Plastic Hinge Length of Reinforced Concrete Columns

290 ACI Structural Journal/May-June 2008

ACI Structural Journal, V. 105, No. 3, May-June 2008.MS No. S-2006-224.R3 received April 19, 2006, and reviewed under Institute

publication policies. Copyright © 2008, American Concrete Institute. All rights reserved,including the making of copies unless permission is obtained from the copyright proprietors.Pertinent discussion including author’s closure, if any, will be published in the March-April 2009 ACI Structural Journal if the discussion is received by November 1, 2008.

ACI STRUCTURAL JOURNAL TECHNICAL PAPER

As a column experiences earthquake-induced lateral displacementswhile supporting gravity loads, severe damage is observed inregions subjected to large moments. These regions are commonlyreferred to as plastic hinges and they experience large inelasticcurvatures. The inelastic curvatures in plastic hinges are typicallyassumed to be constant over the plastic hinge length, lp , to simplifythe estimation of the tip displacement of a column. Therefore, if theplastic hinge length is known, the tip displacement of a column caneasily be obtained by integrating curvatures, and vice versa. Aspart of the research reported in this paper, the effects of axial loadand shear span-depth ratio (L/ h) on lp are evaluated experimen-tally. Based on the experimental observations, a new analyticalapproach that can be used to estimate lp is presented. Finally, theresearch findings are synthesized into a simple expression that canbe used to estimate lp.

Keywords: axial load; deformation capacity; plastic hinge length.

INTRODUCTIONNumerous researchers1-10 have suggested various

expressions that can be used to estimate the plastic hinge lengthlp of concrete members. Large differences exist among differentexpressions, however, and the performance of different expres-sions in estimating lp has not been evaluated systematically.

This paper presents test results from four full-scaleconcrete columns. In addition, the effect of axial load andshear span-depth ratio (L/h, where L is the height of a canti-lever column and h is the overall depth of a column) on thelp is investigated. A concrete compression strain method toestimate lp is proposed. Finally, a simple lp expression isproposed and the influence of the estimated lp on the accuracyof lateral load-drift predictions is studied.

RESEARCH SIGNIFICANCEResearch conducted over the past three decades has

resulted in various constitutive relationships for concrete andreinforcing bars. As a result, the sectional behavior ofreinforced concrete columns can be reliably estimated. For agiven sectional performance, the member behavior of acolumn can be estimated if lp is known. As such, estimatingthe length of a plastic hinge establishes a key step inpredicting the lateral load-drift response of a column. In thiscontext, the work presented herein advances the state ofknowledge: 1) by providing experimental evidence on theeffects of axial load and L/h on lp; 2) by introducing a newconcrete compression strain-based method to estimate lp;and 3) by synthesizing the research findings into a simpleexpression that can be use to estimate lp.

DEFINITION OF PLASTIC HINGE LENGTHPlastic hinges form at the maximum moment regions of

reinforced concrete columns. If lp is known, the tip displacementof a column can be easily obtained by integrating curvatures.Therefore, accurate assessment of lp is important in relatingsection-level response to member-level response of a

concrete column. The length of a plastic hinge depends on manyfactors. The following is a list of important factors that influencethe length of a plastic hinge: 1) level of axial load; 2) momentgradient; 3) level of shear stress in the plastic hinge region; 4)mechanical properties of longitudinal and transverse reinforce-ment; 5) concrete strength; and 6) level of confinement and itseffectiveness in the potential hinge region.

In the 1950s and 1960s, researchers1-5 studied lp to estimatethe flexural deformation capacity of reinforced concretebeams. To estimate the flexural deformation capacity, theplastic rotation capacity and the lp are used

(1)

Park and Paulay11 extended this concept to a cantilevercolumn. They simplified the curvature distribution along thelength of a column using a plastic hinge (Fig. 1). Using thesecond moment area theorem, they calculated the tipdisplacement of a column (Eq. (2))

(2)

By further simplifying Eq. (2), Park and Paulay11 obtainedthe relationship between curvature and displacement ductilities(Eq. (3))

(3)

Equations (2) and (3) have been commonly used to estimatethe lp of concrete columns.6-9,12,13 It is interesting to notethat the curvature profile along the column is often related tothe tip displacement rather than flexural displacement.6-9,12,13

θpεcu εce–

c-------------------- lp×=

Δtip Δy Δp+φyL2

3----------- φ φy–( )lp L 0.5lp–( )+= =

μΔ 1 3 μφ 1–( )lp

L--- 1 0.5

lp

L---–⎝ ⎠

⎛ ⎞+=

Title no. 105-S28

Plastic Hinge Length of Reinforced Concrete Columnsby Sungjin Bae and Oguzhan Bayrak

Fig. 1—Definition of plastic hinge length.11

291ACI Structural Journal/May-June 2008

In such a case, the lp in Eq. (2) and (3) can include the effectsof bar slip and shear displacements.

