99-s41

ACI Structural Journal/July-August 2002 399

ACI Structural Journal, V. 99, No. 4, July-August 2002.MS No. 01-231 received August 9, 2001, and reviewed under Institute publication

policies. Copyright © 2002, American Concrete Institute. All rights reserved, includ-ing the making of copies unless permission is obtained from the copyright proprietors.Pertinent discussion will be published in the May-June 2003 ACI Structural Journal ifreceived by January 1, 2003.

ACI STRUCTURAL JOURNAL TECHNICAL PAPER

Comparisons of strengths determined from 354 physical tests ofrectangular reinforced concrete columns available in the literaturewith the strengths calculated from selected computational proce-dures were conducted. The computational procedures compared inthis study include ACI 318-99, Eurocode 2, and a commerciallyavailable nonlinear finite element modeling software. The physicaltests used for comparison were conducted on tied, reinforcedconcrete columns that were pinned at both ends and subjected toshort-term loads, producing pure axial force or axial force com-bined with symmetrical single-curvature bending. The studyincluded only those columns for which the complete informationrequired for analysis was available from the published physicaltest data and for which the compressive strength of normal densityconcrete ranged from approximately 2500 to 8500 psi (17 to 58 MPa).No further physical tests were conducted as part of this study.Major variables include the concrete strength, the end eccentricityratio, the slenderness ratio, the reinforcing steel index, and thetransverse reinforcement (tie/hoop) volumetric ratio. The comparativestudy provides a critical review of the reliability of the computationalmethods examined. Most of these methods are affected to some degreeby at least some of the major variables studied. A recommendation forimproving the ACI 318-99 procedure is also presented.

Keywords: column; reinforced concrete; strength; test.

INTRODUCTIONA comparative study of selected methods used for computing

the strengths determined from physical tests of rectangularreinforced concrete columns was undertaken. Physical testsincluded in this study involve tied columns made of normaldensity concrete. The computational methods comparedinclude ACI 318-99 (ACI Committee 318 1999), Eurocode 2(European Committee for Standardization 1992), and acommercially available nonlinear finite element modelingsoftware (ABAQUS 1994a,b).

The ACI design method is strongly influenced by the columneffective flexural rigidity (EI), which varies due to cracking,creep, and the nonlinearity of the concrete stress-strain curve. Inan attempt to account for some of these variables, Mirza (1990)proposed a refined equation for calculating the flexural rigidityof reinforced concrete slender columns designed accordingto the ACI procedure. This equation is also included in thecomparative study reported herein.

During the past 10 to 15 years, commercial finite elementmodeling (FEM) software has become more readily available,and its use by design engineers has been steadily increasing.Presently, there are several FEM programs that are able tomodel the reinforced concrete column strength. In an attemptto examine the applicability of FEM for computing thestrength of reinforced concrete columns, the results from acommercially available nonlinear FEM software (ABAQUS

1994a,b) were also compared with the physical tests.

To determine the influence of a full range of variables onthe computational methods examined, 354 physical tests ofrectangular reinforced concrete columns were taken from25 studies reported in the literature. Due to practical reasons andconcrete stress block modeling implications, it was decided notto include columns with concrete strengths of less than 2500 psi(17 MPa) or more than 8500 psi (58 MPa). Furthermore,only those columns for which the complete informationrequired for strength calculations was available in thepublished literature were included. The columns investigatedwere pinned at both ends and subjected to short-term loads,producing pure axial force or axial force combined withsymmetrical single-curvature bending. No new physicaltests were conducted for this study.

The major variables investigated in this study included theconcrete strength fc′, the end eccentricity ratio e/h, the slen-derness ratio l/h, the reinforcing steel index ρrs fy /fc′, and thetransverse reinforcement (tie/hoop) volumetric ratio ρ′′,where e = eccentricity of the axial load at column ends; h =overall depth of the flexural rigidity area taken perpendicularto the axis of bending; l = length of the column; fy = yieldstrength of reinforcing bars; and ρrs = ratio of the cross-sectional area of longitudinal reinforcing bars to the grossflexural rigidity area (longitudinal reinforcement ratio).Based on statistical analyses of the ratios of tested-to-computed strengths (strength ratios), comparisons of differentcomputational methods used, as well as evaluations of majorvariables affecting the strength, were conducted. Most of thesemethods were affected to some degree by at least some of themajor variables studied. The evaluations and comparisonsprovide an insight for critical review of the variability andother statistics related to the computational methods examined,and are presented and discussed in this paper. The results ofthe study are limited to normal-strength concrete. A similarstudy of high-strength concrete columns is currently in itsinitial stages at Lakehead University.

RESEARCH SIGNIFICANCEThis paper provides a critical review of procedures used for,

and related statistics of strength computations of, reinforcedconcrete columns. The evaluations and comparisons ofcomputational procedures presented in the paper provideguidance for the reliability of several design methods usedfor reinforced concrete columns. A recommendation forimproving the ACI procedure is presented.

