Engineering Structures 71 (2014) 73–87

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Engineering Structures

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Cm-factor for RC slender columns under unequal eccentricities and skewangle loads at the ends

http://dx.doi.org/10.1016/j.engstruct.2014.04.0200141-0296/� 2014 Elsevier Ltd. All rights reserved.

⇑ Tel.: +34 96 387 7007x75615; fax: +34 96 387 7569.E-mail address: jlbonet@cst.upv.es (J.L. Bonet).

Luiz C. Leite, José L. Bonet ⇑, Luis Pallarés, Pedro F. MiguelInstituto de Ciencia y Tecnología del Hormigón, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain

a r t i c l e i n f o

Article history:Received 29 May 2013Revised 4 March 2014Accepted 8 April 2014Available online 3 May 2014

Keywords:Cm-factorSlender columnHigh-strength concreteReinforced concreteBiaxial bendingUnequal load eccentricities

a b s t r a c t

Reinforced concrete slender columns, in general, are subjected to a non-uniform bending moment law,due to the occurrence of unequal eccentricities and skew angles at the ends. Design codes suggest calcu-lating an equivalent moment factor Cm in each bending direction with the aim of reducing the analysis ofthe column to another equivalent one with equal eccentricities and angles at the ends.

This paper presents a new equation to compute the equivalent moment factor Cm derived from thenumerical model. This model has been calibrated and checked with 255 experimental tests from the lit-erature. The experiments correspond to RC slender columns, pinned–pinned and subjected to short-termultimate loads and unequal eccentricities at the ends, for both uniaxial bending and biaxial bending. InHSC slender columns, a factor k3 is proposed to be applied along the concrete cover region in order to takeinto account the concrete cover spalling off.

From the comparison with the current formula of the design codes (ACI-318 and Eurocode 2), the pro-posed Cm increases the safety factor for low and medium slenderness of the column (kg < 20), low levels ofaxial load (N/Ncr < 0.50), high-strength concrete (50 < fc < 100 MPa) and double curvature bending (b < 0).The proposed formula allows reinforced concrete slender columns to be both checked and designed withenough accuracy for practice engineering, and is simple to apply.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Reinforced concrete slender columns, whether in buildingstructures or in civil works, are subjected to combined axial loadsand biaxial bending due to their position in the structure, the shapeof the section or the nature of the external actions. Moreover,depending on the load distribution and the support conditions atits ends, the eccentricities of the axial load applied at the endscan be either equal in direction, dimension and orientation, orunequal, in direction, dimension or orientation (Fig. 1a).

Different design codes (ACI-318 [1], EC-2 [15], EHE-08 [10])reduce the calculation of the original distribution of moments(Fig. 1a) to the calculation of an equivalent distribution ofmoments (Fig. 1b) where the equivalent eccentricity (ee, ae)applied at the ends must cause the same total bending momentin the column which would correspond to the original distributionof moments.

In combined axial loads and uniaxial bending, the design codescalculate the equivalent eccentricity ee by means of the equivalentmoment factor Cm, where:

ee ¼ Cm � e2 ð1Þ

where e2 is the maximum first-order eccentricity in absolute valueat the ends of the column.

In the case of combined axial loads and biaxial bending withunequal eccentricities and skew angles at the ends, the applicationof the equivalent moment factor Cm is generalized at each maininertia axis of the cross section of the column (x, y), see Fig. 1.

The components of the equivalent biaxial eccentricity ee in eachdirection (eex, eey) are calculated as follows:

eex ¼ Cmx � e2x ð2Þ

eey ¼ Cmy � e2y ð3Þ

where: Cmx, Cmy are the equivalent moment factors in each maindirection; e2x, e2y are the components of the first-order eccentricityof the highest absolute value in each main direction.

The modulus (ee) and the skew angle (ae) of the equivalent biax-ial eccentricity with respect to the bending x-axis are:

Nomenclature

List of symbolsAc gross area of the cross sectionb overall width of a cross-sectionCm equivalent uniform uniaxial bending moment diagram

factorCm,NS equivalent uniform uniaxial bending moment diagram

factor computed by the numerical modelCm,prop equivalent uniform uniaxial bending moment diagram

factor proposed by each authorcmec mechanical cover of the longitudinal reinforcementCmx; Cmy equivalent moment factor in each main directioncnom nominal concrete cover, distance between the surface of

the closer reinforcement and the nearest concrete sur-face

EI effective flexural stiffness of columne1 smallest first-order eccentricity at column ende2 largest first-order eccentricity at column end in abso-

lute valuee2x; e2y components of the first-order eccentricity of the highest

absolute value in each main directioneb load eccentricity at bottom end of columnee modulus of the equivalent biaxial eccentricityeex; eey component of the equivalent biaxial eccentricityet load eccentricity at top end of columnetb; etb components of the eccentricity at top end of columnexb; eyb components of the eccentricity at bottom end of columnfc concrete compressive strengthfcm mean concrete cylinder strengthfy yield strength of steelh overall thickness of cross section perpendicular to the

