Two-plastic-hinge and two dimensional finite element models for post-tensioned precast concrete segmental bridge columns

(1)

Two-plastic-hinge and two dimensional ﬁnite element models

for post-tensioned precast concrete segmental bridge columns

Chung-Che Chou

a,b,⇑

_{, Hao-Jan Chang}

c

_{, Joshua T. Hewes}

d

a

Department of Civil Engineering, National Taiwan University, Taipei, Taiwan b

National Center for Research on Earthquake Engineering, Taipei, Taiwan c

Department of Civil Engineering, National Chiao Tung University, Hsinchu, Taiwan d

Department of Civil and Environmental Engineering, Northern Arizona University, Flagstaff, AZ, USA

a r t i c l e

i n f o

Article history:

Received 11 October 2010 Revised 25 June 2012 Accepted 17 July 2012

Available online 13 September 2012 Keywords:

Precast concrete segmental bridge column Unbonded strands

Cyclic tests

Two-plastic-hinge model Finite element model

a b s t r a c t

Recent studies have confirmed that unbonded post-tensioned (PT) precast concrete segmental bridge col-umns are capable of undergoing large lateral deformation with negligible residual drift. To provide a clear guideline for the modeling of the columns for practicing engineers as well as researchers, this paper pre-sents two types of numerical models: (i) a two-plastic-hinge model using the sectional moment–curva-ture analysis procedure at two segment interfaces and (ii) a two-dimensional (2D) finite element model using truss and beam-column elements in the computer program PISA. Three unbonded PT precast con-crete-filled tube segmental bridge column specimens are cyclically tested. Two specimens have mild steel bars crossing to different column heights for studying the effects of anchorage position on the hysteretic energy dissipation (ED) capacity. The test results show that (1) the mild steel bars (‘‘ED bars’’) can increase hysteretic energy dissipation, and Specimens 1–3 have equivalent viscous damping of 6.5– 8.8%, (2) a plastic hinge length in the first or second segment varies with anchorage position of ED bars and lateral displacement, and (3) an equivalent unbonded length along which the strain in the ED bar is assumed uniformly distributed on each of the two sides is 5–6 bar diameter. A 2D finite-element model is utilized to predict the cyclic behavior of the specimens. Parametric studies using finite-element models are also conducted to investigate the effects of ED bar area, initial strand force, and aspect ratio on the cyclic behavior.

1. Introduction

Innovative precast concrete segmental bridge columns that incorporate unbonded post-tensioning elements to provide self-centering capacity and with devices to dissipate seismic input en-ergy have recently been proposed in the United States as well as in other countries. Because of the limited knowledge regarding the seismic behavior of such columns, this type of bridge column has been used only in regions of low seismicity[1]. Hewes’s studies

[2,3]confirmed that the columns are easily assembled in the labo-ratory and capable of undergoing large lateral deformations with small residual drift upon unloading, exhibiting a ‘‘flag-shaped’’ hysteretic behavior. The most significant characteristic of the behavior is bilinear-elastic but with hysteretic energy mainly added to the post-elastic portion of response. The resulting

hyster-etic energy dissipation capability is low in comparison with conventional monolithic reinforced concrete columns or con-crete-ﬁlled tube columns[4]. Past studies showed that the addition of longitudinal mild steel bars crossing column segment joints or external steel plates at the column base can improve hysteretic en-ergy dissipation capacity[5–7]. It has also been shown that incor-porating ductility-dependent stiffness degradation in the ﬂag-shaped hysteretic model can improve prediction of the columns subjected to seismic loads[8].

The lateral deformation of a PT segmental column is attributed primarily to gap opening at segment interfaces (joints). For conve-nience, it is assumed that the segmental column behaves like a conventional reinforced concrete column with a plastic hinge (gap opening) at the segment joint. Conventional moment curva-ture analysis, which is used to obtain the moment-rotation rela-tionship of the self-centering connections [9,10], can be used to obtain the pushover curve of the segmental column[3,11]. How-ever, when segment joints have unequal strengths, the pushover response of the segmental column cannot be determined based on a single plastic hinge at the column base. For a PT segmental

http://dx.doi.org/10.1016/j.engstruct.2012.07.009

⇑ Corresponding author at: Department of Civil Engineering, National Taiwan University, Taipei, Taiwan. Tel.: +886 2 3366 4349; fax: +886 2 2739 6752.

E-mail addresses: [email protected] (C.-C. Chou), [email protected]

(2)

column without energy dissipation devices at the base, the column lateral deformation can be characterized by rotation of the column about the bottom two segment interfaces and the corresponding lengths of two plastic hinges are[6]:

Lp1¼ 0:5d ð1Þ

Lp2¼ 0:2d ð2Þ

where Lp1is the plastic hinge length in the ﬁrst segment; Lp2is the

plastic hinge length in the second segment, and d is the cross-sectional diameter of the segmental column. Note that Eqs. (1) and (2)are obtained based on experimental curvatures along the column height, and a larger plastic hinge length is caused by a larger gap opening at the segment interface. Therefore, providing an energy dissipation device at the column base changes gap opening at segment interfaces and associated plastic hinge lengths.

