1/10
BOND-SLIP BEHAVIOR OF REINFORCED CONCRETE MEMBERS Halil SEZEN
1
Jack P. MOEHLE
2
1
Civil & Environmental Engineering & Geodetic Science, The Ohio State University 470 Hitchcock Hall, 2070 Neil Ave., Columbus, OH 43210-1275 USA
2
Pacific Earthquake Engineering Research Center, University of California, California, Berkeley th 1301 South 46 St., Richmond, CA 94804-4698 USA Keywords: Keywords : bond-slip model, reinforced concrete, seismic, bond stress, development length
ABSTRACT Experimental research indicates that elongation and slip of the tensile reinforcement at the beamcolumn interface could result in significant fixed-end rotations that are not included in the flexural analysis. These additional rotations at beam-column fixed ends can increase the total member lateral displacement significantly. This paper presents examples of experimental cyclic end rotations and resulting lateral displacements due to longitudinal bar slip for the four full-scale reinforced concrete column specimens tested at the University of California, Berkeley. The columns were subjected to constant and varying axial loads, and uni-directional lateral load resulting in contraflexure at column midheight. Test results showed that the contribution of bar slip deformations to total member lateral displacement was considerably large and almost constant throughout the tests. Based on experimental results and previous theoretical investigations, an analytical procedure is developed to characterize the bond-slip behavior. The procedure is used to compute column endrotations and corresponding lateral displacement due to longitudinal bar slip at beam-column interfaces under monotonic lateral load. The relation between the bending moment and longitudinal bar slip at the interface is obtained from the moment-curvature moment-curvature analysis using a fiber cross-section and uniaxial material properties. A bi-uniform relationship is adopted to model the bond stress distribution along the development length of the anchored reinforcing bar. The proposed bond-slip model is evaluated using column test results and compared with other analytical models. Calculated lateral load-slip displacement relations are compared with the measured cyclic test results.
1
INTRODUCTION AND BACKGROUND INFORMATION
Longitudinal bar slip at the end of reinforced concrete members results in additional member end rotation that is not part of flexural deformations. The contribution of these member end-rotations to total lateral deformations can be relatively large. Experimental results from four double-curvature columns tested by [13] indicated that, in some cases, the bar slip deformations may be as large as flexural deformations. Figure 1 shows the contribution of flexure, longitudinal bar slip, and shear displacement components to the total member lateral displacement for the columns tested by [13]. For all columns, slip deformations contributed between twenty five and forty percent of the total lateral displacement. The contribution of slip deformations did not appear to vary too much with increasing lateral displacement or damage in the column. The results suggests that if the deformations resulting from longitudinal bar slip are ignored in the analysis or member modeling, the predicted member deformations deformations may be significantly significantly smaller or the predicted lateral member stiffness may be larger than the existing member stiffness. Numerous researchers have investigated the anchorage behavior of reinforcing bars experimentally, and a number of analytical bond-slip models have been developed over the years. One of the most widely used bond stress-slip relationships proposed by [3] and [4] is based on an experimental program at the University of California, Berkeley (Figure 2 i). Within last twenty years, various refined computer models were proposed to implement this model, and a few similar models were developed to represent bond-slip deformations in reinforced concrete members under monotonic and cyclic loads ([8], [5], [6], [11], and [14]). A detailed analysis of bar slip under monotonic loading and a description of available analytical procedures for force-slip deformation relationships are summarized by [2]. By assuming an average uniform bond stress, u b along the development length, l d d of a reinforcing bar, from equilibrium of forces in the bar: F bar = f A = u p l . Substituting the bar perimeter, pb=πd b and bar s s b b d d 2 As=πd b /4 yields