TECHNICAL LITERATURE: PLASTIC HINGE LENGTHBaker1

To investigate the moment-curvature relationship ofbeams and columns, 94 beam/column tests were conductedby six laboratories under the auspices of CEB in the 1960s.The main test variables included concrete strength, yieldstrength or amount of tension reinforcement, amount ofcompression reinforcement, single or double concentratedloads, and axial load. Based on the test results, Baker1

proposed the following equation to calculate the lp

(4)

where z is the distance between the critical section and thepoint of contraflexure and d is effective depth of a beam.

Baker1 reported that the lp ranged from 0.4d to 2.4d forpractical values of z/d. The z/d ratio is a similar parameterto L/h and was used to represent the effect of moment gradient.Baker and Amarakone2 simplified Eq. (4) to obtain thefollowing

(5)

where c is the neutral axis depth at collapse.

Mattock3

Mattock3 conducted 37 beam tests and investigated theeffect of various parameters on the behavior of reinforcedconcrete beams. The parameters studied by Mattock3 wereconcrete strength ( fc′ = 4000 to 6000 psi [28 to 41 MPa]),effective depth of beam (d = 10 to 20 in. [254 to 508 mm]),moment gradient (z/d = 2.75 to 11), amount (ρl = 1 to 3%),and yield strength ( fy = 47 to 60 ksi [324 to 414 MPa]) oftension reinforcement. Based on the test results, Mattock3

concluded that the spread of plasticity along a beam lengthincreased as z/d increased, and as the net tension reinforcement(q – q′)/qb decreased. Mattock3 proposed the followingempirical relationship to calculate lp.

(6)

where both d and z are in inches.

lp k1k2k3zd---⎝ ⎠⎛ ⎞ 0.25

d=

lp 0.8k1k3zd---⎝ ⎠⎛ ⎞ c=

lpd2--- 1 1.14 z

d--- 1–⎝ ⎠

⎛ ⎞ 1 q q′–qb

-------------⎝ ⎠⎛ ⎞ d

16.2----------–

⎩ ⎭⎨ ⎬⎧ ⎫

+=

Corley5

To expand Mattock’s3 work, Corley5 tested 40 simplysupported concrete beams subjected to single point loads.The confinement and size effects were the primary variablesinvestigated. In addition, the effects of moment gradient andamount of tension reinforcement were studied. Corley5

reported that the length of the plastic hinge region wasprimarily a function of the geometry of a concrete beam andthat the size of a beam did not have a significant influence onthe rotational capacity. Based on the scatter in the measuredvalues of the plastic hinges, he concluded that the effectof (q – q′)/qb in Eq. (6) could be ignored. Corley5 suggestedthe use of a simple expression for calculating lp

(7)

where both d and z are in inches.

Mattock4

Mattock4 simplified Eq. (6) in 1967. He stated that eventhough there was considerable scatter, the trend in theobserved lp could be represented reasonably well by thefollowing simple expression

(8)

Park et al.6Park et al.6 tested four full-scale concrete columns with

square sections of 22 x 22 in.2 (550 x 550 mm2) and an L/hof 2. The axial loads applied to the column specimens werein the range of 0.2fc′Ag to 0.6fc′Ag. Park et al.6 estimated thelp of the test specimens using Eq. (2). They concluded thatthe experimentally obtained lp were comparatively insensitive tothe axial load level and had an average value of 0.42h, whereh is the overall depth of the column. They suggested using asimple lp of 0.4h for concrete columns.

Using a similar approach, Priestley and Park7 proposed thefollowing equation for the calculation of lp in reinforcedconcrete columns

lp = 0.08L + 6db (9)

The suggested lp (Eq. (9)) has two components. Priestleyand Park7 stated that the first term mainly accounted forcolumn bending, while the second accounted for bar slip dueto the elongation of longitudinal bars beyond the theoreticalbase (tensile strain penetration into the joint or foundation).

Paulay and Priestley8 revised Eq. (9) to account fordifferent grades of flexural reinforcement. The revisedexpression is given in Eq. (10)

(10)

Paulay and Priestley8 reported that Eq. (10) resulted inlp ≈ 0.5h for typical columns.

Sakai and Sheikh14

Based on an extensive review of the literature, Sakai andSheikh14 reported that the lp increased as the aspect ratio (L/h or

lpd2--- 0.2 z

d-------+=

lpd2--- 0.05z+=

lp 0.08L 0.15db fy (fy in ksi)+=

lp 0.08L 0.022db fy (fy in MPa)+=

ACI member Sungjin Bae is a Structural Engineer at Bechtel Corporation, Frederick,MD. He received his BS and MS from Hanyang University, Seoul, Korea, and his PhDfrom the University of Texas at Austin, Austin, TX. He is a Member of ACI Committee 209,Creep and Shrinkage of Concrete, and Joint ACI-ASCE Committee 441, ReinforcedConcrete Columns. His research interests include the behavior of concrete columnssubjected to combined axial load and flexure, testing of large-scale reinforcedconcrete columns, and performance-based design of concrete columns.