This study extends the concept of ‘‘two plastic hinges’’ to de-velop an analytical model for predicting pushover responses of unbonded PT concrete segmental columns, where joint opening is mainly located at the bottom two segments. For veriﬁcation purposes, cyclic tests were conducted on three columns. Each column segment was encased in a steel tube to raise the concrete compressive strength and ultimate compression strain. Two spec-imens included energy-dissipating (ED) bars with different

anchorage location for studying the effects of bar anchorage loca-tion on the plastic hinge length and energy dissipaloca-tion. While Ou et al.[11] demonstrated that a detailed three-dimensional (3D) ﬁnite-element (FE) model utilizing solid elements in segments and contact elements between segment interfaces can predict the cyclic response of PT segmental columns in tests, this current work aimed instead to develop a simpliﬁed two-dimensional (2D) FE model using truss and beam-column elements to predict the cyclic behavior of PT segmental columns. This simpler mod-eling approach saves computation effort by reducing model com-plexity and serves as a reasonable alternative to 3D analyses. Based on the 2D FE model, a parametric study on unbonded PT concrete segmental columns was conducted to evaluate the opti-mum area and moment ratios of ED bars for a circular PT column section.

(a) Column Deformation

(b) Strand Movement in Segment

Fig. 1. Unbonded PT concrete segmental columns under lateral load.

Table 1 Material properties. Specimen No. Concrete (MPa) Cement grout (MPa) ED bar Yield strength (MPa) Tensile strength (MPa) 1 53 – – – 2 48 63 307 497 3 51 72 307 497

(3)

2. Two-plastic-hinge model for precast concrete segmental bridge column

A PT concrete column is composed of a load stub, four segments and a footing, which are post-tensioned together by using unbond-ed strands (Fig. 1a). Eight longitudinal reinforcing bars (#8 and #5 reinforcement in the ﬁrst and other segments, respectively) that do not cross segment joints are used to reduce compressive strains of concrete, while four ED bars are incorporated to increase energy dissipation of the column. The behavior of the unbonded PT con-crete segmental column under a lateral load is characterized by three stages (Fig. 1a). Stage one corresponds to column response prior to decompression of the column at any section. Stage two commences when the PT force, Fs, begins increasing due to joint

opening at the column base; a crack forms at the base and propa-gates to mid section depth. A crack also forms at the bottom of the second segment but does not reach mid depth, therefore not signif-icantly affecting the strand elongation. This stage represents the beginning of signiﬁcant nonlinearity in the pushover response. With a further decrease of the neutral axis depth within segments, the strand is stretched and the strand force increases. Stage three initiates when a crack at the bottom of the second segment prop-agates to mid section depth. The large opening of the ﬁrst and sec-ond segment joints causes an increase of the PT force,DFs1and

DFs2, respectively. The conventional moment–curvature analysis

procedure proposed by Hewes and Priestley[3]can be applied to

obtain the pushover relationship of the column before stage three. A two-plastic-hinge model proposed in this study is then used to obtain the remainder of column response with an explicit account-ing for openaccount-ing at the bottom two segment joints. In this analysis, the tensile strength of concrete is set to zero to account for the ef-fect of joint opening, and a linear strain distribution is assumed only for the regions in concrete compression zone.

ED bars that are continuous across lower segment joints are used to enhance the hysteretic energy dissipation. The unbonded portion of an ED bar is inserted into a tube to prevent buckling un-der compression, and the interior diameter of the tube slightly ex-ceeds the diameter of the bar to allow for free axial deformation. The ends of an ED bar are bonded in the footing and a column seg-ment. Elongation of the ED bar due to joint opening causes a uni-form distribution of strain in the unbonded length, which penetrates into the bonded regions on both sides of the bar for a certain length. For simplicity, an additional unbonded length Lua

along which the strain in the bar is uniformly distributed is as-sumed on each of the two sides of the bar in the model. The value of Luawas assumed to be one bar diameter (1db) based on the

re-search by Raynor et al.[12]. Bars tested by Raynor et al.[12]were conﬁned by corrugated steel ducts with ﬁber reinforced grout, and the value of Luain this study was different for DYWIDAG

corru-gated ducts grouted with non-shrinkage high-strength cement. Moreover, no ﬁbers were used to increase the tensile strength of grout, so the value of Lua(5–6 bar diameter) obtained from this

(4)

study was used, indicating that the non-shrinkage high-strength cement was not a good material in transferring bond forces. This value will be explained in the section of the test program.

2.1. Strand and ED bar strains

The change of strand axial force after decompression is a result of gap openings at the column base and the bottom of the second segment. Therefore, the tensile strain developed in the strands is:

est

¼

ein

þ1 L hp1 D 2 c1 þ hp2 D 2 c2 ð3Þ

where hp1and hp2are the angles of rotation in the ﬁrst and second

segments, respectively; c1and c2 are the positions of the neutral

axis at the column base and top of segment one, respectively;

e

in

is the initial tensile strain in the strands, and L is the unbonded length of strands.

The tensile strain developed in the ED bar,

e

ED, is also a result of

gap openings at the column base and bottom of segment two. The value of

e

EDis:

eED

¼

D

Laþ

D

Lb LEDþ 2Lua¼

½hp1d1þ hp2d2

LEDþ 2Lua ð4Þ

whereDLais the elongation of the ED bar due to gap opening at the

column base;DLbis the elongation of the ED bar due to gap opening

at the bottom of segment two; LEDis the unbonded length in the ED

bar; d1is the distance between the position of the ED bar and the

neutral axis at the column base, and d2 is the distance between

the position of the ED bar and the neutral axis at the bottom of seg-ment two. If the ED bar is anchored in the ﬁrst segseg-ment, the term hp2d2(andDLb) in the numerator of Eq.(4)is omitted.

Fig. 3. Specimen 3 details.

Table 2

Plastic hinge lengths of each specimen.