Title no. 99-S41

Comparative Study of Strength-Computation Methods for Rectangular Reinforced Concrete Columnsby S. Ali Mirza and Edward A. Lacroix

ACI Structural Journal/July-August 2002400

SUMMARY OF PHYSICAL TESTS USEDThe experimental data used for this study were taken from

354 physical tests reported by Bresler (1960); Bresler andGilbert (1961); Bunni (1975); Chang and Ferguson (1963);Cusson and Paultre (1992); Drysdale and Huggins (1971);Ernst, Hromadik, and Riveland (1953); Fang, Hong, and Wu(1994); Gaede (1958); Goyal and Jackson (1971); Green andHellesland (1975); Heimdahl and Bianchini (1975); Hognestad(1951); Hudson (1965); Kim and Yang (1995); Martin andOlivieri (1965); Mehmel et al. (1969); Pfister (1964); Ramuet al. (1969); Razvi and Saatcioglu (1989); Roy and Sozen(1964); Scott, Park, and Priestley (1982); Sheikh andUzumeri (1980); Todeschini, Bianchini, and Kesler (1964);and Viest, Elstner, and Hognestad (1956). These tests were con-ducted on reinforced concrete column specimens subjected topure axial load or axial load combined with symmetrical single-curvature bending. The failure strength of a test column wastaken as the peak strength reached on the load-deflectionresponse. The geometric and material properties of the testspecimens are summarized in Table 1 and cover significantlylarge ranges of the column cross section size, fc′, e/h, l/h, fy, ρrs,and ρ′′. This helped to examine the effects of different variableson the strength of reinforced normal-strength concrete columns.

In this study, the concrete strength fc′ was defined as thestrength obtained from the standard (6 in. [150 mm] diameterby 12 in. [300 mm] high) cylinder tests, or as the equivalentstandard cylinder strength computed from cube tests. Forsome of the physical tests, the cube test strengths instead ofthe cylinder test strengths were reported. In such cases, thereported strengths were converted to the equivalent standardcylinder strengths by employing the following procedure.Equation (1) was used to first convert the strength of a cubeof a given size to the strength of a 4 in. (100 mm) cube, andthen to convert the strength of the 4 in. (100 mm) cube to thestrength of a 6 in. (150 mm) cube

(1)

in which fo and vo are the concrete strength and volume of a4 in. (100 mm) cube, respectively; and f and v represent theconcrete strength and volume of the cube of a given size,respectively. Equation (1), which is based on the statisticaltheory of brittle fracture of solids, was first presented byBolotin (1969) and is also documented by Mirza, Hatzinikolas,and MacGregor (1979). Once the strength of an equivalent6 in. (150 mm) concrete cube was obtained, it was convertedto the equivalent standard cylinder strength by using Eq. (2),proposed by L’Hermite (1955)

(2)

f fo 0.58 0.42vo

v----

1 3⁄

+=

fc′ 0.76 0.2log10fcu

2840------------

+ fcu=

in which fcu is the strength of a 6 in. (150 mm) cube. For SIunits, replace 2840 psi with 19.6 MPa.

The longitudinal reinforcing steel yield strengths fy weretaken as those reported for bar sample tests in individualstudies available from the literature. The transverse reinforce-ment yield strengths were not reported for some studies. Inthose instances, the yield strength of the transverse reinforce-ment was assumed to be equal to the yield strength of the longi-tudinal reinforcing bars.

DESCRIPTION OF COMPUTATIONALMETHODS EXAMINED

The strengths of column specimens were computed usingACI 318-99, Eurocode 2, and a nonlinear FEM procedure. Inthese computations, the understrength factors φ for ACI 318-99and the so-called partial safety factors (material resistancefactors) for Eurocode 2 were taken equal to 1.0. Similarly,no understrength factors, material resistance factors, or both,were applied for computing strengths from the nonlinearFEM procedure. The computed strengths, therefore, representthe unfactored strengths in this study.

ACI 318-99 procedureThe ACI unfactored axial load strengths for given (test) e/h

ratios were computed from the cross section and column axialforce-bending moment-strength interaction curves, similarto the ones shown in Fig. 1. These strength-interactioncurves were generated for each test column used in this studyand were based on the equilibrium of forces and the compat-ibility of strains. The assumptions given in ACI 318-99 (ACICommittee 318 1999), including the maximum usable strainof 0.003 at the extreme concrete compression fiber, theequivalent rectangular compressive stress block with a stressordinate of 0.85fc′ for concrete, the elastic perfectly plasticstress-strain relationship for reinforcing steel, and the momentmagnifier equations for slenderness effects, were used incomputations of the ACI strengths. The following ACI lim-itations, however, were ignored in this study:

1. ACI 318-99 specifies that the maximum axial load actingon a tied reinforced concrete column shall be limited to 0.8φPo ,where Po is the column cross section pure axial loadstrength, computed using

Po = 0.85fc′ (Ag – Ast) + fyAst (3)

in which Ag = gross area of the column cross section; andAst = total cross-sectional area of the longitudinal reinforcement.The ACI upper limit on axial load is intended primarily toaccount for accidental eccentricities not considered in the

S. Ali Mirza, FACI, is a professor of civil engineering at Lakehead University, ThunderBay, Ontario, Canada. He is a member and past chair of Joint ACI-ASCE Committee441, Reinforced Concrete Columns; and is a member of ACI Committees 335, Com-posite and Hybrid Structures; and 340, Design Aids for ACI Building Codes. He was arecipient of the ACI Structural Research Award in 1990.

Edward A. Lacroix is a project engineer, Walters Inc., Hamilton, Ontario, Canada.He received his MSc from the University of Manitoba. His research interests includereinforced concrete and composite columns.

