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Shear-strength degradation model for RC columns subjected to cyclic loading
Engineering Structures 34 (2012) 187–197
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Shear-strength degradation model for RC columns subjected to cyclic loading Hong-Gun Park a, Eun-Jong Yu b, Kyung-Kyu Choi c,⇑ a
Dept. of Architecture & Architectural Engineering, Seoul National Univ., San 56-1, Shinlim-dong, Kwanak-gu, Seoul 151-744, South Korea Dept. of Architectural Engineering, Hanyang Univ., 17 Haengdang-dong, Seongdong-gu, Seoul 133-791, South Korea c School of Architecture, Soongsil Univ., Sangdo-dong, Dongjak-gu, Seoul 156-743, South Korea b
a r t i c l e
i n f o
Article history: Received 22 February 2011 Revised 30 August 2011 Accepted 30 August 2011 Available online 4 November 2011 Keywords: Concrete columns Seismic design Shear strength Deformation Shear failure Cyclic loads
a b s t r a c t An analytical model was developed to estimate the shear-strength degradation and the deformation capacity of slender reinforced concrete columns subjected to cyclic transverse loading. The shear capacity of the concrete compression zone was defined as a function of the inelastic flexural deformation of the column, based on the material failure criteria of concrete. The shear capacity is degraded as the inelastic flexural deformation increases. The deformation capacity of a column is determined when the degraded shear capacity reaches the shear force demanded by flexural yielding of the column. Other failure mechanisms including rebar-buckling and -fracture and flexural failure were also considered to estimate the deformation capacity. The proposed model was applied to test specimens possessing various design parameters. The result showed that the proposed model estimated the shear-strength degradation and deformation capacity of the test specimens with reasonable precision. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction The use of a performance-based design method for ensuring the safety of structures subjected to earthquakes provides a strong impetus for an accurate estimation of the deformation capacity of reinforced concrete members. Particularly, an accurate evaluation of the deformation capacity is required for columns, because they have a relatively low deformation capacity, due to axial compression; their failure frequently brings about catastrophic damage to the overall structure. According to evidence from past strong earthquakes, reinforced concrete columns are susceptible to diagonal tension cracking that frequently leads to a brittle shear failure. Therefore, a major portion of previous studies for the earthquake resistance of columns has focused on investigating their shear strength. Experimental studies by Ang et al. [1], Aschheim and Moehle [2], Wong et al. [3], Moretti and Tassios [4], Ho and Pam [5], and Lee and Watanabe [6] showed that columns subjected to cyclic lateral loading may fail early, in shear, after flexural yielding. Based on test results, these studies reported that the shear strength of columns is heavily dependent on their inelastic deformations, and the shear strength degrades more quickly than flexural strength under cyclic loading. Priestley et al. [7] reported the shear-strength degradation and early shear failure is attributed to the development of diagonal tension cracks in the plastic hinge regions. ⇑ Corresponding author. E-mail address: [email protected] (K.-K. Choi). 0141-0296/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2011.08.041
In previous studies, several shear-capacity models have been proposed to account for the shear-strength degradation of columns subjected to cyclic lateral loading. The ATC seismic design guideline [8] proposed a shear-capacity curve, and it describes shearstrength degradation in terms of displacement ductility (Fig. 1). Martin-Perez and Pantazopoulou [9] proposed a shear-capacity curve similar to the ATC model considering the effect of bond, aggregate interlock, and dowel action on the shear strength degradation of RC columns. Priestley et al. [7] proposed an improved shear-capacity curve for columns by considering the contributions of concrete, transverse reinforcement, and axial load (Fig. 2). In the latter model, the shear strength of concrete and the shear contribution of the truss mechanism are defined as functions of the member’s inelastic deformation demand. In FEMA 273 [10], a ductility-related factor was introduced to describe the degradation of concrete’s shear capacity. Sezen and Moehle [11] followed a similar approach, but they applied the ductility-related factor to reinforcing bars as well as concrete (Fig. 2). Mullapudi and Ayoubm [12], and Sima et al. [13] developed a fiber beam-column element formulation and a constitutive model using smeared crack approach to simulate the seismic behavior of concrete columns subjected to cyclic load, respectively. In current design codes, decrease in shear strength under cyclic loading has been well recognized. However, strength degradation is not explicitly defined in terms of member’s deformation level, and the strength degradation is expressed in a more conservative way. ACI318-08 [14] neglects concrete shear strength for the members subjected to low compressive force in Special Moment
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of reinforced concrete columns (shear span ratio, 2.0 < a/d < 4.0) addressing the various failure mechanisms: the concrete shear failure, the buckling and fracture of longitudinal reinforcing bars, and flexural failure (i.e., concrete crushing in the compression zone). 2. Shear demand of columns
Fig. 1. ATC model [8] for shear-strength degradation.
Resisting Frame defined in the special provisions for seismic design. The NZC [15] also ignores the concrete contribution to the members’ shear resistance. Similarly, FEMA 273 [10] ignores the shear contribution of concrete members at moderate or high ductility demand levels. Due to the complexity of shear-strength degradation, which varies with flexural deformation, most previous models have estimated degradation of shear capacity, depending on practical experiences obtained from laboratory testing and field observations of earthquake-damaged buildings. If a more rigorous model is developed, based on the fundamental failure mechanism of reinforced concrete, it may improve the understanding of the mechanism behind shear-strength degradation; such understanding may eventually enable a more accurate assessment of the deformation capacity of reinforced concrete columns. Recently, Park et al. [16] developed a strain-based shearstrength model based on the material failure criteria of concrete. This model reasonably describes variations in the shear capacity of reinforced concrete beams, according to their flexural deformation. Originally, it was developed to estimate the shear strength of beams that fail in shear, before flexural yielding. However, this model is also applicable to estimating the shear strength of members after flexural yielding. In addition, Choi and Park [17] expanded this approach to concrete beams subjected to cyclic loading, and successfully the developed and verified the analytical model to evaluate the envelope of the load-deformation behavior up to the failure by using test results with a variety of range of test parameters. Based on their approach, in the present study, an analytical model was developed for estimating the seismic shear behavior
In a slender column where shear failure occurs after flexural yielding, the load-carrying capacity of the column is determined by its flexural yield strength. Therefore, the shear demand, which the column should resist, is determined as the shear force required for flexural yielding. The shear demand can be calculated from flexural moment-curvature analysis, using geometric data, material properties, and applied axial load. In a column confined by lateral ties, the confinement effect should be considered when describing the post-yielding flexural behavior of the column. In the present study, the compressive stress-strain relationship of the confined concrete was defined with an ascending branch of a second-order parabolic function and a linearly descending branch (Fig. 3) (Hognestad [18]; Vecchio and Collins [19]; Collins et al. [20]; Légeron and Paultre [21]).
"
ra ðeÞ ¼ fcc0 2
2 #
e e e1 e1
for
e 6 e1
rd ðeÞ ¼ fcc0 Z m ðe e1 Þ for e1 < e 6 eult ;
ð1aÞ
ð1bÞ
where ra ðeÞ and rd ðeÞ represent the stress-strain relationships for the ascending and descending branches, respectively; Z m ½¼ 0:15fcc0 =ðe85 e1 Þ denotes the slope of the descending branch; fcc0 is the compressive strength of the confined concrete; e1 is the compressive strain corresponding to the peak compressive strength, fcc0 ; and e85 is the compressive strain corresponding to 85% of the peak compressive strength on the descending branch. fcc0 , e1 , and e85 are defined according to Saatcioglu and Razvi’s model [22]:
fcc0 ¼ fc0 þ k1 fle
ð2Þ
e1 ¼ e01 ð1 þ 5KÞ
ð3Þ
e85 ¼ 260qe1 þ e085
ð4Þ
In these equations, e01 , and e085 denote material parameters for unconfined concrete: the peak compressive strength, strain at the peak strength, and post-peak strain corresponding to 85% of the peak strength, respectively. The parameters k1 , fle , K, and q, as used in the above equations, are calculated from the geometry fc0 ,
Fig. 2. Existing models for estimating shear capacity degraded by inelastic deformation.
H.-G. Park et al. / Engineering Structures 34 (2012) 187–197
V d ¼ ðM PdÞ=a
189
ð7Þ
where d is the tip displacement of a cantilever column and a is the shear span of the column. For a cantilever column, the shear span, a, is the distance between the point of lateral load application to the critical section at the fixed end. For a column subjected to a double curvature, the shear span is defined as the half span length. 3. Shear capacity of concrete
Fig. 3. Stress-strain relationships for unconfined concrete and confined concrete.
and material properties of the column cross-section, according to Saatcioglu and Razvi [22]. The calculation procedure was presented elsewhere [22]. In a cross-section of a column subjected to flexural deformation, the stress distribution over the compression zone in Eq. (1) can be expressed as a function of the distance from the neutral axis, z, by assuming a linear strain distribution.
2 az az for z 6 minðc; c=aÞ c c
ra ðzÞ ¼ fcc0 2
ð5aÞ
rd ðzÞ ¼ fcc0 Z m ðaz=c 1Þ P 0 for minðc; c=aÞ < z 6 min½c; ðc=aÞðeult =e1 Þ;
ð5bÞ
where c is the distance from the extreme compression fiber to the neutral axis in the cross-section of the core concrete (after spalling of concrete cover); and að¼ e=e1 Þ indicates the normal strain e at the extreme compression fiber of core concrete, normalized by e1 . Usually, the concrete cover spalls out under cyclic loading. Accordingly, the contribution of the concrete cover was ignored in evaluations of shear demand and shear capacity (see Fig. 4). The resultant moment at the cross-section is calculated by summing the moment contributions of all internal forces about the centroid of the cross-section.
