Plastic hinge analysis of FRP confined circular concrete columns

Construction and Building Materials 27 (2012) 223–233

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Plastic hinge analysis of FRP confined circular concrete columns Dong-Sheng Gu a,b, Yu-Fei Wu a,⇑, Gang Wu c, Zhi-shen Wu c a

Department of Building and Construction, City University of Hong Kong, Hong Kong College of Environmental and Civil Engineering, Jiangnan University, PR China c College of Civil Engineering, Southeast University, PR China b

a r t i c l e

i n f o

Article history: Received 7 July 2010 Received in revised form 25 July 2011 Accepted 26 July 2011 Available online 23 August 2011 Keywords: Concrete Column Confinement FRP Plastic hinge length Deformation capacity

a b s t r a c t Experimental tests have identified that FRP confinement affects the plastic hinge length of reinforced concrete columns. Some tests found that the confinement increased the plastic hinge length; whereas others showed otherwise. The plastic hinge length as well as the drift capacity of FRP confined circular concrete columns are studied in this work through a combination of numerical simulation, experimental study, and mechanism analysis. Data regressions are employed to formulate the plastic hinge length, which is found to be closely related to the confinement ratio of FRP. The obtained plastic hinge model shows that FRP confinement increases the plastic hinge length at low confinement ratio; however, it has an opposite effect when the confinement ratio is high. The accuracy of the proposed model is verified by test results from 29 large-scale FRP-confined circular concrete columns. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Deformation capacity is critical for structures expected to withstand seismic activity and must be carefully designed to satisfy explicit deformation demands in performance-based design [1]. Reinforced concrete (RC) columns are often required to undergo a large number of inelastic deformation cycles for a design earthquake while maintaining a certain strength to ensure the stability of the structure. A sufficient deformation capacity for RC columns can usually be achieved by providing adequate confining reinforcement at a potential plastic hinge region [2,3]. The deformation-based approach to the design of confining steel for RC columns has been extensively investigated [4], and is accepted by many current design codes. In recent years, fiber-reinforced polymer (FRP) jackets have become popular in providing confinement to RC columns [5]; however, extensive analytical modeling of the plastic hinge deformation capacity of FRP-confined RC columns under seismic load is limited [6]. Although nonlinear numerical simulation can be used for calculating the deformation capacity of FRP-confined RC columns [7,8], a simple design-based procedure is necessary for engineering use. Curvature capacity at the cross-sectional level and drift capacity at the member level are often used as criteria for evaluating the deformation capacity of columns. A procedure for calculating the curvature capacity of FRP-confined columns has been proposed [9,10], but studies calculating deformation capacity at the member ⇑ Corresponding author. Tel.: +852 34424259; fax: +852 34427612. E-mail address: [email protected] (Y.-F. Wu). 0950-0618/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.conbuildmat.2011.07.056

level are scarce in the open literature. Calculation of that deformation capacity requires a model for the plastic hinge length, which can be combined with the model for ultimate curvature to give the drift capacity of the member [11,12]. The literature reveals contradictory conclusions about the plastic hinge length of FRP-confined columns. One suggestion is that the plastic hinge length of an FRP-confined RC column is smaller than that of a normal RC column [13]. This conclusion derives from the test observation of steel jacketed RC columns in which the jacket restricted the spread of plastic yielding [14]. Other researchers have accepted this position [15]. However experimental tests have also shown that the plastic hinge lengths of most FRP-confined columns are larger than those of normal RC columns [16]. Some researchers have proposed that to ensure simplicity, the plastic hinge length of an FRP-confined column be considered equal to that of a normal RC column [10,17]. Given these findings, it is clear that the plastic hinge length of FRP-confined RC columns needs further investigation. The parameters that affect the ultimate curvature and plastic hinge length of FRP-confined circular concrete columns are extensively studied in this paper, and models are proposed to predict the ultimate drift ratio of those columns. 2. Deformation relationships Using the well-known plastic hinge concept (Fig. 1), Park and Paulay [3] proposed an expression for the ultimate displacement, Du, at the tip of a cantilever column

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c 1 ¼ : D 3:8  5:0n c ¼ 0:19 þ 0:9n: D

N

F

Moment L

Actual Curvature

H Lp

φy

φp φu

Fig. 1. Plastic-hinge analysis.

c ¼ 0:12 þ ð1:07  1:05K e xs Þ  n; D

2

Du ¼ Dy þ Dp ¼

/y L þ ð/u  /y Þlp ðL  0:5lp Þ; 3

ð1Þ

where Dy is the yield displacement, Dp is the plastic displacement, /u is the ultimate curvature at the column base, /y is the yield curvature, and L and lp are the lengths of the cantilever column and the plastic hinge, respectively. The ultimate drift ratio, hu, can be expressed as:

hu ¼

Du /y L ð/u  /y Þlp ðL  0:5lp Þ ¼ þ : L 3 L

/y ¼ k

D

;

ð8Þ

where xs is the confinement ratio of transverse steel, and Ke is the shape factor related to the effective area of confinement. In this work, the same approach is adopted to derive a model similar to Eq. (8) for the calculation of the compression zone depth of FRP-confined circular columns. As the confinement provided by FRP is different from that by steel stirrups, and also the concrete cover does not spall in FRP confined columns, the details of the model will be different from those for RC columns.

