Analysis Bridge Column

314 ACI Structural Journal/May-June 2007

ACI Structural Journal, V. 104, No. 3, May-June 2007.MS No. S-2006-140 received March 30, 2006, and reviewed under Institute publication

policies. Copyright © 2007, American Concrete Institute. All rights reserved, includingthe making of copies unless permission is obtained from the copyright proprietors. Pertinentdiscussion including author’s closure, if any, will be published in the March-April2008 ACI Structural Journal if the discussion is received by November 1, 2007.

ACI STRUCTURAL JOURNAL TECHNICAL PAPER

A study was conducted at University of Nevada, Reno, Nev., on theseismic behavior of double interlocking spirals columns to determinethe effects of the shear level, spiral distance, and crossties. Theexperimental studies consisted of shaketable testing of six large-scalecolumn models until failure. The effect of different parameters wasevaluated based on damage progression and the measured data.The analytical studies included the evaluation of plastic hingelength, lateral force and displacement relationships, shear capacity,development of a nonlinear shear stiffness model, and designrecommendations for supplementary crossties. The test resultsindicated that an upper limit on the distance of spiral sets of 1.5times the spiral radius is appropriate, but supplementary crossties areneeded when the shear level is moderate or high. The study alsoshowed that the proposed analytical method to estimate the post-yielding shear stiffness provides a close estimate when it iscompared with the experimental data.

Keywords: column; shear; stiffness; test; ties.

INTRODUCTIONSeismic performance of reinforced concrete columns is

largely controlled by the level of confinement provided bytransverse reinforcement. In rectangular bridge columns,interlocking spirals are used frequently as transversereinforcement because spirals require less amounts of steelto provide the same level of confinement as that providedby rectilinear ties. Another advantage of spirals is that theyare often easier to construct than rectilinear ties. Theoreticalestimation of the column lateral force-displacement relationshipand shear capacity are frequently used to evaluate theseismic performance of columns. The California Departmentof Transportation’s (Caltrans) “Bridge Design Specifications(BDS)”1 and “Seismic Design Criteria (SDC)”2 are the onlycodes in the U.S. that include provisions for columns withinterlocking spirals. Due to the limited research in the past,Caltrans’ provisions are driven mainly by those of singlespirals and constructability considerations.

Previous studies3-5 on columns with interlocking spiralshave been performed on specimens subjected to pseudo-static loading. Tanaka and Park3 performed the first reportedexperimental and analytical studies on columns with doubleinterlocking spirals. Based on analytical studies, Tanaka andPark3 recommended that the horizontal distance betweencenters of the spirals di be limited to 1.2 times the radius ofthe spiral R. The authors stated that limiting the di and addingcrossties connecting the spirals will provide sufficient inter-locking and adequate shear transfer between the spirals. TheCaltrans upper limit on di is 1.5R. Shear-critical rectangularreinforcement concrete columns with interlocking spiralswere tested in a study by Benzoni et al.4 Most of the researchwas focused on analysis of the shear strength of the columnsfor the case of variable axial load. Different approaches to

determine the shear capacity for interlocking spirals werecompared with the experimental results, and a methodwas proposed.

The first dynamic tests of columns with interlockingspirals using shaketables were performed at the University ofNevada, Reno, Nev., to determine if the maximum limit of diestablished by the Caltrans provisions is satisfactory underdifferent levels of shear stress and if the supplementalcrossties connecting adjacent spiral sets are necessary.The experimental investigation and conclusions werereported elsewhere.6 This paper presents the results ofstudies on: 1) the validity of analytical models; 2) developmentof a new nonlinear shear stiffness model; and 3) development ofnew simple design recommendations for horizontal crosstiesconnecting interlocking hoops.

RESEARCH SIGNIFICANCEInterlocking spirals are often used in bridge columns with

rectangular or oval cross sections. The amount of data on theseismic performance of these columns from past research islimited. The analytical studies presented in this article usenew experimental data to address several important aspectsof the modeling and design of interlocking spiral columns.The correlation studies with experimental data demonstratethe applicability and limitations of analytical models. Theproposed plastic shear stiffness model allow for a more accuratemodeling of the lateral load response of columns. Designrecommendations for crossties proposed in the study providea simple tool to ensure sufficient interaction between spiral sets.

SUMMARY OF EXPERIMENTAL STUDIESDetails of experimental studies are presented elsewhere.6

A brief summary of the shaketable tests is presented hereinfor completeness.

The spread between the spiral sets, the average shearstress, and horizontal crossties connecting the spirals werethe test parameters included in the experimental study. Thelevel of shear stress was determined by a shear index. Theaverage shear stress was calculated as the maximummeasured shear force divided by 0.8 times the gross area.The shear index is found by dividing the average shear stressby 0.083 (MPa) (1 [psi]). A low index of 3 and ahigh index of 7 were selected to cover the range in actualbridge columns. Even in columns with a high shear index,flexure rather than shear dominates the behavior, although

fc′ fc′

Title no. 104-S31

Analytical Evaluation of Bridge Columns with Double Interlocking Spiralsby Juan F. Correal, M. Saiid Saiidi, David Sanders, and Saad El-Azazy

315ACI Structural Journal/May-June 2007

the extent of shear cracking might be relatively high. Two1/4-scale specimens with di of 1.0R and 1.5R (SpecimensISL1.0 and ISL1.5) subjected to low shear, and two 1/5-scalespecimens with di of 1.0R and 1.5R (Specimens ISH1.0 andISH1.5) subjected to high shear were initially tested at the JamesE. Rogers and Louis Wiener Large-Scale Structures Laboratoryat the University of Nevada, Reno, Nev. Based on the testresults of the first two high-shear columns, two additionalvariables, one with an intermediate level of di of 1.25R(Specimen ISH1.25) and the other with supplementarycrossties and di of 1.5R (Specimen ISH1.5T) were tested.The cross sections of all the specimens were oval.

