tension, deformed bar

1. Introduction

Although, the bond-slip relationship which represents the actual behavior of reinforced concrete members should be utilized in the analysis of crack width, the bond-slip relationships based on the results of pull-out tests with a short embedment length using the average bond stress have been used. In such test method, test results does not give the appropriate bond-slip relationship because the concrete is located in compressive stress fields and internal cracks of surrounding concrete relative to a reinforcing bar does almost not occur. But in the reinforced concrete member, where cracks have occurred, the concrete is located in tensile stress fields and internal cracks therefore easily occur. It may be regarded that utilizing bond-slip relationship derived from the pull-out tests contains inevitable problems.

In general, the analysis of crack width, which is composed of the crack spacing and difference of the mean bar and concrete strains, is an establishment of the crack spacing in response to the load variation. One of the main factors deciding the crack spacing when predicting the crack width is the bond stress. Therefore, the prediction formula of the crack width may be considered as a function of bond stress. To obtain the exact crack spacing, maximum bond stress is needed. But maximum bond stress could not be obtained in most of the proposed bond-slip relationships. The analysis of crack width in reinforced concrete members requires an appropriate bond-slip relationship. It is, therefore, necessary to establish a realistic bond-slip relationship, which takes into account the effect of long embedment length and concrete stress condition.

This paper presents a relationship between bond stress and slip based on the axial tension test that can be used to estimate the bond-slip relationship of reinforced concrete members, while considering the relative rib area of the deformed bar. The relationships that have been suggested for the estimation of the bond-slip relationship are the JSCE specification

1, Ikki

2,3 and Shima

4. These

have been extended in order to analyze the bond-slip behavior of reinforced concrete members. An analytical equation is formulated in order to estimate the bond-slip behavior of reinforced concrete members under axial tension loading as an exponential function of the relative rib area and the non-dimensional slip. The validity, accuracy and efficiency of the proposed relationship are established by comparing the results of the present study with the results obtained from the experimental data as well as analytical relationships provided by eminent researchers

2-4 and design codes

1.

2. Bond-slip Relationship 2. 1 Existing relationships provided by eminent researchers

In 1986, Shima4 reported a bond-slip relationship that can consider the tension stiffening effect

and that can be maintained under any boundary conditions. Equation (1) was proposed for the bar with a long embedment, and Equation (2) was proposed for the bar with a short embedment, while considering the strain effect in accordance with the boundary conditions, as:

[ ]})/(40exp{19.0 6.03/2'scb dSf −−×=τ (1)

[ ] )101/(1)}/(50001ln{73.0 533/2'sscb dSf ετ +×+×= (2)

where, 'cf is the compressive cylinder strength of concrete, sε is the strain of the bar is the ,

sd is the diameter of bar, S and sdS / are the relative slip in the direction of bar and the non-

dimensional relative slip, respectively.

Here, the boundary condition of Equation (1) is 0=sε when 0=S , with a sufficient embedment

length more than 25 sd . This relationship is beneficial when the concrete is under the field of

compressive stress and can be applied to the analysis of the behavior of a reinforced concrete member without any experiment in order to determine experimental factors for a bond stress-slip relationship. For this reason, this relationship is the representative bond stress-slip relationship that is currently being used in the analysis and design of reinforced concrete members.

However, the relationship given by Shima4 was derived from the experimental data of a pull-out

test for an anchored bar embedded in the footing of a reinforced concrete pier or column. The applicability of this relationship may therefore be limited because restrained cracks occurred in the specimens. In addition, the bond-slip behavior may be influenced by many factors such as the concrete cover, the amount of confining bar or the axial force of the column. While taking into consideration the aforementioned problems, Ikki

2.3 performed axial tension tests and suggested the

following relationship, which is modified from the relationship given by Shima4, as:

]})/(40exp[1{9.0 6.03/2'scdsfb dSfkk −−×××=τ (3)

where, sfk is the coefficient of concrete stress condition and is 1.0 when the concrete is

compressed, and 0.7 when the concrete is tensioned. dk is a coefficient of the bar direction during

concrete casting and is 1.0 for a vertically cast reinforcing bar, and 0.9 for a horizontally cast reinforcing bar.