ð2Þ 3.1. Numerical simulations

The yield curvature of a reinforced concrete column can be given as [13,18]:

ey

ð7bÞ

For RC columns with a significant confinement by steel stirrups, the ultimate failure strain is greater than 0.004 and closely related to the degree of confinement. Therefore, apart from the parameter n in the above equations, another parameter involving confinement should be included in the model. From moment–curvature analyses, the following equation was proposed by Jiang et al. [21] for calculation of the compression zone depth at ultimate failure:

Assumed Curvature

Mu

ð7aÞ

ð3Þ

where k ¼ 2:45 for spiral-reinforced columns, ey is the yield strain of the longitudinal reinforcement, and D is the diameter of the column. The confinement of FRP does not significantly affect the column yield state, and thus the equation is also applicable to FRPconfined columns [10]. In Eq. (2), /u and lp are the two parameters that need to be determined.

In this section a numerical procedure using the conventional layered method [18–22] is adopted to study the moment–curvature relationships of FRP-confined circular sections. The moment–curvature relationship is generally insensitive to the constitutive model for confined concrete, and the change in the yield curvature and ultimate moment is insignificant when different constitutive models are used [23,24]. The constitutive model for FRP-confined concrete proposed by Lam and Teng [25] is adopted for the numerical simulation. No tensile strength of the concrete is considered. The Giuffre–Menegotto–Pinto model is used as the stress–strain relationship for the longitudinal steel bars [26].

3. Ultimate curvature 3.2. Numerical simulation parameters The ultimate curvature /u can be expressed as

/u ¼ ecu =c;

ð4Þ

where c is the depth of the neutral axis (or the depth of the compression zone) when the strain at the extreme compressive fiber reaches the ultimate strain, ecu . The neutral axis depth c of RC sections has been comprehensively studied in the literature. Although many factors such as axial load, longitudinal steel and transverse reinforcement can affect the value of c in general, the following simplified equation has been adopted by many researchers for evaluation of c [19,20]:

/y D D /c D G0 ¼ ¼ ; þ c ec 1 þ G1 n ec

ð5Þ

where /c is the curvature at a given extreme compression fiber strain ec , n is the axial load ratio (defined as n ¼ N=Ag fc0 , where Ag is the gross cross-sectional area, fc0 is the concrete compressive strength, and N is the axial force), and G0 and G1 are two parameters depending on the value of ec . By substituting Eq. (3) into Eq. (5), the following equation is obtained for circular columns:

D G0 ey ¼ þ 2:45 : c 1 þ G1 n ec

ð6Þ

From the moment–curvature analyses of flexure-dominant columns, G0 and G1 are calculated to be 5.3 and 9.4, respectively, for ec ¼ 0:004 [19,20]. Similar expressions have been adopted by others to calculate c/D at ec ¼ 0:004, such Eq. (7a) by Kowalsky [18] and Eq. (7b) by Jiang et al. [21]:

A typical bridge pier with a diameter of one meter, a clear cover of 40 mm, and a cylinder compressive strength fc0 of 30 MPa is used for the analysis. The yield stress of the longitudinal reinforcement, fy, is 420 MPa. The axial load ratio n varies from 0.1 to 0.4 in increments of 0.1. The longitudinal reinforcement ratio qs (defined as qs = As/Ag, where As is the area of longitudinal reinforcement) changes from 1% to 4% in 1% increments. The confinement ratio, 0 kf (defined as kf ¼ fl =fc0 ¼ 2Ef t f ef =Df c , where fl is the lateral confining pressure exerted by the FRP, Ef is the modulus of the FRP, tf is the thickness of the FRP jacket, and ef is the FRP rupture strain from flat coupon test), varies between 0.1 and 0.3 in increments of 0.1. Two types of FRP are considered in the numerical simulation: carbon FRP (CFRP) and glass FRP (GFRP). The ultimate tensile strength and rupture strain of the CFRP are 3500 MPa and 0.013, respectively, while those for the GFRP are 1380 MPa and 0.023, respectively. As non-dimensional parameters are adopted in the analyses, the conclusions derived are more general. 3.3. Numerical results The ratio of the neutral axis depth to the sectional diameter, c/D, is plotted versus the extreme compressive fiber strain for one group of CFRP confined sections in Fig. 2. Clearly, the compression zone depth approaches a stable value when ec is greater than 0.004. As the ultimate strain of FRP confined columns is generally much greater than 0.004, the calculation of compression zone depth is insensitive to the ultimate strain. Therefore, the compression zone depth is

D.-S. Gu et al. / Construction and Building Materials 27 (2012) 223–233

225

calculated using the ultimate strain of concrete corresponding to the ultimate failure of FRP confined columns under concentrate loading which is provided in Ref. [25]. Fig. 3 shows the results of c/D for sections with different types of FRP and different confinement ratios. The stress–strain model in Ref. [25] is not only related to the confinement ratio but also related to the confinement stiffness ratio and the strain capacity of FRP at hoop rupture which are different for different types of FRP. It can be seen that FRP type has an insignificant effect on the compression zone depth under an identical confinement ratio. Therefore, the type of FRP can be ignored in the modeling. Linear regression of the numerical results including 48 data points for GFRP-confined sections and another 48 points for CFRP-confined sections leads to the following equation:

c ¼ 0:19 þ ð0:72  0:67kf Þ  n: D

ð9Þ

Comparing Eq. (9) with Eq. (8), it can be seen that these two equations are very similar. For externally confined circular columns, the shape parameter Ke equals to 1. The ratio of the numerical result to that calculated with Eq. (9) for the study above has a mean of 1.01 and a coefficient of variation (COV) of 10.5%, which shows that Eq. (9) has a good computational accuracy. It should be noted that kf in Eq. (9) is calculated from the ultimate strain of flat coupon test. The actual rupture strain of FRP jacket measured in column tests usually fall significantly below the value from flat coupon tests. Therefore, a strain efficiency factor is usually multiplied to the ultimate flat coupon strain to calculate the actual effective strain. However, based on Ref. [25], there is another more convenient way to allow for this factor: to include the strain efficiency factor directly in the design equation. If the definition of kf is changed to actual effective strain, the coefficient of 0.67 in Eq. (9) is simply changed to another value. The advantage of using the flat coupon strain directly is that designers do not need to care about the strain efficiency factor in their calculation as it is automatically allowed for in the equation.