All the columns were designed based on Caltrans’ “BridgeDesign Specifications (BDS)”1 and “Seismic DesignCriteria (SDC).”2 The longitudinal steel ratios were 2.0 and2.8% in the columns with low and high shear, respectively.These values were chosen because they are typical for lowand high shear columns. The transverse steel ratios weredesigned for a target displacement ductility of 5, whilemeeting the limitations of the Caltrans provisions. Figures 1and 2 show the overall dimension for the specimens with lowand high shear.

An axial load index (the ratio of column axial load and theproduct of the specified concrete compressive strength andgross cross-sectional area) of 10% was used to representaxial load level in new bridge columns. All the columns weretested under increasing amplitudes of the Sylmar recordfrom the 1994 Northridge earthquake in the strong direction ofthe columns until failure. The columns were instrumentedto measure strains, acceleration, axial force, lateral force,lateral displacement, and curvature. The low-shearcolumns were tested in single-curvature cantilever modeand the high-shear columns were tested in double curvature.

The overall seismic performance of all the columns wassatisfactory. Although the measured displacement ductilitycapacity did not reach the target in some cases, it met theminimum Caltrans requirement. Vertical cracks adjacent tothe interlocking region, however, were observed even undersmall earthquakes in the column with high shear and di of1.5R. The relatively large area of plain concrete outside theinterlocking zone increases was found to be vulnerable tocracking when the column shear was high. The test results

for a similar column that was reinforced with additionalhorizontal crossties connecting the interlocking spiralsshowed that the crossties not only reduced and delayedvertical cracks adjacent to the interlocking region but alsoreduced the strength degradation. Based on the experimentalresults, recommendations for the design of crossties weredeveloped and are presented in subsequent sections.

ANALYSIS OF SPECIMENSMoment curvature analysis

A cross section analysis of the specimens was required todetermine the theoretical plastic hinge length, the lateral loaddisplacement capacity, and the shear capacity. Program SPMC,7

specifically developed for columns with interlocking spirals,was used to perform moment curvature analyses of thespecimens. Table 1 lists the measured yield stress of thelongitudinal reinforcement and the measured concrete

Juan F. Correal is an Assistant Professor of civil engineering at the University ofAndes, Colombia. He received his BS and MSCE from the University of Los Andes,and his PhD from the University of Nevada, Reno, Nev., in 2004. He is a licensed CivilEngineer in the state of California. His research interests include the seismic design ofbridges and applications of innovative materials for design and repair of structures.

M. Saiid Saiidi, FACI, is a Professor of civil and environmental engineering and theDirector of the Office of Undergraduate Research at the University of Nevada. He isPast Chair and a member of ACI Committee 341, Earthquake-Resistant ConcreteBridges, and is a member of ACI Committee 342, Evaluation of Concrete Bridgesand Bridge Elements, and Joint ACI-ASCE Committee 352, Joints and Connections inMonolithic Concrete Structures. His research interests include analytical andexperimental studies of reinforced concrete bridges under seismic loading withconventional and innovative materials.

David Sanders, FACI, is a Professor of civil engineering at the University of Nevada.He is Past Chair of ACI Committee 341, Earthquake-Resistant Concrete Bridges; isChair of Joint ACI-ASCE Committee 445, Shear and Torsion; and is a member of ACICommittees 318, Structural Concrete Building Code, 369, Seismic Repair andRehabilitation, and 544, Fiber Reinforced Concrete. His research interests includeshaketable studies of reinforced concrete bridges.

Saad El-Azazy is the Seismic Research Program Manager at the California Department ofTransportation (Caltrans), Sacramento, Calif. He received his BS from Cairo University,Giza, Egypt, and his MS and PhD from Ohio State University, Columbus, Ohio. Hisresearch interests include bridge seismic retrofit and the performance of new bridges.

Fig. 1—Cross sections of test specimens.

Fig. 2—Evaluation of test specimens.


compression strength used in the M-φ analysis after adjustedfor the strain rate effect for each specimen. The average valueof the measured axial load was used in the M-φ analyses. TheHognestad model8 and the modified Mander et al.9 modelwere used in the SPMC7 to determine unconfined andconfined concrete stress-strain relationships, respectively.The latter was assumed to be applicable to double-spiralcolumns, even though it is commonly used for single-spiralcolumns. This is because most of the confined concrete indouble-spiral columns is confined by a single spiral, and theinterlocking region constitutes a relatively small area.

The stress-strain relationship for the steel was idealizedusing the parabolic strain hardening model recommended byPriestley et al.10 The values of the strain at concrete strength,crushing strain of unconfined concrete, and strain at thebeginning of strain hardening were taken according to therecommendations in SDC.7 Measured ultimate strains of0.16 and 0.08 were used for the longitudinal and spirals,respectively. The M-φ curves were idealized by elasto-plasticrelationships to quantify the ductility capacity of the columns.This was done by passing the elastic portion of the idealizedcurve to pass through the point that corresponds to the firstlongitudinal bar yielding. The idealized plastic momentcapacity was obtained by balancing the areas between theactual and the idealized M-φ curves beyond the first reinforcingbar yield point.