Specimen 1 2 3

Lp1 0.5d 0.5d 0.5d

Lp2 0.2d 0.5d 0.2d

(5)

2.2. Column lateral displacement

The column top ﬂexural displacement,D, can be expressed as

D

¼

D

eþ

D

p1þ

D

p2 ð5Þ

whereDeis the elastic displacement of the column. The plastic

dis-placementDp1, resulting from rigidly rotating the entire column

about the column base, is expressed as

D

p1¼ /b /0y1 M1 M0 y1 ! Lp1H1¼ hp1H1 ð6Þ

where Lp1is the plastic hinge length in the ﬁrst segment (Eq.(1)); /b

is the curvature at the base section; /0

y1 is the theoretical ‘‘ﬁrst

yield’’ curvature at the base section, corresponding to the neutral axis position at the centroidal axis of the section; M0

y1is the

theoret-ical ‘‘ﬁrst yield’’ moment at the base; M1is the computed moment

at the base, and H1is the height between the column base and the

point of lateral loading. The column above segment one further ro-tates about the interface at the bottom of segment two, resulting in an additional plastic displacementDp2:

D

p2¼ /2 /0y2 M2 M0y2

!

Lp2H2¼ hp2H2 ð7Þ

where Lp2is the plastic hinge length in the second segment (Eq.(2));

/2is the curvature at the bottom of segment two; /0y2is the

theoret-ical ‘‘ﬁrst yield’’ curvature at the bottom of segment two, corre-sponding to the neutral axis position at the centroidal axis of the section; M0

y2is the theoretical ‘‘ﬁrst yield’’ moment at the bottom

of segment two, M2is the computed moment at the bottom of

seg-ment two; and H2is the height between the bottom of segment two

and the point of lateral loading.

Fig. 5. Test setup.

(a) Specimen 1

(b) Specimen 2

(c) Specimen 3

(6)

2.3. Iterative procedure

The position of the neutral axis and the concrete extreme ﬁber compressive strain at the two interfaces are parameters to deter-mine the angle of segment rotation, strand strain, ED bar strain and lateral displacement of the column. Thus, an iterative proce-dure has to be carried out in the calculation of the pushover re-sponse of the column. This can be done by applying the moment–curvature analyses at the two interfaces with increasing concrete compressive strain at the base. The tensile strength of concrete is set to zero to account for the effect of joint opening, and a linear strain distribution is assumed for concrete and longi-tudinal bars in compression. For a given concrete extreme ﬁber compressive strain ec1at the base of the column, at step n, the

pro-cedure to calculate the corresponding lateral displacement at the top of the column is described as follows. At an iteration i, knowing the lateral force F at an iteration i 1, one can obtain the corre-sponding curvature at each interface by interpolating the mo-ment–curvature analysis:

1. Assume a position of the neutral axis, c1, at the column base, a

position of the neutral axis, c2, and the concrete extreme ﬁber

compressive strain, ec2, at the bottom of segment two.

2. Calculate the angle of rotation hp1and hp2based on the

respec-tive plastic hinge length and the linear normal strain proﬁle in the compression zone characterized by the concrete extreme ﬁber compressive strain and zero strain at the neutral axis at each interface.

3. Compute the tensile strain

e

stin the strands, the tensile strain

e

EDin the ED bars, and the compressive strain in the

longitudi-nal reinforcement in the segment.

4. Compute the resulting normal stresses using the individual stress–strain relationships for each of the components. The con-crete compressive stress is computed based on the conﬁned concrete model proposed by Mander et al. [13]. The stress in the longitudinal reinforcement and the ED bar are calculated based on a bi-linear steel stress–strain relationship.

5. Integrate the normal stresses over the respective areas to obtain the corresponding normal force in each component.

6. Sum the normal forces; check for vertical force equilibriums at the base and the interface above segment one, respectively, and the ratio of computed moment M1/M2(=H1/H2). The ratio of M1/M2is

always constant since H1and H2do not vary throughout loading.

7. Iterate over the positions of the neutral axis and concrete com-pressive strain by returning to step 1 until two vertical force equilibriums and one moment ratio are satisﬁed.

Specimen 2

Specimen 3

(a) Minimal Opening at the Bottom of Segment Three

(b) Opening at the Bottom of Segment Two

(c) Opening at the Base

Specimen 2

Specimen 3

Specimen 2

Specimen 3

(7)

8. The lateral force, F, is then updated based on the moment at the column base. The iterations are continued until the lateral forces, F, in two consecutive iterations are close (within 5% dif-ference). The column top displacement is then calculated using Eqs.(5)–(7).

The procedure is repeated for increasing values of the concrete extreme ﬁber compressive strain at the base until the complete pushover relationship is determined. Termination of the analysis occurs when the conﬁned concrete compressive strain reaches the ultimate strain

e

cu[13]. The maximum axial strain in the ED

bar is limited to 12 times the yield strain, which is about 2%. This strain level is much less than the fracture strain of ASTM A615M Grade 40 (280) reinforcement [14]. The past study [15] also showed that the ED bar within the strain limit of 2% sustained many cycles of inelastic loading before fracture. As long as the strain limit (2%) was adopted, the Grade 60 steel performed as well as the Grade 40 steel. Fracture of the PT strands did not occur prior to the failure of concrete due to the use of unbonded strands. Based on past studies on PT structural systems [15–19], a maximum strand force limited to 70–80% of the ultimate strand force pre-vented fracture of strands. The maximum strand force in this study was conservatively limited to 0.5fpuAstwhere Astis the area of the

strands and fpu (=1860 MPa) is the ultimate tensile strength of

the strand. A conservative value, 50% of the ultimate, was adopted in this study to maintain the same PT load as Specimen 1, which was previously tested[6].

Note that in the development of the pushover curve, the strand is assumed at the center of segment interfaces. However, as the top of the column displacesD, the position of strands moves away

(a) Gap Opening

(b) ED Bar Strain

(c) L

ua

Variation

Fig. 8. Comparison between test and analysis.