ACI member Oguzhan Bayrak is an Associate Professor of civil, architectural, andenvironmental engineering at the University of Texas at Austin, Austin, TX, and aFellow of the Clyde E. Lee Endowed Professorship. He is Chair of Joint ACI-ASCECommittee 441, Reinforced Concrete Columns; a member of ACI Committees 341,Earthquake-Resistant Concrete Bridges; E803, Faculty Network CoordinatingCommittee; and Joint ACI-ASCE Committee 445, Shear and Torsion.


L/D), which is nearly equivalent to the L/h, increased.Bilinear curves illustrating the relationship between the lpand the aspect ratio were reported as trend lines by theauthors. They concluded that the lp was affected by theamount of transverse reinforcement, axial load level, andaspect ratio.

Sheikh and Khoury9

Sheikh and Khoury,9 Sheikh et al.,12 and Bayrak andSheikh13 reported that the measured lp were approximatelyequal to 1.0h in their column tests. It is important to notethat most of their column tests were conducted under highaxial loads.

Mendis10

Mendis10 conducted tests on 13 simply supported concretebeams subjected to single point loads and examined their lp.He reported that the lp increased as the L/h or the longitu-dinal reinforcement ratio increased, but decreased as theamount of lateral reinforcement increased. Based on fourcolumn tests where the axial load levels were low (P/fc′Ag =0.06 to 0.20), Mendis concluded that the lp was not sensitiveto the level of axial load.

PREVIOUS RESEARCH INTO PLASTIC HINGE LENGTH: DISCUSSION

Various expressions, recommended for use in lp estimations,were briefly discussed in the previous section. A comparisonof the previously reported lp expressions is provided in Fig. 2.This figure clearly illustrates that large variations existamong the suggested lp.

It is interesting to note that most of the lp expressions donot consider axial load as a parameter, except for the

expressions proposed by Baker1 and Baker and Amarakone.2

This is because the behavior of concrete beams was themajor focus in investigating the lp in early research.1-5 In themore recent research,6-9 the behavior of concrete columnshas been the focus of lp investigations.

In addition, although the effect of axial load on the lp ofconcrete columns has been reported by several researchers,the reported results can be considered contradictory. Asdiscussed previously, Park et al.6 and Mendis10 concludedthat the lp were insensitive to axial load levels based on theirtest results. In contrast, Atalay and Penzien15 reported thatthe spread of plasticity or the length of the damaged regionincreased as the axial load level increased. Tanaka andPark16 reported that the lp increased from 0.46h to 0.75h asthe applied axial load level increased from 0.1fc′Ag to0.3fc′Ag. The test results of Thomson and Wallace17 andLégeron and Paultre18 indicated that the increase of lp withthe axial loads could also be observed for high-strengthconcrete columns. Therefore, an investigation into the lp ofreinforced concrete columns is needed to: 1) reconcile differ-ences encountered in the previous research; and 2) developan expression that can be used to estimate lp more accuratelyfor various axial load levels.

As discussed previously, Eq. (2) and (3) have beencommonly used to estimate the lp of concrete columns.6-9,12,13

Using these equations, the lp can be estimated from measuredtip displacements and curvatures. Tip displacements of aconcrete column include flexural, bar slip, and sheardisplacement components. Therefore, the lp estimated usingEq. (2) and (3) includes the effect of bar slip and sheardisplacements. As shear deformations can be ignored forcolumns with moderate L/h, the lp measured in this mannerwill be influenced mainly by flexural and bar slip displace-ments. The influence of various displacement componentsother than flexural displacements can be observed in Eq. (9)and (10), which were proposed by Priestley and Park7 andPaulay and Priestley.8 These equations have two separateterms that take into account the effect of flexural and bar-slipdisplacements. By relating the curvature distribution alongthe column shown in Fig. 1 to the tip displacement ratherthan flexural deformations, the lp in Eq. (2) and (3) includesthe effects of bar slip and shear displacements. Conversely,the displacements calculated through the use of the lp expressionderived in this manner implicitly account for displacementcomponents other than the flexural displacements.

As flexural displacements are directly related to curvatures,it is more appropriate to relate the lp to the flexural displacementcomponent only. Bar slip and shear displacements have to beestimated independently. Subsequently, the tip displacementcan be estimated by summing these displacement compo-nents. Following this logic, a new lp expression is proposedin this study.

PLASTIC HINGE LENGTHS: COLUMN TESTSTo investigate the effect of L/h and axial load (P/Po), four

full-scale concrete columns were tested under moderate tohigh axial load levels and reversed cyclic displacementexcursions. Figure 3 illustrates the test setup used in thisresearch. The test frame, shown in Fig. 3, can be used toapply and maintain an axial load up to 2000 kips (8900 kN)and reversed cyclic moments as large as 4000 ft·kips(5400 kN·m). White arrows in Fig. 3 illustrate the axial forceapplied by the hydraulic ram and the loads applied by twoactuators that were used to impart reversed cyclic base

Fig. 2—Comparison of plastic hinge length expressions.

Fig. 3—Test setup.


rotations. The shear forces at hinges are shown with blackarrows. Table 1 shows the details of the specimens tested inthis research program and the magnitude of the axial loadthat was applied to each specimen during testing. Adetailed discussion of this experimental research can befound elsewhere19 and is considered to be beyond the scopeof this paper.

