effective stiffness of reinforced concrete columns

476 ACI Structural Journal/July-August 2009

ACI Structural Journal, V. 106, No. 4, July-August 2009.MS No. S-2007-399.R1 received July 8, 2008, and reviewed under Institute publication

policies. Copyright © 2009, American Concrete Institute. All rights reserved, including themaking of copies unless permission is obtained from the copyright proprietors. Pertinentdiscussion including author’s closure, if any, will be published in the May-June 2010 ACIStructural Journal if the discussion is received by January 1, 2010.

ACI STRUCTURAL JOURNAL TECHNICAL PAPER

Existing and proposed models of the effective stiffness of reinforcedconcrete columns subjected to lateral loads are assessed using theexperimental response of 329 concrete columns. Existing modelsappropriate for design applications tend to overestimate themeasured effective stiffness and are unacceptably inaccurate,because they generally neglect the influence of anchorage slip onthe effective stiffness of the column. A three-component model thatexplicitly accounts for deformations due to flexure, shear, andanchorage-slip is shown to provide a more accurate estimate of themeasured effective stiffness for the database columns. This modelis simplified by neglecting small terms and approximating theresults of moment-curvature analysis to obtain an accurate andrational effective stiffness model appropriate for design applications.For this model, the ratio of the measured stiffness to the calculatedstiffness had a mean and coefficient of variation of 1.02 and 22%for circular columns and 0.95 and 25% for rectangular columns.

Keywords: bond slip; column; flexural stiffness; lateral loads; reinforcedconcrete; yield.

INTRODUCTIONThe assumptions made in estimating the stiffnesses of

structural members dominate the computed performance ofa building or bridge subjected to earthquake ground motions.If these assumptions are used in a linear analysis, they controlpredictions of the period of the structure, the distribution ofloads within the structure, and the deformation demands. Themember stiffnesses also control the yield displacement,which in turn affects the displacement ductility demandscalculated as part of a nonlinear analysis.

The consequences of overestimating or underestimatingthe actual stiffnesses of structural members depend on thetype of structural system and the response parameter ofinterest. For example, a low estimate of the effective stiffnessesof columns in a moment-resisting frame usually leads to aconservative (high) estimate of the displacement demands.In contrast, a low estimate of the effective stiffnesses forcolumns in a shear-wall building would lead the designer tounconservatively underestimate the elastic shear demands onthe columns. The need for an accurate estimate of effectivestiffnesses is even more crucial for time-history analyses, inwhich the peaks and valleys of the ground-motion responsespectrum significantly influence the computed performance.

To assist engineers in developing numerical models for theestimation of lateral deformation demands, most codes andstandards provide recommendations for member effectivestiffness. The Federal Emergency Management Agency(FEMA) 356 seismic rehabilitation guidelines (ASCE 2000)specify the most commonly used procedure for estimatingcolumn stiffness in the U.S. This procedure has been adoptedinto the Seismic Rehabilitation Standard, ASCE 41 (ASCE2007a). It recently has been superseded by a new procedurespecified in ASCE 41 Supplement 1 (ASCE 2007b),described in Elwood et al. (2007). A similar model isincluded in the commentary to the New Zealand concrete

code, NZS 3101-06 (NZS 2006), based on the commonlyused recommendations by Paulay and Priestley (1992). ACI318-08 (ACI Committee 318 2008) will be the first editionof ACI 318 to provide stiffness recommendations specifi-cally for lateral-load analysis. These code proceduresare convenient for preliminary analysis, because theycan be implemented without performing a moment-curvatureanalysis and without knowing the details of the columnreinforcement. Simple effective stiffness models forapplication in design have also been proposed by Mehanny etal. (2001) and Khuntia and Ghosh (2004).

This paper uses data from the Pacific Earthquake EngineeringResearch Center (PEER) Structural Performance Database,developed by the second author and his students (Berry et al.2004), to assess the accuracy of these practical methodologies andto propose a new procedure.

RESEARCH SIGNIFICANCEThe assumed stiffness of a column dominates the results of

linear and nonlinear analyses of buildings and bridgessubjected to ground motions. Currently, most designprofessionals assume that the column effective stiffness is afixed proportion (say 50 or 100%) of the gross-section stiffness.Using a database of 329 columns with rectangular andcircular cross sections, this paper shows that existing proceduresfor estimating column stiffness are inaccurate. Based onsimplifications of a three-component model, the paperproposes a new procedure that is more rational, practical,and accurate. The proposed procedure could be used immediatelyby design professionals, and it could be incorporated intodesign provisions, such as ASCE 41 (ASCE 2007a) orACI 318 (ACI Committee 318 2008).