(a) Specimen 2

(b) Specimen 3

Fig. 9. Curvature along column height.

(a) Plastic Hinge Length versus Drift

(b) Lateral Displacement Ratio

(8)

from the segment center (Fig. 1b) with strands anchored in the load stub (point A) and the footing (point B). Assuming a linear de-formed shape of strands between the points A and B, the displace-ments of strands at the bottom two joints are given by:

D

s1¼

D

H3 H1Hh1 ð8Þ

D

s2¼

D

H3 H1Hh2 ðhp1hsÞ ð9Þ

where H is the height between the top and bottom strand anchor-ages; h1is the height between the bottom strand anchorage and

the column base; h2 is the height between the bottom strand

anchorage and the bottom of segment two; H3is the height of the

top strand anchorage measured from the column base, and hs is

the segment height. Since the column top displacementDaffects the position of strands in segments, a modification of the iterative procedure has to be carried out after obtaining the displacement of the column. Repeat the iterative procedure from step 3 using the updated position of strands to find out the updated strand force, moment, and column top displacement,D(Eq.(5)). The iterations are continued until the difference between two consecutive itera-tions in terms ofDis sufficiently small.

3. Experimental program

3.1. Specimen details

The experimental program[20]had three unbonded PT precast concrete segmental bridge columns which had a footing, four

concrete-ﬁlled tube (CFT) segments, and a load stub. Each column was post-tensioned with nineteen 15-mm diameter seven wire, uncoated, low-relaxation ASTM A416 Grade 270 strands placed at the mid-depth of the cross section. The total initial PT force after losses was 2365 kN, 2321 kN and 2300 kN for Specimens 1, 2 and 3, respectively. Specimen 1 was previously tested by Chou and Chen

[6], and Specimens 2 and 3 were tested in this study. The speci-mens were identical except that Specispeci-mens 2 and 3 had a total of four #6 ED bars to enhance hysteretic energy dissipation and eight longitudinal reinforcing bars in the segments (but not across seg-ment joints) to reduce concrete compressive strains. The ratio of the area of ED bars to concrete sectional area was

q

= 0.66%. The ED bars conformed to ASTM A615M Grade 40 (280) steel reinforce-ment (Table 1) and each had a circular steel plate welded on one end, which was anchored in the footing. For Specimen 2 (Fig. 2), the other end of the ED bar was bonded in the first segment using high-strength non-shrinkage grout (Table 1). The bonded length was 490 mm, larger than 380 mm specified in Section 6.5.1 of PCI Design Handbook[21]. The upper end of ED bars in Specimen 3 was anchored in the second segment (Fig. 3). Note that the max-imum strain in the ED bars depends on not only the unbonded length but also the location of bar anchorage. Elongation of ED bars in Specimen 3, which is caused by joint opening at the base and the interface above the first segment, is much larger than that in Spec-imen 2, which is caused by joint opening at the base only. Although the unbonded lengths of ED bars were 250 and 500 mm in Speci-mens 2 and 3, maximum tensile strains of SpeciSpeci-mens 2 and 3 were 1.2% and 2.1%, respectively, at a 6% drift.

Table 2lists plastic hinge lengths of each specimen in the anal-ysis. Because the ED bars in Specimen 2 were anchored in the ﬁrst

(a) 2D finite element model

Strain (%) -80 -60 -40 -20 0 20 S tr es s (M P a) -2 0 2 4 6 -1 0 1 2 3 Strain (%) -400 -200 0 200 400 St re ss ( M P a)

(b) Concrete

(c) ED Bar

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 18 17 16 19 20 21 24 23 22 25 26 36 34 32 30 28 27 Segment 1 Segment 2 Segment 3 Segment 4 Load Stub Lateral Displacement Initial PT Force Rigid Rod Concrete Fiber ED Bar PT Element Beam-column element

(9)

segment, thus increasing the ﬁxity of the ﬁrst segment and leading to large column lateral deformations with respect to the interface at the top of segment one, Lp2 = 0.5d was used instead of Eq.(2). Plastic hinge lengths were obtained based on the test conducted earlier for similar PT precast concrete segmental bridge columns

[6]. The plastic hinge length, which is deﬁned as the region where the experimental curvature is larger than the ideal yield curvature when the column is under lateral loading, increases with drift and approximates to certain values in segments one and two after 3% drift (Table 2). These values listed inTable 2might apply for only the type of columns in this study and might change for columns

with different height-diameter ratio, column shape, and initial PT stress in the cross-sectional area.Fig. 4shows the predicted push-over response of three specimens based on the two-plastic-hinge model; the peak strength when considering the eccentricity of strands is lower than that without considering strand eccentricity.

3.2. Test setup

A 500-kN actuator was placed at the load stub (Fig. 5), and the specimen was then tested quasi-statically with a pre-deﬁned dis-placement history, consisting of one drift cycle with amplitudes Lateral Displacement (mm) -300 -200 -100 0 L at era l F -150 -100 -50 0 50 100 150

(a) Hysteretic Response (Specimen 2)

Lateral Displacement (mm) -300 -200 -100 0 100 200 300 L at era l F o rc e (k N ) -150 -100 -50 0 50 100 150 -6 -4 -2 0 2 4 6 Drift (%) Test Model

(b) Hysteretic Response (Specimen 3)

(c) Gap Opening

(d) ED Bar Strain

(10)

of 0.1%, 0.15%, 0.2%, and 0.3%, followed by three drift cycles with amplitudes of 0.4%, 0.6%, 0.9%, 1.5%, 2%, 3%, 4%, 5%, and 6%. Since the ultimate concrete compression strain calculated based on the conﬁned concrete model [13] was 0.029, which was exceeded when the column drift was 6%, the test was stopped after complet-ing two cycles at a drift of 6%.