The damage observed within the plastic hinge region ofeach test specimen and the corresponding tie strains alongthe columns are shown in Fig. 4. Specimens S24-2UT andS24-3UT were tested under an axial load level of 0.5Po,while Specimens S24-4UT and S24-5UT were tested undera lower axial load level of 0.2Po. Specimens S24-2UT,S24-4UT, and S24-5UT had 24 in. (609.6 mm) squaresections (L/h = 5), whereas Specimen S17-3UT had a 17.25 in.

(438.15 in.) square section (L/h = 7). Figure 4 shows that theseverely damaged regions of Specimens S24-2UT andS17-3UT are longer than those of Specimens S24-4UT andS24-5UT. The tie strains measured along the column specimensat the 20th loading cycle in Fig. 4 also illustrate thatSpecimens S24-2UT and S17-3UT experienced inelastic tiestrains over a longer length than Specimens S24-4UT andS24-5UT. Many ties in Specimens S24-2UT and S17-3UTexperienced larger inelastic strains after the 20th loadingcycle than the measurement limits of strain gauges. Therefore, itcan be concluded that, for the specimens tested in this study,lp increased as the level of axial load increased.

Figure 5 illustrates the sectional and member performances ofSpecimens S24-2UT and S24-4UT. Specimen S24-2UT andS24-4UT have the same section size (h = 24 in. [610 mm]) and

Table 1—Details of test specimens

Specimenb x h,

in. x in. (mm x mm)fc′ ,

ksi (MPa)

Longitudinal steel Transverse steel

P/Po†

Size ρl , %fyl,

ksi (MPa) Size sh, in. (mm) ρs,* %

fyh, ksi (MPa) Ash/Ash,ACI

S24-2UT 24 x 24 (610 x 610) 63 (43.4) No. 7-1 (22M-1) 1.25 73 (50.3) No. 4-1 (13M-1) 3-3/4 (95) 2.04 62 (42.7) 1.09 0.5

S17-3UT 17.25 x 17.25 (440 x 440) 6.3 (43.4) No. 5 (16M) 1.25 72 (49.6) No. 3-1 (10M-1) 3-3/8 (86) 1.76 72 (49.6) 1.12 0.5

24-4UT 24 x 24 (610 x 610) 5.3 (36.5) No. 7-2 (22M-2) 1.25 58 (40.0) No. 3-2 (10M-2) 6 (152) 0.72 66 (45.5) 0.44 0.2

S24-5UT 24 x 24 (610 x 610) 6.0 (41.4) No. 7-2 (22M-2) 1.25 58 (40.0) No. 4-2 (13M-2) 6 (152) 1.30 63 (43.4) 0.74 0.2*ρs = volumetric ratio of transverse reinforcement to concrete core.†Po = 0.85fc′Ac + fylAs.

Fig. 4—Strain distribution.


L/h = 5. These specimens were tested under different axial loadlevels (P/Po = 0.5 for Specimen S24-2UT and P/Po = 0.2 forSpecimen S24-4UT). Figure 5(a) illustrates that the sectionalperformance of Specimen S24-4UT is somewhat better thanthat of Specimen S24-2UT with respect to deformationcapacity. The opposite trend can be observed from the memberperformances, however, as shown in Fig. 5(b). The reversal inthe trends is attributed to the difference observed in the lp ofSpecimens S24-2UT and S24-4UT.

PLASTIC HINGE LENGTH: CONCRETE COMPRESSION STRAIN METHOD

Estimating the length of a plastic hinge that forms in areinforced concrete column through the use of a sophisti-cated computer program is complicated. Because of thisdifficulty, the estimation of lp has typically been based onexperimental data. A plastic hinge is defined as the zonewhere severe damage of concrete sections occurs. Byobserving that large cracks typically occur at curvaturesgreater than the yield curvature, some researchers20,21

attempted to relate the yield moment to the lp. This approachis practical when the curvature increases with moment (thatis, in the ascending branch of the response), but the applicationof this approach to members that experience post-peakstrength degradation is questionable.

In this section, a concrete compression strain method thatcan be used to estimate the lp of concrete columns is introduced.Following the description of this concrete compression strainmethod, the key factors that influence the length of plastichinges are identified by conducting sensitivity analyses.Based on the results of the sensitivity analyses, a simpleexpression that can be used to estimate the length of theplastic hinges is proposed. Finally, the influence of the lpestimations on the lateral load-displacement responsepredictions is discussed.