DATABASE OF MEASUREDEFFECTIVE STIFFNESSES

The PEER Structural Performance Database (Berry et al.2004) provided the data needed to evaluate the accuracy ofvarious models of column stiffness. The database contains thecyclic force-deformation response, geometry, axial load, andmaterial properties for more than 400 tests of reinforced concretecolumns. A total of 366 of these columns were tested incantilever, double-curvature, and double-cantileverconfigurations, which makes it possible to isolate thecolumn’s stiffness/flexibility from other sources of flexibility,such as a flexible supporting beam. To limit the analyses tocolumns typical of practice, the axial load was limited to amaximum of 0.66Ag fc′ , and the shear-span-to-depth ratio

Title no. 106-S45

Effective Stiffness of Reinforced Concrete Columnsby Kenneth J. Elwood and Marc O. Eberhard

477ACI Structural Journal/July-August 2009

was limited to a minimum of 1.4 (all variables are defined inthe Notation section).

The selected specimens include 221 columns with rectangularcross sections with rectangular transverse reinforcementand 108 columns with circular, octagonal, and square crosssections with spiral transverse reinforcement (for brevity,these database subsets will be referred to as “rectangularcolumns” and “circular columns,” respectively). Theminimum, maximum, mean, and median properties of theselected column databases are reported in Table 1. Both setsof column data have wide ranges of column parameters; andfor both sets of data, the strength of over 90% of the includedcolumns was limited by their flexural strength. This statisticis a consequence of the limit on the aspect ratio, whicheliminated many shear-critical columns. The distribution ofsome parameters varies greatly between the two sets. Forexample, compared with the rectangular column dataset, thecircular column set includes fewer columns with axial loadsabove 0.3Ag fc′ or fc′ above 60 MPa (8700 psi). The circularcolumns also tend to have lower values for db/D whencompared with the rectangular columns. These differences

are consistent with the observation that many of the tests ofcircular columns were proportioned to represent typicalbridge practice, whereas most of the rectangular columnswere proportioned to reflect construction practice for buildings.

For each column, the envelope of the measured lateralload-displacement relationship was corrected for P-deltaeffects to give the effective lateral force envelope for eachcolumn. The yield displacement and effective stiffness ofeach column were determined as shown in Fig. 1. Forapproximately 90% of the columns, the effective stiffnesswas defined based on the point on the measured effectiveforce-displacement envelope that corresponded to the calculatedforce at first yield, Ffirst yield (Fig. 1(a)). Adopting the samedefinition of yield used by Benzoni et al. (1996) and others,the yield force was defined as the first point at which thetension reinforcement yielded or the maximum concretestrain reached a value of 0.002, whichever came first. Thedatabase does not include strain measurements, so the forceat first yield was determined with a moment-curvatureanalysis, using a linear model for the steel and the Mander etal. (1988) constitutive relationship for the concrete.

This definition could not be used for columns whosestrength did not substantially exceed the yield force (forexample, shear failures). For these columns, defined as thosewhose maximum measured effective force Fmax was not atleast 7% larger than the calculated force at first yield, theeffective stiffness was defined based on the point on themeasured force-displacement envelope with an effectiveforce equal to 0.8Fmax (Fig. 1(b)).

Assuming the column is fixed against rotation at both ends andhas a linear variation in curvature over the height of the column,

ACI member Kenneth J. Elwood is an Associate Professor at the University of BritishColumbia, Vancouver, BC, Canada. He received his PhD from the University ofCalifornia, Berkeley, Berkeley, CA. His research interests include the behavior andperformance-based design of reinforced concrete structures under seismic loading. Heis Chair of ACI Committee 369, Seismic Repair and Rehabilitation, and a member ofACI Committee 374, Performance-Based Seismic Design of Concrete Buildings, andJoint ACI-ASCE Committee 441, Reinforced Concrete Columns.

Marc O. Eberhard, FACI, is a Professor at the University of Washington, Seattle,WA. He received his BS in civil engineering, and materials science and engineeringfrom the University of California, Berkeley, and his MSCE and PhD from the University ofIllinois, Chicago, IL. He is a member of Joint ACI-ASCE Committee 445, Shearand Torsion.

Fig. 1—Definition of yield displacement and effectivestiffness from test data for: (a) yielding columns; and (b)columns that did not yield.