axis bendingk3 factor applied to cylinder stress to obtain concrete stress

in column

Le effective length of columnM1; M2 smallest and largest factored moments applied at col-

umn ends, respectivelyN applied axial load of the columnNcr critical axial load of the columnNmethod maximum axial load applying the simplified methodNtest maximum axial load from the experimental testn number of data pointsae skew angle of the equivalent biaxial eccentricity with

respect to the bending-x axisatop skew angle at top end of column measured with respect

to the x-axisabottom skew angle at bottom end of column measured with re-

spect to the x-axisb end eccentricities ratiodns moment magnifier factor for columns that are part of

braced (nonsway) frameskg geometrical slenderness (Le/h)lb=1 relative maximum first-order bending moment in the

case of equal eccentricities at the endslb–1 relative maximum first-order bending moment due to

the load with different eccentricities at the endsle relative equivalent uniform bending momentm relative axial loadn accuracy degree of the application of the Cm factornmethod accuracy degree of simplied method proposed by design

codesnNS ratio of the maximum axial load obtained in the exper-

imental test (Ntest) and the maximum axial load com-puted by the numerical model (NNS)

x mechanical reinforcement ratioql longitudinal reinforcement ratio

74 L.C. Leite et al. / Engineering Structures 71 (2014) 73–87

ee ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie2

ex þ e2ey

qð4Þ

ae ¼ ataneey

eex

� �ð5Þ

Fig. 1. Equivalent

In the case of columns whose material behavior is elastic and linear,such generalization of the Cm factor to biaxial bending provides acondition on the safe side, since for each main bending direction,the section of maximum total moment is not necessarily locatedat the same position throughout the column. Nevertheless, the

eccentricities.

Table 1Formulae of the equivalent moment factor Cm for braced column under uniaxialbending.

Authors Proposes: Equivalent moment factor (Cm)

Campus and Massonnet[8]

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:3 � 1þ b2

� �þ 0:4 � b

rAustin [3] 0.6 + 0.4 � b P 0.4MacGregor et al. [19] 0.75 + 0.25 � bRobinson et al. [26] 1.45 � 0.05 � (4 � b)2

Trahair [31] 1þb2 þ 0:4� 0:23 � að Þ � 1�b

2

� �3

Duan et al. [14] 1 + 0.25 � a � 0.6 � a1/3 � (1 � b)Sarker and Rangan [27] aþ ð1� aÞ � b! a ¼ 0:975� 0:00375 � km; b � �0:5Tikka and Mirza ([28]

and [29]Eq. (1): 0.6 + 0.4 � b P 0.4Eq. (2): 0.2 + 0.8 � (0.5 + 0.5 � b)1.1 P 0.3

Tikka and Mirza [30] Eq. (1): 0.6 + 0.4 � b P 0.3Eq. (2): 0.2 + 0.8 � (0.5 + 0.5 � b)1.1 P 0.3

EC-2 [15]; CM-90 [12];EHE-08 [10]

0.6 + 0.4 � b P 0.4

ACI-318 [1] 0.6 + 0.4 � bPallarés et al. [23] Theoretical – linear elastic:

Cm ¼ g � aþ n � Cming ¼ 0:4 � b� 1ð Þ ; n ¼ 0:15 � bþ 0:85Cmin ¼ 0:5 � bþ 1ð Þ � 0:4

8<:Nonlinear:

lCm ¼ 0:25 � bþ 0:75ð Þ � k2g �m� 1�bð Þ

2500 � CminCmin ¼ 0:45 � bþ 0:55 � 0:4

(

L.C. Leite et al. / Engineering Structures 71 (2014) 73–87 75

application of this procedure in reinforced concrete slender col-umns whose behavior is non-linear does not necessarily entail asolution on the safe side, since there is interaction between thebending axes.

Table 1 shows the expressions for the Cm factor proposed by dif-ferent authors and codes for braced columns under uniaxial bend-ing. As can be seen, there is no agreement among the differentproposals, neither regarding the different variables nor the influ-ence on the Cm factor. Austin’s formula [3], whereCm = (0.60 + 0.40 � b) P 0.40, is widely assumed by Design Codes(ACI-318 [1], MC-90 [12], EC-2 (2004) [15], EHE-08 [10]) and wasderived from the differential equation obtained from the equilib-rium of the column in the deformed shape and assuming a linearelastic behavior of the material subjected to any eccentricities ratioat both ends (b = e1/e2, where e1 and e2 are the column first orderend eccentricities, and e2 is the largest in absolute value). Austin’sformula does not consider the nonlinear behavior of concrete. Fur-thermore, most of the Cm factors were obtained for normal strengthconcretes. Therefore, the mechanical behavior of high strengthconcrete cannot be directly extrapolated from the normal strengthone, and it is necessary to update the applicability of such expres-sions to any range of strength, for both uniaxial bending and biax-ial bending.

