Model for the Lateral Behaviorof Reinforced Concrete ColumnsIncluding Shear Deformations

8/12/2019 Model for the Lateral Behaviorof Reinforced Concrete ColumnsIncluding Shear Deformations

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Model for the Lateral Behaviorof Reinforced Concrete ColumnsIncluding Shear Deformations

Eric J. Setzlera)

and Halil Sezena)

This research is focused on modeling the behavior of reinforced concretecolumns subjected to lateral loads. Deformations due to flexure, reinforcement

slip, and shear are modeled individually using existing and new models.Columns are classified into five categories based on a comparison of their

predicted shear and flexural strengths, and rules for combining the threedeformation components are established based on the expected behavior ofcolumns in each category. Shear failure in columns initially dominated by

flexural response is considered through the use of a shear capacity model. The

proposed model was tested on 37 columns from various experimental studies.In general, the model predicted the lateral deformation response envelopereasonably well. DOI: 10.1193/1.2932078

INTRODUCTION

Numerous reinforced concrete buildings exist in the United States and around theworld that do not have sufficient detailing to ensure satisfactory performance in earth-quakes. Past earthquakes have caused widespread damage to reinforced concrete struc-tures that were not designed according to modern seismic design codes. Many of these

structures still exist in seismically active areas, and they may be susceptible to majordamage or collapse within their expected lifetimes. It is possible to retrofit an existingstructure to improve its shear strength and flexural deformation capacity, allowing it to

perform satisfactorily in an earthquake. This requires the ability to model the as-builtcapacity of the structure, so that additional strength and/or deformation capacity require-ments can be determined.

Columns are often the most critical components of earthquake damage-prone struc-tures. The goal of the research reported here is to develop a model that can serve as aresponse envelope for the behavior of a reinforced concrete column subjected to axialand cyclic lateral loading. While the primary motivation and focus of this research is themodeling of lightly reinforced columns that experience flexural yielding followed by

shear failure such as those described above, the model is general enough such that it isalso applicable for columns failing in shear (e.g., very short columns) or columns de-veloping plastic hinges and failing in flexure (e.g., well-reinforced long columns).

a)Department of Civil and Environmental Engineering and Geodetic Science, The Ohio State University, 470

Hitchcock Hall, 2070 Neil Ave., Columbus, OH 43210

493

Earthquake Spectra,Volume 24, No. 2, pages 493511, May 2008; 2008, Earthquake Engineering Research Institute


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LATERAL DEFORMATION COMPONENTS

The total lateral deformation of a column subjected to lateral loads at its ends iscomprised of deformations due to three response mechanisms: flexure, reinforcement

slip at the column end(s), and shear. The interaction of these components determines theoverall behavior of a column. These deformations are depicted schematically in Figure 1.Each deformation component can be modeled individually.

FLEXURAL DEFORMATIONS

The flexural response of a reinforced concrete section can be calculated through amoment-curvature analysis. The moment-curvature analysis uses constitutive models for

the concrete and reinforcing steel that can capture the nonlinear behavior of each mate-rial accurately. Moment-curvature analysis is used to calculate sectional response in thisresearch. A steel constitutive model with a yield plateau and nonlinear strain-hardeningregion is specified to match the experimentally observed behavior of the reinforcingsteel. Concrete confinement is considered by using the constitutive model by Mander etal. (1988) to model the core and cover concrete separately. The full strain capacity of the

confined and unconfined concrete was used, as calculated by the Mander et al. model.The specified maximum unconfined concrete strain was 0.006.

For an applied lateral load at the column end, the moment can be determined at anypoint in the column. Then, the moment-curvature relationship can be used to determinethe curvature distribution over the column height. The lateral displacement of a column

due to flexural deformations, f, can be calculated by integrating the curvature distribu-

tion over the height of the column as follows:

Figure 1. Components of lateral deformation in a reinforced concrete column.

494 E. J. SETZLER AND H. SEZEN


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f= 0

L

xxdx 1

wherexis the curvature distribution, x is measured along the axis of the column, andLis the height of the column. In this research, Equation 1 is used to compute the flexuraldeformations for lateral loads up to the load that initiates yielding in the longitudinalsteel. After yielding has occurred, the flexural deformations are calculated from a plastichinge model as

f= f,y+yLpa Lp2 2

where f,y is the flexural deformation at yield, calculated using the integration method

(Equation 1), is the curvature at the column end, andy is the curvature at yield. a is

the shear span, and is equal to L for a cantilever column orL / 2 for a column with fixedsupports at both ends. For columns with fixed supports at both ends, the second term on

the right side of Equation 2 must be multiplied by 2. The plastic hinge length, Lp, istaken as one half of the total section depth per the recommendations of Moehle (1992).Complete details of this flexural deformation model, including the constitutive models

and moment-curvature analysis, are given in Setzler (2005).