3.3. Results of experiments

Fig. 6shows the hysteretic response of three specimens during the test. The peak strengths are close to the predicted values as seen in the ﬁgure. Hysteretic energy dissipation of Specimen 1 was associated with the plastic straining of concrete in compres-sion, and that of Specimens 2 and 3 were associated with the plas-tic straining of the ED bar and of concrete in compression. The hysteretic energy increased slightly with drift, and the equivalent viscous damping[22] for Specimens 2 and 3 at a 6% drift were 7.5% and 8.8%, higher than 6.5% of Specimen 1 at the same drift.

Fig. 7shows the opening of segment joints for Specimens 2 and 3 in the test. The joint opening at the bottom of segment three (Fig. 7a) was small compared to that of bottom two segments (Fig. 7b and c). Although the gap at the base of Specimen 3 was very large, ED bar buckling or fracture was not observed during the test. This can be conﬁrmed by stable hysteretic responses of Specimen 3 throughout the test (Fig. 6c). For Specimen 2 with ED bars anchored at the footing and segment one, opening at the bot-tom of segment two was larger than that at the column base (Figs. 7 and 8a), leading to small elongation of ED bars. For Specimen 3 with ED bars anchored at the footing and segment two, a larger opening was observed at the column base than at the bottom of segment two throughout the test (Figs. 7 and 8a). Elongation of ED bars in Specimen 3 was primarily caused by joint opening at these two interfaces. Although the unbonded length of ED bars in Specimen 3 was double that in Specimen 2, the tensile strain of ED bars was larger in Specimen 3 than Specimen 2 (Fig. 8b). The maximum strain in the ED bar predicted by the two-plastic-hinge model shows agreement with that obtained from the tests. The va-lue of Luaon both sides of the original unbonded length was 5db

and 6dbin Specimens 2 and 3, respectively. These two values were

determined by equating the strain gauge reading in the unbonded length of the ED bar to a strain value calculated based on elonga-tion of the ED bar from the gap-opening angle and the posielonga-tion of the neutral axis during the tests.Fig. 8c shows variation of Lua,

approaching to a constant value (5–6db) at a medium-to-high

col-umn drift.

Fig. 9shows the distribution of curvature along the column height for both the push and pull directions. The experimental cur-vatures were calculated as

/¼

D

t

D

c

DLg ð10Þ

whereDtis the elongation of a displacement transducer on the

ten-sion side of the segment; Dc is the shortening of a displacement

transducer on the compression side of the segment at the same height level; D is the horizontal distance between these two dis-placement transducers, and Lgis the gage length. Note that the

cur-vature at the column base and the bottom of segment two is larger than that at the bottom of segment three, indicating that joint open-ing is most signiﬁcant at the bottom two joints. This is correspond-ing to the observed performance in the test (Fig. 7). Moreover, for Specimen 2, the gap opening at the base was smaller than that at the bottom of the second segment, leading to the curvature at the base smaller than that at the bottom of second segment. Stiffening of segment 2 in Specimen 3 was provided by the bonded ED bars, so

the curvature of Specimen 3 (Fig. 9b) at the base was always larger than that at the bottom of the second segment.

The plastic hinge lengths Lp1and Lp2in segments 1 and 2 were

calculated based on the method by Chou and Chen[6].Fig. 10a shows that plastic hinge length increases with drift and approxi-mates to half the section diameter in the first and second segments in Specimen 2. The plastic hinge length in Specimen 3 approxi-mates to half the section diameter in first segment and one-fifth the section diameter in second segment, as observed in Specimen 1. This indicates that the anchorage position of ED bars affects the curvature in the plastic hinge region, the plastic hinge length in segments, the elongation of ED bars, and corresponding energy dissipation capacity. This is logical since stiffness is inversely pro-portional to curvature. The plastic hinge length, a constant value obtained at a high drift in each segment, was used for simplicity in the iterative procedure in predicting the pushover curve of col-umns.Fig. 6shows the analytical results close to peak values of the hysteresis response of columns in the tests.Fig. 10b shows ratios of the flexural displacementsDe,Dp1, andDp2, which were calculated

based on experimental curvatures (Fig. 9) and plastic hinge lengths (Fig. 10a), to the imposed displacementsDtotalby the actuator. Each

drift has three bars: the ﬁrst represents ratios of Specimen 1, while the second and third represent those of Specimens 2 and 3, respec-tively. The contribution of the ﬂexural displacement due to the sec-ond segment rotation is small in Specimens 1 and 3. Except for the drift of 6%, the summation of measured displacements is close to the actuator displacement. Concrete at the base crushes at a drift of 6%, leading to more deformation measured by the displacement transducer placed in the concrete crush zone. Therefore, the curva-ture calculated at the base is overestimated and the associated dis-placement is larger than the test result.

4. Finite element analysis

A two-dimensional (2D) ﬁnite-element model for unbonded PT concrete segmental bridge columns was developed in this re-search. The model was created and analyzed using the computer program, PISA[23]. Different analytical models for investigating seismic performances of PT structures can be found elsewhere

[24–26]. The modeling techniques of unbonded PT columns are de-scribed as follows and schematically illustrated inFig. 11.