Plastic hinge lengthDifferent equations for estimating the plastic hinge length

lp have been proposed. The expression developed by Paulay

and Priestley,11 Baker and Amarkone,12 Dowell andHines,13 and the equation used by Caltrans2 were comparedwith the measured plastic hinge lengths. The cross-sectionalproperties obtained from the SPMC7 analysis were used. Inestimating lp based on Dowell and Hines’ method,13 theshear capacities were determined using three differentmethods: Caltrans,2 Tanaka and Park,3 and Benzoni et al.4 A45-degree shear crack and the measured material propertiesincluding amplification due to the strain rate effect wereused in all the equations.

The calculated and measured plastic hinge lengths, as afraction of column depth, are listed in Table 2. The measuredplastic hinge length was determined using the measuredplastic curvatures and plastic displacements of the columnspecimens. All the theoretical expressions underestimated lp.When calculating the ductility capacity, the lower lp lead tounderestimation of ductility capacity, which is conservative.Dowell and Hines’ equation13 showed the closest correlationwith the measured lp, with an average difference of 41 and31% for low and high-shear columns, respectively. In columnswith low shear, the equations by Baker and Amarkone12 andDowell and Hines13 show that an increase in spread of thespirals increases the plastic hinge length. The same patternwas observed in the experimental results. A direct relationshipbetween the aspect ratio and the plastic hinge length isrecognized by the Baker and Amarkone12 and Dowell andHines13 formulations. As a result, the increase of the calculatedplastic hinge length from these methods is attributed mainlyto the higher aspect ratio of 3.6 for the column with di = 1.5Rcompared with the lower aspect ratio of 3.3 for the columnwith di = 1.0R.

Longer plastic hinge lengths were measured in high-shearcolumns compared with the low-shear columns. This isbecause the amount of shear deformation in high-shearcolumns is relatively large, which leads to higher plasticdeformations. Dowell and Hines’ equation13 also showslarger values of lp in high-shear columns. This equationrecognizes that the plastic hinge length is affected by theincreasing of the longitudinal bar stress due to the diagonalshear cracking, known as the tension shift phenomenon.Even though the aspect ratio is less in high-shear columns, itappears that the tension shift has a more significant effect onthe plastic hinge length than the aspect ratio.

The experimental results in high-shear columns show thatthe increase in the distance between the spiral sets increasesthe plastic hinge length. It appears that the spread of thespirals increased the displacements due to shear, thusincreasing the apparent plastic hinge length. A larger post-yield shear deformation was calculated in columns with di =1.5R compared with a column with di = 1.0R by using theproposed nonlinear shear model described in subsequent

Table 1—Material and cross section propertiesof specimens

Properties Units

Specimens

Low shear High shear

ISL1.0 ISL1.5 ISH1.0 ISH1.5 ISH1.25 ISH1.5T

fc′kPa 42,044 41,836 33,643 33,913 50,448 50,417

psi 6098 6068 4879 4919 7317 7312

fyMPa 485 498 466 467 449 456

ksi 70 72 68 68 65 66

Mp

kN-m 241 339 160 200 221 238

kips-in. 2133 2999 1419 1771 1957 2107

φy

rad/m 0.0142 0.0118 0.0159 0.0144 0.0141 0.0134

rad/in. 0.00036 0.00030 0.00040 0.00036 0.00036 0.00034

φu

rad/m 0.1237 0.1094 0.0987 0.1070 0.1082 0.0990

rad/in. 0.00314 0.00278 0.00251 0.00272 0.00275 0.00251

Table 2—Calculated and measured plastic hinge length expressed as fraction of column depth

Specimen Paulay and Priestley11 Baker and Amarkone12 Caltrans2

Dowell and Hines13

Measured

Vs based on

Caltrans2 Tanaka and Park3 Benzoni et al.4

ISL1.0 0.49 0.43 0.49 0.59 0.50 0.50 0.75

ISL1.5 0.49 0.45 0.49 0.53 0.53 0.54 0.83

ISH1.0 0.42 0.33 0.52 0.76 0.81 0.87 0.98

ISH1.25 0.40 0.27 0.47 0.69 0.72 0.80 0.96

ISH1.5 0.39 0.32 0.45 0.69 0.70 0.80 1.12

ISH1.5T 0.39 0.28 0.44 0.72 0.72 0.80 1.27

ACI Structural Journal/May-June 2007 317

sections. None of the theoretical expressions of the plastichinge length takes into account this effect explicitly.

Lateral force-displacement relationshipsThe idealized analytical moment-curvature relationships

were used to calculate the lateral force-displacementrelationships. The lateral force was found by dividing theidealized plastic moment and the height of the columns. Forcolumns with low shear, the height was taken as thecantilever length shown in Fig. 2. High-shear columns bendin double curvature with rotationally fixed ends. Therefore, theplastic moment was divided by one half of the clear height ofthe column (Fig. 2) to obtain the plastic lateral force. In high-shear columns, the lateral displacement was determined atthe inflection point (midheight of the column) then doubledto calculate the total lateral displacement at the top of thecolumn. The total deformation of all the columns wascalculated as the summation of flexural, shear,8 and bondslip deformations.14 The Dowell and Hines’ lp equation13

using the Benzoni et al.4 shear equation was used in theanalysis to estimate plastic deformation.