Table 1—Range of properties for database columns

Parameter

Circular columns (108 specimens)

Rectangular columns (221 specimens)

Mini-mum

Maxi-mum Mean Median

Mini-mum

Maxi-mum Mean Median

a/D 1.5 10.0 4.0 4.0 1.5 7.6 3.6 3.2

fc′, MPa (ksi)

18.9 (2.7)

90.0(13.1)

37.9(5.5)

34.4(5.0)

21.0(3.0)

118.0(17.1)

52.3(7.6)

36.5(5.3)

fy, MPa (ksi)

240(34.8)

565(82.0)

420(61.0)

446(64.7)

318(46.2)

587(85.2)

456(66.2)

453(65.7)

ρ 0.005 0.056 0.024 0.021 0.010 0.060 0.024 0.021

D/db 12 48 27 28 12 32 18 16

, MPa (psi)

0.05(0.6)

0.99(11.9)

0.27(3.3)

0.19(2.3)

0.09(1.1)

0.71(8.6)

0.32(3.9)

0.30(3.6)

P/Agfc′ –0.1 0.58 0.15 0.10 0.00 0.63 0.23 0.20

v fc′⁄

Fig. 2—Effect of key parameters on measured effectivestiffness.


the measured effective modulus of rigidity (for simplicity,referred to here as stiffness) can be defined as

(1)

where F0.004 is the calculated effective force on the columnwhen the extreme concrete fiber reaches a maximumcompressive strain of 0.004, and Δy is the displacement at yieldaccording to Fig. 1 for an equivalent cantilever of length a.

Figure 2 demonstrates the influence of several key parameterson the measured effective stiffness, expressed as a fraction ofthe gross-section stiffness, EIg. Within the dataset, the

EIeffmeasF0.004a3

3Δy

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

measured effective stiffness ranges from 10 to 122% of thegross-section stiffness (Table 2). The measured effectivestiffness can exceed EIg because the effect of the longitudinalreinforcement on the transformed section is not accounted forwhen determining EIg. Correlation coefficients for the rectangularand circular columns (Rr and Rc, respectively) are also shown inFig. 2. The normalized effective stiffness increases mostconsistently with increasing axial-load ratio (P/Agfc′) and aspectratio (a/D), with correlation coefficients ranging from 0.58 to0.79. The normalized effective stiffness also increases withincreases in concrete compressive strength ( fc′). The normalizedstiffness decreases with an increase in the ratio of the steelyield stress to concrete compressive strength ( fy /fc′). Thenormalized stiffness correlates only weakly with the normalizedbar size and longitudinal reinforcement ratio.

EVALUATION OF EXISTING MODELSThe models implemented in many of the structural codes

are similar in form to each other. Chapter 8 of ACI 318-08(ACI Committee 318 2008) provides three options forapproximating member stiffnesses for the determination oflateral deflection of building systems subjected to factoredlateral loads: (a) 0.35EIg for flexural members (P < 0.1Ag fc′)and 0.7EIg for compression members (P ≥ 0.1Agfc′ ); (b)0.5EIg for all members; or (c) as determined by “a moredetailed analysis considering the reduced stiffness of allmembers under the loading conditions.” Figure 3 and Table 2compare options (a) and (b) with the measured effectivestiffnesses from the column databases. Option (a) generallyoverestimates the effective stiffness for axial loads below0.4Ag fc′. Option (b) overestimates the stiffness for columnswith low axial loads and underestimates the stiffness forcolumns with high axial loads.

The figure and table also include evaluations of the effectivestiffness models from FEMA 356 (ASCE 2000), ASCE 41Supplement 1 (ASCE 2007b), and Paulay and Priestley(1992), all of which allow for interpolation between effectivestiffness values at low and high axial loads. The FEMA 356(ASCE 2000), and the Paulay and Priestley (1992) recommen-dations also tend to overestimate the measured effectivestiffness for the columns with low axial loads, particularlyfor the rectangular column dataset. Of these existingprocedures, ASCE 41 Supplement 1 (ASCE 2007b) providesthe best average estimate of the measured effective stiffness(refer to Table 2). But none of these models are accurate. Thecoefficient of variation for all of these models ranges from35 to 58%, depending on the particular model and dataset.As will be demonstrated in the following, this scatter can be

Table 2—Statistics for ratio of measured to calculated effective stiffnessfor existing effective stiffness models

Model

Circular columns (108 specimens) Rectangular columns (221 specimens)

Minimum Maximum Mean MedianCoefficient of variation, % Minimum Maximum Mean Median

Coefficient of variation, %

Gross section 0.13 1.21 0.39 0.34 55.1 0.10 1.22 0.37 0.33 58.1

FEMA 356 0.25 1.96 0.74 0.64 48.1 0.19 1.95 0.68 0.64 48.8

ASCE 41 Supplement 1 0.42 2.11 1.02 0.91 39.1 0.27 1.95 0.82 0.83 36.0

ACI 318-08 (a) 0.24 1.81 0.76 0.69 46.7 0.14 1.74 0.59 0.53 49.5

ACI 318-08 (b) 0.25 2.42 0.78 0.67 55.1 0.19 2.43 0.75 0.65 58.1

Paulay and Priestley (1992) 0.26 1.62 0.68 0.61 41.3 0.17 1.56 0.58 0.56 44.4

Mehanny et al. (2001) 0.26 1.24 0.61 0.56 37.7 0.16 1.29 0.49 0.49 41.2

Khuntai and Ghosh (2004) 0.21 1.58 0.68 0.62 41.8 0.16 2.03 0.66 0.62 54.3

Fig. 3—Measured effective stiffness from database comparedwith existing code models (ASCE 41-S1 = ASCE Supplement 1[ASCE 2007b]; PP92 = Paulay and Priestley [1992]).