Fig. 3. Effect of FRP type on neutral axis depth.

Experimental results of moment–curvature response for FRPconfined circular columns show similar behavior. The relationship between the ultimate curvature and confinement ratio of four columns (Specimens ST-2NT, ST-3NT, ST-4NT and ST-5NT) from the work by Sheikh and Yau [27] is presented in Fig. 4b, as are the test results of two unconfined columns from the same study. For the two FRP-confined columns and the corresponding unconfined columns under an axial load level of n = 0.64, a linear relationship is observed between the ultimate curvature and the confinement ratio. This result agrees with the trend for the square columns. The measured ultimate curvature of Specimen ST-5NT was relatively

3.4. Influence of FRP type upon deformation capacity Test results have indicated that the ultimate curvature of an FRP-confined square concrete column under simulated seismic load is closely related to the confinement ratio, and the influence of the type of confining FRP is insignificant [9], as illustrated in Fig. 4a. The curvature capacity increases linearly with an increase in the confinement ratio. Difference in the type of confining FRP does not cause a significant difference in the ultimate curvature, as long as the confinement ratio is maintained at the same value. The curvature capacity increases linearly with an increase in the confinement ratio.

Fig. 2. Neutral axis depth.

Fig. 4. Relationship between ultimate curvature and confinement ratio.

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D.-S. Gu et al. / Construction and Building Materials 27 (2012) 223–233

Table 1 Details of test specimens. Specimen

FRP Tp

J0 J1 J2 J3c J4c J5c J6b,c J7b,c b,c

J8

CH0 CH1b CH2 CH3 CL0 CL1 CL2 CL3

– D C C D C C D C D C D – C D D C – D C C

a

n

N (kN)

fc0 (MPa)

D (mm)

L (mm)

Spiral

Longitudinal bar

ff (MPa)

t (mm)

ply

ef

kf

ds (mm)

fys (MPa)

fy (MPa)

x  db (mm)

– 1832 4232 4232 1832 4232 4232 1832 4232 1832 4232 1832 – 3945 1832 1832 3945 – 1832 3945 3945

– 0.258 0.111 0.111 0.258 0.111 0.111 0.258 0.111 0.258 0.111 0.258 – 0.167 0.258 0.258 0.167 – 0.258 0.167 0.167

– 1 1 1 2 2 1 1 0.5d 1 1 2 – 0.5d 1 2.5d 1.5d – 4 2.5d 3.5d

– 0.031 0.018 0.018 0.031 0.018 0.018 0.031 0.018 0.031 0.018 0.031 – 0.015 0.031 0.031 0.015 – 0.031 0.015 0.015

– 0.113 0.111 0.111 0.225 0.222 0.223

0.05 0.05 0.05 0.05 0.05 0.05 0.05

100 100 100 100 100 100 100

28 28 28 28 28 28 28

300 300 300 300 300 300 300

850 850 850 850 850 850 850

/6@160 /6@160 /6@160 /6@160 /6@160 /6@160 /6@160

350 350 350 350 350 350 350

400 400 400 400 400 400 400

12  19 12  19 12  19 12  19 12  19 12  19 12  19

0.168

0.05

100

28

300

850

/6@160

350

400

12  19

0.336

0.05

100

28

300

850

/6@160

350

400

12  19

– 0.127

0.36 0.36

1200 1200

34.9 34.9

360 360

1100 1100

/6@150 /6@150

320 320

382 382

12  25 12  25

0.188 0.157 – 0.3 0.261 0.366

0.36 0.36 0.36 0.36 0.36 0.36

1200 1200 1200 1200 1200 1200

34.9 34.9 34.9 34.9 34.9 34.9

360 360 360 360 360 360

1100 1100 800 800 800 800

/6@150 /6@150 /6@150 /6@150 /6@150 /6@150

320 320 320 320 320 320

382 382 382 382 382 382

12  25 12  25 12  25 12  25 12  25 12  25

Notes: x – number of bars; ds – diameter of spiral; fys – yield strength of spiral. a Type, D for DFRP, C for CFRP. b Hybrid FRP. c Different loading protocol. d For 0.5 ply of CFRP fabric, 20 mm wide bands were provided at a clear spacing of 20 mm. For DFRP, one-half of the fibers were removed from the sheet.