4.1. PT segmental column model

Fig. 11a illustrates a typical column model with four segments, each of which is composed of 50 concrete ﬁbers. Each ﬁber is mod-eled using one dimensional truss element which consists of two nodes, each with three degrees of freedom: translations in the x and y-directions and rotation about the z-direction. The truss ele-Table 3

Comparison of the FM model prediction to the experimental response. Drift (%) 0.9 1.5 2 3 4 5 6 Specimen 1 Model (kN) 147 164 171 183 196 202 202 Test (kN) 153 179 179 190 196 196 194 Ratio 0.96 0.92 0.96 0.96 1 1.03 1.04 Specimen 2 Model (kN) 158 180 191 202 208 212 216 Test (kN) 155 178 187 200 204 201 194 Ratio 1.02 1.01 1.02 1.01 1.02 1.05 1.11 Specimen 3 Model (kN) 171 194 202 214 224 229 233 Test (kN) 155 179 191 206 211 210 208 Ratio 1.1 1.08 1.06 1.04 1.06 1.09 1.12

(11)

is used in the center of the section and is linked at each segment joint to transfer column shears. No flexural and axial stiffnesses are specified to the beam-column elements. The ultimate concrete compressive strength and strain are calculated based on Mander’s confined concrete stress–strain model [13]. Degradation of the concrete under compressive cyclic loading is described using the three-parameter concrete model [27] in the computer program PISA. The three parameters in the model, which describes stiffness degradation, strength deterioration, and pinching of concrete un-der cyclic loading, are 10, 0.88, and 1.0, respectively. The concrete tensile strength and stiffness are assumed to be zero. Fig. 11b shows a typical response of a concrete truss element in cyclic loading.

The strands are modeled as a tension-only truss element (PT), which is anchored between the top of the load stub and bottom of the footing. The tensile force in the PT element increases when gap opening at segment interfaces extends beyond the center of the segment. Since the strands are within the elastic range in the tests, elastic behavior is assigned to the strands. Each ED bar has an unbonded length to dissipate seismic energy and a bonded length for anchorage. The bonded length of the ED bar is modeled as a rigid element and its end is anchored to the horizontal rigid rod at the segment interface. The other end of the ED bar is ﬁxed the base. The ED bar is loaded whenever the node of a horizontal rigid rod at the segment interface deforms relative to the base. The stress–strain response of the ED bar under cyclic loading is approximated by a bi-linear kinematic hardening model (Fig. 11c). Longitudinal reinforcement in the segment for reducing concrete compressive strain is modeled as a truss element with compression only properties.

and the corresponding lateral force was determined by the shear force developed in a rigid bar representing the load stub.Fig. 12a and b shows comparisons of the cyclic responses obtained from the FE models along with the experimental results of Specimens 2 and 3. The ratio of the FE model prediction to the experimental response is listed inTable 3. The response of the FE model is in good agreement with the cyclic response of the specimen within the target drift. The series of distributed concrete truss elements allows for measurement of the gap opening: the extension of the extreme concrete truss element in each segment reasonably pre-dicts the gap opening at the segment interface (Fig. 12c). Because larger gap opening obtained from the FE model results in larger strains in the ED bars and PT strands, the peak strength predicted by the FE model is larger than that from the experimental re-sponse. The maximum strains in the ED bars predicted by the FE models show agreement with those obtained by Specimens 2 and 3 tests (Fig. 12d). Signiﬁcant differences exist in the results in terms of the maximum ED bar strain when the equivalent unb-onded length Luaon each side of the original unbonded length of

the ED bar is assumed as one bar diameter (1db) in the model.

However, the discrepancy of the gap-opening due to the assump-tion of one bar diameter in the model is minor (Fig. 12c).

4.3. Parametric study

An analytical study using the modeling techniques described earlier was conducted for 24 unbonded PT segmental column mod-els. Three parameters were investigated in this study: the aspect ratio, the amount of initial PT force, and ED bars. As listed in

Table 4

Column model details.

Model Column size Initial PT force ED bar

Segment size Segment

number Column height (mm) A ¼ 0:25f0 cAcB ¼ 0:35fc0Ac (kN) q (%) Unbonded length (mm) Anchorage position (segment) emax (%) No. Diameter (mm) Height (mm) 3 500 500 4 2450 A 2300 0.66 740 2 2.4 4 500 500 4 2450 A 2300 1.2 740 2 2.4 5 500 500 4 2450 A 2300 1.8 740 2 2.3 6 500 500 4 2450 A 2300 2.4 740 2 1.9 7 500 500 8 4900 A 2300 0.66 740 4 2.5 8 500 500 8 4900 A 2300 1.2 740 4 2.5 9 500 500 8 4900 A 2300 1.8 740 4 2.4 10 500 500 8 4900 A 2300 2.4 740 4 2.0 11 1000 1000 4 4900 A 9048 0.66 1480 2 2.5 12 1000 1000 4 4900 A 9048 1.2 1480 2 2.4 13 1000 1000 4 4900 A 9048 1.8 1480 2 2.4 14 1000 1000 4 4900 A 9048 2.4 1480 2 2.0 15 500 500 4 2450 B 3220 0.66 740 2 2.2 16 500 500 4 2450 B 3220 1.2 740 2 2.2 17 500 500 4 2450 B 3220 1.8 740 2 2.1 18 500 500 4 2450 B 3220 2.4 740 2 1.5 19 500 500 8 4900 B 3220 0.66 740 4 2.3 20 500 500 8 4900 B 3220 1.2 740 4 2.3 21 500 500 8 4900 B 3220 1.8 740 4 2.2 22 500 500 8 4900 B 3220 2.4 740 4 1.9 23 1000 1000 4 4900 B 12,667 0.66 1480 2 2.3 24 1000 1000 4 4900 B 12,667 1.2 1480 2 2.4 25 1000 1000 4 4900 B 12,667 1.8 1480 2 2.2 26 1000 1000 4 4900 B 12,667 2.4 1480 2 1.9

(12)

Table 4, the aspect ratios of columns are 4.9 and 9.8 with segment diameters of either 500 mm or 1000 mm. The initial PT forces are 0:25f0

cAcand 0:35fc0Ac, where fc0¼ 50 MPa is the concrete

compres-sive strength and Acis the gross area of the concrete section. The

ratios of the area of ED bars to concrete sectional area Ac are

q

= 0.66%, 1.2%, 1.8%, and 2.4%. The ED bar in this study is anchored at the base and the mid-height of the column and the maximum tensile strain in its unbonded length,

e

max, ranges 1.9–2.5%.