2/10
l d =
f s d b
(1)
4u b
Using a uniform bond stress along the development length of a reinforcing bar, [10] modeled bar slip deformations at the beam-column interface. The assumed uniform bond stress was 0.54 f c ′ MPa (6.5 f c ′ in psi units). Based on the assumption of linearly decreasing bar stress with the distance from the interface and zero stress at the distance of development length, and using Eq.(1), the slip or the elongation of the reinforcing bar over the development length is given by slip =
ε s l d
2
=
f s l d 2E s
2
=
f s d b 8E s u b
2
=
f s d b 4.3E s f c ′
(2)
This model is expected to be more representative of the elastic range as it uses the elastic modulus of elasticity for steel. Based on experimental results, [12] and [7] demonstrated that at the beam-column interface, the strains in the reinforcing bar could be much larger than the yield strain causing columns to experience significant fixed-end rotations. Based on experimental results from bridge column tests, [7] proposed the bi-uniform bond stress-slip model shown in Figure 2 ii. In this model, for slip values less than the slip corresponding to the yield strain in the bar, the uniform bond stress is approximated as 1 f c ′ MPa (12 f c ′ psi). For slip values exceeding the slip at yield, the bond stress capacity is 0.5 f c ′ MPa (6 f c ′ psi). The analytical procedure proposed by [2] predicts the monotonic force-deformation relationship of a reinforcing bar embedded in concrete. The numerical procedure includes calculation of the displacement due to slippage of the bar with nonlinear strain distribution (Figure 2 iii). According to this model, four regions are developed along a reinforcing bar in tension, namely, elastic region with length Le; yield plateau region with length Lyp; strain-hardening region with length Lsh; and pullout-cone region with length L pc . In the model, an elastic uniform bond stress, u e is assumed along the length of the bar except for the pullout-cone region. In addition to elastic uniform bond stress, a frictional uniform bond stress, u f is assumed in the yield plateau and strain-hardening regions. The elastic uniform bond stress, u e is adopted from [1] and is equal to 0.86 f c ′ MPa (10.4 f c ′ psi) for the specimens tested by [13]. The uniform frictional bond stress, u f is based on the results of an experimental investigation carried out by [11]. The slip is calculated by integrating the axial strains over the length of the bar, which is the area under the strain diagram shown in Figure 2 iii. slip = ε s L pc + 0.5(ε s + ε sh ) Lsh + 0.5(ε sh + ε y Lyp + 0.5ε y Le
2
(3)
PROPOSED BOND-SLIP MODEL
To calculate the member deformations resulting from longitudinal bar slip, it is essential to estimate the relation between the bending moment and section rotation at the beam-column interface. Once the rotation at member ends is known, the member lateral displacement can be calculated easily. A brief summary of the proposed model developed to calculate monotonic force-slip deformation relations is presented here. 2.1 Reinforcing Bar Slip A large crack opening at the beam-column interface may lead to an increase in the axial strain in the longitudinal bar crossing the crack. As a result of bond deterioration between steel and concrete, and penetration and accumulation of axial strains along the tensile reinforcement inside the joint, the extension and slip of the reinforcing bar at the interface can be significant. The slip resulting from accumulated axial strains in the bar embedded in the joint can be calculated by integrating the strains over the portion of the bar between the interface and the point with no axial strain. Using a bilinear strain distribution shown in Figure 3, the slip is determined from Eq.(4).
3/10
l d
slip =
∫ ε dx
=
s l d
ε
0
′ l dy + l d
l dy
slip =
∫ ε dx + ∫ ε dx = l dy
0
y l dy
ε
2
≤ ε y
ε
s
2
+
l d ′ 2
(4)
(ε s + ε y )
ε
s
> ε y
2.2 Development Length The development lengths over the elastic and inelastic portions of the bar ( l d and l’ d, respectively, Figure 3) are based on the assumption of bi-uniform bond stress distribution, and calculated from equilibrium of forces in the bar. The elastic and inelastic development lengths are calculated from Eq.(1) and from the following equation, respectively.