Table 1—Summary of geometric and material properties of reinforced concrete specimens used*

Properties Minimum values Maximum values

b x h, in. x in. 3.0 x 3.0 17.7 x 17.7

fc′, psi 2550 8246

e/h 0.00 1.25

l/h 2 40

fy, psi 39,503 104,500

ρrs, % 0.80 7.06

ρ′′, % 0.04 4.73*Number of specimens = 354; h = depth of concrete cross section perpendicular toaxis of bending; and b = width of concrete cross section parallel to axis of bending.Note: 1.0 in. = 25.4 mm; 1000 psi = 6.895 MPa.


design but that may exist in a real structure. As the columnspecimens used in this study were prepared and tested undercontrolled laboratory conditions, this limit was not used forcomputing the ACI strengths;

2. ACI 318-99 also specifies that the cross-sectional areaof longitudinal reinforcing bars shall be not less that 0.01Ag.The ACI lower limit on Ast is intended primarily to guardagainst yielding of the longitudinal reinforcement due tocreep and shrinkage of the concrete under sustained compres-sive stresses at service loads. Because the column specimensused were subjected to short-term loads, this limit was notapplied to the ACI computations in this study; and

3. ACI 318-99 further specifies that the slenderness ratiol/h of pin-ended columns subjected to symmetrical single-curvature bending shall be not more than 30 when suchcolumns are designed using the moment magnifier method.The ACI upper limit on l/h is intended to represent the upperrange of physical tests of slender columns used in calibrationstudies when the moment magnifier method was developed.As some test results for columns with l/h greater than theACI upper limit are now available in the literature, this limitwas ignored herein to investigate whether the momentmagnifier approach was applicable to columns that exceededthe maximum l/h permitted by ACI.

The cross section strength interaction curve (Fig. 1) foreach test column was represented by 102 points. The firstpoint was determined from Eq. (3) for pure axial loadstrength (e/h = 0), the last point was computed for pure bendingmoment strength (e/h = ∞), and the remaining 100 pointswere distributed uniformly along the interaction curve betweenthese two points. The area of concrete displaced by reinforcingbars was included in computations.

The column (member) axial load-bending moment inter-action curve (Fig. 1) was developed from the cross-sectionalstrength interaction curve. The pure axial load strength of acolumn was first determined as the lesser of the twostrengths computed from Eq. (3) and (4)

Pc = Π2EI/l2 (4)

in which Pc = critical load of a pin-ended column. This estab-lished the maximum axial load that could be applied on thecolumn. The cross-sectional bending moment resistance Mcswas then divided by the moment magnification factor δns toobtain the column bending moment resistance Mcol for eachpoint on the cross-sectional interaction curve with axial loadPu less than the column pure axial load strength, as indicatedin Fig. 1. The δns factor used in this study was developed forpin-ended columns subjected to symmetrical single-curvaturebending from the moment magnifier equations of ACI 318-99,and was taken as

(5)

Note that the stiffness reduction factor of 0.75 used in theACI δns expression was not included in Eq. (5) for reasonsdescribed in a previous part of this section. ACI 318-99permits two alternative EI expressions for use in Eq. (4) and(5). For columns subjected to short-term loads and used inthis study, the ACI EI equations simplify to

δns1

1Pul

2

Π2EI-------------

–

---------------------------- 1.0≥=

EI = 0.4EcIg (6)

EI = 0.2EcIg + EsIse (7)

in which Ec = modulus of elasticity of concrete taken equal to57,000 psi (4733 MPa); Es = modulus of elasticityof reinforcing steel bars taken as 29,000,000 psi (199,955 MPa);and Ig and Ise = moment of inertia of the gross cross sectionand of the reinforcing bars, respectively, taken about thecentroidal axis of the cross section. In an attempt to take intoaccount the cracking and nonlinearity of the concrete stress-stain curve in determining EI, Mirza (1990) proposed thefollowing equation for calculating the flexural rigidity ofreinforced concrete columns subjected to short-term loads

(8)

This equation was also included in the comparative studyreported herein. Hence, the ACI column strengths were com-puted in three different ways, using Eq. (6), (7), and (8).

For short columns (l/h ≤ 6.6 for the type of columns studied),the column strengths were taken equal to the cross-sectionalstrengths as permitted by ACI 318-99. Hence, the axial loadstrength Pu of a short column was computed through inter-polation from the points generated on the cross-sectional in-teraction curve, using the e/h ratio evaluated from the testdata. Similarly, the axial load strength Pu of a slender column(l/h > 6.6 for the type of columns studied) was determinedfrom the column interaction curve using the test e/h ratio, asindicated in Fig. 1. Software was developed for computingthe ACI unfactored strengths based on the ACI 318-99assumptions and procedure given in this section. Furtherdetails are documented elsewhere (Lacroix and Mirza 1998).

Eurocode 2 procedureThe Eurocode 2 (1992) unfactored axial load strengths

were computed in a similar manner as that described in theprevious section. The cross-section and column strength

fc′ fc′

EI 0.3 1 eh---–

EcIg EsIse+ EsIse≥=

Fig. 1—Schematic cross section and column axial load-bending moment interaction diagrams.

402 ACI Structural Journal/July-August 2002

interaction curves were generated for each test specimen andwere based on the assumptions and requirements of Euro-code 2. However, the following exceptions were applied tothese computations. Eurocode 2 places an upper limit of7300 psi (50 MPa) on fc′ , limits the smaller cross section di-mension to a minimum of 8 in. (200 mm), and requires aminimum end eccentricity ratio e/h of 0.05 and 0.10 for designof short columns (l/h < 7.5 for the type of columns studied)and slender columns (l/h ≥ 7.5 for the type of columns studied),respectively. As the column specimens used in this study wereprepared and tested under laboratory conditions, these require-ments were ignored for computing the Eurocode 2 strengths.

The Eurocode 2 cross section strength interaction curve(Fig. 1) for each test column was defined also by 102 pointsgenerated using a similar procedure as that described forACI. The Eurocode 2 assumptions for computing the cross-sectional strengths are identical to those of ACI 318-99, withthe exception that Eurocode 2 uses the maximum strain at theextreme compression fiber of concrete equal to 0.0035, andthe depth of concrete stress block equal to 0.8 times the dis-tance from the extreme compression fiber to the neutral axis.