The measured and calculated force-displacement curvesfor low-shear columns are shown in Fig. 3. The measuredbackbone curves represent the envelopes of the measuredhysteretic curves for the predominate direction of themotion. Each backbone curve was idealized by an elasto-plastic plot to quantify the ductility capacity of the columnsand to allow for comparison with analytical results. Theanalysis overestimated the lateral load by 0.4 and 10% forISL1.0 and ISL1.5, respectively. The 10% differencebetween the analytical and experimental curves in thecolumn with di = 1.5R can be attributed to the strengthdegradation observed in the backbone curve as the columnapproached failure. A difference of 13 and 1% was foundbetween the calculated and measured yield displacementsfor ISL1.0 and ISL1.5, respectively. In low-shear columns,the contribution of the calculated shear8 and bond slip14

deformation was 15 and 14% of the total yield deformation,respectively. Because ductility capacity is inverselyproportional to the yield displacement, the displacementductility capacity can be overestimated if the shear and bondslip deformations are excluded in the calculation of theyield displacement.

The calculated ultimate displacement was substantiallylower than the measured value in both columns. This isattributed to two factors: 1) failure in analytical studies isdefined as the point at which the extreme fiber of theconfined concrete fails in compression, whereas the actualperformance of the columns showed that the columns couldsustain additional deformations even after the edge of theconfined core was damaged; and 2) the ultimate confinedconcrete strain used in the analysis is a lower bound andconservative estimate of the crushing strain of concrete.The combined effect of these two factors is that the actualultimate strain in the extreme edge of the confinedconcrete is considerably larger than the value used in analysis.Consequently, the actual plastic displacement significantlyexceeds the estimated value. Whereas the underestimation ofultimate displacement is desirable in design, in detailedanalytical modeling, it leads to estimates with poor correlationwith the measured data. Because of the two aforementionedfactors, the analysis led to a substantially conservativeestimate of the displacement ductility capacity in the low-shear columns.

Figure 4 shows the measured and calculated force-displacement curves for high-shear columns. The analysisunderestimated the lateral load capacity by 6% for ISH1.0and overestimated the load by 7, 7, and 11% for ISH1.25,ISH1.5, and ISH1.5T, respectively. Differences between theeffective yield displacement in the experimental and analyticalresults were 1, 11, 6, and 13% for ISH1.0, ISH1.25, ISH1.5,and ISH1.5T, respectively, with shear and bond slipdeformation included in the analysis. The calculated sheardeformation8 was 48, 42, 39, and 42% of the total calculatedyield displacement for ISH1.0, ISH1.25, ISH1.5, andISH1.5T, respectively. Considering the small range of thesefigures, it appears that shear deformation was insensitive to di.

The contribution of the calculated bond slip was 15% ofthe total yield deformation in high-shear columns, whichwas approximately the same as that in low-shear columns.As stated in the discussion on low-shear columns, thesedeformations should be included in the calculation of theyield displacement to avoid unconservative estimates ofdisplacement ductility capacities. Similar to the trends inthe calculated ultimate displacement in low-shearcolumns, the calculated ultimate displacements were 28 to51% smaller than the measured values. The lower calculateddisplacements are attributed to the same factors described forlow-shear columns. The underestimation of the displacementductility capacity is generally conservative and desirable instructural design.

Fig. 3—Experimental and analytical force displacementcurves low-shear specimens.


Shear capacityBrittle shear failure in a concrete member is undesirable,

and it is necessary to use a reliable method to determine theshear capacity. Caltrans,2 Tanaka and Park,3 and Benzoni et al.4

shear equations were used to calculate the shear capacity ofthe test specimens and compare the results with the measuredmaximum column shear forces. No strength reduction factorswere used. The contribution of the crossties in ISH1.5T wasincluded in the calculation of the shear capacity. A 45-degreeshear crack angle was selected based on the inclination of theshear cracks observed in the test specimens.6 The measuredmaterial properties (Table 1) modified for the strain rateeffect were used.

A comparison of the experimental results and the shearcapacities calculated using the three methods is shown in

Fig. 5 for high-shear specimens. The bond slip and sheardeformations were included in the analysis. The low-shearspecimens were not included in the comparison because theywere flexure dominated with large margin against shearfailure. The intersection of the experimental and analyticalcurves marks the displacement ductility at which shearfailure is expected. In the specimen with di of 1.0R (ISH1.0),all the analytical methods underestimated the shear capacityof the column. The failure displacement according to theCaltrans,2 Tanaka and Park,3 and Benzoni et al.4 methodswould lead to an estimate of ductility capacity of 1.4, 1.0,and 1.2, respectively, whereas ISH1.0 failed in flexure/shearat displacement ductility of 4.7. For a column with di of1.25R (ISH1.25R), shear failure would be expected atductility of 3.0 based on the Benzoni et al.4 method. The

Fig. 5—Calculated shear capacity and test results.

Fig. 4—Experimental and analytical force displacement curves high-shear specimens.


Caltrans2 and Tanaka and Park3 equations would not showshear failure. However, the specimen failed in flexure/shear atductility of 5.1.

Columns with di of 1.5R, with (ISH1.5T) and without(ISH1.5) crossties, failed in flexural mode at a displacementductility of 3.8 and 4.0, respectively. The Benzoni et al.4

method estimated shear failure in ISH1.5 at a ductility of 2.7.The Caltrans2 and Tanaka and Park3 methods, however,indicated no shear failure, which was consistent with theobserved behavior. The Benzoni et al.4 method wasconservative in this case. None of the three methods indicatedshear failure in ISH1.5T, which is in agreement with theactual performance.