2. 2 Proposed Relationship

In research carried out by Ikeda5 , the effect of slip on the bond-slip relationship is determined

from the relationship between the non-dimensional slip, sdS / and the average bond stress, meanb,τ .

The effect of the bar diameter is also considered by using a non-dimensional slip, sdS / . This is

because the slip is proportional to the bar diameter. The effect of concrete strength should also be considered. The effect of concrete strength on the bond-slip relationship in the case of a long embedment was shown that as the bond stress is proportional to the non-dimensional bond stress,

3/2'/ cb fτ 6. All of these results were obtained from the pull-out test using specimens with an

extremely short embedment. To formulate the bond-slip relationship that successfully holds under an axial tension boundary

condition and a long embedment length, the aforementioned non-dimensional indexes were used to

estimate the bond stress. In addition, the relative rib area, Rf , was introduced in order to consider

the effect of the rib geometry of the deformed bar. The mathematical form7 with some modification

was adopted. The relationship between the non-dimensional bond stress and slip as well as the relative rib area was proposed as:

]5.5)/(5[exp]})/(4500exp[1{ 9.05.045.13/2'Rsscb fdSdSfk +−×−−××=τ (4)

where, k and Rf is a coefficient related to the effects of bond stress and the relative rib area of bar, respectively.

To be able to express in terms of sdS / , an expression for sdS / is needed. Ko8 proposed the

following relationship between the non-dimensional relative slip and maximum crack width as:

[ ] 7.0844.2max /)25ln(0111.0/ Rs fWdS ×= (5)

where, maxW is the maximum crack width.

By substituting equation (5) into equation (4), the relationship between the bond stress and the

maximum crack width can be rewritten as:

]5.5)/)25(ln(0555.0[exp

]])/)25(ln(5907.6exp[1[

9.0844.2max

5.0015.1844.2max

3/2'

RR

Rcfb

ffW

fWfk

+−×

−−××=τ (6)

In both the ACI code

9 and Eurocode 2

10 specifications, a general value of 0.10 ~ 0.40 mm is

suggested for the allowable maximum crack width. In reality, the crack width measurement range that considers the crack width characteristics at the stabilized state is over 0.1 mm. When it is assumed

that maxW is 0.1 mm. It was found that the first square bracket term in equation (4),

5.045.1 ]})/(4500exp[1{ sdS−− , closed down to 1 according to the increase of maximum crack width

and thus could be ignored. The bond stress at the maximum crack width can also be considered as the maximum bond stress. Therefore, in this paper, the relationship between the maximum crack

width and max,bτ is proposed in a simplified form, as:

]5.5)/(5[exp 9.0max

3/2'max, Rscb fdSfk +−××=τ (7)

3. Experimental Program 3.1 Specimen and experimental conditions

In order to obtain verification of the proposed relationship, the results of axial tension tests performed by Ikki

2,3 were used. The experimental conditions and properties of each specimen are

shown in Table 1. Table 1 Properties of test specimens

Type (Series)

Specimens Direction of

bar on casting

Stirrup (D6)

Cross section [mm]

Deformed bar

Young’s modulus [GPa]

Yield stress [MPa]

T-Series

TH1-30 Horizontal Yes 300×300

D19 181.3 357.8

TH1-45 Horizontal Yes 450×450

TH0-30 Horizontal No 300×300

TH0-45 Horizontal No 450×450

TV0-45 Vertical No 450×450

TV0-50 Vertical No φ 500mm

D-Series D-30 Vertical No φ 300mm

D16 192 390 D-45 Vertical No φ 450mm

All specimens are made up of two series with different properties, namely the T-Series and the D-Series. The concrete cover of all specimens is 40 mm, regardless of the concrete cross section. In the T-Series, a square section (300mm×300m, 450mm×450mm) was used with the exception of the TV0-50 specimen, where a circular section was used ( 500mm). The length of each specimen was 764 mm, as shown in Fig. 5. Concrete was cast in a horizontal direction and perpendicular to the embedded reinforcing bars, while the TV0-45 specimen and the TV0-45 specimen were cast in a vertical direction parallel to the embedded reinforcing bars. In addition, a stirrup was placed only in the TH-30 and TH-45 specimens. A tensile reinforcing bar was embedded at the center of the square and circular sections. The test details of the TH1-45 specimen are shown in Fig. 4. In the D-Series, two types of circular sections were used, with diameters of 30 cm and 45 cm. The length of each specimen was 964 mm. Concrete was cast in a vertical direction. A stirrup was not used and the tensile reinforcing bar was embedded at the center of the circular section.