M ¼ C c ðcp yc Þ þ
n X
Asi fsi ðcp dei Þ;
ð6Þ
The shear capacity of a reinforced concrete column is composed by the shear transfer mechanism of the intact concrete in the compression zone, the shear resistance of the transverse reinforcement, the dowel action of the longitudinal reinforcement, and the aggregate interlocking along crack surfaces (ASCE-ACI Committee 426 [23]). After flexural yielding, the tension zone in the cross-section of slender columns is severely damaged by flexural cracking. Therefore, the shear resistance of intact concrete in the compression zone is significantly greater than the contributions of the aggregate interlock and dowel action. Therefore, in this study, to be conservative and simple, the shear resistance of the reinforced concrete columns was assumed to be provided by the compression zone of the intact concrete and transverse reinforcement. The contributions of aggregate interlock and dowel action were disregarded (Kotsovos and Pavlovic´ [24]; and Priestley et al. [7]). As shown in Fig. 5, the concrete in the compression zone is subjected to a combination of compressive normal and shear stresses. Therefore, to evaluate accurately the shear capacity of the compression zone, the interaction between the compressive normal stress and the shear stress should be considered. In the present study, a material failure criterion with a simple form was used to describe the failure mechanism of concrete (Park et al. [16]; Chen [25]). According to Park et al. [16], at a location in the concrete cross-section, the shear stress capacities of the concrete subjected to the normal stress rðzÞ can be derived from the Rankine’s failure criteria [25].
mcc ðzÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi fcc0 fcc0 rðzÞ for failure controlled by compression ð8aÞ
i¼1
where C c is the resultant compressive force of the concrete, cp is the distance from the centroid to the extreme compression fiber of the core concrete, yc is the distance from the location of the resultant compressive force to the extreme compression fiber of the core concrete, Asi is the area of the longitudinal re-bars, fsi is the tensile stress of longitudinal re-bars at the i-th re-bar layer, n is the number of re-bar layers, and dei is the effective depth for the longitudinal rebars at the i-th layer (see Fig. 4). The shear force required by flexural moment M can be calculated, considering the second-order effect (P d effects) of the axial compressive force.
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mct ðzÞ ¼ ft0 ½ft0 þ rðzÞ for failure controlled by tension
ð8bÞ
where ft0 is tensile strength of concrete.The governing shear stress capacity mc at a location in the compression zone is determined as the smaller of the two stress capacities in Eq. (8). As can be easily derived from the equations, when the normal stress rðzÞ is less than r ð¼ fcc0 ft0 Þ, the governing shear stress capacity mc is determined as mct controlled by tension. Otherwise, mc is determined as mcc . When 0 0 the p normal ffiffiffiffiffiffiffiffiffiffiffi stress reaches r ð¼ fcc ft Þ, a is 1 m, where 0 0 m ¼ ft =fcc .
Fig. 4. Effective depth and strain distribution in cross-section of a column subjected to cyclic shear.
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Fig. 5. Rankine’s failure criteria for concrete.
Fig. 6 shows the variation of the shear stress capacity at a crosssection, according to the progress of flexural curvature (or the strain, ae1 , in the extreme compressive fiber). Fig. 6(a) and (b) show the distributions of the normal strain and stress, respectively. Fig. 6(c) and (d) show the shear stress capacities mcc (Eq. (8a)) and mct (Eq. (8b)), respectively. Points A, B, and Y marked in Fig. 6 indicate the initial state, the onset of flexural cracking, and the flexural yielding, respectively. Points C and D indicate the states at which the normal stress at extreme compression fiber reaches r and the peak compressive strength, respectively. As mentioned, the governing shear stress capacity mc at a location in the compression zone is determined as the smaller of mcc and mct (see Fig. 6(e)). In Stages A through C (i.e., when a 6 1 m), in all locations in the compression zone, mc is determined as mct . On the other hand, in Stages C through E (i.e., when a > 1 m), in the region of the compression zone that experiences compression softening, mc is determined as mcc . In particular, after the compressive normal stress at the extreme compression fiber of the cross-section reaches the compressive strength of concrete (a > 1, Stage DE), the failure criteria of the concrete is satisfied by the compressive normal stress alone. As a result, the region of the compression zone experiencing compressive softening no longer provides shear stress capacity (refer to Stage E in Fig. 6(c) and (e)). The shear capacity V c of the concrete compression zone in a cross-section can be calculated by integrating the governing shear stress capacity mc over the compression zone. Figs. 6(f) and 7 show the variation of the shear capacity of the concrete cross-section with the increasing flexural curvature. The shear capacity at each stage can be calculated as follows: In Stage AB prior to flexural cracking, the member behaves elastically and shear force is resisted by the entire section. In this stage,
Fig. 6. Variations of shear stress capacity, in a cross-section with the progress of flexural deformation.
the shear stress capacity at each location of the compression zone is governed by the tension. Thus, the shear capacity in Stage AB is suggested to be calculated as
V AB ¼ b
Z
h=2
h=2
mct ðzÞdz ¼ b
Z
h=2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ft0 ½ft0 þ rðzÞdz;
ð9Þ
h=2
where b is the column width, h is the column depth, and rðzÞ is the normal stress (Eq. (5)). In Stage BC after flexural cracking, the shear stress capacity is provided only by the compression zone, and it is governed by
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kDE ¼ mðm2 þ 1Þðp=4 tan1 ðmÞÞ þ m2 =2;
ð15aÞ
where
kDE is simplified as 0:69m for a typical value of mð0:26—0:33Þ: ð15bÞ As shown in Fig. 6(f), after the yielding of longitudinal bars (Stage Y), the shear capacity continuously decreases, because the depth of the compression zone, c, decreases to satisfy the forceequilibrium in the cross-section. As presented in Eqs. (10), (12), and (14), the shear capacity of the compression zone is inversely proportional to the curvature, /½¼ ða=cÞe1 , of the cross-section, and is affected by the ratio, m2 ¼ ft0 =fcc0 , of the tensile strength to the compressive strength of concrete. A good estimation of the mean tensile strength of concrete, ft0 , is given by MacGregor et al. [27] as
Fig. 7. Shear capacity and demand curve in a column cross-section.
tension. The shear capacity in this stage is calculated by integrating the tension-controlled shear stress capacity over the compression zone:
V BC ¼ b
Z
c
0
mct ðzÞdz ¼ kBC fcc0 bc=a;
ð10Þ
qffiffiffiffi ft0 ¼ 0:292 fc0 ðMPaÞ:
ð16Þ
The total shear capacity of a cross-section is given by the sum of the contributions of concrete and transverse reinforcement.
Vn ¼ Vc þ Vs
ð17Þ
V s ¼ qt bde1 fyt cot h;
ð18Þ
where
kBC ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 h m m þ ða 1Þ a2 þ 2a þ m2 2 2 1 a 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ð1 þ m2 Þ tan1 : þ sin m 1 þ m2
ð11Þ
In Eq. (10), the shear capacity is inversely proportional to the curvature of the cross-section /½¼ ða=cÞe1 . As shown in Fig. 6(f), the shear capacity of the cross-section drops significantly with the onset of flexural cracks. This is because the area resisting the shear force substantially decreases immediately after flexural cracking. The shear capacity stops decreasing when the neutral axis distance is stabilized. In Stage CE, at the extreme compression fiber, when the compressive stress exceeds r ð¼ fcc0 ft0 Þ (i.e., a > ð1 mÞ or z > ð1 mÞc=a), the shear stress capacity mcc by compression becomes smaller than mct by tension. In this case, in the compression zone, some region is governed by tension while the other region is governed by compression. Thus, the shear capacity at Stage CD is obtained by adding the integration of the shear stress capacities in the two regions:
Z
ð1mÞc=a
Z
c
mcc ðzÞdz ¼ kCD fcc0 bc=a
ð12Þ
i.
h kCD ¼ mðm2 þ 1Þ p=4 tan1 ðmÞ þ ða 1Þ2 þ m2 2
ð13Þ
V CD ¼ b
mct ðzÞdz þ b
ð1mÞc=a
0
Mehta and Monteiro [26] reported that the uniaxial tensile strength of concrete is approximately 0.07–0.11 times the compressive pffiffiffiffiffiffiffiffi strength of concrete. For this typical range of mð¼ ft0 fcc0 Þ, Eq. (13) can be simplified as kCD ¼ 0:69m ða 1Þ2 =2. The tensile strength of concrete is defined later (Eq. (16)). After the compressive stress at the extreme compression fiber of core concrete reaches the concrete peak strength (Stage DE, a > 1:0), the part of the compression zone that experiences compressive softening (i.e., concrete in the descending branch) cannot provide shear stress capacity. Therefore, the shear force is resisted by the remainder in the compression zone ð0 6 z 6 c=aÞ. Like the shear capacity at Stage CD, the shear capacity at Stage DE is obtained by adding the integration of the shear stress capacities controlled by tension and compression:
V DE ¼ b
Z 0
ð1mÞc=a
mct ðzÞdz þ b
Z
c=a
ð1mÞc=a
mcc ðzÞdz ¼ kDE fcc0 bc=a
ð14Þ
where V c is calculated from Eqs. (9), (10), (12), (14). qt is the transverse reinforcement ratio in the cross-section. fyt is the yield strength of the transverse reinforcement. The average angle of tensile cracking h is assumed to be 35° (CSA [28]). In the proposed model, the shear capacity of concrete V c is not affected by the angle of tensile cracks h, because the shear capacity of concrete is assumed to be provided only by the intact concrete in the compression zone. Since the concrete cover of the reinforced concrete column spalls out under repeated cyclic loading, the distance from the extreme compression fiber of core concrete to the longitudinal tension re-bars at the extreme tension fiber, de1 , is used in Eq. (18), instead of the effective depth, d (see Fig. 4). Fig. 7 shows the variations in shear demand and shear capacity. The yielding of shear reinforcement can be determined by comparing the capacity of shear reinforcement V s and the shear demand V d;max . When the shear demand V d;max is less than V s , shear reinforcement does not yield. When V d;max is greater than V s , shear reinforcement yield. In the figure, as flexural deformation increases, the shear contribution of concrete decreases; ultimately, only transverse reinforcement contributes to the shear capacity of the column section: V n V s . Therefore, ultimately, only the effective amount of the transverse reinforcement corresponding to [V s V d;max ðP 0Þ] can contribute to lateral confinement. Therefore, for the evaluation of compressive strength, fcc0 , and correspondent compressive strain, e1 , of confined concrete, the effective transverse reinforcement is used. 4. Determination of failure mode and deformation capacity Shear failure must be examined at all potential critical sections of a column, since both the shear capacity and shear demand differ at each cross-section along the column height. The analytical method for determining the location of the critical section in a simply supported beam can be found in Park et al. [16]. However, for a cantilever column or a column with double curvature that fails in shear after yielding, the fixed end is the critical section of the column. This is because at the fixed end, the shear capacity decreases due to flexural yielding, while the shear demand is uniform along the column height. For a column with a low or moderate amount of shear reinforcement, failure of the column may be initiated by shear failure at the
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Fig. 8. Determination of failure mode and deformation capacity. (a) shear failure of the compression zone (3CMD 12 of Lynn et al.), (b) flexural failure (H-2-15 of Esaki); and (c) bar fracture (TP2 of Takemura and Kawashima).
compression zone. In this case, the shear capacity curve intersects the shear demand curve (Fig. 8(a)). The columns, laterally confined by ties, are expected to maintain their ductile behavior following the shear capacity curve for a time, even after reaching the intersection point. Therefore, in the present study, the deformation capacity is defined as the deformation corresponding to 80% of the maximum shear demand after the intersection of the shear capacity and shear demand curves, according to Williams and Sexsmith [29] (see Fig. 8(a)). By using the shear capacity and shear demand curves (Eqs. (7) and (17)), the strain ratio a0 ð¼ e=e1 Þ corresponding to the deformation capacity is determined at 80% of the maximum shear demand.