ACI Structural Journal/July-August 2009 479

attributed mainly to the fact that these procedures are basedon consideration of expected flexural deformations.

Using a computed moment-curvature relationship, asshown in Fig. 4, the effective flexural stiffness of the columnEIflex can be determined based on the moment at first yieldof the column, Mfirst yield. The moment-curvature responsewas determined for each column in the database based onplane-section analysis and using the concrete constitutivemodel by Mander et al. (1988) and a linear constitutivemodel for steel. Figure 5 compares the code-based modelswith the calculated flexural stiffnesses of the columns in thedatabase expressed as a fraction of the gross-section stiffness(EIflex /EIg). For many of the columns considered, the modelsprovide an adequate estimate of the flexural stiffness.Comparing the results in Fig. 3 and 5, however, it is apparent thatother sources of flexibility must be taken into account toaccurately estimate the total effective stiffness.

Other effective stiffness models incorporating the influenceof variables beyond axial load have been proposed in theliterature. For example, Mehanny et al. (2001) accounts forthe influence of the longitudinal reinforcement by intro-ducing a model based on the transformed moment of inertiaand the balanced axial load

(2)

The statistics reported in Table 2 indicate that theMehanny model consistently overestimates the measuredeffective stiffnesses and has large coefficients of variation (38%for circular columns and 41% for rectangular columns).

Khuntai and Ghosh (2004) recommend an effective stiffnessmodel for lateral-load analysis of reinforced concreteframes, with and without slender columns, accounting forthe influence of longitudinal reinforcement and effectiveeccentricity of the axial load. They propose the followingequation for compression members (P > 0.1Agfc′ )

(3)

The effective stiffness from Eq. (3) is limited to greaterthan the effective stiffness for flexural members determinedbased on a similar model included in Khuntai and Ghosh(2004). Compared with the Mehanny model, the Khuntai andGhosh model better predicts the average stiffness, but thecoefficient of variation for the ratio of the measured to calculatedeffective stiffnesses exceeds 40% for the circular columnsand exceeds 50% for the rectangular column dataset (Table 2).

With the exception of ASCE 41 Supplement 1 (ASCE2007b), all of the models considered were developedprimarily to provide an estimate to the flexural effectivestiffness (determined based on moment-curvature analyses)and, hence, ignore additional flexibility due to bar slip andshear deformations. Consequently, these models ignore theimportant dependence of the effective stiffness on the aspectratio of the column, evident in Fig. 2(b). Rather than relyingon purely statistical models, it would be preferable to developa simple model, whose form is based on the theoreticalcalculation of the yield displacement accounting for theflexibility due to flexure, shear, and bar slip. As shown in thefollowing sections, this approach can provide a moreaccurate estimate of the measured effective stiffnesses of thedatabase columns.

EI effcalc EIg tr,

⁄ 0.4 P 2.4Pb⁄+( ) 0.9≤=

EIeffcalc EIg 0.80 25ρ+( ) 1 e D 0.5P Po⁄–⁄–( ) 1.0≤=⁄

THREE-COMPONENT MODELOF YIELD DISPLACEMENT

Several researchers (Sozen 1974; Priestley et al. 1996;Lehman and Moehle 1998; Berry and Eberhard 2007) haveproposed estimating the yield displacement of an equivalentcantilever column of length a as the sum of the displacementcomponents due to flexure, shear, and bar slip

(4)Δy Δflex Δshear Δslip+ +=

Fig. 5—Comparison of flexural stiffness with code models(ASCE 41-S1 = ASCE Supplement 1 [ASCE 2007b]; PP92 =Paulay and Priestley [1992]).

Fig. 4—Definition of yield curvature and flexural stiffness(modulus of rigidity).


In this section, a similar model is developed based on thecolumn datasets. Then, the three components of deformationare combined into a single, nondimensional equation, whichcan serve as the basis for the development of practicaleffective-stiffness models based on gross-section properties.

Flexural deformationsThe calculated flexural curvatures in a reinforced concrete

column can be integrated directly to estimate the columndeformations attributable to flexure. Alternately, assuming alinear variation in curvature over the height of the column,the contribution of flexural deformations to the displacementat yield can be estimated as follows

(5)

where M0.004 is the flexural moment at a maximum concretecompressive strain of 0.004, and φy is the yield curvature, asdefined in Fig. 4.