′ = l d
f s − f y d b
(5)
4u b′
As illustrated in Figure 3, for all strains exceeding ε y , in Eq.(5) the elastic strain increment is approximated as the difference between the current stress in the bar and the yield stress (i.e., f s-f y). It should be noted that the model proposed here results in zero inelastic slip if the stress, f s does not increase while the strain increases along the yield plateau of the steel stress-strain relation. This behavior seems unrealistic. Therefore [13] suggests a modest strain-hardening (e.g., at least one percent) when the strain in the bar is within the yield plateau of the stress-strain relation. 2.3 Bond Stress-Slip Relationship Using equilibrium at first yielding in the longitudinal bar and assuming a linear strain distribution along the bar, by inserting Eq.(1) into Eq.(4) for l d (similar to derivation of Eq.(2)), the average uniform 2
bond stress at yielding, u by can be calculated in terms of slip, u by = f y d b /(8E s slip) . The slip was measured at the ends of twelve column specimens tested by [8] and [13]. Using the measured slip values at yield displacement, uniform bond stresses, u by are calculated. The calculated bond stresses are normalized by f c ′ and presented in Figure 4. For the twelve columns considered, the average bond stress is 0.95 f c ′ MPa (11.4 f c ′ psi), and the standard deviation is 0.2 f c ′ MPa (2.5 f c ′ psi). In this study, a uniform bond stress of 1.0 f c ′ MPa is assumed in the elastic range (Figure 3). In the portion of the reinforcing bar over which the yield strain is exceeded, a uniform bond stress of 0.5 f c ′ MPa is used as suggested by [7]. 2.4 Deformations due to Bar Slip If Eqs.(1) and (5) are substituted into Eq.(4) for the elastic and inelastic development lengths, and using the assumed average bi-uniform bond stresses, the slip is given by slip = slip =
ε s f s d b
ε s
8 f c ′ ε s f s d b
8 f c ′
+
(ε s + ε y )(f s − f y ) d b 4 f c ′
≤ ε y (6)
ε s
> ε y
In the proposed model, the rotation due to bar slip is assumed to be concentrated at the ends of a column in the form of rigid body rotation (Figure 5). It is assumed that the section would rotate about its neutral axis. As illustrated in Figure 6, which is the close-up view of the beam-column interface shown in Figure 5, the section rotation due to bar slip can be calculated by dividing the bar slip by the width of the open crack, which is the difference between the depth of the section, d , and the neutral axis depth, c . θ slip
=
slip d − c
(7)
4/10
Substitution of Eq.(6) into Eq.(7) yields θ slip
θ slip
= =
ε s f s d b
ε s
8 f c ′ (d − c ) ε s f s d b
8 f c ′ (d − c )
≤ ε y (8)
[ε y f y + 2(ε s + ε y )(f s − f y )]
ε s
> ε y
As illustrated in Figure 5, if the slip rotation at the top and bottom of a double-curvature column is known, total lateral displacement due to bar slip can be calculated from
∆ slip =
3
θ slip,top
+ θ slip,bottom L
(9)
COMPARISON OF MODELS AND EXPERIMENTAL RESULTS
The moment-curvature analysis of the column cross-section used in [13] is carried out using uniaxial material models and a fiber cross-section model. The longitudinal steel and concrete were modeled using the data from steel coupon tests and concrete cylinder tests [13]. The reinforcing bar stress-slip relations from Eq.(2) ([10]), Eq.(3) ([2]), and the proposed model (Eq.(6)) are compared in Figure 7. For steel strains smaller than the yield strain, all models use the same elastic slip equation given in Eq.(4) where l d is calculated from Eq.(1). The only difference between the three models is the magnitude of the assumed average elastic bond stress, u b, which is 0.54 f c ′ , 0.86 f c ′ and 1.0 f c ′ MPa in [10], [2], and the proposed models, respectively. After the first yielding in the bar, the difference between the bar stress-slip behavior from [2] and the proposed models is the result of different assumptions for the development lengths and the average bond stresses in the inelastic portion of the bar. In the fiber section moment-curvature analysis, the stress in the tension reinforcement can be monitored and recorded as a function of moment. Then, using the analysis results and following the analytical procedures presented above, the section moment-bar slip relations can be obtained under different axial loads. Figure 8 shows the moment-slip relations under two different axial loads. Note that the slip from [10] (Eq.