V n ¼ 0:8 V d;max 0
kf cc bc=a þ V s ¼ 0:8ðM=a Pd=aÞ kf cc bc 0:8ðM=a Pd=aÞ V s
Ks dbi
ð22Þ
ð19bÞ
5. Verification
ð20Þ
2:5 ;
c 0:04 P e1 de1 c
ð19aÞ
where k is defined in Eqs. (11), (13), and (15), depending on the range of a. In reality, since M, d, c, and k are functions of a, the solution of Eq. (20) requires iterative calculations. When a sufficient amount of transverse reinforcement is used, the shear capacity (Eq. (17)) exceeds the shear demand (Eq. (7)), and thus the shear failure of concrete can be avoided (Fig. 8(b)). In this case, the column fails in flexure due to the concrete crushing in the compression zone. The deformation capacity corresponding to the flexural failure is defined as the deformation corresponding to 80% of the maximum shear demand in the descending branch of the shear demand curve. Under cyclic loading, spalling of the concrete cover occurs after a yielding of the longitudinal reinforcement. Subsequently, buckling or fracture of longitudinal bars may occur, which causes failure of the column (Matamoros and Sozen [30]; Lehman et al. [31]) (Fig. 8(c)). In the proposed model, relevant empirical equations are used to predict such failure modes. Moyer and Kowalsky [32] proposed an equation for the buckling strain of a longitudinal compression bar, from existing test results.
esb ¼ 3
ecf ¼
In summary, a column fails due to one of the following failure modes: shear failure of the concrete compression zone, flexural failure, or the buckling or fracture of longitudinal bars.
0
a0 ¼
Wood [33] observed that, for members subjected to reverse cyclic loading, the fracture of longitudinal tension bars can occur when the tensile strain of the re-bars reaches the fracture strain ef . In the present study, the fracture strain of re-bars ef ¼ 0:04 was used, according to the study by Wood [33] and Chen [25]. Compressive strain at the extreme fiber corresponding to the fracture of longitudinal re-bars is calculated assuming linear strain distribution.
ð21Þ
where s is the vertical spacing of the transverse ties, dbi is the diameter of the longitudinal re-bar, and K is the effective length factor for buckling (=1). It is assumed that bar buckling occurs when the compressive strain of concrete ae1 at the extreme compression fiber of the core concrete equals the buckling strain esb .
An extensive database of previously conducted lateral-load tests of reinforced concrete columns is available at the Pacific Earthquake Engineering Research Center (PEER) website [34] and Lee and Watanabe [35]. In the database, a total of 29 rectangular columns (a=d > 2:0) subjected to repeated cyclic loadings were used to verify the proposed model. All test specimens (nine columns) reported to have failed in shear was used, and 21 more specimens reported to have failed in flexure-shear or flexure were also selected for comparison. The geometry and material properties of the specimens are listed in Appendix (Table A1). The specimens had a broad range of design parameters: 0:0158 6 ql 6 0:0321, 0:0007 6 qt 6 0:0089, 20:2 6 fc0 6 86 MPa, 2:22 6 a=d 6 3:74, and 0 6 P=ðAg fc0 Þ 6 0:284, where ql and qt are the longitudinal and transverse reinforcement ratios, respectively. The tip displacement, d, of cantilever columns calculated by the proposed method was transformed to the rotational angles, by dividing it by the shear span: R ¼ d=a. Assuming yielding of longitudinal re-bars, the rotation of a column is calculated as follows:
R ¼ /ða=3Þl for / 6 /y R ¼ /y ða=3Þl þ ð/ /y Þlh
ð23aÞ for /y < /;
ð23bÞ
where the curvature / ¼ ae1 =c; /y ½¼ ey =ðde1 cÞ is the curvature at yielding; and lh indicates the effective plastic hinge length. Based on the results of the JCI Colloquium [36] and Lee and Watanabe [35], lh is estimated as lh ¼ 0:5d½M=ðVhÞ ¼ 0:5ad=h, where 0:75d 6 lh 6 d. In test specimens, bond-slip and shear deformations as well as flexural deformation occur. As a result, the yield stiffness of the specimens decreases. To address the effects of the bond-slip and shear deformations, l was used in Eq. (23).
H.-G. Park et al. / Engineering Structures 34 (2012) 187–197
193
Fig. 9. Comparison of estimation by the proposed model, with test results.
lð¼ EIflex =EIy;eff P 1:0Þ, which is the ratio of the flexural yield stiffness to the effective yield stiffness, can be estimated according to Elwood and Eberhard [37].
The experimental rotational capacity was determined as the rotation corresponding to 80% of the maximum applied shear force. When the load-deformation relationships were different in posi-
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form yield strength of the tension re-bars, the depth of the compression zone (c) should increase. However, for a given curvature, the increase in the depth of the compression zone increases the strain in the extreme compression fiber, which causes further expansion of the region experiencing compressive softening. Consequently, the compressive force of the compression zone cannot be maintained and, as a result, the flexural capacity decreases suddenly and the concrete members fail in flexure (Park et al. [38]). Other specimens with low longitudinal reinforcement ratios (e.g., TP2 and TP1 of Takemura and Kawashima [34]) were estimated to fail by the bar fracture, as shown in Fig. 9(l) and (m). In these specimens, because of the shallow depth of the compression zone, a significant tensile strain exceeding the fracture strain developed in the longitudinal tension re-bars. When the fracture strain of re-bars ef ¼ 0:04 was used, the test results for the specimens TP2 and TP1 in Fig. 9(l) and (m) were not well predicted by the proposed method. For these specimens, neither shear failure nor flexural failure was expected to occur. Therefore, further studies are required for the bar-fracture and -buckling mechanisms under cyclic loading. In Figs. 9 and A1, the mean ratio of the estimated rotational capacity to the test result and its coefficient of variation were 1.08 and 0.217, respectively, while the maximum and minimum values of the ratio were 0.72 and 1.89, respectively. The mean ratio of the estimated shear strength to the test result and its coefficient of variation were 1.04 and 0.055, respectively. This result indicates that the proposed method estimated the shear strength and rotational capacity of the columns, with acceptable accuracy. Except
tive and negative directions (i.e., pushing and pulling), the smaller of the rotational capacities of the specimen measured in the two directions was chosen as the rotational capacity. Since the experimental rotational capacity is defined on the envelope curve, the failure point may deviate slightly from the experimental cyclic curves. In Figs. 9 and A1 (Appendix), the shear force-rotation relationships of the test specimens are compared to the estimation results by the proposed analytical method. As seen in the figure, the proposed method estimated the shear force-rotation relationship, shear strength, and rotation capacity, all with reasonable precision. According to the estimation results, most of the specimens failed by shear failure of the compression zone, after flexural yielding of the longitudinal re-bars at the extreme layer. However, in the specimen CUW of Umehara and Jirsa(PEER [34]) (Fig. 9(n)), because of a high longitudinal reinforcement ratio, the longitudinal re-bars did not yield. Shear failure of the specimen was defined at 80% of the shear strength, after the intersection of the shear capacity and shear demand curves. On the other hand, in the specimens with sufficient shear reinforcement—as seen in Fig. 9(b)–(e) – since the shear reinforcement increased the shear capacity, the shear capacity and shear demand curves did not intersect. Thus, specimens H-2-13, H-2-15, HT-2-13, and HT-2-15 of Esaki [32] were estimated to fail in the flexural failure mode, with large inelastic rotations. According to Park et al. [38], as the region experiencing compressive softening of concrete in the cross-section expands, the resultant compressive force of the compression zone starts decreasing. As a result, to satisfy the force-equilibrium to the uni-
Table A1 Dimensions and material properties of existing test specimens [34,35]. Specimens
a b c
b (mm)
h (mm)
a (mm)
db1a (mm)
s (mm)
ql (%)
qt (%)
fc0 (MPa)
0 fcc (MPa)
fylb
fyt
P Ag fc0
ac d
Aboutaha Esaki
SC3 H-2-13 H-2-15 HT-2-13 HT-2-15
914 200 200 200 200
457 200 200 200 200
1219 400 400 400 400
25 12.7 12.7 12.7 12.7
406 40 50 60 75
1.88 2.53 2.53 2.53 2.53
0.096 0.64 0.51 0.65 0.52
21.9 23 23 20.2 20.2
21.9 24.3 23 21.7 20.2
434 363 363 363 363
400 364 364 364 364
0 0.067 0.041 0.067 0.041
3.07 2.28 2.28 2.28 2.28
Lynn et al.
2CLH18 2SLH18 3CLH18 3CMD12 3CMH18 3SLH18 3SMD12
457 457 457 457 457 457 457
457 457 457 457 457 457 457
1,473 1,473 1,473 1,473 1,473 1,473 1,473
25.4 25.4 31.8 31.8 31.8 31.8 31.8
457 457 457 305 457 457 305
1.94 1.94 3.03 3.03 3.03 3.03 3.03
0.07 0.07 0.07 0.17 0.07 0.07 0.17
33.1 33.1 26.9 27.6 27.6 26.9 25.5
33.1 33.1 26.9 27.6 27.6 26.9 25.5
331 331 331 331 331 331 331
400 400 400 400 400 400 400
0.073 0.073 0.089 0.262 0.262 0.089 0.284
3.74 3.71 3.74 3.74 3.74 3.71 3.71
Ohue et al.