Compressive strain profile of core concreteAs reinforced concrete columns are subjected to earth-

quake-induced lateral displacements while supporting

gravity loads, plastic hinges form within the core concrete.The magnitude of damage in concrete members is closelyrelated to the intensity of concrete compressive strainsmeasured at critical sections. Under increasing curvatures,concrete strains on the compression side of the memberincrease. As the concrete compressive strains increase andreach a critical value, the cover concrete spalls off. Subse-quently, yielding of longitudinal bars on the compressionside occurs. This is followed by crushing of the coreconcrete. With accumulation of damage under increasingcompressive strains, plastic hinges form. Experimentalobservations6,19 indicate that the plastic hinges start to formafter the maximum moment capacity is reached. Therefore,the magnitude of concrete compressive strains experiencedby the core concrete when the maximum moment capacity isreached at the critical section of a column can give a goodindication on the formation of a plastic hinge. To estimate thelength of a plastic hinge that may form at the base of a reinforcedconcrete column, the following procedure is proposed:

1. The moment-curvature response of a typical columnsection within the potential plastic hinge region is obtainedfrom a sectional analysis (Fig. 6(a)). In the sectional analysis,core concrete is modeled as confined concrete, while coverconcrete is modeled as unconfined concrete. Bar bucklingbehavior in compression reinforcement is not consideredbecause the ascending branch of the moment-curvaturerelationship is of concern. Reinforcing bar buckling is notexperienced within the ascending branch of the response;

2. Neglecting the second order effects, the bending momentdiagram is drawn along the column height (Fig. 7(c)). Themoment value at the center of the potential plastic hinge(~0.5h away from the base) is assumed to be equal to themaximum moment capacity of the section obtained from thesectional analysis in Step 1. This assumption would imply thatthe sections between the center of the plastic hinge and thebase of the column have a larger flexural capacity thanpredicted using the sectional analysis. The shift in the locationof the critical section (Fig. 7(d)) is due to the confinementprovided by the concrete stub to the neighboring sections. Thiseffect, commonly referred to as the stub confinement effect,has been observed in previous experimental research6,9,13,18,22;

3. Using the bending moment diagram from Step 2 and thesectional analysis’ results (Fig. 6(b)) from Step 1, thecompressive strain experienced by the outer fiber of coreconcrete can be determined for a sufficient number ofsections along the length of a column. If these compressivestrains are plotted along the length of a column, the compressiveconcrete strain profile along the length of a column can beobtained (Fig. 7(d));

4. The compressive strains at the outer fiber of the coreconcrete are equal to the reinforcing bar strains. In effect, thecompressive strain profile obtained in Step 3 represents thevariation of compressive longitudinal bar strain along thelength of a column. By examining the compressive strainprofile, such as the one shown in Fig. 7(d), the length of theregion in which longitudinal bars are yielding in compressioncan be estimated. The identification of this length establishes akey step in estimating the lp of a column; and

5. As indicated in Step 2, the critical section shifts awayfrom the face of the stub due to additional confinementeffects provided by the stub. Because of the additionalconfinement provided by the stub to adjacent sections,sections within a distance of approximately 0.25h from thestub remain nearly undamaged. Therefore, to estimate theFig. 6—Sectional analysis for Specimen S24-2UT.

Fig. 5—Comparison of Specimens S24-2UT and S24-4UT.

ACI Structural Journal/May-June 2008 295

length of the plastic hinge region, where columns areexpected to dissipate large amounts of inelastic energy byundergoing large inelastic deformations, 0.25h is subtractedfrom the overall length in which compressive reinforcing barstrains greater than the yield strain are calculated.

The curvature profile depicted in Fig. 7(e) also shows asimilar trend in the compressive strain profile shown inFig. 7(d). Hence, the use of curvature distribution along theheight of a column can be considered as an alternative foridentifying the zone in which inelastic curvatures are localized.While this curvature profile has been used by someresearchers,20,21 the use of the compressive strain profilefacilitates a clearer observation of the effect of axial loads onthe lp. In addition, because the damage in concrete is moreclosely related to the magnitude of the concrete strains thanthe curvatures, compressive strains were used in the methoddescribed previously.

In this approach, it is assumed that plastic hinges start toform when the maximum moment capacity is reached and,therefore, important information on the plastic hinge formationcan be obtained by analyzing the compressive strain profilewhen the maximum moment is reached. It is important tonote that the lp predicted in this manner can be used toestimate flexural deformations only because sectionalanalysis is used as the basis of calculations. To calculatethe tip deformation of columns, deformations due to bar slipand shear deformations need to be calculated separately andadded to the flexural deformations.

The lengths of the potential plastic hinge regions ofSpecimens S24-2UT through S24-5UT are estimated byusing the analysis procedure described previously. By usingthe compressive strain profiles along heights of the testspecimens, the lp were estimated and the results of theseanalyses are shown in Fig. 8. This figure illustrates that theproposed method provides good estimations for the lp of thecolumn specimens tested in this study.

PARAMETRIC STUDY ON PLASTIC HINGE LENGTHIt was shown that the proposed concrete compression

strain method provided good estimations for the lp of thecolumns tested in this study. Using the proposed method, theinfluence of various parameters on the lp is studied. The

effect of axial load level (P/Po), L/h, and the amount oflongitudinal reinforcement (ρl = As/Ag) are investigated.