Proposed nonlinear shear stiffness modelExpressions to calculate the uncracked and cracked shear

stiffness for reinforced concrete members were developedby Park and Paulay8 and have been used because of theirsimplicity. Shear deformation was measured in high-shearcolumns by using a series of vertical, horizontal, anddiagonal displacement transducers in the plastic hingeregions described in detail in Reference 6. The measuredlateral force shear deformation relationship was nonlinear.Figure 6 shows a bilinear curve used to idealize the nonlinearbehavior of the lateral load and shear deformation for IH1.25.The trend in the measured curve in this plot represents theobserved trend in all the high-shear columns. In the availableliterature, there are no expressions for calculating the post-yield shear stiffness or the second slope shown in Fig. 6.Priestley et al.15 suggested that the shear stiffness drops inproportion to the ratio of post-yield and elastic flexuralstiffness. Using this approach for the test columns, however,led to post-yield shear stiffness values that were 40 to 73%lower than the measured results. Thus, a method to determinethe nonlinear shear stiffness was developed in the present study.

Diagonal cracks increase the shear deformation of reinforcedconcrete members. The first diagonal cracks were observedin the test specimens in the plastic hinge zone as an extensionof flexural cracks. The experimental data reported inReference 6 showed that 60 to 70% of the total sheardeformation occurs in the plastic hinge zones of thecolumn. As a result, it is reasonable to calculate the shearstiffness of the overall member taking into account the highshear deformation in the plastic hinge and relatively smalldeformation elsewhere. Accordingly, the following expressionfor the member shear stiffness was proposed (Fig. 7)

(1)

where Kv equals the shear stiffness of member; npr equals thenumber of potential plastic regions: 1 for bending in singlecurvature and 2 for bending in double curvature; Kd equalsthe stiffness in plastic zone, assumed to be equal to the effectivecolumn depth d; Kd–L equals the stiffness outside the plasticzone(s); d equals the effective column depth; and L equalsthe clear length of the column.

The initial slope in Fig. 6 represents the column stiffnessbefore the effective flexural yield point. The initial stiffnessis controlled by cracked and uncracked shear stiffnessdescribed in Park and Paulay8

Kv1

nprd

Kd

----------L nprd–

Kd L–

-------------------+

-------------------------------------=

(2)

where KvE equals the elastic shear stiffness (Fig. 6); Kv,45equals the cracked stiffness = [ρv /(1 + 4nρv)]Esbwd; Kv′ =uncracked stiffness = Ecbwd/3; ρv = Av /sbw; Av equals crossarea of the stirrups; s equals spiral pitch; Es equals elasticmodulus of steel; bw equals the width of the concretemember; d equals the effective column depth; n = Es/Ec; andEc equals the elastic modulus of concrete.

After yielding, the shear stiffness is a combination of theplastic shear stiffness and the stiffness of the remainder ofthe column, which is assumed to be cracked

(3)

where KvPY equals the post-yield shear stiffness, and Kpequals plastic shear stiffness.

The plastic shear stiffness Kp was based on truss analogy andwas calibrated using experimental results from Reference 6and two other studies.15,16 The Park and Paulay8 crackedshear stiffness was developed based on the shear distortion of areinforced concrete element using the truss analogy. The shear

KvE1

nprd

Kv 45,

------------L nprd–

Kv′-------------------+

--------------------------------------- 0.1Kv′≥=

KvPY1

nprd

Kp

----------L nprd–

Kv 45,

-------------------+

-------------------------------------=

Fig. 6—Lateral force versus shear deformation.

Fig. 7—Modified shear stiffness model.


distortion depends of the elongation of the transverse steel Δs andthe shortening of the concrete compression strut Δc.Considerable shear distortion occurs at the post-yield stage.The elastic moduli of steel and concrete are the only variablesthat control the shear distortion in a reinforced concreteelement. The effective moduli of the steel and concrete weredetermined from the experimental results for the inter-locking spiral columns.

The measured strains in the spirals in all the test specimensshowed that yielding of transverse steel is limited and extensiveyielding does not occur until column failure.6 Therefore, thespiral steel was assumed to be in the elastic range even afteryielding of the longitudinal bars, and the elastic modulus wasused for the spirals. The measured strains in the direction ofthe concrete strut ranged from 0.007 to 0.0146. These strainscorrespond to the nonlinear range of concrete stress strainrelationship. Therefore, the modulus of the concrete wasmodified to account for the large shear distortion at the post-yield stage. Past research has shown that the compressivestress-strain relationship of concrete is affected by the transversestrain.17 Figure 8 shows the stress-strain relationship foruniaxial loading idealized by Hognestad.8 The Hognestadstress-strain relationship was idealized by a bilinear relationship(Fig. 8). The slope of the elastic portion of the idealizedcurve was assumed to be the same as the concrete modulusof elasticity recommended by ACI 318.18 The stress at theultimate point on the idealized curve was assumed to be theaverage of the peak and the ultimate stresses in theHognestad model.8 The second slope was established byequalizing the area between the Hognestad model8 and theidealized curve. A linear relationship was found between thesecond slope of the idealized curve Ecp and the concretecompression strength fc′

Ecp = 12fc′ (4)

Experimental studies by Vecchio and Collins17 haveestablished that the principal compressive stress in theconcrete fc2 is not only a function of the principal compressivestrain ε2, but also of the principal tensile strain ε1. As a result,cracked concrete subjected to high tensile strains perpendicularto the direction of the compression is softer and weaker thanconcrete in a standard cylinder test (ε1 = 0). If the effect ofthe principal tensile strain ε1 is incorporated in theHognestad model,8 the second slope of the idealized model

Ecp (Fig. 8) is reduced considerably. A factor βp representingthe softening of the cracked concrete due to the transversetensile strains and shear cracking was introduced into Eq. (4)as follows