2D16RS 4D13RS
200 200
200 200
400 400
16 13
50 50
2.01 2.65
0.48 0.48
32 29.9
32 29.9
369 370
316 316
0.143 0.153
2.28 2.28
Saatcioglu and Ozcebe
U1 U2 U3 U4
350 350 350 350
350 350 350 350
1,000 1,000 1,000 1,000
25 25 25 25
150 150 75 50
3.21 3.21 3.21 3.21
0.3 0.3 0.6 0.89
43.6 30.2 34.8 32
43.6 30.2 37.6 40.7
430 453 430 438
470 470 470 470
0 0.033 0.029 0.031
3.27 3.27 3.27 3.27
Sezen and Moehle
No. 1 No. 4
457 457
457 457
1,473 1,473
28.7 28.7
305 305
2.47 2.47
0.17 0.17
21.1 21.8
21.1 21.8
434 434
476 476
0.152 0.147
3.74 3.74
Takemura and Kawashima
TP2 TP1
400 400
400 400
1,245 1,245
12.7 12.7
70 70
1.58 1.58
0.2 0.2
35.7 35.9
35.7 35.9
363 363
368 368
0.027 0.027
3.46 3.46
Umehara and Jirsa
CUW
410
230
455
19
89
3.01
0.16
34.9
34.9
441
414
0.162
2.43
Xiao and Martirossyan
HC4-8L16-T6-0.1P HC4-8L16-T6-0.2P
254 254
254 254
508 508
15.9 15.9
51 51
2.46 2.46
0.75 0.75
86.0 86.0
86.0 86.0
510 510
449 449
0.096 0.192
2.24 2.24
Lee and Watanabe [35]
BA2 BA3 BA4
200 200 200
300 300 300
600 600 600
13 13 13
53 71 49
1.45 1.45 1.45
0.27 0.2 0.8
29.3 29.3 29.3
29.3 29.3 33.5
400 400 400
696 836 692
0 0 0
2.22 2.22 2.22
Wight and Sozen (East)
No.25.033 152
305
876
19
127
2.45
0.3
33.6
33.6
496
345
0.071
2.87
Yield strength of the longitudinal reinforcement. Diameter of longitudinal reinforcement. Shear span to depth ratio obtained from PEER data base.
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H.-G. Park et al. / Engineering Structures 34 (2012) 187–197
(1) Based on the material failure criteria of concrete, the shear capacity of the concrete in the compression zone can be defined as a function of the flexural deformation of the beam. (2) After flexural yielding, the shear capacity of the compression zone decreases with inelastic flexural deformation. This is because inelastic deformation reduces the depth of the compression zone and increases the area of the concrete subjected to compressive softening. When the degraded shear capacity reaches the shear demand, the shear failure of the compression zone is initiated. (3) When a sufficient amount of shear reinforcement is used, shear failure of the compression zone can be avoided; however, flexural failure occurs at a large inelastic deformation. A low longitudinal reinforcement ratio increases the tensile strain of the re-bars, which causes bar fracture failure. The wide spacing of lateral ties can cause buckling of the longitudinal re-bars. (4) High axial compressive force applied to columns significantly decreases the rotational capacity of columns, though it does not cause significant variations in shear strength.
for the specimens with high compressive strengths (e.g., HC48L16-T6-0.1P and HC4-8L16-T6-0.2P of Xiao and Martirossyan [34] in Fig. A1), the estimations by the proposed method generally showed good results. Fig. 8 presents the examples of the estimations by previous models (Priestley et al. [7]; and Sezen and Moehle [11]). In Appendix (Table A2), the reported failure modes are compared with the predicted failure mechanisms. For several specimens, the reported failure mode is significantly different from the predicted failure mechanism. However, it should be noted that the proposed method predicts only the initial failure mechanisms of columns, and after the initiation of failure, the columns can show complicated post-failure deformation modes, which may be different from the initial failure mechanisms. 6. Conclusions An analytical model was developed, to estimate the load-deformation relationships of slender columns that are significantly affected by shear-strength degradation after flexural yielding under cyclic lateral loading. To estimate the deformation capacity of columns, the proposed model addressed degradation in shear capacity, according to inelastic flexural deformation. The shear capacity of concrete was assumed to be provided by the intact concrete in the compression zone. Failure mechanisms of flexure, bar buckling, and bar fracture were also considered. For verification, the proposed method was applied to existing test specimens. The failure mechanisms used in the proposed method are summarized as follows:
The validity of the proposed method was verified for the columns with the following range of parameters: 0:0158 6 ql 6 0:0321, 0:0007 6 qt 6 0:0089, 20:2 6 fc0 6 86 MPa, 2:22 6 a=d 6 3:74, and 0 6 P=ðAg fc0 Þ 6 0:284. In the future, the applicability of the proposed method for a wider range of design parameters (e.g. shear span to depth ratio higher than 3.74) should be verified.
Table A2 Comparison of rotational capacity and shear strength, between estimation results and test results. Specimens
a b
Measured rotational capacity(rad)
Estimated rotational capacity (rad)
Ratio (Measured/ Estimated)
Measured shear strength (kN)
Estimated shear strength (kN)
Ratio (Measured/ Estimated)
Reported failure modea
Estimated failure mechanisma
Aboutaha Esaki
SC3 H-2-13 H-2-15 HT-2-13 HT-2-15
0.022 0.018 0.025 0.022 0.024
0.017 0.016 0.023 0.019 0.022
1.29 1.13 1.09 1.16 1.09
509 102 97 96 92
503 103 95 97 90
1.01 0.99 1.02 0.99 1.02
S -b -b -b -b
S F F F F
Lynn et al.
2CLH18 2SLH18 3CLH18 3CMD12 3CMH18 3SLH18 3SMD12
0.026 0.02 0.011 0.017 0.011 0.01 0.018
0.024 0.02 0.013 0.016 0.011 0.011 0.018
1.08 1 0.85 1.06 1 0.91 1
228 228 311 367 367 311 340
222 222 297 349 349 297 339
1.03 1.03 1.05 1.05 1.05 1.05 1
F-S F-S S S S S F-S
S S S S S S S
Ohue et al.
2D16RS 4D13RS
0.039 0.018
0.044 0.017
0.89 1.06
92 107
92 109
1 0.98
F-S F-S
S S
Saatcioglu and Ozcebe
U1 U2 U3 U4
0.048 0.038 0.044 0.09
0.047 0.032 0.042 0.08
1.02 1.19 1.05 1.13
238 299 301 298
233 289 293 300
1.02 1.03 1.03 0.99
F S F F
S S F F
Sezen and Moehle
No. 1 No. 4
0.023 0.026
0.023 0.024
1 1.08
320 323
279 281
1.15 1.15
F-S F-S
S S
Takemura and Kawashima
TP2 TP1
0.039 0.037
0.051 0.054
0.77 0.72
149 149
144 144
1.03 1.03
F F
Bf Bf
Umehara and Jirsa
CUW
0.015
0.01
1.5
314
307
1.02
S
S
Xiao and Martirossyan
HC4-8L16-T6-0.1P HC4-8L16-T6-0.2P
0.04 0.036
0.032 0.019
1.25 1.89
271 349
267 336
1.01 1.04
F-S F-S
F F
Lee and Watanabe [35]
BA2 BA3 BA4
0.036 0.033 0.042
0.035 0.034 0.039
1.03 0.97 1.08
142 140 144
124 123 122
1.15 1.14 1.18
S S S
F F F
Wight and Sozen Mean COV
No.25.033(East)
0.035
0.034
1.03 1.08 0.217
85.3
83.6
1.02 1.04 0.055
S
S
S = shear failure (of the compression zone), F = flexural failure, F-S = flexural-shear failure, and Bf = bar fracture. Failure mode was not reported for these test specimens.
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H.-G. Park et al. / Engineering Structures 34 (2012) 187–197
Fig. A1. Additional comparison of estimation by the proposed model, with test results.
H.-G. Park et al. / Engineering Structures 34 (2012) 187–197
Acknowledgment This research was financially supported by the Ministry of Construction and Transportation of Korea (04 R&D C02-02). The authors are also grateful to the Prof. Moehle and the PEER Center at the University of California at Berkeley for providing us with the test data base, which is essential for the verification of this research. Appendix A See Tables A1 and A2. See Fig. A1. References [1] Ang BG, Priestley MJN, Paulay T. Seismic shear strength of circular reinforced concrete columns. ACI Struct J 1989;86(1):45–59. [2] Aschheim M, Moehle JP. Shear Strength and Deformability of RC Bridge Columns Subjected to Inelastic Cyclic Displacements. Rep. No. UCB/EERC-92/ 04, Earthquake Engineering Research Center, University of California at Berkeley, Berkeley, CA, 1992. [3] Wong YL, Paulay T, Priestley MJN. Response of circular reinforced concrete beams to multi-directional seismic attack. ACI Struct J 1993;90(2):180–91. [4] Moretti M, Tassios TP. Behaviour of short columns subjected to cyclic shear displacements: experimental results. Eng Struct 2007;29:2018–29. [5] Ho JCM, Pam HJ. Inelastic design of low-axially loaded high-strength reinforced concrete columns. Eng Struct 2003;25:1083–96. [6] Lee J-Y, Watanabe F. Predicting the longitudinal axial strain in the plastic hinge regions of reinforced concrete beams subjected to reversed cyclic loading. Eng Struct 2003;25:927–39. [7] Priestley MJN, Verma R, Xiao Y. Seismic shear strength of reinforced concrete columns. J Struct Eng 1994;120(8):2310–29. [8] Applied Technology Council (ATC 32). Improved Seismic Design Criteria for California Bridges, Provisional Recommendations, 1996. [9] Matin-Perez B, Pantazopoulou SJ. Effect of bond, aggregate interlock and dowel action on the shear strength degradation of reinforced concrete. Eng Struct 2001;23:214–27. [10] BSSC. NEHRP Commentary on the Guidelines for the Seismic Rehabilitation of Buildings (FEMA Publication 273). Applied Technology Council (ATC-33 Project), Washington, DC, 1997. [11] Sezen H, Moehle JP. Shear strength model for lightly reinforced concrete columns. J Struct Eng 2004;130(11):1703–962. [12] Mullapudi TR, Ayoub A. Modeling of the seismic behavior of shear-critical reinforced concrete columns. Eng Struct 2010;32:3601–15. [13] Sima JF, Roca R, Molins C. Cyclic constitutive model for concrete. Eng Struct 2008;30:695–706.