Fig. 13shows that with the same amount of ED bars, the in-crease of the initial PT force inin-creases the lateral strength of the column models and decreases residual displacement. As the ED bar ratio

q

increases, the hysteretic energy dissipation increases as well as does residual drift. If the ED bar ratio does not exceed a certain value, the columns exhibit a ﬂag-shaped hysteretic behavior with almost zero residual displacement upon unloading. An optimum ED bar ratio can result in an optimum ﬂag-shaped hysteretic behavior with the unloading branch approaching the ab-scissa while keeping small residual displacement.Fig. 14a shows the relationship between the equivalent viscous damping and residual displacement for all models. Based on minimizing residual displacement irrespective of aspect ratio or initial PT force, the optimum ED bar ratio

q

is about 1.2%, corresponding to the opti-mum equivalent viscous damping of 12–13%. The figure shows that the residual displacement increases significantly when the equiva-lent viscous damping is larger than 13%.Fig. 14a also shows that with the same ED bar ratio and initial PT force, the column models 7–10 with a high aspect ratio (=8) tend to have larger equivalent viscous damping than the column models 3–6 with a low aspect ratio (=4). This behavior can be attributed to the fact that the col-umns with a high aspect ratio have smaller peak strength at the target drift.Fig. 14b shows the ratios of the moment provided by ED bars to the total maximum column moment capacity; this ratio is proportionally dependent on the equivalent viscous damping (or ED bar ratio). To reach the optimum flag-shaped hysteretic behav-ior, the maximum moment provided by ED bars is about one-quar-ter of the total column moment based on the equivalent viscous damping of 12–13%.

5. Conclusions

The study of the cyclic performance of unbonded PT precast concrete segmental bridge columns with circular cross section is described in this paper. Longitudinal mild steel bars (ED bars) were

-200 -100 0 100 200 -200 -100 0 100 200 -200 -100 0 100 200 -200 -100 0 100 200 Lateral Displacement (mm) -400 -200 0 200 400 Lateral F orce (kN) Pin=0.25fc'Ac ρ=1.2% LateralDisplacement (mm) -400 -200 0 200 400 Lateral F orce (kN) Pin=0.25fc'Ac ρ=2.4% Lateral Displacement (mm) -400 -200 0 200 400 Lateral F orce (kN) Pin=0.35fc'Ac ρ=1.2% LateralDisplacement (mm) -400 -200 0 200 400 Lateral F orce (kN) -6 -3 0 3 6 Drift (%) Pin=0.35fc'Ac ρ=2.4%

(a) Model 4

(b) Model 6

(c) Model 16

(d) Model 18

-6 -3 0 3 6 Drift (%) -6 -3 0 3 6 Drift (%) -6 -3 0 3 6 Drift (%)

Fig. 13. Effects of ED bar ratio and PT force on cyclic behavior.

(a) Equivalent viscous damping versus

residual displacement relationship

(b) equivalent viscous damping versus

moment ratio relationships

Fig. 14. Equivalent viscous damping versus residual displacement and ED bar moment ratio relationships (6% drift).

(13)

sponse of the column. In order to capture the cyclic response of the column, a 2D FE model was developed using one-dimensional truss and beam-column elements. The concrete truss elements were modeled as zero tensile strength and stiffness to capture the gap-opening mechanism at segment interfaces.

Test results show that the ED bar can increase hysteretic energy dissipation, and Specimens 1–3 have equivalent viscous damping of 6.5–8.8%. An equivalent unbonded length along which the strain in the ED bar is assumed uniformly distributed on each of the two sides of the original unbonded length is 5–6 bar diameter. The plastic hinge length in the first or second segments varies with anchorage position of the ED bar and lateral displacement of the column. For Specimens 1 and 3 with larger joint opening and con-crete damage at the base than the bottom of the second segment, the plastic hinge length approaches to half the section diameter in the first segment and one-fifth the section diameter in the sec-ond segment. However, for Specimen 2 with larger joint opening and concrete damage at the bottom of the second segment than at the base, the plastic hinge length approaches to half the section diameter in these two segments. By comparing the results of push-over analyses using the two-plastic-hinge model to those from the tests, it is evident that given the constant plastic hinge lengths in the bottom two segments the simplified analytical model is capa-ble of predicting the pushover curve, the segment joint opening and the strain in the ED bars.