large, as can be observed by comparing Fig. 4b with Fig. 4a, possibly because of scattering of test result. To further study the influence of the types of FRP for circular columns, the authors of this paper undertook an extensive experimental investigation into the deformation capacity of FRP-confined circular concrete columns that involved 14 column specimens. Two types of FRP were used in the test: CFRP and Dyneema fiber reinforced polymer (DFRP). DFRP has a much larger fracture strain and a lower elastic modulus compared with those of CFRP. Details of the material properties and the test specimens are provided in Table 1. The failures of all columns were dominated by FRP rupture at the base of the columns under flexure. More details of the experimental program can be found elsewhere [28]. The behavior of Specimens J1 and J2 in Table 1 can be compared to evaluate the relative effectiveness of CFRP and DFRP in strengthening deformation capacity of deficient columns. These two columns were identical in every aspect except for the type of confining FRP. From the response hysteresis loops in Fig. 5, it can be seen that the two columns behaved in a very similar manner, and the drift capacities of them are essentially identical. It is worth noting that the confinement ratio of these two columns is identical, and the strain capacity of DFRP is about twice as much as that of CFRP. More detailed discussions about the influence of the type of confining FRP upon the deformation capacity of FRP confined columns can be found in the work by Gu et al. [28]. From all of the study above, we can conclude that the type of confining FRP has an insignificant effect on the ultimate deformation capacity of FRP confined concrete columns which is mainly controlled by the confinement ratio. 3.5. Ultimate strain The ultimate concrete strain, ecu , can be calculated from the measured ultimate curvature multiplied by the depth of the neutral axis. The calculated results of ecu for the square columns in Fig. 4a are given in Fig. 6. The stress–strain model of FRP-confined concrete by Lam and Teng [29] is used in the calculation of the neutral axis depth

Fig. 5. Cyclic responses.

D.-S. Gu et al. / Construction and Building Materials 27 (2012) 223–233

for FRP-confined rectangular columns. The figure shows a linear relationship between the ultimate concrete strain and the confinement ratio. By data regression, the ultimate concrete strain can be expressed as:

ecu ¼ 0:0068 þ 0:035kf :

ð10Þ

When the confinement ratio is zero, Eq. (10) gives 0.0068, which is the ultimate strain of non-ductile RC columns. The above result agrees well with the measured test results of concrete spalling strain. The concrete spalling strain of square columns was measured to be 0.0057 by Sheikh and Khoury [30] and 0.0066 by Watson and Park [31]. The measured value for circular columns was in the range of 0.004–0.011 with a mean value of 0.0066 [32]. Ref. [19] gave the mean value of 0.008 and 0.005 for circular and rectangular column, respectively, from a database containing 40 circular columns and 102 rectangular columns. It is observed from the above results that the spalling strain for circulars column is larger than that for rectangular columns. This is reasonable because there is only one point at the extreme compression fiber that can reach the ultimate strain in a circular section and this point is restrained by adjacent points, causing a delay in spalling. The above results also agrees with the conclusion derived by Wu et al. [33] from their theoretical study, which suggests that the onset of the complete crushing of concrete at the extreme compression fiber, i.e. r = 0 at e = ecu (ecu  0.006 for square columns), is the ultimate failure point for concrete rectangular columns. The mechanism behind this definition of ultimate curvature is that when e = ecu at the extreme fiber, the concrete compression zone will reach such a critical area that a further increase in curvature will result in a compressive resistance of the concrete that is smaller than the axial force. That resistance will cause a sudden collapse in a plain concrete section or an accelerated drop in moment resistance in an RC section [33]. For circular column sections, a similar accelerated drop in moment resistance will occur after e = ecu (ecu  0.008), although a theoretical sudden collapse does not exist when the compressive strain reaches ecu at the extreme compression fiber. As the confinement effectiveness for circular sections and concrete spalling strain are different from those for square sections, Eq. (10) should be modified for circular sections to take the form

ecu ¼ 0:008 þ a0 kf ;

ð11Þ

where a0 is a constant to be determined, and a value of 0.008 is taken for the spalling strain of concrete cover, as proposed in Ref. [19]. The difference in the following results is insignificant when a value of 0.006 is used in Eq. (11), as the difference between 0.008 and 0.006 is relatively small compared with the second term in Eq. (11) that is contributed by FRP confinement.

By substituting Eqs. (9) and (11) into Eq. (4), the ultimate curvature of FRP-confined circular section is expressed as

/u ¼

0:008 þ a0 kf : ð0:19 þ ð0:72  0:67kf Þ  nÞD

ð12Þ

The coefficient a0 can be determined from the test results in Fig. 4b. A least squares analysis shows that a0 = 0.09, leading to

/u ¼

0:008 þ 0:09kf : ð0:19 þ ð0:72  0:67kf Þ  nÞD

ð13Þ

The ratios of the test results to the calculated results have a mean value of 1.03 and a COV of 25.5%. 4. Plastic hinge length Estimating the length of a plastic hinge is a key step toward the prediction of the drift capacity. Numerous models have been proposed to calculate the plastic hinge length of steel reinforced concrete columns. However, very limited work has been reported in the literature for the plastic hinge length of FRP-confined concrete columns, and as mentioned previously, the conclusions are sometimes contradictory. 4.1. Test results and observations It was observed from the tests conducted by the authors that the length of the damaged region of FRP confined columns is significantly influenced by the amount of confining FRP. The failure zones of Specimens CH3, CL2 and CL3 in Table 1 are depicted in Fig. 7. Specimens CL2 and CL3 were identical in every respect except that an additional layer of CFRP was used for Specimen CL3. It can be seen from the figure that one more layer of CFRP resulted in a smaller length of the damaged region in Specimen CL3 compared with that of Specimen CL2. Specimen CH3 had a smaller confinement than that in Specimen CL2, and hence, a greater length of the damaged region than that in Specimen CL2. The observation that larger amount of confining FRP results in a smaller damaged region has been reported by others [16,34]. In the test program of Ref. [16], the confinement ratio of Specimen RC-2 was about 82% of that of Specimen RC-3, which resulted in a 120% increase in the length of the damaged zone in RC-2. Because the length of the damaged region is closely related to the plastic hinge length [12], it can be concluded that the plastic hinge length is a function of the amount of confining FRP. 4.2. Expression of plastic hinge length As the ultimate drift ratio is related to the plastic hinge length through Eqs. (2) and (13), the plastic hinge length is only affected by the confinement ratio, and the type of confining FRP should have no significant effect. A further and detailed analysis of the effect of FRP type on plastic hinge length is given later in Section 5.3. It has been proposed by Paulay and Priestley [35] that for steel reinforced columns, the plastic hinge length, lp, is the sum of two components,

lp ¼ aL þ bfy db ; but lp > 0:044 f y db ;

Fig. 6. Relationship between ultimate concrete strain and confinement ratio.