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[14] American Concrete Institute. Building Code Requirements for Structural Concrete. ACI 318–08, 2008. [15] Standards Association of New Zealand. Code of Practice for the Design of Concrete Structures. NZS-3101, Wellington, New Zealand, 1982. [16] Park H, Choi K, Wight JK. Strain-based shear strength model for slender beams without web reinforcement. ACI Struct J 2006;103(6):783–93. [17] Choi K, Park H. Evaluation of inelastic deformation capacity of slender beams subjected to cyclic loading. ACI Struct J 2010;107(5):507–15. [18] Hognestad E. A Study of Combined Bending and Axial Load in Reinforced Concrete Members. Bulletin 399, Engineering Experimental Station, Univ. of Illinois, USA; 1951. [19] Vecchio FJ, Collins MP. The modified compression-field theory for reinforced concrete elements subjected to shear. ACI J 1986;83(2):219–31. [20] Collins MP, Mitchell D, Vecchio FJ. A general shear design method. ACI Struct J 1996;93(1):36–45. [21] Légeron F, Paultre P. Uniaxial confinement model for normal- and highstrength concrete columns. J Struct Eng 2003;129(2):241–52. [22] Saatcioglu M, Razvi S. Strength and ductility of confined concrete. J Struct Eng 1992;118(6):1590–607. [23] ASCE-ACI Joint Task Committee 426. Shear strength of reinforced concrete members. J Struct Eng 1973;99(6):1091–187. [24] Kotsovos MD, Pavlovic´ MN, Ultimate limit-sate design of concrete structures: a new approach. Thomas Telford, London; 1998. p. 208. [25] Chen WF. Plasticity in reinforced concrete. New York: McGraw-Hill; 1982. p. 474. [26] Mehta PK, Monteiro PJM. Concrete, microstructure, properties, and materials. 3rd ed. McGraw Hill; 2004. pp. 659. [27] MacGregor JG, Sozen MA, Siess CP, Strength and behavior of prestressed concrete beams with web reinforcement. University of Illinois Civil Engineering Studies, Structural Research Series 210, Urbana, 1960. [28] CSA Technical Committee on Reinforced Concrete Design. ‘‘A23.3-04 Design of Concrete Structures,’’ Canadian Standards Association, Rexdale, Ontario, 2004. [29] Willams MS, Sexsmith RG. Seismic damage indices for concrete structures: a state-of-the-art review. Earthquake Spectra 1995;11(2):319–49. [30] Matamoros AB, Sozen MA. Drift limits of high-strength concrete columns subjected to load reversals. J Struct Eng 2003;129(3):297–313. [31] Lehman D, Moehle J, Mahin S, Calderone A, Henry L. Experimental evaluation of the seismic performance of reinforced concrete bridge columns. J Struct Eng 2004;130(6):869–79. [32] Moyer MJ, Kowalsky MJ. Influence of tension strain on buckling of Re-bar in concrete columns. ACI Struct J 2003;100(1):75–85. [33] Wood SL. Minimum tensile reinforcement requirements in walls. ACI Struct J 1989;86(4):582–91. [34] PEER, Structural Performance Database, http://nisee.berkeley.edu/spd/. [35] Lee J, Watanabe F. Shear deterioration of reinforced concrete beams subjected to reversed cyclic loading. ACI Struct J 2003;100(4):480–9. [36] Japan Concrete Institute. JCI Colloquium on Ductility of Concrete Structures and Its Evaluation, 1988, p. 466. [37] Elwood K, Eberhard MO. Effective stiffness of reinforced concrete columns. ACI Struct J 2009;106(4):476–84. [38] Park H, Kang S, Chung L, Lee D. Moment-curvature relationship of flexuredominated walls with partially confined end-zone. Eng Struct 2007;29(1):33–45.
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Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Shear-strength degradation model for RC columns subjected to cyclic loading Hong-Gun Park a, Eun-Jong Yu b, Kyung-Kyu Choi c,⇑ a
Dept. of Architecture & Architectural Engineering, Seoul National Univ., San 56-1, Shinlim-dong, Kwanak-gu, Seoul 151-744, South Korea Dept. of Architectural Engineering, Hanyang Univ., 17 Haengdang-dong, Seongdong-gu, Seoul 133-791, South Korea c School of Architecture, Soongsil Univ., Sangdo-dong, Dongjak-gu, Seoul 156-743, South Korea b
a r t i c l e
i n f o
Article history: Received 22 February 2011 Revised 30 August 2011 Accepted 30 August 2011 Available online 4 November 2011 Keywords: Concrete columns Seismic design Shear strength Deformation Shear failure Cyclic loads
a b s t r a c t An analytical model was developed to estimate the shear-strength degradation and the deformation capacity of slender reinforced concrete columns subjected to cyclic transverse loading. The shear capacity of the concrete compression zone was defined as a function of the inelastic flexural deformation of the column, based on the material failure criteria of concrete. The shear capacity is degraded as the inelastic flexural deformation increases. The deformation capacity of a column is determined when the degraded shear capacity reaches the shear force demanded by flexural yielding of the column. Other failure mechanisms including rebar-buckling and -fracture and flexural failure were also considered to estimate the deformation capacity. The proposed model was applied to test specimens possessing various design parameters. The result showed that the proposed model estimated the shear-strength degradation and deformation capacity of the test specimens with reasonable precision. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction The use of a performance-based design method for ensuring the safety of structures subjected to earthquakes provides a strong impetus for an accurate estimation of the deformation capacity of reinforced concrete members. Particularly, an accurate evaluation of the deformation capacity is required for columns, because they have a relatively low deformation capacity, due to axial compression; their failure frequently brings about catastrophic damage to the overall structure. According to evidence from past strong earthquakes, reinforced concrete columns are susceptible to diagonal tension cracking that frequently leads to a brittle shear failure. Therefore, a major portion of previous studies for the earthquake resistance of columns has focused on investigating their shear strength. Experimental studies by Ang et al. [1], Aschheim and Moehle [2], Wong et al. [3], Moretti and Tassios [4], Ho and Pam [5], and Lee and Watanabe [6] showed that columns subjected to cyclic lateral loading may fail early, in shear, after flexural yielding. Based on test results, these studies reported that the shear strength of columns is heavily dependent on their inelastic deformations, and the shear strength degrades more quickly than flexural strength under cyclic loading. Priestley et al. [7] reported the shear-strength degradation and early shear failure is attributed to the development of diagonal tension cracks in the plastic hinge regions. ⇑ Corresponding author. E-mail address: [email protected] (K.-K. Choi). 0141-0296/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2011.08.041
In previous studies, several shear-capacity models have been proposed to account for the shear-strength degradation of columns subjected to cyclic lateral loading. The ATC seismic design guideline [8] proposed a shear-capacity curve, and it describes shearstrength degradation in terms of displacement ductility (Fig. 1). Martin-Perez and Pantazopoulou [9] proposed a shear-capacity curve similar to the ATC model considering the effect of bond, aggregate interlock, and dowel action on the shear strength degradation of RC columns. Priestley et al. [7] proposed an improved shear-capacity curve for columns by considering the contributions of concrete, transverse reinforcement, and axial load (Fig. 2). In the latter model, the shear strength of concrete and the shear contribution of the truss mechanism are defined as functions of the member’s inelastic deformation demand. In FEMA 273 [10], a ductility-related factor was introduced to describe the degradation of concrete’s shear capacity. Sezen and Moehle [11] followed a similar approach, but they applied the ductility-related factor to reinforcing bars as well as concrete (Fig. 2). Mullapudi and Ayoubm [12], and Sima et al. [13] developed a fiber beam-column element formulation and a constitutive model using smeared crack approach to simulate the seismic behavior of concrete columns subjected to cyclic load, respectively. In current design codes, decrease in shear strength under cyclic loading has been well recognized. However, strength degradation is not explicitly defined in terms of member’s deformation level, and the strength degradation is expressed in a more conservative way. ACI318-08 [14] neglects concrete shear strength for the members subjected to low compressive force in Special Moment
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of reinforced concrete columns (shear span ratio, 2.0 < a/d < 4.0) addressing the various failure mechanisms: the concrete shear failure, the buckling and fracture of longitudinal reinforcing bars, and flexural failure (i.e., concrete crushing in the compression zone). 2. Shear demand of columns
Fig. 1. ATC model [8] for shear-strength degradation.
Resisting Frame defined in the special provisions for seismic design. The NZC [15] also ignores the concrete contribution to the members’ shear resistance. Similarly, FEMA 273 [10] ignores the shear contribution of concrete members at moderate or high ductility demand levels. Due to the complexity of shear-strength degradation, which varies with flexural deformation, most previous models have estimated degradation of shear capacity, depending on practical experiences obtained from laboratory testing and field observations of earthquake-damaged buildings. If a more rigorous model is developed, based on the fundamental failure mechanism of reinforced concrete, it may improve the understanding of the mechanism behind shear-strength degradation; such understanding may eventually enable a more accurate assessment of the deformation capacity of reinforced concrete columns. Recently, Park et al. [16] developed a strain-based shearstrength model based on the material failure criteria of concrete. This model reasonably describes variations in the shear capacity of reinforced concrete beams, according to their flexural deformation. Originally, it was developed to estimate the shear strength of beams that fail in shear, before flexural yielding. However, this model is also applicable to estimating the shear strength of members after flexural yielding. In addition, Choi and Park [17] expanded this approach to concrete beams subjected to cyclic loading, and successfully the developed and verified the analytical model to evaluate the envelope of the load-deformation behavior up to the failure by using test results with a variety of range of test parameters. Based on their approach, in the present study, an analytical model was developed for estimating the seismic shear behavior
In a slender column where shear failure occurs after flexural yielding, the load-carrying capacity of the column is determined by its flexural yield strength. Therefore, the shear demand, which the column should resist, is determined as the shear force required for flexural yielding. The shear demand can be calculated from flexural moment-curvature analysis, using geometric data, material properties, and applied axial load. In a column confined by lateral ties, the confinement effect should be considered when describing the post-yielding flexural behavior of the column. In the present study, the compressive stress-strain relationship of the confined concrete was defined with an ascending branch of a second-order parabolic function and a linearly descending branch (Fig. 3) (Hognestad [18]; Vecchio and Collins [19]; Collins et al. [20]; Légeron and Paultre [21]).