The Eurocode 2 column (member) strength interactioncurves (Fig. 1) were developed also from the cross-sectionalstrength interaction curves. The Eurocode 2 method for com-puting second order effects in columns, however, is differentfrom that of ACI 318-99. Eurocode 2 uses a model column,which is an isolated cantilever fixed at the base and free atthe top, as illustrated in Fig. 2. The model column is assumedto be bent in single curvature under the applied axial load andbending moment, producing the maximum bending momentat the base. The column is designed for a total eccentricityetot, which consists of: a) the equivalent first order eccentricitythat includes the effect of moment gradient; b) the eccentricitycaused by initial imperfections; and c) the second ordereccentricity acting on the failure cross section. For the typeof columns studied, etot was taken as

etot = e + ea + e2 (9)

where e = first order eccentricity of a column subjected toequal and opposite end moments; ea = additional eccentricitythat accounts for initial imperfections such as dimensionalinaccuracies and the uncertainty in the position of the line ofaction of the axial load; and e2 = second order eccentricity as

indicated in Fig. 2. For the purpose of analysis, Eq. (9) wasrearranged as

e = etot – (ea + e2) (10)

Equation (10) represents the relationship between the bend-ing moment strength of a column and that of its cross sectionin the sense that the column and cross-sectional bending mo-ment strengths (Mcol and Mcs in Fig. 1) can be computed asPue and Puetot, respectively. Hence, for each point on thecross section interaction curve, etot was computed as Mcs/Pu,and the sum of ea and e2 was subtracted from etot to obtainthe column end eccentricity e. The column bending momentstrength was then computed as Pue. Note that any points onthe cross section interaction curve that generated negativevalues of e were not considered for developing the columninteraction curve. The Eurocode 2 equations for ea and e2were simplified to Eq. (11) and (12) for the pin-ended columnsused in this study

(11)

(12)

where εy = yield strain of reinforcing steel; d = effectivedepth of the cross section in the direction of stability failure;k1 = coefficient that applies correction to e2 for columns withlow slenderness ratio = (l – 4.5h)/6h ≤ 1.0; and k2 = coefficientthat takes into account the decrease in curvature with increasingaxial force = (Po – Pu)/(Po – Pbal) ≤ 1.0; Po = pure axial loadstrength of the cross section computed from Eq. (3); and Pbal =axial load that maximizes the bending moment strength ofthe cross section computed from 0.4fc′ (Ag – Ast), as permittedby Eurocode 2. For /31.9 in Eq. (11), ea and l are in in.For SI conversion (ea and l in m), replace 31.9 by 200.

Eurocode 2 does not provide a specific equation for deter-mining the pure axial load strength of a column (member). Inthis study, the Eurocode 2 column pure axial load strengthwas defined as the condition when the column end eccentricitye, which is computed from Eq. (10), equals zero, and it wasdetermined by interpolating between two adjacent points onboth sides of the column pure axial load strength (Fig. 1).

For a short column (l/h < 7.5), second order effects wereneglected, and the axial load strength was determined for thetest e/h ratio from the cross section interaction curve, as per-mitted by Eurocode 2. Similarly, the axial load strength of aslender column (l/h ≥ 7.5) was computed for the test e/h ratiofrom the column interaction curve, as indicated in Fig. 1. Acomputer program was developed for computing the Euro-code 2 unfactored strengths based on the procedure andassumptions outlined previously. Further details are docu-mented by Lacroix and Mirza (1998).

Finite element modeling (FEM) methodThe FEM of the strengths and load-deflection responses of

reinforced concrete test columns was carried out by using acommercially available nonlinear FEM software (ABAQUS1994a,b). The software was capable of performing a staticsecond order strength analysis involving both the materialand geometric nonlinearities. It included an extensive libraryof predefined three-dimensional (space) beam sections that

eal

31.9---------- l

400---------≥=

e2 k1l

2

10------

2k2εy

0.9d-------------

=

l

Fig. 2—Model column used by Eurocode 2 for determiningsecond-order eccentricities.


were used—this greatly simplified the data input required tomodel the reinforced concrete column cross section.

A reinforced concrete column cross section was assumedto consist of three different materials, each represented by adifferent stress-strain curve. These materials included theunconfined concrete outside the transverse tie reinforce-ment, partially confined concrete within the transverse ties,and longitudinal reinforcing steel bars, as indicated in Fig. 3(a),(b), and (c), respectively. The unconfined concrete wasmodeled by using a thin-walled box beam section with theinner wall of the box beam section coinciding with thecenterline of the transverse tie reinforcement. The partiallyconfined concrete was modeled by using a rectangular beamsection with the outer edge of the rectangular beam sectioncoinciding with the centerline of the transverse tie reinforce-ment and the inner edge of the box beam section. The longi-tudinal reinforcing bars were modeled by superimposing barelements within the rectangular beam section mesh. The twobeam sections and bar elements were superimposed at commonnode points to fully define the column cross section, as illus-trated in Fig. 3(d). The FEM software accounted for the areaof concrete displaced by longitudinal reinforcing bars andnumerically integrated the cross section to obtain the gener-alized force/moment-strain/curvature relationships. As theintegration points are used to discretize a beam section andto define its mesh for numerical computations, an increase inintegration points leads to a denser mesh, resulting in a moreaccurate solution but a higher computational time. For thisstudy, the number of integration points was increased sub-stantially from the software default condition. The locationsand number of integration points used are shown in Fig. 3(a)through (c) for the three materials noted previously.