227

ð14Þ

where db is the diameter of the longitudinal tension reinforcement, a = 0.08, and b = 0.022. The first term of the equation accounts for the moment gradient along the height of the cantilever, and the second term accounts for additional rotation at the base resulting from strain penetration of the longitudinal reinforcement into the supporting base. The general form of Eq. (14) can be adopted for FRPconfined columns, but variation should be made to reflect the influence of FRP confinement. Based on the discussion in the previous

228

D.-S. Gu et al. / Construction and Building Materials 27 (2012) 223–233

(a) CH3

(b) CL2

(c) CL3

Fig. 7. Damaged zone of columns.

section, the first term of Eq. (14) should be related to the confinement ratio. As a result, we assume that the plastic hinge length is given by

lp ¼ ðb0 þ b1 kf þ b2 k2f ÞL þ 0:022 f y db ;

ð15Þ

where the coefficients b0, b1, and b2 are constants to be determined. For the second term of Eq. (15) we adopt the original form of Eq. (14). This is because the strain penetration of the longitudinal steel bar into the supporting base is independent of the column properties and depends only on the properties of the reinforcing bars [36]. In fact, Eq. (14) is a special case of Eq. (15). By substituting Eqs. (3) and (13) into Eq. (2), the ultimate drift ratio is obtained by

0:008 þ 0:09kf 2:45ey L   þ hu ¼  2:45ey 3D 0:19 þ 0:72  0:67kf  n     lp lp L ; 1  0:5 D L L

!

ð16Þ

in which the plastic hinge length lp is given by Eq. (15). 5. Ultimate drift ratio 5.1. Database To determine the coefficients in Eq. (15), test results from FRPconfined circular concrete columns under simulated seismic load reported in the open literature are collected. The test results are screened according to the following criteria: (1) the columns were flexure-critical, and the failure mode was controlled by the rupture of FRP at the base of the column; (2) there was no lap splice in the plastic hinge region; (3) the fiber of the FRP was in the lateral direction; and (4) the columns were retrofitted with full length jacket. It is believed that the plastic hinge length is related to the extent of wrapping to some degree. Because of this reason, the effect of confinement on plastic hinge length will be further complicated if the length of jacketing is also considered in the modeling. For simplicity, only cases where full length jacket was provided are considered in this work. Fifteen FRP-confined circular columns that meet the above criteria are collected. Table 2 provides the details of these columns. Combined with the 14 columns tested by the authors as listed in Table 1, a total of 29 columns are used to determine the coefficients in Eq. (15). Because little lateral-steel reinforcement was used in the test columns, the confinement from the steel reinforcement is negligible compared with that of FRP. Half height is considered as the column length for columns tested under the double fixed end condition, so that all columns can be treated as cantilevers.

It is worthwhile to note that the ultimate displacement is related to its definition and the loading protocol. In this work, rupture of FRP is considered the ultimate failure of columns. The ultimate displacement is defined as the maximum drift before unstable and accelerated drop in strength of column caused by significant rupture of FRP. This definition of failure provides not only a uniform benchmark but also a qualitative turning point that reflects an intrinsic physical change (such as the loss of axial load carrying capacity) in the columns in the course of deformation. Although the loading history is less sensitive to the ultimate displacement under the failure mode of FRP rupture, the loading history does affect the ultimate displacement. Therefore, ideally the ultimate displacements should be obtained under an identical loading protocol. However, very few data can be collected from the literature under an identical loading protocol. Even for the tests undertaken by the authors in Table 1, six columns were tested with a loading protocol (two cycles of particular drift ratios and a subsequent monotonic loading) that is different from the standard three cycles of drift ratios for other columns to identify the effect of loading history. The ultimate displacements of these six columns are multiplied by a factor of 0.8 that is determined by equating the ultimate displacement of Specimens J2 with that of J3. These two columns are identical in all aspects except for the loading histories. More details for the calculation of the ultimate displacement of the columns can be found in Ref. [28]. It is also noted that before the ultimate failure the drop in strength from the peak point is not greater than 15% for all of the columns selected in the database. 5.2. Regression analyses By minimizing the sum of the error between the theoretical result from Eq. (16) and the test results of all 29 columns, coefficients b0, b1, and b2 in Eq. (15) are calculated to be 0.59, 2.30, and 2.28, respectively. Table 3 gives the theoretical results from Eq. (16) in comparison with the test results. The ratios of the test results to the calculated results have a mean value of 1.01 and a COV of 14.0%. The following equation was proposed to calculate the ultimate drift ratio for FRP-confined circular concrete columns in Ref. [10]:

h¼

    2:45 L l l L ; ey þ ðKðI; n; e; kf Þ  2:45ey Þ p 1  0:5 p 3 D D L L

ð17Þ

where

 KðI; n; e; kf Þ ¼ 0:0034 

4:44ek2f þ ð0:65 þ 3:84eðI þ 0:59ÞÞkf þ 0:56I þ 0:33 ; 0:44n þ 0:32I þ 0:04kf þ 0:02 ð18Þ