"
ra ðeÞ ¼ fcc0 2
2 #
e e e1 e1
for
e 6 e1
rd ðeÞ ¼ fcc0 Z m ðe e1 Þ for e1 < e 6 eult ;
ð1aÞ
ð1bÞ
where ra ðeÞ and rd ðeÞ represent the stress-strain relationships for the ascending and descending branches, respectively; Z m ½¼ 0:15fcc0 =ðe85 e1 Þ denotes the slope of the descending branch; fcc0 is the compressive strength of the confined concrete; e1 is the compressive strain corresponding to the peak compressive strength, fcc0 ; and e85 is the compressive strain corresponding to 85% of the peak compressive strength on the descending branch. fcc0 , e1 , and e85 are defined according to Saatcioglu and Razvi’s model [22]:
fcc0 ¼ fc0 þ k1 fle
ð2Þ
e1 ¼ e01 ð1 þ 5KÞ
ð3Þ
e85 ¼ 260qe1 þ e085
ð4Þ
In these equations, e01 , and e085 denote material parameters for unconfined concrete: the peak compressive strength, strain at the peak strength, and post-peak strain corresponding to 85% of the peak strength, respectively. The parameters k1 , fle , K, and q, as used in the above equations, are calculated from the geometry fc0 ,
Fig. 2. Existing models for estimating shear capacity degraded by inelastic deformation.
H.-G. Park et al. / Engineering Structures 34 (2012) 187–197
V d ¼ ðM PdÞ=a
189
ð7Þ
where d is the tip displacement of a cantilever column and a is the shear span of the column. For a cantilever column, the shear span, a, is the distance between the point of lateral load application to the critical section at the fixed end. For a column subjected to a double curvature, the shear span is defined as the half span length. 3. Shear capacity of concrete
Fig. 3. Stress-strain relationships for unconfined concrete and confined concrete.
and material properties of the column cross-section, according to Saatcioglu and Razvi [22]. The calculation procedure was presented elsewhere [22]. In a cross-section of a column subjected to flexural deformation, the stress distribution over the compression zone in Eq. (1) can be expressed as a function of the distance from the neutral axis, z, by assuming a linear strain distribution.
2 az az for z 6 minðc; c=aÞ c c
ra ðzÞ ¼ fcc0 2
ð5aÞ
rd ðzÞ ¼ fcc0 Z m ðaz=c 1Þ P 0 for minðc; c=aÞ < z 6 min½c; ðc=aÞðeult =e1 Þ;
ð5bÞ
where c is the distance from the extreme compression fiber to the neutral axis in the cross-section of the core concrete (after spalling of concrete cover); and að¼ e=e1 Þ indicates the normal strain e at the extreme compression fiber of core concrete, normalized by e1 . Usually, the concrete cover spalls out under cyclic loading. Accordingly, the contribution of the concrete cover was ignored in evaluations of shear demand and shear capacity (see Fig. 4). The resultant moment at the cross-section is calculated by summing the moment contributions of all internal forces about the centroid of the cross-section.
M ¼ C c ðcp yc Þ þ
n X
Asi fsi ðcp dei Þ;
ð6Þ
The shear capacity of a reinforced concrete column is composed by the shear transfer mechanism of the intact concrete in the compression zone, the shear resistance of the transverse reinforcement, the dowel action of the longitudinal reinforcement, and the aggregate interlocking along crack surfaces (ASCE-ACI Committee 426 [23]). After flexural yielding, the tension zone in the cross-section of slender columns is severely damaged by flexural cracking. Therefore, the shear resistance of intact concrete in the compression zone is significantly greater than the contributions of the aggregate interlock and dowel action. Therefore, in this study, to be conservative and simple, the shear resistance of the reinforced concrete columns was assumed to be provided by the compression zone of the intact concrete and transverse reinforcement. The contributions of aggregate interlock and dowel action were disregarded (Kotsovos and Pavlovic´ [24]; and Priestley et al. [7]). As shown in Fig. 5, the concrete in the compression zone is subjected to a combination of compressive normal and shear stresses. Therefore, to evaluate accurately the shear capacity of the compression zone, the interaction between the compressive normal stress and the shear stress should be considered. In the present study, a material failure criterion with a simple form was used to describe the failure mechanism of concrete (Park et al. [16]; Chen [25]). According to Park et al. [16], at a location in the concrete cross-section, the shear stress capacities of the concrete subjected to the normal stress rðzÞ can be derived from the Rankine’s failure criteria [25].
mcc ðzÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi fcc0 fcc0 rðzÞ for failure controlled by compression ð8aÞ
i¼1
where C c is the resultant compressive force of the concrete, cp is the distance from the centroid to the extreme compression fiber of the core concrete, yc is the distance from the location of the resultant compressive force to the extreme compression fiber of the core concrete, Asi is the area of the longitudinal re-bars, fsi is the tensile stress of longitudinal re-bars at the i-th re-bar layer, n is the number of re-bar layers, and dei is the effective depth for the longitudinal rebars at the i-th layer (see Fig. 4). The shear force required by flexural moment M can be calculated, considering the second-order effect (P d effects) of the axial compressive force.
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mct ðzÞ ¼ ft0 ½ft0 þ rðzÞ for failure controlled by tension
ð8bÞ
where ft0 is tensile strength of concrete.The governing shear stress capacity mc at a location in the compression zone is determined as the smaller of the two stress capacities in Eq. (8). As can be easily derived from the equations, when the normal stress rðzÞ is less than r ð¼ fcc0 ft0 Þ, the governing shear stress capacity mc is determined as mct controlled by tension. Otherwise, mc is determined as mcc . When 0 0 the p normal ffiffiffiffiffiffiffiffiffiffiffi stress reaches r ð¼ fcc ft Þ, a is 1 m, where 0 0 m ¼ ft =fcc .
Fig. 4. Effective depth and strain distribution in cross-section of a column subjected to cyclic shear.
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Fig. 5. Rankine’s failure criteria for concrete.
Fig. 6 shows the variation of the shear stress capacity at a crosssection, according to the progress of flexural curvature (or the strain, ae1 , in the extreme compressive fiber). Fig. 6(a) and (b) show the distributions of the normal strain and stress, respectively. Fig. 6(c) and (d) show the shear stress capacities mcc (Eq. (8a)) and mct (Eq. (8b)), respectively. Points A, B, and Y marked in Fig. 6 indicate the initial state, the onset of flexural cracking, and the flexural yielding, respectively. Points C and D indicate the states at which the normal stress at extreme compression fiber reaches r and the peak compressive strength, respectively. As mentioned, the governing shear stress capacity mc at a location in the compression zone is determined as the smaller of mcc and mct (see Fig. 6(e)). In Stages A through C (i.e., when a 6 1 m), in all locations in the compression zone, mc is determined as mct . On the other hand, in Stages C through E (i.e., when a > 1 m), in the region of the compression zone that experiences compression softening, mc is determined as mcc . In particular, after the compressive normal stress at the extreme compression fiber of the cross-section reaches the compressive strength of concrete (a > 1, Stage DE), the failure criteria of the concrete is satisfied by the compressive normal stress alone. As a result, the region of the compression zone experiencing compressive softening no longer provides shear stress capacity (refer to Stage E in Fig. 6(c) and (e)). The shear capacity V c of the concrete compression zone in a cross-section can be calculated by integrating the governing shear stress capacity mc over the compression zone. Figs. 6(f) and 7 show the variation of the shear capacity of the concrete cross-section with the increasing flexural curvature. The shear capacity at each stage can be calculated as follows: In Stage AB prior to flexural cracking, the member behaves elastically and shear force is resisted by the entire section. In this stage,
Fig. 6. Variations of shear stress capacity, in a cross-section with the progress of flexural deformation.
the shear stress capacity at each location of the compression zone is governed by the tension. Thus, the shear capacity in Stage AB is suggested to be calculated as
V AB ¼ b
Z
h=2
h=2
mct ðzÞdz ¼ b
Z
h=2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ft0 ½ft0 þ rðzÞdz;
ð9Þ
h=2
where b is the column width, h is the column depth, and rðzÞ is the normal stress (Eq. (5)). In Stage BC after flexural cracking, the shear stress capacity is provided only by the compression zone, and it is governed by
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kDE ¼ mðm2 þ 1Þðp=4 tan1 ðmÞÞ þ m2 =2;
ð15aÞ
where
kDE is simplified as 0:69m for a typical value of mð0:26—0:33Þ: ð15bÞ As shown in Fig. 6(f), after the yielding of longitudinal bars (Stage Y), the shear capacity continuously decreases, because the depth of the compression zone, c, decreases to satisfy the forceequilibrium in the cross-section. As presented in Eqs. (10), (12), and (14), the shear capacity of the compression zone is inversely proportional to the curvature, /½¼ ða=cÞe1 , of the cross-section, and is affected by the ratio, m2 ¼ ft0 =fcc0 , of the tensile strength to the compressive strength of concrete. A good estimation of the mean tensile strength of concrete, ft0 , is given by MacGregor et al. [27] as
Fig. 7. Shear capacity and demand curve in a column cross-section.
tension. The shear capacity in this stage is calculated by integrating the tension-controlled shear stress capacity over the compression zone:
V BC ¼ b
Z
c
0
mct ðzÞdz ¼ kBC fcc0 bc=a;
ð10Þ
qffiffiffiffi ft0 ¼ 0:292 fc0 ðMPaÞ:
ð16Þ
The total shear capacity of a cross-section is given by the sum of the contributions of concrete and transverse reinforcement.
Vn ¼ Vc þ Vs
ð17Þ
V s ¼ qt bde1 fyt cot h;
ð18Þ
where
kBC ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 h m m þ ða 1Þ a2 þ 2a þ m2 2 2 1 a 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ð1 þ m2 Þ tan1 : þ sin m 1 þ m2
ð11Þ
In Eq. (10), the shear capacity is inversely proportional to the curvature of the cross-section /½¼ ða=cÞe1 . As shown in Fig. 6(f), the shear capacity of the cross-section drops significantly with the onset of flexural cracks. This is because the area resisting the shear force substantially decreases immediately after flexural cracking. The shear capacity stops decreasing when the neutral axis distance is stabilized. In Stage CE, at the extreme compression fiber, when the compressive stress exceeds r ð¼ fcc0 ft0 Þ (i.e., a > ð1 mÞ or z > ð1 mÞc=a), the shear stress capacity mcc by compression becomes smaller than mct by tension. In this case, in the compression zone, some region is governed by tension while the other region is governed by compression. Thus, the shear capacity at Stage CD is obtained by adding the integration of the shear stress capacities in the two regions:
Z
ð1mÞc=a
Z
c
mcc ðzÞdz ¼ kCD fcc0 bc=a
ð12Þ
i.