Axial load levelTo study the effect of axial load on the lp, a square

column (24 x 24 x 120 in.3 [610 x 610 x 3048 mm3]) isstudied. The percentage of longitudinal reinforcement is keptconstant at 1% (ρl = 0.01) and the ratio of the center-to-centerdistance of extreme reinforcement layers to the overall columndepth (γ) is assumed to be 0.8. A concrete strength of 6000 psi(41.4 MPa) is used. The yield and ultimate strength ofreinforcing bars are assumed to be 60 and 90 ksi (414 and621 MPa), respectively. The curvature and compressiveconcrete strain profiles along the length of the columns areinvestigated for various axial load levels. Figure 9 illustratesa summary of the results of the analyses. As can be observedin this figure, the curvature profiles do not effectively showthe effect of axial load level. Unlike moment-curvaturerelationships, the curvature profiles in Fig. 9(a) show curvatureprofiles along the column height. In moment-curvaturerelationships, the effect of axial load influences the strengthand stiffness. As these curvature profiles provide only

Fig. 8—Comparison of predicted plastic hinge lengths.

Fig. 7—Estimation of plastic hinge length (Specimen S24-2UT).


curvatures but not corresponding moments, however, theeffect of axial load is not reflected in these curvature profiles.In contrast, the compressive strain profiles clearly illustratethe influence of axial load.

For each case studied in Fig. 9, the length of the plastichinge is estimated using the procedure described previously.In other words, the spread of reinforcing bar yielding incompression is examined to establish the length of the plastichinges. Figure 10 summarizes the results of this analysis,shown in Fig. 9(b). As can be seen in this figure, the lengthof the plastic hinge is nearly constant for low axial loads(P ≤ 0.2Po). For low axial loads, the lp is approximatelyequal to 0.25h. Starting at an axial load of approximately0.2Po, the lp increases with increasing axial loads. It isinteresting to note that Mendis10 reported that the lpmeasured in the columns tested in his research were notsensitive to the axial load level. Considering the fact that allof the columns studied by Mendis10 were tested under lowaxial load levels (P ≤ 0.2Po) and bearing in mind the findingsof this parametric study (summarized in Fig. 10), the conclusionreported by Mendis10 can be better appreciated.

The lp estimate of 0.25h (Fig. 10) can be compared with0.4h recommended by Park et al.6 and 0.5h recommended byPaulay and Priestley.8 The differences observed in the lpestimates can be attributed to the displacement componentsused to estimate the lp. Only flexural displacements areconsidered in the proposed analysis, as the strains experiencedby compression bars are obtained from the moment-curvaturerelationships. On the other hand, the tip displacements areused in estimating the lp by Park et al.6 and Paulay andPriestley.8 As discussed previously, tip displacementsinclude bar slip and shear deformation components in additionto the flexural deformations.

Chapter 21 of the ACI Building Code23 requires closely-spaced transverse reinforcement be provided over a length(lo) from a joint face where flexural yielding is likely tooccur as a result of large inelastic lateral displacement ofcolumns. The required length (lo) of the ACI Code is thelargest of the overall depth of a column, 1/6 of the clearheight of a column, or 18 in. (457 mm). Figure 10 shows thatthe estimated lp approaches 0.8h at an axial load of 0.52Po(= φPn,max = φ × 0.8Po = 0.65 × 0.8Po), which is themaximum axial load permitted in the ACI Building Code.23

It is important to note that in the process of estimating the lp,a distance of 0.25h from the face of the stub is subtracted, asdiscussed in the procedure of concrete compression strainmethod. Therefore, the total length from a joint face has tobe considered as the sum of the lp and a distance of 0.25hfrom the joint face. Accordingly, the total length from a jointface is 1.05h at an axial load of 0.52Po, which is close to theACI required length (lo) of 1.0h. An examination of Fig. 1,however, shows that the length of the region where largeinelastic curvatures develop will be longer than the equivalentplastic hinge region. Therefore, the required length ofclosely-spaced transverse reinforcement needs to be largerthan the estimated plastic hinge region of 1.0h from the jointface. In summary, the potential lp specified by ACI 318-0523

is likely to be unsatisfactory for columns supporting highaxial loads.

Shear span-depth ratio (L/h)Researchers1-5,7,8,10 recognized that the length of a plastic

hinge is influenced by the L/h. To investigate the influenceof the L/h on the lp, a series of analyses were conducted. At thisstage of the parametric study, a 24 x 24 in.2 (610 x 610 mm2)square column with varying L/h is considered. The longitudinalreinforcement is kept constant (ρl = 0.01) and a γ value of 0.8is used. The results of the analyses are summarized in Fig. 11.

As can be observed in the figure, the lp increases withincreasing L/h for a given axial load level. For low axialloads (≈ 0.2Po), however, the increases observed in lp withincreasing L/h are insignificant.

For a given L/h, the lp increases with increasing axialloads. The increases in lp observed at small L/h (2 < L/h < 3)are less pronounced than those observed at a large L/h. It isinteresting to note that Park et al.6 tested a series of columnswith an L/h of 2.2. Based on the test results, they concludedthat the lp was not affected by the axial loads and recommendeda lp of 0.4h. Figure 11 clearly illustrates that for an L/h of 2.2,the effect of axial load on the lp is relatively small in comparisonwith the effects seen at high L/h.