Shear deformationsThe column deformation due to shear within the elastic

range of response is small for most columns, but it can belarge (relative to others sources of deformation) for stockycolumns with high levels of shear demand. Before shearcracking, this contribution can be estimated by assuming thatthe effective shear modulus is equal to the gross-section,isotropic elastic value (G = Ec/2.4). As the shear crackingincreases, the effective shear modulus reduces significantly.

For many applications, it is convenient to estimate theshear displacement of an equivalent cantilever column byidealizing the column as a homogeneous, isotropic materialwith a constant, reduced shear modulus

(6)

where Av is the effective shear area of the column crosssection (5/6 of the gross-section area of a rectangular columnand 85% of the gross area of a circular column). The expectedeffects of concrete cracking suggest that the effective shearmodulus should decrease as a function of the nominalprinciple tensile stress. Nonetheless, for application inengineering practice, the effective shear modulus Geff can beapproximated as one half the elastic value for all levels ofdeformation. This value of the effective shear modulus wasselected to optimize the statistics for the effective stiffnessmodel developed below (Eq. (12)).

Δflexa2

3-----φy

a2

3-----

M0.004

EIflex

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

ΔshearM0.004

AvGeff

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

Bar slip deformationsSlip of the reinforcing bars within the beam-column joints

or foundations further increases the lateral displacements.This section derives an expression to estimate the lateraldisplacement of a column due to bar slip prior to yielding ofthe longitudinal reinforcement.

Moments at the ends of a reinforced concrete column tendto cause tension in the longitudinal reinforcing bars, asshown in Fig. 6. This tension force Ts must be resisted by thebond stress u between the reinforcement and the footing orjoint concrete. If the bond stress is assumed to be constant,equilibrium considerations lead to the following expressionfor the length of bar required to resist Ts

(7)

Using Eq. (7) and integrating the triangular strain diagramshown in Fig. 6, the slip of the reinforcing bar δslip can beexpressed as

(8)

The rotation at the end of the column due to slip of thereinforcing bars θslip is given by the ratio of δslip to thedistance from the reinforcement to the neutral axis, c. UsingEq. (8), and recognizing that (εs/c) is equal to the curvatureat the section, the lateral displacement of an equivalentcantilever column of length a due to slip of the reinforcement at“first yield” can be expressed as follows

(9)

As shown in Fig. 1, the yield displacement (and each of itscomponents) is defined as the displacement at an effectiveforce of F0.004; hence, Δslip from Eq. (4) can be determined bymultiplying Eq. (9) by the ratio F0.004 / Ffirst yield. Noting fromFig. 4 that φy = φfirst yield (M0.004/Mfirst yield) = φfirst yield(F0.004/Ffirst yield), the following expression for Δslip is derived

(10)

The average bond stress values recommended in the literaturefor elastic response range from u = 0.5 to 1.0 MPa(u = 6 to 12 psi) (Otani and Sozen 1972; ACICommittee 408 1979; Alsiwat and Saatcioglu 1992; Sozenet al. 1992; Lehman and Moehle 1998). For the purpose ofthis study, the average bond stress was taken as u = 0.8 MPa(u = 9.6 psi). At first yield of the column, the stress inthe tension reinforcement, fs , used in Eq. (9) and (10) willvary depending on the axial load on the column. Forcolumns with low axial loads, the tension reinforcement willyield; hence, fs can be taken as equal to the yield stress, fy.The stress in the tension reinforcement will decrease as theaxial load on the column increases, reaching zero when thedepth of the neutral axis is equal to the effective depth of thecolumn. Consequently, it is expected that the displacementdue to bar slip will increase with decreasing axial load.

ldb fs

4u---------=

δslipεsdbfs

8u--------------=

Δ slip first yield aθslip first yieldadbfsφfirst yield

8u-----------------------------------= =

Δslipadbfsφy

8u------------------=

fc′ fc′fc′ fc′

fc′fc′

Fig. 6—Deformations due to bar slip.


To capture the effect of bar slip within the linear range ofresponse, it is possible to include rotational springs at theends of the column elements to directly model the additionalflexibility from the slip of the longitudinal bars. The springstiffness can be determined as

(11)

According to this approach and neglecting shear deformations,the effective stiffness of the column element, acting in serieswith the bond element, can be taken as EIflex from a moment-curvature analysis (Fig. 4).

Contribution of components to total yield displacement

As shown in Fig. 7, the contributions of flexure (Eq. (5)),shear (Eq. (6)), and bar slip (Eq. (10)) varied consistentlywith the axial-load ratio and aspect ratio. For both the rectangularand circular columns, the flexural mode of deformationcontributed approximately 50 to 100% of the total deformation,depending on the level of axial load and the aspect ratio. Theslip contribution ranged from 0% for columns with high

kslip8udb fs---------

M0.004

φy

-------------- 8udbfs

---------EIflex==

axial loads to approximately 40% for stocky columns withlow axial loads. The results shown in Fig. 7 also indicatethat, except for stocky columns with high axial loads,shear deformations contribute less than 15% of the yielddisplacement for the columns in the database.