229

D.-S. Gu et al. / Construction and Building Materials 27 (2012) 223–233 Table 2 Test specimens collected from the literature. fc0 (MPa)

D (mm)

ds (mm)

fys (MPa)

fy (MPa)

db (mm)

654 654

35.9 35.9

610 610

915 915

/6@150 /6@150

303 303

303 303

19 19

0.64 0.64 0.31

2570 2630 1380

40.4 40.4 44.8

356 356 356

1470 1470 1470

/10@300 /10@300 /10@300

510 510 510

500 500 500

25 25 25

0.244 0.163

0.17 0.17

1400 1400

18.6 18.6

760 760

1750 1750

/10@300 /10@300

426 426

426 426

19 19

0.017 0.017 0.017

0.206 0.124 0.187

0.31 0.34 0.52

1580 1480 1480

90.1 75.2 49.7

270 270 270

2000 2000 2000

– – –

– – –

500 500 500

16 16 16

0.025 0.025 0.025 0.025 0.025

0.430 0.430 0.332 0.332 0.332

0.43 0.52 0.43 0.52 0.62

640 780 840 1010 1210

59.2 59.2 76.7 76.7 76.7

180 180 180 180 180

630 630 630 630 630

/4@60 /4@60 /4@60 /4@60 /4@60

402 402 402 402 402

353 353 353 353 353

12 12 12 12 12

Refs.

Specimen

hu

FRP

n

Tp

ef

kf

[37]

CSJ-RT ISJ-RT

0.053 0.053

G G

0.015 0.015

0.384 0.512

0.06 0.06

[27]

ST-2NT ST-3NT ST-4NT

0.046 0.046 0.089

G C C

0.020 0.014 0.014

0.144 0.112 0.105

[38]

FCS-1 FCS-2

0.063 0.054

C C

0.018 0.018

[16]

RC-1 RC-2 RC-3

0.120 0.110 0.090

C C C

[39]

C60N1-F C60N2-F C80N1-F C80N2-F C80N3-F

0.059 0.057 0.068 0.063 0.059

C C C C C

N (kN)

L (mm)

Spiral

Longitudinal bar

Note: Tp – Type, G for GFRP, C for CFRP, ds – diameter of spiral reinforcement, fys – yield strength of spiral reinforcement.

and

lp ¼ 0:077L þ 8:16db ; ef

in which e ¼ ð0:002 Þ (I ¼

qs fy fc0

ð19Þ

0:45

, and I is longitudinal reinforcement index

). The ultimate drift ratios of the 29 columns calculated from

Eq. (17) are also listed in Table 3. The ratio of measured results to the calculated results from Eq. (17) has a mean value of 2.61 and a COV of 40.0%. The theoretical results from Eqs. (16) and (17) are compared with the test results in Fig. 8, which clearly shows the

Specimen

5.3. Mechanisms affecting plastic hinge length From the study above, the plastic hinge length is derived as

lp ¼ ð0:59  2:30kf þ 2:28k2f ÞL þ 0:022f y db ; when kf > 0:1:

Table 3 Experimental and predicted hu. Refs.

better correlation between the test results and the proposed model. The adoption of Eq. (19) for the plastic hinge length which is applicable only for normal RC columns without external confinement, is a major difference between Binici’s model (Eq. (17)) and the proposed model of Eq. (16).

Measured hu

Calculated hu Eq. (16)

Eq. (17)

[37]

CSJ-RT ISJ-RT

0.053 0.053

0.049 0.054

0.049 0.067

[27]

ST-2NT ST-3NT ST-4NT

0.046 0.046 0.089

0.054 0.048 0.073

0.016 0.014 0.019

[38]

FCS-1 FCS-2

0.063 0.054

0.057 0.055

0.022 0.016

[16]

RC-1 RC-2 RC-3

0.120 0.110 0.090

0.139 0.117 0.103

0.049 0.032 0.034

[39]

C60N1-F C60N2-F C80N1-F C80N2-F C80N3-F

0.059 0.057 0.068 0.063 0.059

0.070 0.064 0.068 0.061 0.055

0.057 0.050 0.043 0.037 0.033

[28]

J1 J2 J3 J4 J5 J6 J7 J8 CH1 CH2 CH3 CL1 CL2 CL3

0.085 0.086 0.086 0.126 0.112 0.125 0.109 0.100 0.050 0.090 0.080 0.068 0.060 0.060

0.092 0.091 0.091 0.114 0.114 0.114 0.106 0.120 0.054 0.063 0.059 0.062 0.059 0.068

0.024 0.021 0.021 0.042 0.037 0.042 0.033 0.063 0.019 0.026 0.019 0.036 0.026 0.036

Statistics of hMe/hCa

Mean COV (%)

1.01 14.0

2.61 40.0

ð20Þ

The confinement ratio kf for a common and practical column is between 0.1 and 0.5. Within this range, the relationship between kf and the value of a (a ¼ 0:59  2:30kf þ 2:28k2f ) is shown in Fig. 9, with a decreasing with an increase in kf (a = 0.38 when kf = 0.1). It should be noted that Eq. (20) is not applicable when kf is less than 0.1, because there is no test data in this range in the database. However, when kf = 0, the plastic hinge length should be given by Eq. (14), i.e. a = 0.08. Thus, a monotonically increasing curve should be adopted when 0 < kf < 0.1, as shown by the dotted line in Fig. 9. Fig. 9 shows that the plastic hinge length is not a constant but a function of the confinement ratio. The ascending part between 0 < kf < 0.1 is not difficult to understand. The compressive strength of confined concrete is mainly related to the confinement ratio, and the strength enhancement of the compression zone causes an in-

Fig. 8. Comparison of models with test results.