βpEcp = 12βp fc′ (5)

The reduced modulus of elasticity of the cracked concretefrom Eq. (5) was used to determine the shortening of thecompression concrete struts. Thus, the expression for shorteningof the diagonal strut Δcp at the post-yield stage becomes

(6)

The shortening of the diagonal strut Δcp was used to calculatethe shear distortion per unit length at the post-yield stage θvp

(7)

where Vs equals the shear force and np = Es /Ecp.The plastic shear stiffness of a reinforced concrete member

per unit length, based on the truss model with 45-degreediagonal cracks, is the value of the Vs when θvp = 1

(8)

Substituting Eq. (8) into Eq. (3), and replacing the post-yieldshear stiffness KvPY by the measured post-yield stiffness KvPYEand solving for βp, the following equation is derived

βp = 4nKvPYEnpd (9)

The measured post-yield stiffness from two columnstested at the University of Nevada,16 one column from thestudy by Priestley et al.15 and the four high-shear columnsfrom Reference 6, were used to calculate βp. An averagevalue of 0.288 and a standard deviation of 0.067 wereobtained. As a result, a βp factor of 0.3 was selected.

To verify the value of βp, the Hognestad model8 wasmodified assuming a transverse tensile strain of ε1 and usingthe adjustments proposed by Vecchio and Collins.17 Aniterative process was used to obtain the tensile strains ε1,which would reduce the idealized post-yield stiffness ofcracked concrete to 30% of the original value (reflecting a βpfactor of 0.3). A transverse tensile strain of 0.0149 wasfound. The diagonal displacement transducers of the panelinstrumentation for high-shear columns were used to determinethe magnitude of the actual tensile diagonal strains. Asexpected, the diagonal strain within the plastic hinge zoneincreased as the amplitude of the motion was increased. Themeasured peak strain for the post-yield range averaged forall the high-shear columns was 0.0112. This strain is 25%lower than the estimate from Reference 17. However, thestrain appears reasonable considering that the expression

Δcp2 2Vs

βpEcpbw

--------------------=

θvpVs

Esbwd--------------- 1

ρv

-----4np

βp

--------+⎝ ⎠⎛ ⎞=

Kpβpρv

βp 4npρv+--------------------------Esbwd=

ρv

KvPYEL 4KvPYELnρv 4+ nKvPYEnpdρv ρv+ Esbwd––( )------------------------------------------------------------------------------------------------------------------------------------

Fig. 8—Hognestad model and idealized curve with tensilestrain ε1 of 0 and 0.015.


developed in Reference17 was for cases with diagonal tensilecracks in only one direction, whereas the test specimensexperienced diagonal cracks in two directions due tocyclic loading.

Equation (5) with βp = 0.3 was substituted in Eq. (8), andthe plastic shear stiffness of the member per unit of length,based on a truss model with 45-degree diagonal cracks wasfound as

Esbwd (10)

The measured shear stiffness was compared with theproposed and existing shear stiffness models. Table 3 liststhe elastic measured shear stiffness with the cracked shearstiffness and the proposed elastic shear stiffness (Eq. (2)).When the columns were treated as being fully cracked, theelastic shear stiffness was underestimated by as much as37%. When the higher stiffness of the uncracked portion ofthe columns was taken into account, the maximum differencewas reduced to 24%.

Table 4 compares the measured post-yield shear stiffnesswith the post-yield shear stiffness determine by Eq. (3) andthe post-yield shear stiffness suggested by Priestley et al.15

The post-yield stiffness proposed by Priestley et al.15 underesti-mated the measured stiffness by 73, 46, 36, and 68% forISH1.0, ISH1.25, ISH1.5, and ISH1.5T, respectively, withan average difference of 56% and a standard deviation of18%. Significant improvement was achieved using theproposed post-yield shear stiffness model. The differencesbetween the calculated and experimental results were 13, 2,19, and 15% for ISH1.0, ISH1.25, ISH1.5, and ISH1.5T,respectively, with an average difference of 5% and a standarddeviation of 15%.

The proposed post-yield shear stiffness was used todetermine the effect of the ultimate shear deformation on thecalculated displacement ductility capacity of the high-shearcolumns. Approximately 58% more shear deformation wascalculated for columns with di of 1.5R compared with thecolumn with di of 1.0R. On average, an increase in thecalculated displacement ductility capacity of 20% wasobtained when the ultimate shear deformation was includedin the ultimate deformation for high-shear columns.

Design procedure for crosstiesCurrently, there is no procedure available to design

crossties connecting interlocking spirals. The study inReference 6 reported vertical cracks located in the inter-locking region in the specimen with di of 1.5R even under arelatively small drift and a lateral load of 58% of the yieldforce. Long interlocking spiral distance can make thecolumn vulnerable to large vertical shear stress at middepth ofthe column (Fig. 9). This vertical stress is proportional to thecolumn shear, at least during the linear response. Becausethe area of plain concrete adjacent to the interlocking regionbecomes large when di = 1.5R, the concrete is susceptible tocracking. The measured response of a similar column withsupplementary horizontal crossties showed that the crosstiesnot only delayed vertical cracks adjacent to the interlockingregion but also helped reduce strength degradation.