Although the gap opening predicted by the FE model differs by 30% when compared to the experimental value, the effect of bar anchorage location on the segment gap opening can be captured by this proposed model. Moreover, the peak strength and flag-shaped hysteretic response of the PT column can also be captured by this FE model. Through parametric studies, a higher ED bar ratio results in more hysteretic energy dissipation. If the equivalent vis-cous damping (or ED bar ratio) is below a certain value, the column exhibits an optimum flag-shaped hysteretic behavior with large hysteretic energy dissipation while keeping small residual dis-placement upon unloading. The optimum equivalent viscous damping is about 12–13%, corresponding to the optimum ED bar ratio of 1.2%, quite larger than the 0.66% of Specimens 2 and 3. For the columns examined, this amount of ED bars contributes about one-quarter of the total column moment. Note that due to arrangement of ED bars in the circular section, only half of them are effective in dissipating seismic energy in one loading direction. The plastic hinge lengths determined from the tests are specific to the column details in this study, and experimental work is fur-ther needed to assess if the two-plastic-hinge model is applicable to different details of PT concrete segmental columns. Although 2D simplified FE model can reasonably predict the test results, a verification of the 2D model with 3D model, to prove that the dif-ference resulting from the simplification can be ignored, is still needed in the future analysis. Moreover, the effect of tensile strength and stiffness of concrete on the column behavior may also be considered in the analytical model.

segmental bridge columns. Report No. SSRP 2001/25. La Jolla (CA): University of California San Diego; 2002.

[4] Usami T, Ge H. Ductility of concrete-ﬁlled steel box columns under cyclic loading. J Struct Eng ASCE 1994;120(7):2021–40.

[5] Chang KC, Loh CH, Chiu HS, Hwang JS, Cheng CB, Wang JC. Seismic behavior of precast segmental bridge columns and design methodology for applications in Taiwan. Taipei (Taiwan): Area National Expressway Engineering Bureau; 2002 [in Chinese].

[6] Chou CC, Chen YC. Cyclic tests of post-tensioned precast CFT segmental bridge columns with unbonded strands. Earthquake Eng Struct Dynam 2006;35:159–75.

[7] Kim TH, Lee HM, Kim YJ, Shin HM. Performance assessment of precast concrete segmental bridge columns with a shear resistant connecting structure. Eng Struct 2010;32(5):1292–303.

[8] Chou CC, Hsu CP. Hysteretic model development and seismic response of unbonded post-tensioned precast CFT segmental bridge columns. Earthquake Eng Struct Dynam 2008;37:919–34.

[9] Pampanin S, Priestley MJN, Sritharan S. PRESSS phase 3: the ﬁve-story precast test building-frame direction response. Report No. SSRP 2000/08. La Jolla (CA): University of California San Diego; 2000.

[10] Chou CC, Chen JH, Chen YC, Tsai KC. Evaluating performance of post-tensioned steel connections with strands and reduced ﬂange plates. Earthquake Eng Struct Dynam 2006;35(9):1167–85.

[11] Ou YC, Chiewanichakorn M, Aref AJ, Lee GC. Seismic performance of segmental precast unbonded posttensioned concrete bridge columns. J Struct Eng ASCE 2007;133(11):1636–47.

[12] Raynor DJ, Lehman DE, Stanton JF. Bond-slip response of reinforcing bars grouted in ducts. ACI Struct J 2002;99(5):568–76.

[13] Mander JB, Priestley MJN, Park P. Theoretical stress–strain model for reinforced concrete. J Struct Eng ASCE 1988;114(8):1804–23.

[14] Paulay T, Priestley MJN. Seismic design of reinforced concrete and masonry buildings. New York: John Wiley & Sons; 1992.

[15] Chou CC, Tsai KC, Yang WC. Self-centering steel connections with steel bars and a discontinuous composite slab. Earthquake Eng Struct Dynam 2009;38:403–22.

[16] Chou CC, Weng CY, Chen JH. Seismic design and behavior of post-tensioned connections including effects of a composite slab. Eng Struct 2008;30:3014–23.

[17] Chou CC, Chen JH. Column restraint in post-tensioned self-centering moment frames. Earthquake Eng Struct Dynam 2010;39(7):751–74.

[18] Chou CC, Chen JH. Seismic design and shake table tests of a steel post-tensioned self-centering moment frame with a slab accommodating frame expansion. Earthquake Eng Struct Dynam 2011;40(11):1241–61.

[19] Chou CC, Chen YC, Pham DH, Truong VM. Experimental and analytical validation of steel dual-core self-centering braces for seismic-resisting structures. In: 9th International conference on urban earthquake engineering/4th Asia conference on earthquake engineering, Tokyo, Japan; 2012.

[20] Chang HJ. Seismic effects of energy dissipation bars on post-tensioned precast concrete segmental bridge columns. Chou CC, Liou GS, thesis advisors. Hsinchu (Taiwan): Department of Civil Engineering, National Chiao Tung University; 2009.

[21] PCI design handbook: precast and prestressed concrete, 5th ed. Chicago (IL): Precast/Prestressed Concrete Institute; 1999.

[22] Priestley MJN, Seible F, Calvi GM. Seismic design and retroﬁt of bridges. New York: John Wiley & Sons, Inc.; 1996.

[23] Tsai KC, Lin BZ. Development of an object-oriented nonlinear static and dynamic 3D structural analysis program. CEER/R92-04. Taiwan: Center for Earthquake Engineering Research, National Taiwan University; 2003. [24] ElGawady MA, Dawood HM. Analysis of segmental piers consisted of concrete

ﬁlled FRP tubes. Eng Struct 2012;38:142–52.

[25] Chou CC, Chen JH. Analytical model validation and inﬂuence of column bases for seismic responses of steel post-tensioned self-centering MRF systems. Eng Struct 2011;33(9):2628–43.

[26] Wang Z, Song W, Wang Y, Wei H. Numerical analytical model for seismic behavior of prestressing concrete bridge column systems. Eng Struct 2011;14:2333–40.

[27] Kunnath SK, Reinhorn AM, Park YJ. Analytical modeling of inelastic seismic response of R/C structures. J Struct Eng ASCE 1990;116(4):996–1017.