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crease in the lever arm of the moment and hence an increase in the cross-sectional moment capacity. In the meantime, the yield moment does not change and hence the section, where first yielding occurs, moves up along the column with the increase in the moment capacity at the base. As a result, the plastic hinge length increases with an increase in the confinement ratio. The fast increase in the plastic hinge length at a low confinement level can be further explained with Fig. 10, in which the typical moment–curvature relationships for unconfined and confined columns are shown. Consider the cantilever column in Fig. 1. For an unconfined column, when the peak moment M0 is reached at the base (bottom) and a further displacement is imposed at the top, only the base section can pass the peak point A of the moment–curvature curve and proceed onto the descending part AB of the curve, as shown in Fig. 10. When the base moment drops, the moment of all other cross-sections above the base will reduce due to the linear relationship of moment for the cantilever column, and hence will go along their unloading paths, such as DE, causing a smaller deformation. Therefore, further increases in deformation of the cantilever can only come from the deformation at the base section with a zero plastic hinge length. There will be a sharp drop in curvature for sections immediately above the base, e.g. from j1 at the base section to j2 at a section above in Fig. 10. Although a sudden drop of curvature at two infinitely close sections at the bottom is theoretical, practically it can cause the rapid drop of curvature above the base and result in a short plastic hinge length [40]. Confined columns, however, always experience a moment enhancement at all sections as shown by CF in Fig. 10, as long as the confinement is not too small [24]. This enhancement in moment results in a continuous increase in the curvature of sections at and above the base, which significantly increases the plasticity zone and hence the plastic hinge length of the confined column. As a result, the confinement leads to an increase in the length of the plastic hinge. In the meantime, there is another mechanism that will simultaneously affect the plastic hinge length: the strain variation of the longitudinal bars at the plastic hinge zone. The confinement increases the frictional bond between the concrete and the longitudinal bars. As shown in Fig. 11a, the confinement from FRP produces an additional frictional bond at the surface of the longitudinal bars which is in direct proportion to the confinement pressure. Even a small increase in the bond stress will produce a significant reduction in the strain of the longitudinal bar at the upper cross-section B–B after the longitudinal bar yields, because a large change of strain is required to produce a small change of steel stress after bar yielding. This is evident from the test results reported in Ref. [41], as shown in Fig. 11b, which illustrate the strains of the longitudinal bar at the base section and the section 135 mm above the base. Clearly, a larger

Moment M0

D

A

F

C G

E

Reloading path

Unconfined column

B

Unloading path

κ2

κ1 Curvature

Fig. 10. Moment–curvature relationship.

confinement causes a significantly larger difference in the strain of the longitudinal bar between the base and the upper section. When the strain variation is large, the difference of curvature between adjacent sections is also large. As a result, the base curvature is applicable to a smaller length for columns with a larger confinement ratio, indicating a smaller plastic hinge length. The strain variation De at the two adjacent sections (A–A and B– B) is directly related to the stress variation in the longitudinal bars that is caused by the bond stress. Comparing columns with different jackets, the additional bond force due to an FRP jacket comes from the frictional bond that is in direct proportion to the confinement pressure. Thus, the additional bond, and hence the variation of curvature at the column’s base, is only related to the confinement pressure or the strength of the FRP jacket, not the type of confining FRP, as long as the FRP jacket ruptures at failure. As a result, the plastic hinge length is only affected by the strength or confinement ratio of the FRP jacket, as pointed out in Section 4.2. The foregoing analysis is consistent with the experimental observation of the effect of confinement on the curvature distribution near the column base for steel jacketed columns by Chai et al. [14] that steel jacketing can restrict the spread of plastic yielding. The same mechanism can also be used to explain why the curvature of confined columns with a lap splice of reinforcement bars in the plastic hinge region is concentrated in a smaller length than that of columns with continuous bars [42]: the existence of more reinforcement bars in the plastic hinge region enhances the bond stress between the concrete and the longitudinal bars. In the ascending part of Fig. 9, the first mechanism dominates and the strain variation of the longitudinal bars due to the small confinement is relatively small, as shown by the case with kf = 0.124 in Fig. 11b. When the confinement ratio increases beyond kf = 0.1, the increase in moment capacity due to confinement will slow down, and hence the increase in the plastic hinge length due to the increase in moment capacity will be relatively small as compared to the effect of the second mechanism. Thus, the second mechanism will dominate. The plastic hinge length can be directly calculated from test results when the moment–curvature relationship for the plastic hinge region and the lateral force–deflection relationship at the top of a column are both available. Eq. (1) can be rewritten as [14]:

lp ¼ L 

Fig. 9. Effect of confinement ratio on plastic hinge length.