Three methods were studied to provide background for thedesign of horizontal crossties. The first was based on theshear reinforcement capacity defined in SDC,1 taking into

Kp9ρv fc′

9 fc′ 10Esρv+--------------------------------=

account the component of the spiral tension force at themiddepth of the column section in the direction of the shearforce. The second method was based on the equilibrium ofthe horizontal spiral force at the middepth of the section. Inthese two methods, the spiral steel shear capacity atmiddepth of the column with di of 1.0R was taken as thetarget, and the crossties were designed to make up for thereduction in the horizontal shear capacity as di exceeded1.0R. The target shear capacity was selected based on thecolumn with di = 1.0R because of the satisfactory seismicperformance of this column. Shear friction was the thirdmethod used to design crossties to resist the vertical shear atmiddepth of the section. The ratio of the area of ties and theirspacing At/st is shown in Eq. (11), (12), and (13) for shear

Fig. 9—Horizontal component of spiral force at middepthand vertical stress due to separate two-column action.

Table 3—Comparison of measured and calculated elastic shear stiffness

Units ISH1.0 ISH1.25 ISH1.5 ISH1.5T

KvMkn/m 37,398 66,653 56,325 49,162

kips-in. 214 381 322 281

Kv,45

kn/m 28,598 42,148 40,594 41,454

kips-in. 163 241 232 237

Diff., % –24 –37 –28 –16

KvE

kn/m 46,467 57,583 46,887 57,165

kips-in. 265 329 268 326

Diff., % 24 –14 –17 16

Note: KvM = measured elastic shear stiffness; Kv,45 = calculated fully cracked shearstiffness; KvE = calculated partially cracked shear stiffness; and Diff. = differencebetween experimental and analytical results.

Table 4—Comparison of measured and calculated post-yield shear stiffness

Units ISH1.0 ISH1.25 ISH1.5 ISH1.5T

KpsMkn/m 3146 4225 3590 3771

kips-in. 18.0 24.1 20.5 21.5

Kps

kn/m 847 2274 2315 1209

kips-in. 4.8 13.0 13.2 6.9

Diff., % –7.3 –46 –36 –68

Kvpy

kn/m 2745 4121 2905 4336

kips-in. 15.7 23.5 16.6 24.8

Diff., % –13 –2 –19 15

Note: KpsM = measured post-yield shear stiffness; Kps = post-yield shear stiffness by

Priestley et al.15; Kvpy = post-yield shear stiffness proposed; and Diff. = differencebetween experimental and analytical results.


capacity, equilibrium of spiral forces at middepth, and shear-friction methods, respectively. Details of the derivation ofthese equations are presented in Reference 6.

(11)

(12)

(13)

where n equals the number of interlocking spirals or hoopcore sections; α equals the index of the horizontal distancebetween spiral centers in terms of spiral radius; α = 1.0 fordi = 1.0R and α = 1.5 for di = 1.5R; Ab equals the area of onespiral steel bar; s equals spiral pitch; θ = sin–1(α/2); V equalsplastic shear demand; t equals the width of the membercross-sectional area; μ equals the coefficient of frictionbetween materials along potential crack (μ = 1.4 for concretecast monolithically)18; fy equals the nominal yield stress ofsteel reinforcement; and Ag equals the gross area of section.

Figure 10 shows the required spacing of crossties using theshear capacity with n = 2, the equilibrium of spiral forces atmiddepthmiddepth, and the shear friction methods. Thecrossties were assumed to be of the same bar size as that ofthe spirals. The required cross tie spacing was nearly the samefor the three methods when α exceeded 1.35. The requiredcrosstie spacing is twice the spiral pitch when α = 1.5,which corresponds to the highest amount of crossties for all

At

st

----- 12---nπ 3 4 α2––

α--------------------------------

⎝ ⎠⎜ ⎟⎛ ⎞ Ab

s-----=

At

st

-----4Ab

s--------- cos 30( ) cos– θ( )( )=

At

st

-----3Vt

2μfyAg

-----------------2Ab 4 α2–

s-----------------------------–

⎝ ⎠⎜ ⎟⎛ ⎞

=

three methods. Larger cross tie spacing was obtained inMethod 2 for α of less than 1.3. Crosstie spacing of fourtimes the spiral pitch is needed for columns with α of 1.25,according to Methods 1 and 3.

Because vertical cracks were caused by vertical stressesalong the interlocking region, it is reasonable to include theshear force in the expression to design the horizontalcrossties. Even though the shear friction method accounts forthe shear force, a negative ratio of At/st (Eq. (13)) can beobtained for columns subjected to a shear index lower than 7. Itshould be noted that no column with a moderate shear indexof 5 and di of 1.5 was tested to determine if crossties can beeliminated for this level of shear. Taking into account theresults of the aforementioned discussions, the experimentalresults, and the relatively low cost of crossties, thefollowing conservative design recommendation are proposed:• The crosstie bars should be of the same size as the spiral

reinforcement and their spacing should be no more thantwice the spirals pitch. Crossties should be detailedwith a 135-degree hook in one end and 90-degree hookat the other; and

• The shear index should be used as a control designparameter to determine the need of crossties in columnsreinforced with interlocking spirals according to Table 5.

CONCLUSIONS1. The proposed method to determine the nonlinear shear

stiffness led to close agreement with the experimentalresults. With nonlinear shear deformations included usingthis method, the estimated displacement ductility capacitymay increase by approximately 20% in columns under relativelyhigh shear;

2. The proposed method to design crossties in interlockingspiral columns is a simple, practical, and conservativemethod that can be easily adopted in practice. Crossties thatmet the proposed design requirement improved the performanceof a column model tested under simulated earthquakes;

3. All the theoretical estimates of the plastic hinge lengthlp were lower than the measured lengths. The method byDowell and Hines used in combination with the Benzoni et al.’sshear capacity led to the closest estimate. The measuredplastic hinge lengths were, on average, 39% larger than thoseestimated using this method. Proper consideration of theeffects of the aspect ratio and the transverse steel ratio in theDowel and Hines method appears to have contributed to theclose correlation;

4. The displacement ductility capacity is significantlyoverestimated if the effects of the shear and bond slipdeformations are not included in the calculation of the yielddisplacement; and

5. Among the methods that were used in this study, theBenzoni et al. equation to estimate the shear capacity led toresults that were closest to the experimental data.