REINFORCEMENT SLIP

When a reinforcing bar embedded in concrete is subjected to a tensile force, strainaccumulates over the embedded length of the bar. This strain causes the reinforcing barto slip relative to the concrete in which it is embedded. Slip of a columns reinforcing

bars in the anchoring concrete (i.e., the footing or beam-column joint) will cause rigid-body rotation of the column, as shown in Figure 1. This rotation is not accounted for ina flexural analysis, where the column ends are assumed to be fixed.

The bar slip model used in this study was originally developed by Sezen and Moehle(2003), and includes further developments by Sezen and Setzler (2008). It is illustratedin Figure 2. This model assumes a stepped function for bond stress between the concreteand reinforcing steel over the embedment length of the bar. Based on experimental ob-

servations (Sezen 2002), the bond stress is taken as 12fcpsi for elastic steel strains (ubin Figure 2) and6fc psi for inelastic steel strains ub, where fc is the concrete com-

pressive strength. The rotation due to slip, s, is calculated as

s=slip

dc 3

whereslipis the extension of the outermost tension bar from the column end (Figure 2),

anddandc are the distances from the extreme compression fiber to the centroid of thetension steel and the neutral axis, respectively. The column lateral displacement is equalto the product of the slip rotation and the column length.

MODEL FOR T HE LATERAL BEHAVIOR OF REINFORCED CONCRETE COLUMNS 495


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SHEAR DEFORMATIONS

Shear displacements are calculated in the proposed model using a combination oftwo existing models. The computer program Response-2000 (Bentz 2000) uses ModifiedCompression Field Theory (Vecchio and Collins 1986) to compute the monotonic shear

behavior of cracked concrete. It is used in the proposed model to compute the shear

force-deformation relationship up to the attainment of peak strength. After this point, ashear model by Patwardhan (2005) is adopted. The proposed shear model is shown in

Figure 3. The peak strength, Vpeak, is the maximum strength from Response-2000. The

Figure 2. Proposed slip rotation model.

Figure 3. Proposed shear model.



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drift at the onset of shear failure, v,u, is adopted from Gerin and Adebar (2004), and iscalculated as

v,u=4 12 vnfc

v,n 4wherevnis the shear stress at peak strength (vn = Vpeak/ bd, whereb is the column width)andv,nis the maximum drift at peak strength computed by Response-2000. The drift at

axial load failure, v,f, is calculated as

v,f= ALF f,f s,fv,u 5

where ALFis the total drift at axial load failure, andf,fands,fare the flexural and slipdeformations, respectively, at the point of axial load failure. The total drift at axial load

failure under cyclic loading is calculated using the axial capacity model by Elwood and

Moehle (2005a):

ALF

L =

4

100

1 + tan2

tan+P sAsvfyvdctan

6

where is the angle of the shear crack, P is the axial load, Asv is the area of transversesteel with yield strength fyv at spacing s, anddc is the depth of the core concrete, mea-sured to the centerlines of the transverse reinforcement. In the derivation, was assumed

to be 65 degrees. The rules governing the post-peak column behavior, given in the next

section, define the values forf,fands,f. Complete details of the proposed shear modelare available in Setzler (2005).

TOTAL LATERAL RESPONSE

The total lateral response of a reinforced concrete column can be modeled using aset of springs in series. The flexure, bar slip, and shear deformation models discussedabove are each modeled by a spring. Each spring is subjected to the same force, and thetotal displacement response is the sum of the responses of each spring. Figure 4 illus-trates the spring model schematically. Up to the peak strength of the column, the three

deformation components are simply added together to predict the total response. Rulesare established for the post-peak behavior of the springs based on a comparison of the

shear strength Vn, the yield strength Vy, and the flexural strength Vp. The yieldstrength is defined as the lateral load corresponding to the first yielding of the tension

bars in the column during flexural analysis. The flexural strength is the lateral load cor-

responding to the maximum moment sustainable by the column cross section. Both val-ues are obtained from the moment-curvature analysis. The shear strength is calculated

from the equation proposed by Sezen and Moehle (2004):



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Vn=k6fca/d1 + P

6fcAg0.80Ag+ Asvfyvd

s 7

whereAgis the gross cross sectional area anda / dis the aspect ratio.kis a factor relatedto the displacement ductility, which is the ratio of the maximum displacement to the

yield displacement.kis equal to 1.0 for displacement ductilities less than 2, it is equal to0.7 for displacement ductilities greater than 6, and it varies linearly for intermediate dis-

placement ductilities.kis taken as 1.0 in the proposed model for classification purposes,because the classification system outlined below is based on the initial, or low-ductility

shear and flexural strengths.Figure 5 plots the flexural response and shear strength for five fictitious columns.

These columns all have the same shear strength, but different flexural and yield

strengths. By comparingVn, Vy, andVp, these columns can be classified into one of thefollowing five categories.