(2)) is a function of moment and can be calculated without momentcurvature analysis. The moment-slip relations from all three models are most similar in the elastic range under zero axial load. Because the model developed by [10] is largely based on the assumptions of elastic material behavior, the slip is smaller in the inelastic range. The difference between the other two models is a result of assumed inelastic development lengths and average bond stresses. The effect of axial load on the section behavior is included in the moment-curvature analysis. Under higher axial loads, the axial strain in the tension reinforcement decreases resulting in smaller slip as illustrated in Figure 8. The calculated development length was slightly smaller than the actual embedment length in the column specimens tested by [13] indicating that Eq.(8) can be evaluated using the test results from [13]. The neutral axis depth of the cross-section ( c in Figure 6), reinforcing steel stress, f s and strain, ε s are obtained as a function of section moment from the cross-section moment-curvature analysis. In Figure 9, the monotonic moment-slip rotation relations calculated from Eq.(8) are compared with the hysteretic moment-slip rotation relations measured at the top and bottom of a specimen tested by [13]. Using the computed moment-slip rotation relations, the monotonic lateral load-slip displacement relations are calculated from the proposed model (Eq.(9)). The calculated and measured lateral loadslip displacement relations compare relatively well for the four column specimens tested by [13] (Figure 10).
4
CONCLUSIONS
Test results from [13] show that the contribution of bar slip deformations to total member lateral displacement could be significant. In addition to flexural deformations, bar slip deformations should be considered in modeling and analysis of reinforced concrete members. A monotonic bond-slip model is proposed and compared with other analytical models. Calculated lateral load-bar slip displacement relations compared reasonably well with the measured test results. The bond-slip model proposed in this paper is different from similar monotonic models proposed in [2], [7], and [10]:
5/10
• •
•
[10] appears to represent elastic behavior relatively well. The proposed model uses the same approach for modeling elastic bond-slip behavior using a larger bond stress. In addition to different inelastic development lengths and elastic bond stresses used by [2] and the proposed model, the bond stress distribution over the inelastic portion of the bar is represented differently in the two models. In the proposed model, the inelastic bond stress is reduced and is equal to one half of the elastic bond stress. On the other hand, the inelastic bond stress in [2] is increased and equal to the summation of elastic bond stress and a frictional bond stress. The proposed model and [7] use the same bond stress-slip model. However, in [7] the member end rotation includes the effect of slip in both compressive and tensile longitudinal reinforcement. According to [7], the slip rotation is the difference in the slips in compressive and tensile longitudinal bars divided by the distance between the bars. The proposed model considers the slip deformations in the tensile reinforcement only, and the slip rotation is calculated from Eq.(7).
ACKNOWLEDGEMENTS This work was funded by the Pacific Earthquake Engineering Research (PEER) Center through the Earthquake Engineering Research Centers Program of the National Science Foundation. The experiments by [13] were carried out in the research laboratories of PEER at the University of California, Berkeley.