ACKNOWLEDGMENTSThe research presented in this paper was sponsored by the California

Department of Transportation. N. Wehbe of South Dakota State University isthanked for developing the computer program SPMC and assisting in itsapplication. The assistance of P. Laplace, J. Pedroarena, and P. Lucas ofthe University of Nevada, Bridge Structures Laboratory, is gratefullyacknowledged.

REFERENCES1. California Department of Transportation, “Bridge Design Specifications,”

Engineering Service Center, Earthquake Engineering Branch, Calif.,July 2000, pp. 8-1 to 8-54.

Fig. 10—Comparison of three methods to design horizontalcrossties.

Table 5—Design recommendations for crossties

Shear index di Crossties

<3 1.0R to 1.5R Not required

3 to 71.0R to 1.25R Not required

>1.25R Required

>7 1.0R to 1.5R Required

Note: di equals the horizontal distance between centers of spirals as function of radiusof spirals R.

323ACI Structural Journal/May-June 2007

2. California Department of Transportation, “Seismic Design CriteriaVersion 1.2,” Engineering Service Center, Earthquake Engineering Branch,Calif., Dec. 2001, 100 pp.

3. Tanaka, H., and Park, R., “Seismic Design and Behavior of ReinforcedConcrete Columns with Interlocking Spirals,” ACI Structural Journal, V. 90,No. 2, Mar.-Apr. 1993, pp. 192-203.

4. Benzoni, G.; Priestley, M. J. N.; and Seible, F., “Seismic ShearStrength of Columns with Interlocking Spiral Reinforcement,” 12th WorldConference on Earthquake Engineering, Auckland, New Zealand,2000, 8 pp.

5. Buckingham, G. C., “Seismic Performance of Bridge Columns withInterlocking Spirals Reinforcement,” MS thesis, Washington State University,Pullman, Wash., 1992, 146 pp.

6. Correal, J.; Saiidi, M.; and Sanders, D., “Seismic Performance of RCBridge Columns Reinforced with Two Interlocking Spirals,” ReportNo. CCEER-04-6, Center for Civil Engineering Earthquake Research,Department of Civil Engineering, University of Nevada, Reno, Nev.,Aug. 2004, 530 pp.

7. Wehbe, N., and Saiidi, M., “Moment-Curvature Analysis forInterlocking Spirals SPMC V 1.0,” Report No. CCEER-03-1, University ofNevada, Reno, Nev., May 2003, 114 pp.

8. Park, R., and Paulay, T., Reinforced Concrete Structures, John Wiley& Sons, 1975, 769 pp.

9. Mander, J.; Priestley, M. J. N.; and Park, R., “Theoretical Stress-StrainModel for Confined Concrete Columns,” Journal of Structural Engineering,ASCE, V. 114, No. 8, Aug. 1988, pp. 1804-1846.

10. Priestley, N.; Seible, F.; and Calvi, G., Seismic Design and Retrofit ofBridges, John Wiley & Sons, New York, 1996, 686 pp.

11. Paulay, T., and Priestley, M. J. N., Seismic Design of ReinforcedConcrete and Masonry Buildings, John Wiley & Sons, 1992, 744 pp.

12. Baker, A. L. L., and Amarkone, A. M. N., “Inelastic HyperstaticFrames Analysis,” Proceedings of the International Symposium on theFlexural Mechanic of Reinforced Concrete, ACI-ASCE, Miami, Fla.,Nov. 10-12, ASCE, New York, 1964, pp. 85-142.

13. Dowell, R., and Hines, E., “Plastic Hinge Length of ReinforcedConcrete Bridge Columns,” Third National Seismic Conference andWorkshop On Bridges and Highways, Apr.- May 2002, Portland, Oreg.,pp. 323-334.

14. Wehbe, N.; Saiidi, S.; and Sanders, D., “Effects of Confinement andFlare on the Seismic Performance of Reinforced Concrete Bridges Columns,”Report No. CCEER-97-2, University of Nevada, Reno, Nev., 1997, 407 pp.

15. Priestley, M. J. N.; Seible, F.; and Benzoni, G., “Seismic Performance ofCircular Columns with Low Longitudinal Steel Ratios,” Report No. SSRP-97/15, University of California-San Diego, La Jolla, Calif., 75 pp.

16. Cheng, Z.; Saiidi, S.; and Sanders D., “Development of a SeismicDesign Method for Reinforced Concrete Two-Way Bridge ColumnHinges,” Report No. CCEER-06-01, Center for Civil Engineering Earth-quake Research, Department of Civil Engineering, University ofNevada, Reno, Nev., Feb. 2006, 497 pp.

17. Vecchio, F. J., and Collins, M. P., “The Modified Compression-FieldTheory for Reinforced Concrete Elements Subjected Shear,” ACI JOURNAL,Proceedings V. 83, No. 2, Mar.-Apr. 1986, pp. 219-231.

18. ACI Committee 318, “Building Code Requirements for StructuralConcrete (ACI 318-02) and Commentary (318R-02),” American ConcreteInstitute, Farmington Hills, Mich., 2002, 443 pp.

Analysis Bridge Column

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