Category I:VnVyThe shear strength is less than the lateral load causing yielding

Figure 4. Spring representation of the proposed model for a column with fixed ends.



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in the tension steel. The column fails in shear while the flexural behavior remains elastic.

Category II:VyVn0.95VpThe shear strength is greater than the yield strength,but less than the flexural strength of the column. The column fails in shear, but inelastic

flexural deformation occurring prior to shear failure affects the post-peak behavior.

Category III:0.95VpVn1.05VpThe shear and flexural strengths are essentiallyidentical. Due to the inherent variability in the models used to predict the strengths, it is

not possible to predict conclusively which mechanism will govern the peak response.Shear and flexural failure are assumed to occur simultaneously, and both mechanismscontribute to the post-peak behavior.

Category IV: 1.05VpVn1.4Vp The shear strength is greater than the flexuralstrength of the column. The column experiences large flexural deformations potentiallyleading to a flexural failure. Inelastic shear deformations affect the post-peak behavior,

and shear failure may occur as displacements increase.

Category V:Vn1.4VpThe shear strength is much greater than the flexural strengthof the column. The column fails in flexure while the shear behavior remains elastic.

Category IV and V specimens are those that are expected to fail in flexure, becausetheir flexural strength is lower than their initial shear strength. However, the shear

strength of a column decreases as displacements increase (Sezen and Moehle 2004). Ifthe initial shear strength is greater than the flexural strength, but shear degradationcauses the shear strength to become less than the flexural strength, shear failure couldoccur in the column after the flexural strength has been reached. Category IV specimens

are those where shear failure could occur at high displacements, while Category V speci-mens are those whose shear strength is high enough that shear failure is not expected

Figure 5. Lateral load-displacement ductility relationship for columns in each category.



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even as the shear strength decreases. The factorkin the Sezen-Moehle shear strengthmodel (Equation 7) can take a minimum value of 0.7. Therefore, if the shear strength is

less than1/0.71.4times the flexural strength, shear failure is possible and the columnwill be in Category IV. It is noted that these response categories are defined for the pro-

posed macro-scale model shown in Figure 4, and are not intended for finite element

analysis.

The classification system given above describes the expected column behavior basedon a comparison of the shear and flexural strengths. From these descriptions, rules gov-erning the flexure, slip, and shear springs are defined. These rules are described below,and are illustrated in Figure 6. As stated above, the total deformation is calculated by

summing the three deformation components for the initial response, up to the peak

strength of the column, for columns in all categories.

Category I:The peak strength of the column is the shear strength, as calculated inthe proposed shear model. After the peak strength is reached, the shear behavior domi-

nates the response. As the column strength decreases, shear deformations continue toincrease according to the shear model (solid line in Figure 6a), while the flexure and slipsprings unload along their initial responses (dashed line in Figure 6b). The post-peakdeformation at any lateral load level is the sum of the post-peak shear deformation andthe pre-peak flexural and slip deformations corresponding to that load.

Category II:The peak strength of the column is the shear strength calculated fromthe proposed shear model. As the column strength decreases, shear deformations con-tinue to increase according to the shear model (solid line in Figure 6a), but the flexure

and slip springs are locked at their values at peak strength (dot-dashed line in Figure 6b).The post-peak deformation at any lateral load level is the sum of the flexural and slipdeformations at peak strength and the post-peak shear deformation corresponding to that

load.

Category III:The peak strength is the smaller of the shear strength and the flexuralstrength. As the column strength decreases, all deformations continue to increase ac-

Figure 6. Behavior of (a) shear spring and (b) flexure and slip springs for each category.



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cording to their individual models (solid line in Figures 6a and 6b). The post-peak de-formation at any lateral load level is the sum of the post-peak flexure, slip, and shear

deformations corresponding to that load.

Category IV:The peak strength of the column is the flexural strength, calculated in

the flexure model. As the column strength decreases, flexural and slip deformations con-tinue to increase according to their models (solid line in Figure 6b), but the shear springis locked at its value at peak strength (dot-dashed line in Figure 6a). The post-peak de-formation at any lateral load level is the sum of the post peak flexural and slip defor-mations corresponding to that load and the shear deformation at peak strength. The col-umn may experience a shear failure after being subjected to large deformations. Shear

failure is predicted through the use of a shear capacity model described below.

Category V:The peak strength of the column is the flexural strength calculated fromthe flexure model. If the column strength decreases, flexural and slip deformations con-

tinue to increase according to their models (solid line in Figure 6b), while the shearspring unloads with an unloading stiffness equal to its initial stiffness (dashed line inFigure 6a). The post-peak deformation at any lateral load level is the sum of the post-

peak flexural and slip deformations and the pre-peak shear deformation correspondingto that load.