Fig. 11—Relationship between plastic hinge length andshear span-depth ratio.

Fig. 10—Relationship between plastic hinge length andaxial load.

Fig. 9—Effect of axial load on curvature and compressivestrain profiles.

ACI Structural Journal/May-June 2008 297

Amount of longitudinal reinforcement (ρl = As /Ag)Mattock3 reported that the lp increased with a decrease in

the net tension reinforcement ((As –As′ )/Ab, where As is thearea of tension reinforcement, As′ is the area of compressionreinforcement, and Ab is the area of balanced reinforcement).The effect of net tension reinforcement was further studiedby Corley.5 In contrast, recently, Mendis10 reported that thelp increased with an increasing amount of tension reinforcement.It is important to note that these conclusions3,5,10 were primarilybased on the results from tests conducted on reinforcedconcrete beams.

Due to the lack of consensus among various researchers,the amount of longitudinal reinforcement has been ignoredin most previous proposals for estimating the lp. To study theeffect of longitudinal reinforcement on lp, a 24 x 24 in.2

(610 x 610 mm2) square column with a column height of120 in. (3048 mm), a L/h of 5, and a γ value of 0.8 is studied.

The results of the parametric study conducted on a 24 x 24 in.2

(610 x 610 mm2) square column are summarized in Fig. 12and 13. As can be observed in Fig. 12(a), the post-crackingstiffness and strength of the column sections increase withincreasing reinforcement ratios.

The curvatures and reinforcing bar strains on the compressionside are evaluated at maximum lateral load capacity andplotted along the height of the column (Fig. 12(c) and (d)).An examination of the curvature and compressive strainprofiles along the height of the column indicates that withincreasing longitudinal reinforcement ratio (ρl), the length ofthe plastic hinge increases. Figure 13 also indicates this trendfor various axial load levels. For all of the axial load levels,lp increases with increasing ρll values.

PLASTIC HINGE LENGTH: NEW EXPRESSIONThe previous discussion using the proposed concrete

compression strain method shows that axial load, L/h, andthe amount of longitudinal reinforcement are main parametersfor estimating the length of a plastic hinge. Based on thisobservation, a linear relationship between these parameters(P/Po, L/h, and As/Ag) is assumed in developing the lp expressionfor simplicity. Least squares analyses were conducted toidentify a coefficient for each parameter using the UW/PEERcolumn database. Key features of the column database usedfor this purpose can be found elsewhere.19 A new lp expressionof Eq. (11) is proposed from a series of least squares analyses

(11)

In the process developing the proposed lp expression, acomputer program is used to estimate the response of columnspecimens. The drift capacities of columns are estimated andcompared with experimentally measured drift capacities.The computational algorithm used in this computer programis illustrated in Fig. 14. As seen in Fig. 14, the tipdisplacements of columns are calculated by estimating

lp

h--- 0.3 P

Po

-----⎝ ⎠⎛ ⎞ 3

As

Ag

-----⎝ ⎠⎛ ⎞ 0.1–+ L

h---⎝ ⎠⎛ ⎞ 0.25 0.25≥+=

various displacement components and summing themtogether. Those displacement components considered incalculating the tip displacement of a column are: 1) flexuraldisplacements; 2) fixed-end rotations resulting from the slipof longitudinal bars out of joints; and 3) shear displacements.The additional displacements due to secondary momentsgenerated by the P-Δ effect are also included.

The lp estimated by using Eq. (11) are used in calculatingflexural displacements to approximate the descending part ofthe lateral load response of concrete columns. A detaileddescription of Eq. (11) can be found elsewhere.19

Table 2—Predicted plastic hinge lengths

Specimen Baker1 Corley5 Mattock4 Park et al.6 Paulay and Priestley8 Sheikh and Khoury9 Measured Eq. (11)

S24-2UT 0.60h 0.49h 0.70h 0.40h 0.80h 1.00h 0.66h 0.69h

S17-3UT 0.65h 0.52h 0.80h 0.40h 0.96h 1.00h 0.91h 0.86h

S24-4UT 0.62h 0.49h 0.70h 0.40h 0.72h 1.00h 0.49h 0.25h

S24-5UT 0.56h 0.49h 0.70h 0.40h 0.72h 1.00h 0.47h 0.25h

Fig. 13—Effect of amount of longitudinal reinforcement.

Fig. 12—Moment and curvature curves with various amountof longitudinal reinforcement (P = 0.3Po).


PLASTIC HINGLE LENGTH:ESTIMATIONS AND EXPERIMENTS

To examine the accuracy of Eq. (11), the lp of four full-scale concrete column specimens are estimated usingvarious expressions and compared with the measured lp, asshown in Table 2. This table illustrates that the use of Eq. (11)results in good estimates for all test specimens. For specimenstested under low axial loads (Specimens S24-4UT andS24-5UT), Eq. (11) gives somewhat smaller lp than themeasured values. This is due to the fact that only flexuraldisplacements are considered in deriving Eq. (11). Thecomparison of damaged specimens after tests (Fig. 8) showsthat the estimated lp by Eq. (11) are similar to sizes ofseverely damaged regions.