Effective stiffnessFor engineering practice, it is convenient to use a single

effective stiffness for a column element. Expressing EIeff calcas fraction of EIg and substituting Eq. (4) through (6) and(10) for Δy, Eq. (1) can be expressed as a function of non-dimensional parameters

(12)

where α =EIflex /EIg and rv is the radius of gyration of thecolumn section in the direction of loading (rv

2 = Ig/Av). For anaverage bond stress value of u = 0.8 MPa (u = 9.6 psi),an effective shear modulus, Geff , equal to one half the elasticvalue, and using moment-curvature analysis to compute α andfs /fy, Fig. 8 and Table 3 show the ratio of the measured effectivestiffness to the effective stiffness determined using Eq. (12)for the column databases.

EFFECTIVE STIFFNESS MODELS FOR PRACTICEFor many practical situations, particularly those in which

the column reinforcement has not yet been selected, it ispreferable to use a version of Eq. (12) that does not requiremoment-curvature analysis. This section evaluates newmodels for effective stiffness that include the influence of barslip. The models correspond to simplifications of Eq. (12), inwhich the results of moment-curvature analysis (that is, αand fs /fy) are approximated and small terms are neglected.

According to Eq. (12), the ratio of the effective stiffness tothe gross-section stiffness is proportional to the normalizedflexural rigidity, α. This normalized flexural rigidity varies

EIeff calc

EIg

-------------------- α

1 38---

db

D-----D

a----

fs

fy

---fy

u--- 18

5------α

rv

D----⎝ ⎠

⎛ ⎞2 D

a----⎝ ⎠

⎛ ⎞ 2 Ec

Geff

---------+ +

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

fc′ fc′

Table 3—Statistics for ratio of measured to calculated effective stiffness for proposed models

Model

Circular columns (108 specimens) Rectangular columns (221 specimens)

Minimum Maximum Mean MedianCoefficient of variation, % Minimum Maximum Mean Median

Coefficient of variation, %

Equation (12) 0.50 1.69 1.04 1.03 21.4 0.45 1.84 0.97 0.92 26.6

Equation (16) 0.63 1.80 1.04 1.00 22.2 0.48 1.47 0.93 0.89 26.6

Equation (17) 0.64 1.76 1.04 0.99 23.5 0.48 1.68 0.92 0.91 26.9

Equation (18) with average bar size 0.57 1.59 1.02 1.00 22.0 0.46 1.63 0.95 0.94 25.5

Fig. 7—Contribution of calculated deformation componentsas function of axial load and aspect ratio.

Fig. 8—Comparison of calculated (Eq. (12)) and measuredeffective stiffnesses.


primarily with the level of axial load, but also with theamount of longitudinal reinforcement. Assuming a linearstress-strain relationship for the concrete and steel, α can beexpressed in terms of the normalized initial strain due to theaxial load, (P/AgEc)/εo, and the relative stiffness of longitudinalreinforcement, ρn. The normalized flexural rigidity can beapproximated as

(13)

The form of Eq. (13) was derived by simplifying (usingthe binomial theorem) an analytical solution for the moment

αapprox 0.2 1.3P AgEc⁄

εo

-------------------⎝ ⎠⎛ ⎞ ρn 1.0≤++=

of inertia of a generic cracked cross section with axial load.The coefficients in Eq. (13) were selected to provide the bestpossible estimate of the flexural rigidity determined frommoment-curvature analysis (Fig. 4). Equation (13) provides areliable substitute for moment-curvature analysis for a widerange of rectangular and circular columns. The ratio αapprox /α(where α is determined from a moment-curvature analysis) hasa mean and coefficient of variation of 0.96 ± 10.6% forrectangular columns and 1.04 ± 9.5% for circular columns.For all of the 329 rectangular and circular columns, this ratioranged from a minimum of 0.69 to a maximum of 1.23.

If the reinforcement ratio has not yet been established, αcan be approximated by substituting an average value of ρnof 0.15 into Eq. (13), which results in the following relationship

(14)

Using Eq. (14), the mean and coefficient of variation forthe rectangular and circular column databases are 0.95 ± 14.4%and 1.04 ± 20.7%, respectively. For the complete database,the ratios ranged from 0.59 to 1.63. Equation (14), whereasless accurate than Eq. (13), provides a good estimate of thenormalized flexural rigidity for cases where the longitudinalreinforcement ratio is not yet known.