Fig. 1. Sectional properties of the specimen TH1-45 and positions of strain gauges 2.2 Determination of local bond stress and slip

In order to measure the strain distribution along the embedded reinforcing bar, foil resistance

strain gauges were installed at an interval of 5 sd . At any location along the bar where there were two

diametrically opposed strain gauges, strain at that location was derived from the average value of these two gauges. In general, the difference between the values of these two gauges was very small. Therefore, the eccentricity of the embedded reinforcing bar was of no concern. In order to obtain the bond stress and slip from the measured strain, the method proposed by Yamao

11 was utilized. At any

point, the bond stress derived from an equilibrium condition can be written as:

dx

dEd

dx

dfd sssssb

ετ44

==

(8)

where, dxds/ε is the slope of the strain distribution curve.

The function of strain )(xs

εε = is assumed to be the strain distribution curve obtained by

connecting every three neighboring points with a second degree polynomial function. In this case, if

the three neighboring points are set as ),(11 −− ii

x ε , ),(ii

x ε , ),(11 ++ ii

x ε , the strain curve can be

expressed as:

2)( xcxbaxiiis

++== εε (9)

where, ia ,

ib ,

ic can be determined using the three neighboring points. The obtained strain

curve shows the strain distribution from point ( 1−i ) to point ( i ).

From equation (3) and equation (4), )(xbi

τ can be calculated as:

4/)2( xcbdE iissbi +=τ (10)

where, biτ is the bond stress corresponding to point .

4. Comparison

In this paper, all the test specimens are calculated by using the proposed relationship and the

results obtained are compared with the experimental data and analytical relationships provided by eminent researchers

2-4 and design codes

1. In Fig. 2, for notational convenience, the results obtained

from the proposed relationship are denoted by the “Present Study”. Also, the analytical relationships have been designated according to the surname of the first authors (Ikki

2,3, Shima

4) and design code

(JSCE1). The concrete design bond strength, bdjf , proposed by the Japan Society of Civil Engineers,

can be expressed as:

3/2'28.0 ckbdj ff = ( MPafbdj 2.3≤ ) (11)

where, 'ckf is the standard compressive strength of design.

The experimental data at different locations along a bar obtained from specimens and analytical relationships are shown in Fig. 2. From this figure, it can be concluded that the bond-slip relationships of a specimen are very similar, independent of the locations along the bar. However, the data obtained from the right sides of the D-30 specimens underestimated the bond stress compared to that of the left sides. For this reason, Ikki

2,3 explained that the local hardness of the concrete layer that is in

contact with the ribs of indentation had decreased due to the bleeding of the concrete.

0 1 2 3

S / ds [%]

0

0.05

0.1

0.15

0.2

0.25

0.3

τ b / f c' TH0-30

10 dsLeft

15 dsLeft

10 dsRight

15 dsRight

Present Study

Shima

JSCE

Ikki

0 1 2 3

S / ds [%]

0

0.05

0.1

0.15

0.2

0.25

0.3

τ b / f c'

TH1-30

15 dsLeft

10 dsRight

15 dsRight

Present Study

Shima

JSCE

Ikki

(a) TH0-30 (b) TH1-30

0 1 2 3

S / ds [%]

0

0.05

0.1

0.15

0.2

0.25

0.3

τ b / f c'

TV0-45

10 dsLeft

15 dsLeft

10 dsRight

15 dsRight

Present Study

Shima

JSCE

Ikki

0 1 2 3

S / ds [%]

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

τ b / f c'

D-30

10 dsLeft

15 dsLeft

10 dsRight

15 dsRight

Present Study

Shima

JSCE

Ikki

(c) TV0-45 (d) D-30

Fig. 2. Comparisons of analytical solutions with the experimental data and the proposed relationship