h kCD ¼ mðm2 þ 1Þ p=4 tan1 ðmÞ þ ða 1Þ2 þ m2 2
ð13Þ
V CD ¼ b
mct ðzÞdz þ b
ð1mÞc=a
0
Mehta and Monteiro [26] reported that the uniaxial tensile strength of concrete is approximately 0.07–0.11 times the compressive pffiffiffiffiffiffiffiffi strength of concrete. For this typical range of mð¼ ft0 fcc0 Þ, Eq. (13) can be simplified as kCD ¼ 0:69m ða 1Þ2 =2. The tensile strength of concrete is defined later (Eq. (16)). After the compressive stress at the extreme compression fiber of core concrete reaches the concrete peak strength (Stage DE, a > 1:0), the part of the compression zone that experiences compressive softening (i.e., concrete in the descending branch) cannot provide shear stress capacity. Therefore, the shear force is resisted by the remainder in the compression zone ð0 6 z 6 c=aÞ. Like the shear capacity at Stage CD, the shear capacity at Stage DE is obtained by adding the integration of the shear stress capacities controlled by tension and compression:
V DE ¼ b
Z 0
ð1mÞc=a
mct ðzÞdz þ b
Z
c=a
ð1mÞc=a
mcc ðzÞdz ¼ kDE fcc0 bc=a
ð14Þ
where V c is calculated from Eqs. (9), (10), (12), (14). qt is the transverse reinforcement ratio in the cross-section. fyt is the yield strength of the transverse reinforcement. The average angle of tensile cracking h is assumed to be 35° (CSA [28]). In the proposed model, the shear capacity of concrete V c is not affected by the angle of tensile cracks h, because the shear capacity of concrete is assumed to be provided only by the intact concrete in the compression zone. Since the concrete cover of the reinforced concrete column spalls out under repeated cyclic loading, the distance from the extreme compression fiber of core concrete to the longitudinal tension re-bars at the extreme tension fiber, de1 , is used in Eq. (18), instead of the effective depth, d (see Fig. 4). Fig. 7 shows the variations in shear demand and shear capacity. The yielding of shear reinforcement can be determined by comparing the capacity of shear reinforcement V s and the shear demand V d;max . When the shear demand V d;max is less than V s , shear reinforcement does not yield. When V d;max is greater than V s , shear reinforcement yield. In the figure, as flexural deformation increases, the shear contribution of concrete decreases; ultimately, only transverse reinforcement contributes to the shear capacity of the column section: V n V s . Therefore, ultimately, only the effective amount of the transverse reinforcement corresponding to [V s V d;max ðP 0Þ] can contribute to lateral confinement. Therefore, for the evaluation of compressive strength, fcc0 , and correspondent compressive strain, e1 , of confined concrete, the effective transverse reinforcement is used. 4. Determination of failure mode and deformation capacity Shear failure must be examined at all potential critical sections of a column, since both the shear capacity and shear demand differ at each cross-section along the column height. The analytical method for determining the location of the critical section in a simply supported beam can be found in Park et al. [16]. However, for a cantilever column or a column with double curvature that fails in shear after yielding, the fixed end is the critical section of the column. This is because at the fixed end, the shear capacity decreases due to flexural yielding, while the shear demand is uniform along the column height. For a column with a low or moderate amount of shear reinforcement, failure of the column may be initiated by shear failure at the
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Fig. 8. Determination of failure mode and deformation capacity. (a) shear failure of the compression zone (3CMD 12 of Lynn et al.), (b) flexural failure (H-2-15 of Esaki); and (c) bar fracture (TP2 of Takemura and Kawashima).
compression zone. In this case, the shear capacity curve intersects the shear demand curve (Fig. 8(a)). The columns, laterally confined by ties, are expected to maintain their ductile behavior following the shear capacity curve for a time, even after reaching the intersection point. Therefore, in the present study, the deformation capacity is defined as the deformation corresponding to 80% of the maximum shear demand after the intersection of the shear capacity and shear demand curves, according to Williams and Sexsmith [29] (see Fig. 8(a)). By using the shear capacity and shear demand curves (Eqs. (7) and (17)), the strain ratio a0 ð¼ e=e1 Þ corresponding to the deformation capacity is determined at 80% of the maximum shear demand.
V n ¼ 0:8 V d;max 0
kf cc bc=a þ V s ¼ 0:8ðM=a Pd=aÞ kf cc bc 0:8ðM=a Pd=aÞ V s
Ks dbi
ð22Þ
ð19bÞ
5. Verification
ð20Þ
2:5 ;
c 0:04 P e1 de1 c
ð19aÞ
where k is defined in Eqs. (11), (13), and (15), depending on the range of a. In reality, since M, d, c, and k are functions of a, the solution of Eq. (20) requires iterative calculations. When a sufficient amount of transverse reinforcement is used, the shear capacity (Eq. (17)) exceeds the shear demand (Eq. (7)), and thus the shear failure of concrete can be avoided (Fig. 8(b)). In this case, the column fails in flexure due to the concrete crushing in the compression zone. The deformation capacity corresponding to the flexural failure is defined as the deformation corresponding to 80% of the maximum shear demand in the descending branch of the shear demand curve. Under cyclic loading, spalling of the concrete cover occurs after a yielding of the longitudinal reinforcement. Subsequently, buckling or fracture of longitudinal bars may occur, which causes failure of the column (Matamoros and Sozen [30]; Lehman et al. [31]) (Fig. 8(c)). In the proposed model, relevant empirical equations are used to predict such failure modes. Moyer and Kowalsky [32] proposed an equation for the buckling strain of a longitudinal compression bar, from existing test results.
esb ¼ 3
ecf ¼
In summary, a column fails due to one of the following failure modes: shear failure of the concrete compression zone, flexural failure, or the buckling or fracture of longitudinal bars.
0
a0 ¼
Wood [33] observed that, for members subjected to reverse cyclic loading, the fracture of longitudinal tension bars can occur when the tensile strain of the re-bars reaches the fracture strain ef . In the present study, the fracture strain of re-bars ef ¼ 0:04 was used, according to the study by Wood [33] and Chen [25]. Compressive strain at the extreme fiber corresponding to the fracture of longitudinal re-bars is calculated assuming linear strain distribution.
ð21Þ
where s is the vertical spacing of the transverse ties, dbi is the diameter of the longitudinal re-bar, and K is the effective length factor for buckling (=1). It is assumed that bar buckling occurs when the compressive strain of concrete ae1 at the extreme compression fiber of the core concrete equals the buckling strain esb .
An extensive database of previously conducted lateral-load tests of reinforced concrete columns is available at the Pacific Earthquake Engineering Research Center (PEER) website [34] and Lee and Watanabe [35]. In the database, a total of 29 rectangular columns (a=d > 2:0) subjected to repeated cyclic loadings were used to verify the proposed model. All test specimens (nine columns) reported to have failed in shear was used, and 21 more specimens reported to have failed in flexure-shear or flexure were also selected for comparison. The geometry and material properties of the specimens are listed in Appendix (Table A1). The specimens had a broad range of design parameters: 0:0158 6 ql 6 0:0321, 0:0007 6 qt 6 0:0089, 20:2 6 fc0 6 86 MPa, 2:22 6 a=d 6 3:74, and 0 6 P=ðAg fc0 Þ 6 0:284, where ql and qt are the longitudinal and transverse reinforcement ratios, respectively. The tip displacement, d, of cantilever columns calculated by the proposed method was transformed to the rotational angles, by dividing it by the shear span: R ¼ d=a. Assuming yielding of longitudinal re-bars, the rotation of a column is calculated as follows:
R ¼ /ða=3Þl for / 6 /y R ¼ /y ða=3Þl þ ð/ /y Þlh
ð23aÞ for /y < /;
ð23bÞ
where the curvature / ¼ ae1 =c; /y ½¼ ey =ðde1 cÞ is the curvature at yielding; and lh indicates the effective plastic hinge length. Based on the results of the JCI Colloquium [36] and Lee and Watanabe [35], lh is estimated as lh ¼ 0:5d½M=ðVhÞ ¼ 0:5ad=h, where 0:75d 6 lh 6 d. In test specimens, bond-slip and shear deformations as well as flexural deformation occur. As a result, the yield stiffness of the specimens decreases. To address the effects of the bond-slip and shear deformations, l was used in Eq. (23).
H.-G. Park et al. / Engineering Structures 34 (2012) 187–197
193
Fig. 9. Comparison of estimation by the proposed model, with test results.
lð¼ EIflex =EIy;eff P 1:0Þ, which is the ratio of the flexural yield stiffness to the effective yield stiffness, can be estimated according to Elwood and Eberhard [37].
The experimental rotational capacity was determined as the rotation corresponding to 80% of the maximum applied shear force. When the load-deformation relationships were different in posi-
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form yield strength of the tension re-bars, the depth of the compression zone (c) should increase. However, for a given curvature, the increase in the depth of the compression zone increases the strain in the extreme compression fiber, which causes further expansion of the region experiencing compressive softening. Consequently, the compressive force of the compression zone cannot be maintained and, as a result, the flexural capacity decreases suddenly and the concrete members fail in flexure (Park et al. [38]). Other specimens with low longitudinal reinforcement ratios (e.g., TP2 and TP1 of Takemura and Kawashima [34]) were estimated to fail by the bar fracture, as shown in Fig. 9(l) and (m). In these specimens, because of the shallow depth of the compression zone, a significant tensile strain exceeding the fracture strain developed in the longitudinal tension re-bars. When the fracture strain of re-bars ef ¼ 0:04 was used, the test results for the specimens TP2 and TP1 in Fig. 9(l) and (m) were not well predicted by the proposed method. For these specimens, neither shear failure nor flexural failure was expected to occur. Therefore, further studies are required for the bar-fracture and -buckling mechanisms under cyclic loading. In Figs. 9 and A1, the mean ratio of the estimated rotational capacity to the test result and its coefficient of variation were 1.08 and 0.217, respectively, while the maximum and minimum values of the ratio were 0.72 and 1.89, respectively. The mean ratio of the estimated shear strength to the test result and its coefficient of variation were 1.04 and 0.055, respectively. This result indicates that the proposed method estimated the shear strength and rotational capacity of the columns, with acceptable accuracy. Except
tive and negative directions (i.e., pushing and pulling), the smaller of the rotational capacities of the specimen measured in the two directions was chosen as the rotational capacity. Since the experimental rotational capacity is defined on the envelope curve, the failure point may deviate slightly from the experimental cyclic curves. In Figs. 9 and A1 (Appendix), the shear force-rotation relationships of the test specimens are compared to the estimation results by the proposed analytical method. As seen in the figure, the proposed method estimated the shear force-rotation relationship, shear strength, and rotation capacity, all with reasonable precision. According to the estimation results, most of the specimens failed by shear failure of the compression zone, after flexural yielding of the longitudinal re-bars at the extreme layer. However, in the specimen CUW of Umehara and Jirsa(PEER [34]) (Fig. 9(n)), because of a high longitudinal reinforcement ratio, the longitudinal re-bars did not yield. Shear failure of the specimen was defined at 80% of the shear strength, after the intersection of the shear capacity and shear demand curves. On the other hand, in the specimens with sufficient shear reinforcement—as seen in Fig. 9(b)–(e) – since the shear reinforcement increased the shear capacity, the shear capacity and shear demand curves did not intersect. Thus, specimens H-2-13, H-2-15, HT-2-13, and HT-2-15 of Esaki [32] were estimated to fail in the flexural failure mode, with large inelastic rotations. According to Park et al. [38], as the region experiencing compressive softening of concrete in the cross-section expands, the resultant compressive force of the compression zone starts decreasing. As a result, to satisfy the force-equilibrium to the uni-
Table A1 Dimensions and material properties of existing test specimens [34,35]. Specimens
a b c
b (mm)
h (mm)
a (mm)
db1a (mm)
s (mm)
ql (%)
qt (%)
fc0 (MPa)
0 fcc (MPa)
fylb
fyt
P Ag fc0
ac d
Aboutaha Esaki
SC3 H-2-13 H-2-15 HT-2-13 HT-2-15
914 200 200 200 200
457 200 200 200 200
1219 400 400 400 400
25 12.7 12.7 12.7 12.7
406 40 50 60 75
1.88 2.53 2.53 2.53 2.53
0.096 0.64 0.51 0.65 0.52
21.9 23 23 20.2 20.2
21.9 24.3 23 21.7 20.2
434 363 363 363 363
400 364 364 364 364
0 0.067 0.041 0.067 0.041
3.07 2.28 2.28 2.28 2.28
Lynn et al.