As with α, the ratio of the steel stress at column yield tothe yield stress of the steel (fs /fy) can also be calculated withmoment-curvature analysis. Alternately, it is possible toapproximate this ratio by taking advantage of its dependenceon the level of axial load. Figure 9, which compares the steelstress ratio determined based on moment-curvature analysiswith the normalized initial strain (P/AgEc)/εo for the databasecolumns, indicates that the steel stress ratio can be approximatedfor both circular and rectangular columns as follows

(15)

Equation (12) can be simplified by eliminating thethird term in the denominator, which is related to sheardeformations and tends to be small (refer to Fig. 7), byapproximating the flexural rigidity with Eq. (13), and byapproximating the steel stress ratio with Eq. (15). Takingadvantage of these relationships, the need for moment-curvature analysis can be eliminated, and the effectivestiffness can be approximated as

(16)

The two coefficients in Eq. (16) (1.5 and 110) were calibratedto compensate for the elimination of the shear term and toachieve a good match with the measured effective stiffnessfor the rectangular and circular column databases. As shownin Table 3, Eq. (16) provides similar levels of accuracy asEq. (12), without requiring moment-curvature analysis.

Using Eq. (14) (instead of Eq. (13)) to estimate α, providesa model that does not require knowledge of the longitudinalreinforcement ratio without a significant decline in themodel accuracy (refer to Table 3).

αapprox 0.35 1.3P AgEc⁄

εo

-------------------⎝ ⎠⎛ ⎞ 1.0≤+=

0.0 fs fy approx43--- 10

3------–

P AgEc⁄εo

-------------------⎝ ⎠⎛ ⎞ 1.0≤=⁄≤

EIeff calc

EIg

---------------------1.5αapprox_Eq . 13

1 110db

D-----⎝ ⎠

⎛ ⎞ Da----⎝ ⎠

⎛ ⎞fs

fy approx_Eq . 15-----------------------------------

⎝ ⎠⎜ ⎟⎛ ⎞

+

---------------------------------------------------------------------------------------- 1.0 and 0.2≥≤=

Fig. 9—Approximation for steel stress as function of axial load.

Fig. 10—Comparison of calculated (Eq. (18)) and measuredeffective stiffnesses.


(17)

The procedure can be simplified further by expressing theaxial load as a fraction of Ag fc′ and recalibrating theconstants, which results in the following relationship

(18)

To avoid the prior selection of the size of the longitudinalbars, the db /D term in Eq. (18) can be estimated prior todesign for a given application. For bridge columns, thelongitudinal bars tend to be well distributed around thecolumn cross section, resulting in a smaller db/D term comparedwith building columns, where higher axial loads andarchitectural constraints tend to result in larger bar sizes relativeto the column dimension. Based on the db/D values for thecircular and rectangular column databases (Table 1), db/D can betaken as 1/25 for bridge columns and 1/18 for building columns.

The form of Eq. (18) is consistent with the theoreticalformulation of effective stiffness based on Eq. (12) and thesummation of deformation components (Eq. (4) through (6) and10). It provides a simple estimate of the effective stiffnesswithout requiring a moment-curvature analysis or selection ofthe longitudinal reinforcement ratio. Figure 10 shows the ratio ofthe measured effective stiffness for the columns in thedatabases to the effective stiffness determined using Eq. (18)and the recommended average values for db/D. The lack oftrends in Fig. 10 suggests that Eq. (18) properly accounts for thedependence of the effective stiffness on column axial load andaspect ratio. Furthermore, the data shows no bias with respect tothe longitudinal reinforcement ratio. As shown in Table 3,Eq. (18) provides accuracy statistics that are consistent withthose found for the much more complex Eq. (12).

CONCLUSIONSThe effective stiffnesses of 108 spiral-reinforced columns

(with circular and octagonal cross sections) and 221 rectangularcolumns were estimated from data in the PEER StructuralPerformance Database (Berry et al. 2004). These data show thatthe normalized effective stiffness of the columns increases withincreasing axial load, a trend that is reflected in many designguidelines and codes. The data also show that EIeff /EIgdecreases with decreasing span-to-depth ratio, particularly forlow axial loads.

Seven existing models for column effective stiffness wereevaluated using the column databases. The existing modelswere generally based on establishing an estimate for theflexural rigidity and ignored the influence of bar slip andshear deformations. All of the models tend to overestimatethe measured effective stiffness and resulted in coefficientsof variation ranging from 35 to 58%.