SHEAR FAILURE DUE TO HIGH DISPLACEMENT DUCTILITY

It has been established that shear strength decreases as displacement ductility in-creases, potentially leading to shear failure in columns initially dominated by flexure(Sezen and Moehle 2004). However, this possibility cannot be accounted for in a simplecombination of the three component models, as outlined above.

Elwood (2004) proposed the idea of imposing a shear failure surface on the lateral

load-total displacement diagram, as shown in Figure 7. In this proposal, if the responseof the column intersects the shear failure surface, shear failure will occur. This model isintended for the prediction of shear failure after the occurrence of flexural yielding. Theshear failure surface is defined by the empirical drift capacity model proposed by El-

wood and Moehle (2005b):

SF

L =

3

100+ 4v

1

500

v

fc

1

40

P

Agfc

1

100 8

where SFis the drift at shear failure, v is the transverse reinforcement ratio, andv is

the nominal shear stress. fc andv have units of psi.

The shear failure surface proposed by Elwood is implemented in the proposed model

to account for delayed shear failure following inelastic flexural response for Category IVspecimens. The drift at shear failure is calculated from Equation 8 using the peak model

strength to calculate the shear stress (i.e.,v = Vp / bd). If the predicted drift in the unmodi-fied model exceeds the calculated drift at shear failure, shear failure is assumed to haveoccurred. The model is modified to decrease linearly from the point of shear failure to

zero strength at the drift at axial load failure, calculated from Equation 6.



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Equation 7 should not be used to predict the displacement at shear failure as it is

intended for calculation ofVn using displacement ductility, because the result is overlysensitive to small variations in the input parameters. In other words, if there is an error

in prediction of displacement ductility in Equation 7, its effect on the predicted shear

strength will be relatively small. However, small changes in Vnmay result in large varia-tions in the corresponding displacement ductility.

COMPARISON OF MODEL AND TEST DATA

Thirty-seven column tests from eight different researchers were assembled from thePacific Earthquake Engineering Research Centers Structural Performance Database(Eberhard 2003). These were selected to cover a wide range of shear and flexuralstrengths, aspect ratios, and transverse reinforcement ratios. Table 1 lists key properties

for the test columns. (In Table 1,fyis the yield strength of the longitudinal reinforcementand l is the longitudinal reinforcement ratio.) Each column was modeled using thecomponent and overall models proposed in this research. Table 2 lists the calculatedshear strength, yield strength, and flexural strength for each column. Each column wasassigned a category from I to V based on the rules outlined previously. The columns are

sorted in order of increasing Vn / Vp ratio, and hence from Category I to Category V, in

Table 2.The first twelve columns in Table 1 were part of a test matrix and research program

at the University of California, Berkeley, to investigate the effect certain parameters on

the seismic response of older building columns. As shown in Table 2, these columns hadfailures marked by shear effects. Similarly, the next set of six specimens, or twelve col-umns on each side of six stubs, tested by Wight and Sozen (1975) was also selected

Figure 7. Shear failure surface model developed by Elwood (2004).



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Table 1. Properties of test columns