For further evaluation, responses of test specimens areestimated by using a computer program, which is describedin Fig. 14. The sectional and member responses of the testspecimens are used as bases of comparison. The lp estimatedby various expressions including Eq. (11) are used ingenerating the member responses. The measured lp are alsoused for this purpose. It is important to note that only theflexural displacements and the magnified displacementsdue to the P-Δ effect are used to get member responseswhen lp expressions other than the proposed are used (asthese lp expressions include the effect of other displacementcomponents and these displacements should not beaccounted for twice).

The estimated moment-curvature and lateral load-driftplots of Specimens S24-2UT and S17-3UT (tested underhigh axial loads) are illustrated in Fig. 15. As can be seen in

Fig. 15—Experimental and predicted response (P = 0.5Po).

Fig. 14—Flowchart of computer program.


Fig. 15, the use of the lp expression proposed by Park et al.6

(lp = 0.4h) results in considerable underestimation of the driftcapacity. In contrast, the expressions proposed by Paulayand Priestley8 (lp = 0.8h) and Sheikh and Khoury9 (lp = 1.0h)result in satisfactory predictions. The same trends can beobserved for Specimen S17-3UT, as shown in Fig. 15.

Results of the analyses conducted on Specimens S24-4UTand S24-5UT are included in Fig. 16. These specimens weretested under low axial load levels. Figure 16 shows that theuse of the lp expression proposed by Park et al.6 (lp = 0.4h)results in a reasonably good estimation for the drift capacity.The expressions proposed by Paulay and Priestley8 (lp =0.72h) and Sheikh and Khoury9 (lp = 1.0h), however, resultin overestimated predictions of the drift capacity. Therefore,it can be concluded that the use of some lp expressions workfor high axial loads, whereas they fail to provide satisfactoryestimates for low axial loads and vice versa. The use of Eq. (11),however, results in satisfactory predictions both for high andlow axial load levels. In performance-based design, predictionof the deformation capacity of reinforced concrete columnsis important. An analysis of Fig. 15 and 16 clearly shows theinfluence of the lp estimations on the drift capacity predictions.This analysis indicates the value of an expression that canprovide reasonable estimations for lp forming in columnssupporting high or low axial loads.

SUMMARY AND CONCLUSIONSBased on the experimental and analytical research into the

seismic behavior of concrete columns reported in this study,the following conclusions can be drawn:

1. In light of the experimental data on four full-scaleconcrete column tests, ACI 318-0523 provisions for thelength of the potential plastic hinge regions were found to beslightly unconservative for columns supporting high axialloads. Although the experimental evidence is not exhaustive,it may be prudent to increase the length of the region inwhich closely-spaced ties are used from 1.0h to 1.5h fromthe joint face until further experimental evidence is gathered;

2. The level of axial load influenced the length of theplastic hinges that formed in the full-scale column specimenstested in this research. Specimens tested under high axialloads developed longer plastic hinges than those tested underlow axial loads; and

3. The following equation, developed in this research, canbe used to estimate the length of the plastic hinges formingin columns supporting a wide range of axial loads.

NOTATIONAb = area of balanced reinforcementAg = gross area of concrete sectionAs = area of tension reinforcementAs′ = area of compression reinforcementb = width of compression face of memberc = distance from extreme compression fiber to neutral axisd = effective depth of beamdb = diameter of longitudinal reinforcementfc′ = compressive strength of concrete

lp

h--- 0.3 P

Po

-----⎝ ⎠⎛ ⎞ 3

As

Ag

-----⎝ ⎠⎛ ⎞ 0.1–+ L

h---⎝ ⎠⎛ ⎞ 0.25 0.25≥+=

Fig. 16—Experimental and predicted response (P = 0.2Po).


fy = yield stress of reinforcementh = overall depth of columnk1 = 0.7 for mild steel

= 0.9 for cold worked steelk2 = 1 + 0.5(P/Pu)k3 = 0.9 – (0.3/23.5)( fc′ – 11.7) ( fc′ in MPa)L = distance from critical section to point of contraflexurelp = plastic hinge lengthP = applied axial forcePo = 0.85fc′ (Ag – As) + fyAs

= nominal axial load capacity as per ACI 318-0523

q = tension reinforcement index (= As/bd × fy /fc′ )q′ = compressive reinforcement index (= As′ /bd × fy/fc′ )qb = balanced tension reinforcement index (= Ab/bd × fy/fc′ )z = distance of critical section to point of contraflexureΔp = plastic displacementΔy = yield displacementεce = elastic concrete compressive strainεcu = maximum concrete compressive strainφ = curvature (Eq. (2))φy = yield curvatureθp = plastic rotationρl = As/Ag

= longitudinal reinforcement ratio

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2. Baker, A. L. L., and Amarakone, A. M. N., “Inelastic HyperstaticFrame Analysis,” Flexural Mechanics of Reinforced Concrete, SP-12,American Concrete Institute, Farmington Hills, MI, 1964, pp. 85-142.

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Plastic Hinge Length of Reinforced Concrete Columns

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