The modeling of a reinforced concrete column (member)was accomplished by using a predefined, three-dimensional,three-node beam (space) element. The length of the columnwas divided into a number of segments, each representing abeam element. Each beam element was connected to adjacentelements at the two outer (common) node points. The centralnode point of the element was generated and used for in-tegration purposes by the software. A typically discretizedcolumn subjected to symmetrical single-curvature bending isillustrated in Fig. 4(a). For the purpose of analysis, however,the symmetry about the midlength permitted the use of anequivalent cantilever column that was 1/2 the length of theoriginal column, as illustrated in Fig. 4(b), leading to sub-stantial savings in computational time. The boundary condi-tions at the top node of the column restrain movements alongthe x- and z- axes, and the rotation about the y-axis, whereasthose at the bottom node of the column restrain movementsalong the x-, y-, and z-axes and the rotation about the y axis(Fig. 4(a)). The node at the column midlength is restrainedfrom movements along and rotations about the y- and z-axes(Fig. 4(b)). These restraints modeled the boundary condi-tions used in physical tests of column specimens analyzed inthis study. To prevent the localization of the element curva-ture, the finite element length was taken equal to, or slightlygreater than, the overall depth of the cross section in theplane of bending, as indicated in Fig. 4(c). A column wasloaded by introducing a set of small applied axial load andbending moment at the top node (Fig. 4(b)), reflecting theend eccentricity used in the physical test of the columnspecimen. The applied axial load and bending momentwere then increased in increments of constant proportions,using a second order analysis procedure, until the failure

occurred. The FEM failure strength of a column was definedas the peak strength reached on the load-deflection responsecurve. For the analysis of columns subjected to pure axialload, an imperfection was added to the initially straight elementmodel to ensure a smooth transition from column stability tocolumn instability. This is due to the fact that a perfectlystraight column subjected to pure axial load remains straightuntil the critical load is reached and then buckles suddenly.The large deflections associated with this sudden bucklingcould not be properly captured using the FEM software. Inthis study, the initial imperfection in pure compressionmembers was introduced by applying a total transverseload equal to 1% of the self-weight of the column distributeduniformly over the entire column length.

For the FEM of cross sections, the elastic perfectly plasticstress-strain curves defined by measured values of yieldstrength fy and modulus of elasticity Es were used for reinforcingsteel bars. These fy and Es values were reported in individualstudies taken from the literature. The local buckling of thelongitudinal reinforcing steel bars was not considered in theFEM analysis because it was assumed to be resisted by thetransverse tie reinforcement and the concrete inside thetransverse ties. The descriptions of both the unconfined andpartially confined concretes in compression outside and

Fig. 3—FEM of reinforced concrete cross section: (a) uncon-fined concrete; (b) partially confined concrete; (c) longitu-dinal reinforcing steel; and (d) entire cross section.

Fig. 4—FEM of reinforced concrete column: (a) column insymmetrical single-curvature bending; (b) 1/2 length ofcolumn used for analysis; and (c) finite element segment.


inside the lateral ties, respectively, were taken from Park,Priestley, and Gill (1982), as illustrated schematically by thestress-strain curves in Fig. 5(a) and (b). The ascending parts,up to the peak stress, and the descending parts, beyond thepeak stress, in these curves are represented by second orderparabolas and straight lines, respectively. It should be notedthat εu in Fig. 5(a) is the ultimate compressive strain of un-confined concrete; and K in Fig. 5(b) is a factor that representsthe degree of concrete confinement provided by lateral ties.The concrete tensile stress-strain relationship used for theFEM of cross sections was represented by a bilinear curveshown schematically in Fig. 5(c). The ascending branch ofthis curve, up to the peak stress, was taken from Mirza andMacGregor (1989), and the descending branch representingtension softening after the cracking of concrete was takenfrom Bažant and Oh (1984). In Fig. 5(c), εfo is the strainwhere the tensile strength of concrete equals zero; and Et isthe tension softening modulus of concrete taken equal to70Ec/(57+ fr) psi (0.48Ec/[0.39 + fr] MPa). The crushing andcracking of concrete were, therefore, modeled using the ulti-mate compressive strain εu and the descending part of thestress-strain curve in Fig. 5(a), and the ultimate tensile strainεfo and the descending part of the stress-strain curve in Fig. 5(c).In an attempt to construct a finite element model as completeas is feasible, the physical properties of concrete were mod-ified to account for in-place conditions and rate-of-loadingeffects. Hence, the compressive strength fc′′ , the modulus ofelasticity Ec , and the modulus of rupture fr used for concrete

stress-strain curves in Fig. 5 were computed from Eq. (13)through (15)

fc′′ = 0.97fc′[0.89(1 + 0.08log10(0.97fc′/t))] (13)

Ec = 60,400 [1.16 – 0.08log10(t)] (14)

fr = 8.3 [0.96(1 + 0.11log10(8.3 /t))] (15)

in which fc′ = compressive strength of standard cylinders (inpsi) as tested or computed from Eq. (2); and t = time (in s)taken by a column specimen to reach its failure load. InEq. (13) through (15), the factor 0.97 represents the effect ofin-place casting, and the term inside the square bracketsrepresents the effect of rate of loading, as documented byMirza, Hatzinikolas, and MacGregor (1979). For SI (MPa)conversion, t should be multiplied by 0.0069 in Eq. (13);60,400 should be replaced by 5015 in Eq. (14); and 8.3should be replaced by 0.69 and t multiplied by 0.0069 inEq. (15). Further details are documented elsewhere (Lacroixand Mirza 1998).

STRENGTH ANALYSES AND COMPARISONSThe axial load strengths of 354 test specimens were calcu-

lated using the five different computational procedures. Thetested strengths were divided by the computed strengths toobtain the nondimensionalized strength ratios. These strengthratios were used for statistical analyses and strength comparisons.