Confined column

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Du  Dy : L2  2 /u  /y

ð21Þ

From the literature, five columns were found for which both the moment–curvature relationship at the plastic hinge region and the lateral force–deflection relationship at the top were reported. One column is Specimen C-5 reported in Ref. [42], and the other four columns were reported in Ref. [27], with the force–deflection relationships reported in Ref. [43]. The main parameters of these

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D.-S. Gu et al. / Construction and Building Materials 27 (2012) 223–233

T- ΔF B-B

ΔF

Δx

Base section A-A

Longitudinal bar T

(a) (b) Fig. 11. Variation of steel strain near column base.

five columns are presented in Table 4. The measured plastic hinge length lpe determined from Eq. (21) and that calculated from Eq. (20), lpc, are also given in Table 4. The ratios of lpe to lpc have a mean value of 0.9 and a COV of 13.3%. It should be noted that Specimen ST-5NT has a low confinement ratio of kf < 0.1, and hence the plastic hinge length was determined from the dotted line in Fig. 9. The calculated result agrees well with the measured result, which demonstrates the rationality of the assumed relationship between a and kf when the confinement ratio is less than 0.1. To further substantiate the relationship shown in Fig. 9, the plastic hinge lengths of FRP-confined square columns reported in the literature are similarly calculated using Eq. (21). The test results reported in Ref. [9], for which the force–deflection relationships were reported in Refs. [34,44], are plotted in Fig. 11. For comparison, the relationship shown in Fig. 9 is also plotted in Fig. 12 with dotted lines. It is clear that the plastic hinge length of FRP-confined square columns decreases with an increase in the confinement ratio, which is similar to the case for circular columns. Compared with Fig. 9, it seems that the slope of descent in Fig. 12 is smaller, which is reasonable because there is less confinement effect for square columns than for circular columns. The confinement effect for square/rectangular columns is much more complicated and is closely related to the corner radius [6,45–47] and cross-sectional aspect ratio [48,49]. Hence, further studies are much needed for the problem. 6. Parametric studies Parametric studies are performed here to offer a general cognition of the effect of various design parameters on the ultimate drift ratio given by Eq. (16). A prototype bridge column with a diameter of 600 mm is selected for the parametric studies. The diameter of the longitudinal bar is assumed to be 19 mm with a yield stress of 420 MPa. The concrete cylinder compressive strength is 36 MPa. The calculated ultimate drift ratio is presented in Fig. 13 for columns with axial load levels of 0.1 and 0.4 and with different aspect

Fig. 12. Plastic hinge length for square columns.

ratios. The figure shows that the ultimate drift ratio does not always increase with an increase in the confinement ratio. The ultimate drift ratio reaches a peak at a critical value of the confinement ratio, after which it decreases with an increase in that ratio. This critical value is around 0.2–0.25, irrespective of the axial load level when the aspect ratio is larger than 1.5. An increase in the axial load level significantly reduces the drift capacity of the columns. For a given confinement ratio, the ultimate drift ratio increases with an increase in the aspect ratio. Fig. 14 shows the test measured relationships between the ultimate drift ratio and the confinement ratio. The specimens in a particular series are identical in every aspect except for the confinement ratio. It can be seen that the ultimate drift ratio significantly increases with an increase in the confinement ratio when that ratio is initially small. However, when the confinement ratio is large, the ultimate drift ratio decreases with an increase in the confinement ratio. This agrees well with the trend found from the parameter studies as presented in Fig. 13.

Table 4 Experimental and predicted lp. Specimen

n

L (mm)

kf

fy (MPa)

db (mm)

Dy (mm)

Du (mm)

/y  106 rad/ mm

/u  106 rad/ mm

lpe Eq. (21) (mm)

lpe/L

lpc Eq. (20) (mm)

lpe/ lpc

C-5 ST-2NT ST-3NT ST-4NT ST-5NT

0.17 0.64 0.64 0.31 0.31

1800 1470 1470 1470 1470

0.478 0.144 0.112 0.105 0.071

358 500 500 500 500

13 25 25 25 25

15.1 16.0 18.5 23.6 25.1

90.7 67.1 67.3 131.5 133.0

55.7 8.7 7.1 10.1 11.8

399.0 78.1 76.5 132.2 141.2

126.8 640.4 601.3 842.7 767.7

0.07 0.44 0.41 0.57 0.52

123.2 724.9 805.7 824.2 919.1

1.03 0.88 0.75 1.02 0.84

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Acknowledgements The work described in this paper was fully supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 122106). Supplementary supports from the National Basic Research Program of China (973 Program) (No. 2007CB714200) and the National Natural Science Foundation of China (Nos. 50608015 and 50908102) are also acknowledged.

References

Fig. 13. Parametric studies.

Fig. 14. Measured relationships between confinement ratio and drift ratio.

7. Conclusions By combining numerical simulations, analytical studies, and regression analyses using a test database including data obtained from the authors’ own tests and those collected from the literature, simple expressions are developed for calculation of the plastic hinge length and the ultimate drift ratio of FRP-confined circular concrete columns. Based on the analytical model obtained, it can be concluded that the plastic hinge length is significantly affected by the confinement ratio, increasing with an increase in the confinement ratio at low confinement and reducing with an increase at high confinement. Although the accuracy of the derived equations may be further improved when more test data is available, the general trend of these results and conclusions derived from this work are creditable, as the test observations, analytical results, and mechanism analyses are consistent. The analytical model indicates that the ultimate drift ratio of a column is affected by its axial load level, confinement ratio, and aspect ratio. Confinement at a low level will increase the drift capacity of the column. However, after a critical value has been exceeded a further increase in confinement will cause a reduction in the deformation capacity of the column. This critical confinement ratio is around 0.2–0.25 when the aspect ratio is greater than 1.5. In terms of enhancing the deformation capacity and ductility of concrete columns by retrofitting FRP jackets, the critical confinement ratio should be treated as an upper bound.

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