Column Ref.a

Typeb Lin. bin. din. a / d f

c

psi fy

ksi fyv

ksil%

v% s / din. Pkip P/Agfc

2CLD12 1 DC 116 18 15.4 3.76 3060 6 3 69 2 .88 0.175 0.78 150 0.151

2CHD12 1 DC 116 18 15.4 3.76 3060 63 69 2.88 0.175 0.78 600 0.605

2CVD12 1 DC 116 18 15.4 3.76 3030 6 3 6 9 2.88 0.175 0.78 c c

2CLD12M 1 DC 116 18 15.4 3.76 3160 63 69 2.88 0.175 0.78 150 0.147

2CLH18 2 DC 116 18 15.6 3.74 4800 4 8 58 2 .23 0.068 1.15 113 0.073

2SLH18 2 DC 116 18 15.6 3.74 4800 4 8 58 2 .23 0.068 1.15 113 0.073

3CLH18 2 DC 116 18 15.5 3.74 3710 4 8 58 3 .63 0.068 1.16 113 0.094

3SLH18 2 DC 116 18 15.5 3.74 3710 4 8 58 3 .63 0.068 1.16 113 0.094

2CMH18 2 DC 116 18 15.6 3.74 3730 4 8 5 8 2.23 0.068 1.15 340 0.281

3CMH18 2 DC 116 18 15.5 3.74 4010 4 8 5 8 3.63 0.068 1.16 340 0.262

3CMD12 2 DC 116 18 15.5 3.74 4010 4 8 5 8 3.63 0.175 0.77 340 0.262

3SMD12 2 DC 116 18 15.5 3.74 3730 48 58 3.63 0.175 0.77 340 0.281

25.033E 3 DE 34.5 6 10 3.45 4880 72 50 2.95 0.327 0.5 25 0.071

25.033W 3 DE 34.5 6 10 3.45 4880 72 50 2.95 0.327 0.5 25 0.071

40.033AE 3 DE 34.5 6 10 3.45 5030 72 50 2.95 0.327 0.5 42.5 0.117

40.033AW 3 DE 34.5 6 10 3.45 5030 72 50 2.95 0.327 0.5 42.5 0.117

40.033E 3 DE 34.5 6 10 3.45 4870 72 50 2.95 0.327 0.5 40 0.114

40.033W 3 DE 34.5 6 10 3.45 4870 72 50 2.95 0.327 0.5 40 0.114

40.048E 3 DE 34.5 6 10 3.45 3780 72 50 2.95 0.467 0.35 40 0.147

40.048W 3 DE 34.5 6 10 3.45 3780 72 5 0 2.95 0.467 0.35 40 0.147

40.067E 3 DE 34.5 6 10 3.45 4840 72 50 2.95 0.654 0.25 40 0.115

40.067W 3 DE 34.5 6 10 3.45 4840 72 5 0 2.95 0.654 0.25 40 0.115

40.092E 3 DE 34.5 6 10 3.45 5150 72 46 2.95 0.920 0.4 40 0.108

40.092W 3 DE 34.5 6 10 3.45 5150 72 46 2.95 0.920 0.4 40 0.108CUS 4 DC 36 9 14.4 1.23 5060 64 60 3.34 0.324 0.24 120 0.162

CUW 4 DC 36 16 7.4 2.40 5060 64 60 3.65 0.364 0.47 120 0.162

U1 5 C 39.4 13.8 12 3.28 6320 62 68 3.80 0.272 0.49 0 0

U3 5 C 39.4 13.8 12 3.28 5050 62 68 3.80 0.544 0.25 135 0.141

U4 5 C 39.4 13.8 12 3.28 4640 64 68 3.80 0.814 0.17 135 0.153

U6 5 C 39.4 13.8 12 3.28 5410 64 62 3.80 0.835 0.22 135 0.131

A1 6 C 91.9 15 22.3 4.12 3950 65 62 2.38 0.308 0.19 138 0.098

A2 6 C 91.9 15 22.3 4.12 3950 65 62 2.38 0.308 0.19 338 0.239

C1-1 7 C 55.1 15.7 13.8 4.00 3610 72 67 2.43 0.643 0.14 101 0.113

C1-2 7 C 55.1 15.7 13.8 4.00 3870 72 67 2.43 0.643 0.14 152 0.158

C1-3 7 C 55.1 15.7 13.8 4.00 3780 72 67 2.43 0.643 0.14 202 0.216

SC3 8 C 48 36 15.6 3.07 3180 63 58 2.23 0.096 1.03 0 0

SC9 8 C 48 18 33.6 1.43 2320 63 58 2.08 0.077 0.48 0 0a

1: Sezen (2002); 2: Lynn (2001); 3: Wight and Sozen (1975); 4: Umehara and Jirsa (1984); 5: Saatcioglu and

Ozcebe (1989); 6: Wehbe et al. (1999); 7: Mo and Wang (2000); 8: Aboutaha et al. (1999)b

C=cantilever, DC=double curvature, DE=double-endedc Axial load varied from 60 kip to 600 kip



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Table 2. Shear and flexural strengths of test columns