NOTATION c = neutral axis depth, d = depth of cross-section, d b = bar diameter, E s = modulus of elasticity of reinforcing bar, f c ′ = specified compressive concrete strength, f s = stress in reinforcing bar, f y = steel yield stress, L = column length, l d = elastic development length, l’ d = development length over the inelastic portion of the bar, l dy = development length corresponding to reinforcing bar yielding at interface, slip = amount of reinforcing bar slip at beam-column interface, u b = elastic uniform bond stress, u’ b = inelastic uniform bond stress, u by = average uniform bond stress at first yielding, ∆ total = total member lateral displacement, ∆ y = member lateral displacement at first yielding of longitudinal reinforcement, ε s = strain in reinforcing bar,
ε y =
yield strain, and
θ slip
= member end rotation due to
longitudinal bar slip.,
REFERENCES [1]
[2] [3]
[4]
[5] [6] [7]
[8]
ACI Committee 408. 1979. Suggested Development, Splice, and Standard Hook Provisions for Deformed Bars. Concrete International, American Concrete Institute , July 1979, Vol. 1, No.7. pp. 44-46 Alsiwat J. M., and Saatcioglu M. 1992. Reinforcement Anchorage Slip under Monotonic Loading. Journal of Structural Engineering, ASCE , Vol.118, No.9, Sept. 1992. pp. 2421-2438 Ciampi V., Eligehausen R., Bertero V. V., and Popov, E. P. 1982. Analytical Model for Concrete Anchorages of Reinforcing Bars under Generalized Excitations. Technical Report UCB/EERC-82/23. Earthquake Engineering Research Center, University of California, Berkeley. Nov. 1982. 121 pages Eligehausen R., Popov E. P., and Bertero V. V. 1983. Local Bond Stress-Slip Relationships of Deformed Bars under Generalized Excitations. Report No. UCB/EERC-83/23. Earthquake Engineering Research Center, University of California, Berkeley. Oct. 1983. 169 pages Filippou F. C., Popov E. P., and Bertero V. V. 1986. Analytical Studies of Hysteretic of R/C Joints. Journal of Structural Engineering, ASCE. Vol. 112, No.7, July 1986. pp. 1605-1622 Hawkins N. M., Lin I., and Ueda T. 1987. Anchorage of Reinforcing Bars for Seismic Forces. ACI Structural Journal , Vol.84, No.5, Sept.-Oct. 1987. pp. 407-418 Lehman D. E., and Moehle J. P. 2000. Seismic Performance of Well-confined Concrete Bridge Columns. PEER-1998/01. Pacific Earthquake Engineering Research Center, University of California, Berkeley. 316 pages Lynn A. C. 2001. Seismic Evaluation of Existing Reinforced Concrete Building Columns. Ph.D. Thesis. Department of Civil and Environmental Engineering. University of California, Berkeley.
6/10
[9]
Morita S., and Kaku T. 1984. Slippage of Reinforcement in Beam-Column Joint of Reinforced th Concrete Frame. Proceedings, 8 World Conference on Earthquake Engineering, San Francisco. Vol. 6. pp. 477-484 Otani S., and Sozen M. A. November 1972. Behavior of Multistory Reinforced Concrete Frames during Earthquakes. Structural Research Series No. 392 , University of Illinois, Urbana. 551 pages Pochanart S., and Harmon T. 1989. Bond-slip Model for Generalized Excitations Including Fatigue. ACI Materials Journal , Vol. 86, 5, Sept.-Oct. 1989. pp. 465-474 Saatcioglu M., Alsiwat J. M., and Ozcebe G. 1992. Hysteretic Behavior of Anchorage Slip in R/C Members. Journal of Structural Engineering, ASCE . Vol.118, No.9, Sept. 1992. pp. 24392458 Sezen, H. 2002. Seismic Behavior and Modeling of Reinforced Concrete Building Columns. Ph.D. Thesis. University of California, Berkeley Soroushian P., and Choi K. -B. 1989. Local Bond of Deformed Bars with Different Diameters in Confined Concrete. ACI Structural Journal . Vol. 86, No.2, March-April 1989. pp. 217-222