2CLH18 2SLH18 3CLH18 3CMD12 3CMH18 3SLH18 3SMD12
457 457 457 457 457 457 457
457 457 457 457 457 457 457
1,473 1,473 1,473 1,473 1,473 1,473 1,473
25.4 25.4 31.8 31.8 31.8 31.8 31.8
457 457 457 305 457 457 305
1.94 1.94 3.03 3.03 3.03 3.03 3.03
0.07 0.07 0.07 0.17 0.07 0.07 0.17
33.1 33.1 26.9 27.6 27.6 26.9 25.5
33.1 33.1 26.9 27.6 27.6 26.9 25.5
331 331 331 331 331 331 331
400 400 400 400 400 400 400
0.073 0.073 0.089 0.262 0.262 0.089 0.284
3.74 3.71 3.74 3.74 3.74 3.71 3.71
Ohue et al.
2D16RS 4D13RS
200 200
200 200
400 400
16 13
50 50
2.01 2.65
0.48 0.48
32 29.9
32 29.9
369 370
316 316
0.143 0.153
2.28 2.28
Saatcioglu and Ozcebe
U1 U2 U3 U4
350 350 350 350
350 350 350 350
1,000 1,000 1,000 1,000
25 25 25 25
150 150 75 50
3.21 3.21 3.21 3.21
0.3 0.3 0.6 0.89
43.6 30.2 34.8 32
43.6 30.2 37.6 40.7
430 453 430 438
470 470 470 470
0 0.033 0.029 0.031
3.27 3.27 3.27 3.27
Sezen and Moehle
No. 1 No. 4
457 457
457 457
1,473 1,473
28.7 28.7
305 305
2.47 2.47
0.17 0.17
21.1 21.8
21.1 21.8
434 434
476 476
0.152 0.147
3.74 3.74
Takemura and Kawashima
TP2 TP1
400 400
400 400
1,245 1,245
12.7 12.7
70 70
1.58 1.58
0.2 0.2
35.7 35.9
35.7 35.9
363 363
368 368
0.027 0.027
3.46 3.46
Umehara and Jirsa
CUW
410
230
455
19
89
3.01
0.16
34.9
34.9
441
414
0.162
2.43
Xiao and Martirossyan
HC4-8L16-T6-0.1P HC4-8L16-T6-0.2P
254 254
254 254
508 508
15.9 15.9
51 51
2.46 2.46
0.75 0.75
86.0 86.0
86.0 86.0
510 510
449 449
0.096 0.192
2.24 2.24
Lee and Watanabe [35]
BA2 BA3 BA4
200 200 200
300 300 300
600 600 600
13 13 13
53 71 49
1.45 1.45 1.45
0.27 0.2 0.8
29.3 29.3 29.3
29.3 29.3 33.5
400 400 400
696 836 692
0 0 0
2.22 2.22 2.22
Wight and Sozen (East)
No.25.033 152
305
876
19
127
2.45
0.3
33.6
33.6
496
345
0.071
2.87
Yield strength of the longitudinal reinforcement. Diameter of longitudinal reinforcement. Shear span to depth ratio obtained from PEER data base.
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(1) Based on the material failure criteria of concrete, the shear capacity of the concrete in the compression zone can be defined as a function of the flexural deformation of the beam. (2) After flexural yielding, the shear capacity of the compression zone decreases with inelastic flexural deformation. This is because inelastic deformation reduces the depth of the compression zone and increases the area of the concrete subjected to compressive softening. When the degraded shear capacity reaches the shear demand, the shear failure of the compression zone is initiated. (3) When a sufficient amount of shear reinforcement is used, shear failure of the compression zone can be avoided; however, flexural failure occurs at a large inelastic deformation. A low longitudinal reinforcement ratio increases the tensile strain of the re-bars, which causes bar fracture failure. The wide spacing of lateral ties can cause buckling of the longitudinal re-bars. (4) High axial compressive force applied to columns significantly decreases the rotational capacity of columns, though it does not cause significant variations in shear strength.
for the specimens with high compressive strengths (e.g., HC48L16-T6-0.1P and HC4-8L16-T6-0.2P of Xiao and Martirossyan [34] in Fig. A1), the estimations by the proposed method generally showed good results. Fig. 8 presents the examples of the estimations by previous models (Priestley et al. [7]; and Sezen and Moehle [11]). In Appendix (Table A2), the reported failure modes are compared with the predicted failure mechanisms. For several specimens, the reported failure mode is significantly different from the predicted failure mechanism. However, it should be noted that the proposed method predicts only the initial failure mechanisms of columns, and after the initiation of failure, the columns can show complicated post-failure deformation modes, which may be different from the initial failure mechanisms. 6. Conclusions An analytical model was developed, to estimate the load-deformation relationships of slender columns that are significantly affected by shear-strength degradation after flexural yielding under cyclic lateral loading. To estimate the deformation capacity of columns, the proposed model addressed degradation in shear capacity, according to inelastic flexural deformation. The shear capacity of concrete was assumed to be provided by the intact concrete in the compression zone. Failure mechanisms of flexure, bar buckling, and bar fracture were also considered. For verification, the proposed method was applied to existing test specimens. The failure mechanisms used in the proposed method are summarized as follows:
The validity of the proposed method was verified for the columns with the following range of parameters: 0:0158 6 ql 6 0:0321, 0:0007 6 qt 6 0:0089, 20:2 6 fc0 6 86 MPa, 2:22 6 a=d 6 3:74, and 0 6 P=ðAg fc0 Þ 6 0:284. In the future, the applicability of the proposed method for a wider range of design parameters (e.g. shear span to depth ratio higher than 3.74) should be verified.
Table A2 Comparison of rotational capacity and shear strength, between estimation results and test results. Specimens
a b
Measured rotational capacity(rad)
Estimated rotational capacity (rad)
Ratio (Measured/ Estimated)
Measured shear strength (kN)
Estimated shear strength (kN)
Ratio (Measured/ Estimated)
Reported failure modea
Estimated failure mechanisma
Aboutaha Esaki
SC3 H-2-13 H-2-15 HT-2-13 HT-2-15
0.022 0.018 0.025 0.022 0.024
0.017 0.016 0.023 0.019 0.022
1.29 1.13 1.09 1.16 1.09
509 102 97 96 92
503 103 95 97 90
1.01 0.99 1.02 0.99 1.02
S -b -b -b -b
S F F F F
Lynn et al.
2CLH18 2SLH18 3CLH18 3CMD12 3CMH18 3SLH18 3SMD12
0.026 0.02 0.011 0.017 0.011 0.01 0.018
0.024 0.02 0.013 0.016 0.011 0.011 0.018
1.08 1 0.85 1.06 1 0.91 1
228 228 311 367 367 311 340
222 222 297 349 349 297 339
1.03 1.03 1.05 1.05 1.05 1.05 1
F-S F-S S S S S F-S
S S S S S S S
Ohue et al.
2D16RS 4D13RS
0.039 0.018
0.044 0.017
0.89 1.06
92 107
92 109
1 0.98
F-S F-S
S S
Saatcioglu and Ozcebe
U1 U2 U3 U4
0.048 0.038 0.044 0.09
0.047 0.032 0.042 0.08
1.02 1.19 1.05 1.13
238 299 301 298
233 289 293 300
1.02 1.03 1.03 0.99
F S F F
S S F F
Sezen and Moehle
No. 1 No. 4
0.023 0.026
0.023 0.024
1 1.08
320 323
279 281
1.15 1.15
F-S F-S
S S
Takemura and Kawashima
TP2 TP1
0.039 0.037
0.051 0.054
0.77 0.72
149 149
144 144
1.03 1.03
F F
Bf Bf
Umehara and Jirsa
CUW
0.015
0.01
1.5
314
307
1.02
S
S
Xiao and Martirossyan
HC4-8L16-T6-0.1P HC4-8L16-T6-0.2P
0.04 0.036
0.032 0.019
1.25 1.89
271 349
267 336
1.01 1.04
F-S F-S
F F
Lee and Watanabe [35]
BA2 BA3 BA4
0.036 0.033 0.042
0.035 0.034 0.039
1.03 0.97 1.08
142 140 144
124 123 122
1.15 1.14 1.18
S S S
F F F
Wight and Sozen Mean COV
No.25.033(East)
0.035
0.034
1.03 1.08 0.217
85.3
83.6
1.02 1.04 0.055
S
S
S = shear failure (of the compression zone), F = flexural failure, F-S = flexural-shear failure, and Bf = bar fracture. Failure mode was not reported for these test specimens.
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Fig. A1. Additional comparison of estimation by the proposed model, with test results.
H.-G. Park et al. / Engineering Structures 34 (2012) 187–197
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