Column Vna,b

Vyb

Vpb

Vn / Vp Category SPD

c

Vmodelb

Vtestb

Vmodel/ Vtest

SC3 88.6 108.7 168.2 0.53 I S 103.9 101.2 1.03

SC9 132.3 143.0 236.2 0.56 I S 109.8 144.5 0.76

3CLH18 46.4 54.7 68.6 0.68 I S 46.8 62.3 0.75

3SLH18 46.4 54.7 68.6 0.68 I S 46.8 60.7 0.77

3CMH18 62.1 73.9 80.4 0.77 I S 57.7 73.7 0.78

CUS 91.7 93.1 115.1 0.80 I S 62.7 73.3 0.86

CUW 59.4 65.6 72.2 0.82 I S 65.8 60.1 1.09

U1 52.7 42.7 65.2 0.81 II F 55.2 62.1 0.89

25.033E 19.3 18.2 20.9 0.92 II S 20.0 19.1 1.05

25.033W 19.3 18.2 20.9 0.92 II S 20.0 20.3 0.99

2CMH18 61.4 59.2 63.6 0.97 III F-S 59.2 68.8 0.86

2CVD12(T)d

56.5 39.3 58.2 0.97 III F-S 53.5 55.4 0.97

3CMD12 79.2 73.9 81.4 0.97 III S 81.3 80.0 1.02

40.033E 20.5 19.8 21.0 0.98 III F-S 21.0 20.4 1.03

40.033W 20.5 19.8 21.0 0.98 III F-S 21.0 22.8 0.92

40.033AE 20.8 20.1 21.3 0.98 III F-S 21.3 22.2 0.96

40.033AW 20.8 20.1 21.3 0.98 III F-S 21.3 22.6 0.94

2CLH18 50.5 40.9 51.7 0.98 III F-S 46.5 54.1 0.86

2SLH18 50.5 40.9 51.7 0.98 III F-S 46.5 52.4 0.89

2CLD12 69.0 58.3 70.4 0.98 III F-S 70.4 70.8 0.99

2CLD12M 69.0 58.3 70.4 0.98 III F-S 70.4 66.2 1.06

3SMD12 78.1 73.1 79.4 0.98 III F-S 79.4 82.5 0.96

40.048E 23.9 19.3 22.1 1.08 IV F-S 20.8 22.6 0.92

40.048W 23.9 19.3 22.1 1.08 IV F-S 20.8 21.3 0.98

40.067E 30.3 19.8 24.3 1.25 IV F-S 21.5 20.7 1.0440.067W 30.3 19.8 24.3 1.25 IV F-S 21.5 20.6 1.04

2CVD12(C)d

92.3 66.7 71.7 1.29 IV F-S 71.7 67.6 1.06

2CHD12 92.3 66.7 71.7 1.29 IV F-S 71.7 80.7 0.89

U3 93.6 56.4 67.7 1.38 IV F 67.7 60.9 1.11

A1 97.0 49.1 67.8 1.43 V F 67.8 75.7 0.90

U6 118.3 57.4 81.3 1.46 V F 81.3 77.1 1.05

40.092E 36.3 19.9 24.3 1.49 V F-S 24.3 25.4 0.96

40.092W 36.3 19.9 24.3 1.49 V F-S 24.3 25.4 0.96

U4 123.2 56.4 80.5 1.53 V F 80.5 73.3 1.10

A2 128.2 62.8 74.7 1.72 V F 74.7 81.7 0.91

C1-3 124.5 51.1 59.5 2.09 V F 59.5 68.6 0.87

C1-2 121.8 47.3 57.6 2.11 V F 57.6 60.2 0.96

C1-1 117.8 43.0 55.2 2.13 V F 55.2 56.2 0.98

aShear strength calculated from Equation 7, assuming k= 1.0 mean: 0.95

b All strengths are given in kips standard deviation: 0.10c

Failure classification reported in Structural Performance Database (F=flexure, S= shear, and

F-S=flexure-shear)dT=axial tension portion of cycle, C=axial compression



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because the failure modes varied widely (Category II through V in Table 2). These speci-mens were very appropriate to study the effect of shear on column behavior. Four col-

umns tested by Saatcioglu and Ozcebe (1989) also covered a wide range of response(Categories II, IV, and V). In addition, these specimens were unique because both thelateral load-shear displacement and lateral load-slip displacement relations were re-

ported. For this study, such cyclic experimental data were valuable in understanding thedeformation components and their contribution to the total deformations. To the authorsknowledge, except for Reference 1, no such data are available for columns included inthe University of Washington database. The remaining nine columns had to be selectedto verify and populate certain failure categories. CUS and CUW (Umehara and Jirsa

1984) and SC3 and SC9 (Aboutaha et al. 1999) were tested by the same research groupat the University of Texas, Austin. These specimens had clear shear failure with verylimited or no flexural effects at failure. Thus, they were perfect candidates to include inCategory I. One of the factors affecting the selection of last five columns was their rela-

tively large aspect ratios (Wehbe et al. 1999; Mo and Wang 2000).Some trends can be observed over the range of data. All columns with aspect ratios,

a / d, less than 2.5 are in Category I, and all those with aspect ratios of 4 or greater arein Category V. All columns in the intermediate categories have aspect ratios between 2.5and 4, although there is not an increasing trend from Category II to III to IV. ColumnsU4, U6, and 40.092E/W are all in Category V despite having aspect ratios less than 4.

These columns have high transverse reinforcement ratios, which appears to preventshear failure and allow high levels of ductility to develop. Conversely, Columns SC3,

3CLH18, 3SLH18, and 3CMH18 have widely spaced transverse reinforcement s / d1.0. This causes them to fail in shear despite aspect ratios greater than 3, and pushesthem into Category I.

The failure mode given in the Structural Performance Database (Eberhard 2003) is

listed in Table 2 for each column. There is generally good agreement between these re-ported failure modes and the category assigned to the columns in the proposed model. It

should be noted that the placement of a column into Category V does not absolutelypreclude the possibility of shear failure under high ductility demand. The division be-

tween these categories ofVn1.4Vp was based on Equation 7 in this study. If the pro-posed model is used in an engineering design application, it may be advisable to in-crease this limit in order to achieve an acceptable factor of safety against shear failure.