[10]
[11] [12]
[13] [14]
t 100 n e m e 80 c a l p s 60 i d f o e 40 g a t n e 20 c r e p
0
100
Specimen−1
flexure
shear slip 0
1
2 ∆
total
100
100
Specimen−2
3
Specimen−3
Specimen−4
80
80
80
60
60
60
40
40
40
20
20
20
0
0
0.5
/ ∆
∆
y
1 / ∆
total
1.5
0
0
1
2 ∆
y
total
3
0
0
1
/ ∆
2 ∆
y
total
3
/ ∆
y
80
0
40
∆
) m60 m ( t n e m40 e c a l p s 20 i d
slip
∆
60
60
40
40
20
20
0
0
slip+shear
∆
30
slip+shear+flexure
flexure
20
10
shear slip
0
1
2
/ ∆y total
∆
3
0
0
0.5
1
/ ∆y total
∆
1.5
0
1
2
/ ∆y total
∆
3
0
1
2
/ ∆y total
∆
Fig. 1. Contribution of flexure, shear and slip displacements to total displacement
3
7/10
(i)
(ii)
(iii)
Fig. 2. Bond stress-slip model proposed by: i) [4], ii) [7], and iii) stress, strain, and bond stress distribution by [2]
Fig. 3. Column deformations due to bar slip and computer model
8/10
2 ) a P 1.8 M ( ) ‘ f c 1.6 √ / y b
u ( 1.4 d l e i y t 1.2 a s s 1 e r t s d n 0.8 o b t i n 0.6 u d e z 0.4 i l a m r o 0.2 n
0
mean + std. dev. mean = 0.95 mean − std. dev.
8 1 H L C 3
8 1 H L C 2
8 1 H L S 3
8 1 H L S 2
8 1 H M C 2
8 1 H M C 3
2 1 D M C 3
2 1 D M S 3
2 1 D L C 2
2 1 D H C 2
2 1 D V C 2
M 2 1 D L C 2
Fig. 4. Calculated bond stresses at yield level
Fig. 5. Column deformations due to bar slip and computer model
Fig. 6. Illustration of slip rotation and forces at the beam-column interface
9/10
80 70 60 ) i s k (
50
f s , s s e r40 t s r a b30
20 Proposed Model Otani and Sozen (1972) Alsiwat and Saatcioglu (1992)
10 0 0
0.05
0.1
0.15
0.2 0.25 slip (in.)
0.3
0.35
0.4
Fig. 7. Calculated reinforcing bar stress-slip relations (1 ksi = 6.895 MPa, 1 in.=25.4 mm) 4500 4000 3500 3000
) . n i − 2500 k ( t n e 2000 m o m
1500 1000
Proposed model, P=150 kips Alsiwat and Saatcioglu (1992), P=150 kips Proposed model, P=0 Alsiwat and Saatcioglu (1992), P=0 Otani and Sozen (1972)
500 0 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
slip (in.)
Fig. 8. Comparison of moment-slip relations (1 k-in.= 0.113 kN m, 1 kip = 4.448 kN, 1 in.=25.4 mm) 5000 ) . n i − k ( t n e m o m
5000 ) . n i − k ( t n e m o m
0
−5000 −0.01
Specimen−1 (top)
−0.005
0 0.005 slip rotation
0.01
measured model
0
−5000 −0.01
Specimen−1 (bottom)
−0.005
0 0.005 slip rotation
0.01
Fig. 9. Example of calculated and measured moment-slip rotation relations (1 k-in.= 0.113 kN m)
10/10
) s 50 p i k ( d a o 0 l l a r e t a−50 l
Specimen −1
experiment model
−1
) s 50 p i k ( d a o 0 l l a r e t a−50 l
0 0.5 −0.5 slip displacement (in.)
1
0 0.5 −0.5 slip displacement (in.)
−1
Specimen −2
0 0.5 −0.5 slip displacement (in.)
1
Specimen −4
Specimen −3
−1
) s 50 p i k ( d a o 0 l l a r e t a−50 l
) s 50 p i k ( d a o 0 l l a r e t a−50 l
1
−1
0 1 2 3 slip displacement (in.)
Fig. 10. Lateral load-slip displacement relations (1 kip = 4.448 kN, 1 in.= 25.4 mm)
4