This paper presented a three-component model, whichexplicitly combines the effects of flexure, bar slip, and shearcomponents of deformation. This model reproduces thetrends observed in the data and it provides an accurate estimateof column stiffness. For this model, the ratio of the measuredeffective stiffness to the calculated stiffness has a mean and

EIeff calc

EIg

---------------------1.5αapprox_Eq. 14

1 110db

D-----⎝ ⎠

⎛ ⎞ Da----⎝ ⎠

⎛ ⎞fs

fy approx_Eq. 15-----------------------------------

⎝ ⎠⎜ ⎟⎛ ⎞

+

---------------------------------------------------------------------------------------- 1.0 and 0.2≥≤=

EIeff calc

EIg----------------------

0.45 2.5P Ag fc′⁄+

1 110db

D-----⎝ ⎠

⎛ ⎞ Da----⎝ ⎠

⎛ ⎞+

-------------------------------------------- 1.0 and 0.2≥≤=

coefficient of variation of 0.97 ± 27% for rectangularcolumns and 1.04 ± 21% for the circular columns.

For the purpose of design, this paper simplifies the three-component model, with little loss of accuracy, by introducing anapproximation to moment-curvature analysis, and byaccounting for the effects of bar slip in terms of span-to-depth ratio and axial-load ratio. According to this procedure, theeffective column stiffness at yield can be estimated as

(18)

where the db/D can be approximated as 1/25 for bridgecolumns and 1/18 for building columns. This model, appropriatefor implementation in design codes for concrete structures, ismore accurate than existing models. Implementation of theproposed model resulted in a mean and coefficient of variationfor the ratio of the measured effective stiffness to the calculatedstiffness of 0.95 ± 25.5% for rectangular columns and 1.02 ±22.0% for the circular columns.

ACKNOWLEDGMENTSThis work was supported in part by the Earthquake Engineering Research

Centers Program of the National Science Foundation, under Award NumberEEC-9701568 through the Pacific Earthquake Engineering Research Center(PEER) and Natural Science and Engineering Research Council (NSERC)of Canada. The views expressed are those of the authors and not necessarilythose of organizations cited here.

NOTATIONAg = gross cross-sectional area of columnAsl = total area of longitudinal reinforcementAv = effective shear area of column cross sectiona = shear spanaF0.004 = moment at maximum concrete compressive strain of 0.004aFfirst yield = moment at first yield (Fig. 4)b = width of rectangular column sectionc = distance from tension reinforcement to neutral axisD = diameter (circular column) or column depth in direction

of loading (rectangular column)db = nominal diameter of longitudinal barsEc = concrete modulus of elasticityEs = reinforcing steel modulus of elasticityEIeff meas = measured effective stiffness (Eq. (1))EIflex = effective flexural stiffness (Fig. 4)EIg = gross bending stiffnessEIg,t = gross transformed bending stiffnesse/D = eccentricity ratio = M0.004 /PDF0.004 = effective force at maximum compressive strain of 0.004

(Fig. 1)Ffirst yield = effective lateral force at first yield (Fig. 1)Fmax = maximum measured effective forcefc′ = concrete compressive strength (150 x 300 mm [6 x 12 in.]

cylinders)fs = longitudinal reinforcement steel stress at column fixed endfy = longitudinal reinforcement yield stressGeff = effective shear modulusl = length over which bond stress acts (Fig. 6)M0.004 = aF0.004 = moment at maximum compressive strain at

0.004Mfirst yield = aFfirst yield = moment at first yield (Fig. 4)n = modular ratio (Es/Ec)P = axial load (positive is compression)Pb = axial compression at balanced failure conditionPo = nominal axial load strength at zero eccentricityRc = correlation coefficient for circular columnsRr = correlation coefficient for rectangular columnsrv = radius of gyration using shear area (rv

2 = Ig/Av)Ts = tension force in one longitudinal bar (Fig. 6)u = constant bond stress

EIeff calc

EIg----------------------

0.45 2.5P Ag fc′⁄+

1 110db

D-----⎝ ⎠

⎛ ⎞ Da----⎝ ⎠

⎛ ⎞+

-------------------------------------------- 1.0 and 0.2≥≤=


v = maximum nominal shear stress (Fmax /Ag)α = normalized effective flexural stiffness = EIflex/EIgΔslip = slip of longitudinal reinforcing bar (Fig. 6)Δflex = displacement at yield due to flexural deformations (Eq. (5))Δshear = displacement at yield due to shear deformations (Fig. (6))Δslip = displacement at yield due to bar-slip deformations

(Eq. (10))Δslip first yield = displacement at first yield due to bar-slip deformations

(Eq. (9))Δy = displacement at yield (refer to Fig. 1)εo = nominal strain at which concrete is assumed to yield

(0.002 in this study)εs = longitudinal reinforcement steel strain at column fixed

endφy = curvature at yield (Fig. 4)φfirst yield = curvature at first yield (Fig. 4)θslip = rotation at end of column due to slip of reinforcing bars

(Fig. 6)ρ = longitudinal reinforcement ratio (Asl /Ag)

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effective stiffness of reinforced concrete columns

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