Table 2 gives the maximum lateral strengths from the proposed model and the test

data. Vmodel is the peak lateral strength in the proposed model. It is the lesser of thestrength predicted by the flexure component model Vp and the strength predicted bythe shear component model (Vpeak, as shown in Figure 3). For shear-dominant columns

or columns failing in shear without significant flexural damage, Vpeak

should theoreti-

cally match Vn from Equation 7. Since Equation 7 and the shear component model usedifferent methods for predicting the shear strength, they do not agree exactly. The dif-

ference is generally small for the test columns; the mean ofVpeak/ Vnis 1.02 with a stan-dard deviation of 0.12. The ratio of predicted strength to observed strength is given for

each column, and the model predictions generally agree quite well with the test data.

The mean ofVmodel/ Vtestis 0.95, and the standard deviation is 0.10.



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Experimental and predicted displacement capacities are given in Table 3 for the test

columns. The displacement ductilities, test, given in Table 3 were taken from the litera-

ture when available, and were calculated graphically for the remaining columns accord-ing to the procedure suggested by Sezen and Moehle (2004). For the yield and ultimate

displacements and the associated displacement ductilities calculated using the graphicalprocedure, the reported values are the average of the two displacements or displacementductilities corresponding to positive and negative load cycles. The mean ratio of pre-

dicted displacement ductility to observed displacement ductility, model/testis 1.90, andthe standard deviation is 1.14. This indicates that the model does not predict the dis-

placements as well as strengths. A trend is noticeable in the ductility values, with higher values typically corresponding to higher categories. However, columns tested underhigh axial load (i.e., 2CHD12, 2CVD12, and 2CMH18) did not follow this trend. Thedecrease in displacement ductility caused by high axial load has been documented in theliterature (e.g., Elwood and Moehle 2005b; Patwardhan 2005).

Lateral force-displacement relationships are shown in Figures 8 and 9 for eight of the37 test columns modeled in this study. Comparisons for the other columns can be foundin Setzler (2005). The plots compare the response envelopes predicted by the proposedmodel to cyclic test data reported in the literature for each column. At least one columnfrom each of the five categories is included in the figures. The model predicts reasonable

response envelopes for the columns examined in the study. For columns in Category IV(Figure 9, plots (a) and (b)), the dashed lines show the proposed model before modifi-cation for delayed shear failure. The solid line is the final model prediction, after con-sideration of the Elwood shear failure surface (Equation 8). The shear failure surfacewas used successfully in predicting the lateral response of Category IV columns. From

moment-curvature analysis, a sudden drop in lateral resistance was calculated for theCategory IV column, 2CHD12 which was subjected to very high axial load. Using the

shear failure surface model, a smoother response beyond shear failure was obtained inFigure 9a. As discussed previously, Category V columns are those whose shear strengthsare high enough such that they are not expected to experience shear failure even at largedisplacements. However, the Elwood shear failure surface and point of axial load failure

were computed for these columns for comparison purposes. As shown in Figure 9, plots(c) and (d), it is not appropriate to modify the model using the Elwood shear failuresurface for Category V columns. The proposed model predicts the behavior of these col-umns well without any shear failure modifications.

SUMMARY AND CONCLUSIONS

The focus of this research was the creation of a model that can predict the monotoniclateral force-displacement relationship for reinforced concrete columns subjected to lat-

eral loading. The research concentrated on lightly reinforced columns that experienceflexure-shear failures. However, the model can be applied to columns with any ratio of

shear and flexural strengths. Therefore, it is applicable to columns that experience shear,flexure, or flexure-shear failures.

The overall lateral deformation of a reinforced concrete column was modeled asthree springs in series, one for each of the deformation components. Column shear



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Table 3. Experimental and predicted displacement capacities of test columns