Comparisons of tested and computed strengthsThe tested strengths are plotted against the strengths com-

puted from the ACI method using Eq. (6), the ACI methodusing Eq. (7), Eurocode 2, the ACI method using Eq. (8), andthe FEM in Fig. 6(a) through (e), respectively. The strengthratio statistics for each computational procedure are alsoshown on the figure, as is the histogram of the strength ratiosin the inset of the figure for that procedure. The diagonal(45-degree) lines in these figures represent the lines of equality.The data for the FEM plotted in Fig. 6(e) indicate the leastscatter about the line of equality, whereas the data for theACI method involving Eq. 6 plotted in Fig. 6(a) indicate thehighest scatter about the line of equality. The data for theremaining three computational procedures plotted in Fig. 6(b)through (d) show less scatter than that in Fig. 6(a) and morescatter than that in Fig. 6(e). These observations are evidentalso by the degree of tightness exhibited by the strength ratiohistograms plotted in the insets of Figs. 6(a) through (e). Themathematical confirmation of these observations is providedby the strength ratio statistics, particularly by the coefficientsof variation given in Fig. 6(a) through (e). The coefficient ofvariation of the strength ratios ranges from the highest valueof 0.22 obtained with the ACI method using Eq. (6), to thelowest value of 0.13 obtained for the FEM, with the coeffi-cients of variation for the other three computational methodsfalling between these two values as indicated in Fig. 6(a)through (e). It should be noted herein that a lower coefficientof variation signifies a more reliable computational methodas long as the average value does not deviate significantlylower than unity. The variabilities associated with the differentcomputational procedures studied can be seen more clearlyby comparing the cumulative frequency curves of strengthratios plotted on a normal probability scale in Fig. 7. Figure 6and 7 were prepared from the data for all 354 columns used,

0.97fc′

0.97fc′ 0.97fc′

Fig. 5—Schematic concrete stress-strain curves used forFEM: (a) unconfined concrete in compression; (b) partiallyconfined concrete in compression; and (c) reinforced concretein tension.


demonstrate the overall performance of computationalmethods examined, and lead to the following conclusions:

1. With an average strength ratio of 0.98 and a coefficientof variation of 0.13, the FEM computes the column strengthsmore accurately than all other computational proceduresstudied. This was expected because of the accuracy of theconcrete stress-strain curves (Fig. 5) used for the FEM aswell as the inherent accuracy of the FEM itself;

2. With an average strength ratio equal to 1.10 and a coeffi-cient of variation equal to 0.22, the ACI method involvingEq. (6) computes the column strengths least accurately ofall computational procedures studied;

3. The accuracy of the ACI method using Eq. (7) and thatof the Eurocode 2 method are approximately the same, asthese procedures produced an average strength ratio of 1.08and 1.12, respectively, with a coefficient of variation of 0.17in both cases; and

4. With an average strength ratio of 1.07 and a coefficientof variation of 0.14, the accuracy of the ACI method isimproved when Eq. (8) is used in place of Eq. (6) or (7). Thisis expected because Eq. (8) computes the flexural rigidity(EI) of reinforced concrete columns more accurately thandoes Eq. (6) or (7) (Mirza 1990).

Effects of major variablesThe effects of the end eccentricity ratio e/h, transverse

reinforcement (tie/hoop) volumetric ratio ρ′′, slenderness ratiol/h, reinforcing steel index ρrs fy /fc′, and concrete strength fc′on strength ratios obtained from different computationalmethods are shown in Fig. 8, 9, 10, 11, and 12, respectively.

Each of these figures was plotted from the data for all 354column tests and illustrates the strength ratios computedfrom: a) the ACI procedure using Eq. (6); b) the ACI methodinvolving Eq. (7); c) Eurocode 2; d) the ACI procedure usingEq. (8); and e) FEM.

As indicated by Fig. 8(e), 9(e), 10(e), 11(e), and 12(e), thestrength ratios obtained from FEM do not appear to be affectedby any of the variables investigated. In fact, the data plottedfor FEM are clustered almost equally on both sides of theline of equality (horizontal line of unity) over the entire

Fig. 6—Comparison of tested strengths with strengths computed from: (a) ACI using Eq. (6); (b) ACI using Eq. (7); (c) Euro-code 2; (d) ACI using Eq. (8); and (e) FEM. (Note: 1 kip = 4.448 kN.)

Fig. 7—Comparison of probability distributions of strengthratios computed from data for all columns studied (n = 354).


range of each of the variables examined. This is expected, asthe variables studied are explicitly or implicitly included inthe FEM analysis. The remaining four computational proce-dures are affected at least to some degree by some of thevariables examined, as indicated by Fig. 8 through 12.

The strength ratios for the ACI and Eurocode 2 procedurestend to decrease as e/h increases, with the highest variationsin strength ratios taking place for columns subjected to purecompression (e/h = 0), as indicated by Fig. 8(a) through (d).This was expected due to inherent inaccuracies of the codeequations in modeling the pure compression failures. Itshould be noted that the effect of e/h on the strength ratioscomputed from the ACI procedure using Eq. (8) is minimal

when e/h > 0 (Fig. 8(d)). This was also expected, as e/h isincluded in Eq. (8) used for computing the column EI. Figure 9(a)through (d) indicate an increase in strength ratios computedfrom the ACI and Eurocode 2 methods as ρ′′ increases. Asimilar trend can be seen in Fig. 10(a) through (d) with regardto l/h, although to a lesser degree. Again, these trends wereexpected because the ACI and Eurocode 2 methods do notaccount for the increase in column stiffness due to theslenderness effect (Mirza 1990) or the increase in columnstrength due to the concrete confinement provided by lateralreinforcement. As expected, the effects of e/h, ρ′′, and l/h arethe most pronounced for the strength ratios computed fromthe ACI method using Eq. (6) (Fig. 8(a), 9(a), and 10(a)),

Fig. 8—Effect of end eccentricity ratio e/h on strength ratios obtained from different computational methods: (a) ACI using Eq. (6);(b) ACI using Eq. (7); (c) Eurocode 2; (d) ACI using Eq. (8); and (e) FEM (n = 354).