The relationship provided by JSCE1 is consistently conservative. It can be seen that when

compared to the proposed relationship in this paper, the relationship given by Shima4 yields an over-

predicted overall bond–slip behavior for all specimens, which is undesirable. Moreover, this relationship predicted the maximum bond stress to be greatly more than the experimental data for all the specimens. It is evident that this relationship predicts an erroneous bond–slip behavior. This may be due to the fact that this relationship was developed for specimens based upon pull-out tests. Also, the relationship given by Ikki

2,3 indicated that this relationship, in general, makes predictions that are

comparatively closer to the experimental data for all the square and circular test specimens. For the square specimens, this relationship slightly underestimated the maximum bond stress, yet the predictions were very close to the experimental data. Conversely, this relationship slightly overestimated values for the circular specimens. However, the maximum bond stress and the associated slip couldn’t be obtained exactly from both the Ikki

2,3 and the Shima

4.

The proposed relationship by the author in this paper is reasonably accurate at predicting the bond-slip behavior of axial tension specimens. Also, the accuracy of this relationship is not affected by changes in test variables. This is important since the test variables change rapidly between different types of bars, concrete strengths etc. Furthermore, an advantage is that the maximum bond stress corresponding to the associated slip can be obtained, which is a core factor in the estimation of crack spacing. This is useful in the analytical and realistic prediction of the crack width and in the load-deformation relationship of reinforced concrete members. This will also make it possible to use a more precise bond model.

5. Conclusion 8 axial tension specimens were analyzed in order to find an accurate and convenient model that

could be used to estimate the bond-slip relationship and the maximum bond stress of axial tension members. The work presented in this paper supports the following conclusions:

(1) The relationship proposed in this paper is that the maximum bond stress corresponding to the relative slip can be obtained. Also, this relationship is useful in the analytical and realistic prediction of the crack width and load-deformation relationship of reinforced concrete members.

(2) By comparing the experimental data and the analytical relationships given by other researcher, it can be concluded that the proposed relationship is a more accurate relationship for estimating the bond-slip behavior than other relationships discussed in this paper. Acknowledgments

This work was supported by the R&D program (05 IPET D04-01, 05 CCT D11) of the Korea Institute of Construction and Transportation Technology Evaluation Planning (KICTTEP). References

1. Japan Society of Civil Engineers (JSCE) (2002). Standard Specifications for Concrete Structures-2002 "Structural Performance Verification"

2. Ikki N., Kiyomiya O. and Yamada M. (1996), “Experimental Study on the effects of numerous factors on

bond-slip relationship”, Proceedings of JSCE, Vol. 33, No. 550, pp. 73～89. 23.

3. Ikki N. and Kiyomiya O. (1999), “Effect of Axial Concrete Stress on Bond Strength of Deformed Bar”,

Proceedings of the Japan Concrete Institute, Vol. 21, No. 3, pp. 373～378.

4. Shima H. (1986), Micro and Macro Models for Bond Behavior in Reinforced Concrete, Doctoral dissertation, Tokyo university, Tokyo. (in Japanese)

5. Ikeda K. (1981), “Study on the Stress Transmission between Reinforcing Bars and Concrete in Reinforced

Concrete Members”, Proceedings of JSCE, No. 307, pp. 85～97.

6. Maekawa, K., Okamura, H. and Pimanmas, A. (2003), Non-linear Mechanics of Reinforced Concrete, Spon Press.

7. Ko W. J. and Park S. K. (2002), “Estimation of Axial Tension Member's Bond Stress-Relative Slip Relationship” Korea Society of Civil Engineering Thesis, Book No.22, Edition 4-A, pp. 815~823.

8. KO W. J. (2002), Flexural Crack Width of Reinforced Concrete Members Estimated Based on Relation of Bond Stress and Relative Slip, Doctoral dissertation, Sungkyunkwan University, Suwon, 2002 (in Korea).

9. American Concrete Institute (2005), Building Code Requirements for Structural Concrete and Commentary. ACI, Michigan, 2005.

10. Eurocode 2 (1992), Design of Concrete Structures – Part 1: General Rules and Rules for Buildings. European Committee for Standardisation, prEN 1992-1 (final draft).

11. Yamao, H., Chou, L. and Niwa, J. (1984), "Experimental Study on Bond Stress-Slip Relationship", Proceedings of JSCE, No. 343, pp. 219-228.