Column Category

y,test

b

u,test

b

test

y,model

b

u,model

b

model model/test

SC3 I 0.27 0.96 3.50 0.42 1.64 3.90 1.11

SC9 I 0.30 0.38 1.27 0.14 1.20 8.46 6.66

3CLH18 I 0.75 1.20 1.58a

0.50 1.13 2.26 1.43

3SLH18 I 0.62 1.15 1.69a

0.50 1.13 2.26 1.34

3CMH18 I 0.89 1.20 2.14a

0.53 0.57 3.40 1.59

CUS I 0.20 0.38 1.95 0.12 0.54 4.50 2.31

CUW I 0.26 0.62 2.40 0.47 1.01 2.15 0.90

U1 II 0.67 2.09 3.12 0.25 1.37 5.43 1.74

25.033E II 0.44 1.23 2.77 0.29 1.59 5.48 1.98

25.033W II 0.42 1.18 2.82 0.29 1.59 5.48 1.94

2CMH18 III 0.65 1.20 1.94a

0.53 1.80 3.40 1.75

2CVD12(T)c

III 1.13 3.41 3.01a

0.90 6.34 7.04 2.34

3CMD12 III 0.77 1.80 2.50a 0.70 1.82 2.60 1.04

40.033E III 0.48 1.27 2.65 0.30 2.13 7.10 2.68

40.033W III 0.44 1.58 3.55 0.30 2.13 7.10 2.00

40.033AE III 0.25 1.20 5.04 0.29 2.08 7.17 1.42

40.033AW III 0.24 1.23 5.46 0.29 2.08 7.17 1.31

2CLH18 III 0.59 3.00 4.17a

0.48 2.28 4.75 1.14

2SLH18 III 0.51 2.40 2.65a

0.48 2.28 4.75 1.79

2CLD12 III 1.03 2.97 2.88a

0.93 3.34 3.59 1.25

2CLD12M III 1.06 3.33 3.14a

0.93 3.34 3.59 1.14

3SMD12 III 0.89 1.80 2.73a

0.72 1.92 2.67 0.98

40.048E IV 0.47 1.68 3.55 0.33 1.34 4.06 1.14

40.048W IV 0.45 1.90 4.44 0.33 1.34 4.06 0.91

40.067E IV 0.42 2.34 5.64 0.30 1.63 5.43 0.9640.067W IV 0.36 2.35 6.55 0.30 1.63 5.43 0.83

2CVD12(C)c

IV 0.82 2.23 2.72a

0.73 1.69 2.32 0.85

2CHD12 IV 0.79 1.02 1.29a

0.62 1.53 2.47 1.91

U3 IV 0.63 1.77 2.81 0.29 1.56 5.38 1.91

A1 V 0.92 4.76 5.3a

0.53 4.36 8.24 1.55

U6 V 0.49 3.47 7.37 0.28 4.01 14.08 1.91

40.092E V 0.38 2.05 5.36 0.28 6.18 22.07 4.12

40.092W V 0.39 2.05 5.59 0.28 6.18 22.07 3.95

U4 V 0.47 3.44 7.43 0.30 4.30 14.14 1.90

A2 V 0.75 3.93 5.2a

0.53 3.22 5.67 1.09

C1-3 V 0.70 3.67 5.26a

0.45 7.03 15.63 2.97

C1-2 V 0.69 3.72 5.38a

0.44 7.33 16.66 3.10

C1-1 V 0.63 3.47 5.50a

0.44 7.56 17.18 3.12

aDuctilities reported in literature mean: 1.90

b All displacements are given in inches standard deviation: 1.14c T=axial tension portion of cycle, C=axial compression



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strength was calculated using the shear strength equation proposed by Sezen and Moehle

(2004), taking the displacement ductility parameter k as 1.0. This shear strength wascompared to the yield and flexural strengths determined from the flexural analysis to

classify columns into one of five categories. Category I columns experience a pure shearfailure. Category II columns also fail in shear, but with flexural effects. Columns in Cat-egory III fail in shear and flexure at nearly the same time. Category IV columns initiallyfail in flexure or develop plastic hinges, but may experience shear failure as displace-ment increases. Columns that experience pure flexural failures are in Category V. For

each category, expected behavior and rules for the combination of the deformation com-ponents were presented.

Category IV specimens are those which are susceptible to shear failure after flexuralcapacity is reached. A shear capacity model (Elwood 2004; Elwood and Moehle 2005b)was used to predict the onset of delayed shear failure for these columns. If shear failure

was predicted, the model was modified and the strength is reduced linearly to the pointof axial load failure.

A database of 37 test columns was assembled, which covered a wide range of col-

Figure 8. Model predictions and test data for the lateral displacement of (a) SC3, (b) 25.033W,

(c) 3CMD12, and (d) 2CLD12.



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umn geometries, properties, and shear to flexural strength ratios. Each of these speci-mens was analyzed using the model proposed in this research. In general, the proposedmodel did an acceptable job of predicting the response envelope for the cyclic test data.

There were several instances where the model predictions were poor, but the behavior ofmost columns was represented well. The model also predicted the maximum strength ofthe columns well overall. The average of the ratio of predicted strength to experimentalstrength was 0.95, with a standard deviation of 0.10.

The classification system used in the combined lateral response model appeared to

represent the shear and flexural behaviors well. The shear capacity model proposed byElwood (2004) was used successfully to predict delayed shear failure in Category IVcolumns. The proposed model was able to predict the experimental behavior best in theflexure-shear failure range (Categories II through IV), the range for which it was in-tended.

Predicted and experimental lateral deformation plots were given for eight of the 37columns examined in this study. The experimental data was modeled reasonably well forthese columns.

Figure 9. Model predictions and test data for the lateral displacement of (a) 40.048W, (b)

2CHD12, (c) A1, and (d) U6.



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ACKNOWLEDGMENTS

This work was partially supported by a National Science Foundation Graduate Re-search Fellowship.

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(Received 15 July 2006; accepted 22 December 2007


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Model for the Lateral Behaviorof Reinforced Concrete ColumnsIncluding Shear Deformations

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