Fig. 9—Effect of transverse reinforcement (tie/hoop) volumetric ratio ρ′′ on strength ratios obtained from different computationalmethods: (a) ACI using Eq. (6); (b) ACI using Eq. (7); (c) Eurocode 2; (d) ACI using Eq. (8); and (e) FEM (n = 354).


followed by those calculated from both the ACI procedureinvolving Eq. (7), and the Eurocode 2 method (Fig. 8(b)through (c), 9(b) through (c), and 10(b) through (c)), and be-come even less significant for the strength ratios computedfrom the ACI method using Eq. (8) (Fig. 8(d), 9(d), and10(d)). No clear effects of ρrs fy/fc′ and fc′ can be establishedfrom the strength ratios plotted in Fig. 11(a) through (d)and 12(a) through (d), respectively, for the ACI and Euro-code 2 procedures. Figure 8 through 12 and related discus-sions lead to the following conclusions:

1. The FEM strengths are not affected by any of the vari-ables examined in this study;

2. Whereas the ACI and Eurocode 2 strengths are affectedto varying degrees by e/h, ρ′′, and l/h, no apparent effects ofρrs fy /fc′ and fc′ on the ACI and Eurocode 2 strengths areevident within the examined ranges of these variables; and

3. The effect of e/h on strengths computed from the ACImethod involving Eq. (8) is minimal.

Strength ratio statistics for columns ofdifferent slenderness

For the type of columns used in this study, the ACI Code(ACI Committee 318 1999) permits the slenderness effectsto be neglected when l/h ≤ 6.6. The Eurocode 2 (EuropeanCommittee for Standardization 1992) raises this limit tonearly 7.5. In this study, the columns were categorized accordingto the ACI Code and are defined as short columns when l/h≤ 6.6, and as slender columns when l/h > 6.6. The strength

ratio statistics (average values and coefficients of variation)for the so-defined short and slender columns are given inTable 2 and 3, respectively. The short and slender columnsare further categorized into four groups each according to theirl/h and e/h ratios, as indicated in Table 2 and 3. It should benoted that Table 3 shows three sets, whereas Table 2 showsonly one set of ACI strength ratio statistics, as Eq. (6)through (8) do not affect the strength of short columns. Table 2and 3 lead to the following conclusions:

1. The ACI and Eurocode 2 methods produce almost iden-tical strength ratio statistics within each of the four groups ofthe short columns (Table 2). This was expected because ACIand Eurocode 2 use very similar analyses for computing thestrengths of column cross sections;

2. The FEM produces a somewhat lower average value buta similar coefficient of variation of the strength ratios than dothe ACI and Eurocode 2 procedures within each of the fourgroups of the short columns (Table 2). This was also expectedbecause the FEM analysis is more comprehensive than theACI or Eurocode 2 analyses, as described previously;

3. The FEM produces the least and the ACI Code involv-ing Eq. (6) produces the greatest variation in strength ratiostatistics (average values and coefficients of variation)among the four groups of the slender columns (Table 3). Thisreinforces a previous conclusion;

4. When Eq. (8) is used instead of Eq. (6) or (7), the co-efficient of variation of the ACI strength ratios reduces, albeit tovarying degrees, within each of the four groups of the slender

Fig. 10—Effect of slenderness ratio l/h on strength ratios obtained from different computational methods: (a) ACI using Eq. (6);(b) ACI using Eq. (7); (c) Eurocode 2; (d) ACI using Eq. (8); and (e) FEM (n = 354).

Table 2—Summary of strength ratio statistics for short columns (l /h ≤≤ 6.6)

Group no. Slenderness ratioEnd eccentricity

ratio No. of specimens

Strength ratio statistics*Tie/hoop

volumetric ratioACI Eurocode 2 FEM

1 l/h ≤ 3 e/h = 0 10 1.07 (0.084) 1.07 (0.084) 0.94 (0.064) ρ′′ = 0.39 – 2.5%

2 l/h ≤ 3 e/h > 0 5 1.10 (0.087) 1.10 (0.085) 1.06 (0.073) ρ′′ = 0.39 – 1.9%

3 3 < l/h ≤ 6.6 e/h = 0 79 1.14 (0.105) 1.14 (0.105) 0.99 (0.110) ρ′′ = 0.04 – 4.7%

4 3 < l/h ≤ 6.6 e/h > 0 19 0.94 (0.133) 0.90 (0.140) 0.92 (0.143) ρ′′ = 0.39 – 1.1%*Statistics shown are average values (and coefficients of variation in parentheses) of ratios of tested-to-computed strengths.


columns (Table 3). Again, this is expected for the reasonsdescribed previously; and

5. The ACI and Eurocode 2 methods are very conservativein computing the pure axial load capacity of the two test col-umns studied with l/h > 30, as indicated by the strength ratiostatistics of Group 3 columns in Table 3. The strength ratiostatistics are somewhat improved for the 21 similar columnssubjected to combined axial load and bending moment (Group 4,with l/h > 30 and e/h > 0, in Table 3). This is particularly validfor Eurocode 2 and the ACI procedure involving Eq. (8). Itshould be noted that the FEM produces a somewhat low averagevalue of the strength ratios for the Group 4 columns (Table 3).

CONCLUSIONSComparisons of physical test strengths of 354 rectangular

reinforced normal-strength concrete columns taken from theliterature with the strengths of the same columns calculatedfrom selected computational procedures are presented in thispaper. The columns examined were tied, pin-ended, and sub-jected to short-term loads, producing pure axial force or axialforce combined with equal and opposite end bending moments.

These comparisons indicate that: a) the strength ratios cal-culated by the ACI and Eurocode 2 procedures are affectedto a varying degree by the column end eccentricity ratio,transverse reinforcement volumetric ratio, and slendernessratio, whereas the strength ratios computed by the FEM are

Fig. 11—Effect of reinforcing steel index ρrsfy /fc′ on strength ratios obtained from different computational methods: (a) ACIusing Eq. (6); (b) ACI using Eq. (7); (c) Eurocode 2; (d) ACI using Eq. (8); and (e) FEM (n = 354).

Fig. 12—Effect of concrete strength fc′ on strength ratios obtained from different computational methods: (a) ACI using Eq. (6);(b) ACI using Eq. (7); (c) Eurocode 2; (d) ACI using Eq. (8); and (e) FEM (n = 354). (Note: 1000 psi = 6.895 MPa.)


not affected by any of the variables investigated; b) the ACImethod involving Eq. (6) computes the column strength ratiosleast accurately, and the FEM computes the column strengthratios more accurately than all other computational proceduresexamined; and c) the computational accuracies of the ACIprocedure using Eq. (7) and the Eurocode 2 method are similar.The comparisons also show that the accuracy of the ACI methodis improved when Eq. (8) is used instead of Eq. (6) or (7).

NOTATIONAg = gross area of column cross sectionAst = total area of longitudinal reinforcementAt = area of cross section of tie/hoop barb = overall width of column cross section taken parallel to axis of

bendingb′′ = outside width of ties/hoopsd = effective depth of cross section in direction of stability failured ′′ = outside depth of ties/hoopsEc = modulus of elasticity of concreteEI = effective flexural rigidity of reinforced concrete columnEs = modulus of elasticity of reinforcing steelEt = tension-softening modulus of concretee = eccentricity of axial load at column ends (first order eccentricity)e2 = second order eccentricity acting on failure cross sectionea = additional eccentricity that accounts for initial imperfectionsetot = total eccentricity of axial load Pu acting on failure cross sectione/h = end eccentricity ratiofc , ft = compressive, tensile stress of concrete that corresponds to given

value of strainfc′ = concrete strength from standard cylinder tests or equivalent

standard cylinder strength from cube testsfc′′ = compressive strength of unconfined concrete used for FEMfcu = strength of 6 in. (150 mm) concrete cubefo = strength of 4 in. (100 mm) concrete cubefr = modulus of rupture of concretefy = yield strength of reinforcing steel barsh = overall depth of column cross section taken perpendicular to

axis of bendingIg = moment of inertia of gross concrete cross section taken about

centroidal axisIse = moment of inertia of reinforcing steel taken about centroidal

axis of cross sectionK = factor representing degree of concrete confinement provided by

lateral ties used for FEMl = column lengthl/h = slenderness ratioMcol = bending moment strength of column (member) at axial load

equal to PuMcs = bending moment strength of column cross section at axial load

equal to Pun = number of test specimensPbal = axial load that maximizes bending moment strength of cross

sectionPc = critical load strength of pin-ended columnPo = pure axial load strength of cross sectionPu = axial load strengthsh = spacing of transverse ties/hoopst = testing time to failure, svo = volume of 4 in. (100 mm) concrete cubeδns = moment magnification factorεc = strain in concreteεfo = strain where tensile strength of concrete equals zeroεo, εot = compressive, tensile strain in concrete at peak stress, respectively

εu = ultimate compressive strain of unconfined concreteεy = yield strain of reinforcing steelφ = understrength (strength reduction) factorρ′′ = tie/hoop volumetric ratio taken equal to 2(b′′ + d′′)At/b′′d ′′shρrs = ratio of cross-sectional area of longitudinal reinforcing bars to

gross area of cross section (reinforcing steel ratio)ρrs fy /fc′= reinforcing steel index

REFERENCESABAQUS Version 5.4 Theory Manual, 1994a, Hibbitt, Karlsson, and

Sorensen, Inc., Pawtucket, R.I.ABAQUS Version 5.4 User’s Manual, 1994b, Hibbitt, Karlsson, and

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Table 3— Summary of strength ratio statistics for slender columns (l/h > 6.6)

Group no.

Slenderness ratio

End eccentricity ratio

No. ofspecimens

Strength ratio statistics*Tie/hoop

volumetric ratioACI using Eq. (6) ACI using Eq. (7) Eurocode 2 ACI using Eq. (8) FEM

1 6.6 < l/h ≤ 30 e/h = 0 72 1.14 (0.339) 1.05 (0.192) 1.17 (0.218) 1.04 (0.172) 0.97 (0.125) ρ′′ = 0.04 – 1.6%

2 6.6 < l/h ≤ 30 e/h > 0 146 1.06 (0.129) 1.06 (0.143) 1.10 (0.145) 1.05 (0.118) 1.01 (0.127) ρ′′ = 0.07 – 2.9%

3 l/h > 30 e/h = 0 2 2.34 (0.042) 2.20 (0.081) 1.90 (0.071) 1.78 (0.074) 0.97 (0.044) ρ′′ = 0.36%

4 l/h > 30 e/h > 0 21 1.12 (0.292) 1.09 (0.203) 1.19 (0.114) 1.05 (0.126) 0.88 (0.146) ρ′′ = 0.12 – 2.1%*Statistics shown are average values (and coefficients of variation in parentheses) of ratios of tested-to-computed strengths.

ACI Structural Journal/July-August 2002410

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