Sarker and Rangan [27] stated that the use of the equivalentmoment factor Cm proposed by most codes, and based on Austin’sproposal [3], leads to non-conservative results for columns withlow and medium slenderness. On the other hand, Tikka and Mirza[28] [30], after conducting a statistical analysis, concluded that theequivalent moment factor Cm may be formulated independentlyfrom the relation between the applied axial load N and the criticalaxial load Ncr of the column, and from the slenderness, with suffi-cient accuracy. The inclusion of the column’s non-linear stiffness EIimproves the accuracy of the Cm factor, although it makes its prac-tical application more difficult. These authors observed that therewas no real advantage when including the N/Ncr factor in theexpression of the Cm factor. Furthermore, it was pointed out thatthe formula used by most codes leads to conservative results.

Lately, Leite et al. [16,17] have studied the behavior of 68 slen-der columns subjected to compression with unequal eccentricitiesat the ends. 32 of these experimental tests were subjected to

uniaxial bending and 36 of them were subjected to biaxial bending,cast in normal and high strength concrete, with geometric slender-ness ranging from 20 to 30. These authors have compared the sim-plified approaches proposed by EC-2 [15] and ACI-318 [1] withtheir experimental results, pointing out that these design codeslead to conservative results, although a high variation coefficientof the results is reached. Furthermore, these authors point out thatHSC columns presented a significant concrete cover spalling-off,which implies a bearing loss of the column. This effect should betaken into account in both the numerical calibration and the sim-plified approaches.

A new equation to compute the equivalent moment factor Cm isproposed in this paper. The equation modifies the one proposed byPallarés et al. [23] since non-conservative results have beenreached for columns bending with double curvature, low and med-ium slenderness and cast in HSC and low axial loads, because theconcrete cover spalling-off was not taken into account in thenumerical simulation carried out by these authors, which wasthe basis for the proposal.

The new proposal for the Cm factor has been derived fromapplying a numerical simulation which considers the non-linearbehavior of slender concrete columns. A modification of thestress–strain concrete formula proposed by Popovics [24] and latermodified by Collins et al. [9] is used to take into account the con-crete cover spalling-off, for the concrete cover area. Moreover,the degree of accuracy for the generalized application of the Cm fac-tor to biaxial bending with unequal eccentricities and skew anglesat the ends, has been analyzed (Eqs. (2) and (3)). In the same way,the Cm factor proposed by Austin [3] has been verified.

The proposal of the Cm factor and the one suggested by Austin[3] have been checked for both NSC and HSC, and for both uniaxialbending and biaxial bending. The Cm factor proposed by Austin [3]leads to non-conservative results in columns bending in doublecurvature (b < 0), with low levels of axial load (N/Ncr < 0.50) andlow or medium slenderness columns (kg < 20). The new proposalsignificantly improves both safety and accuracy in these cases.

2. Numerical model

The new proposal for the Cm factor has been obtained from theapplication of a general method of analysis of non-linear concretestructures based on the finite elements technique. Additionally, thedegree of accuracy for the application of the Cm factor to biaxialbending has been analyzed by means of the same numerical model.This numerical method includes the following issues:

� 1-D finite element with non-constant curvature: the finite ele-ment has 13 degrees of freedom (d.o.f.’s), Marí [20]. This ele-ment has three nodes, with 6 d.o.f.’s in the initial and finalnodes (three rotations and three displacements), while themid-span node has only one degree of freedom in the axialdirection to capture the variable curvature of the element,Fig. 2(a);� The integration of stresses of the cross-section is performed

using the numerical method known as ‘‘modified thick layerintegration’’ (MTLI) proposed by Bonet et al. [6]. This methodis applicable in sections in which the stress field is uniform atleast in one direction (as is the case where the section is underan increasing monotonic load). The concrete for the cross sec-tion was divided into two regions: concrete outside the lateralties (concrete cover) and concrete inside the lateral ties (con-crete core). The section integration was obtained as the additionof two regions of concrete, Fig. 2(b and d). This method is validboth for unconfined core concrete and confined core concrete.The method proposed by Bonet et al. [6] is valid for solid sec-

Fig. 2. Finite element model: (a) general arrangement, (b) cross section integration [6].

Spalling of the concrete cover

e2

e1

AA

Seccion A-A

Fig. 3. Concrete cover spalling-off: HSC column under compression with unequalend eccentricities.

76 L.C. Leite et al. / Engineering Structures 71 (2014) 73–87

tions (non-hollow) and with convex shape. Due to that reason,the integral of the concrete cover is obtained as the differenceof the integral of the whole section minus the integral of thecore of the concrete section. (Fig. 2(c and d)). The integrationprocedure of each region can be divided into three steps:decomposition into ‘‘thick layers’’ (Fig. 2(b)); decompositioninto quadrilaterals (Fig. 2(c)) and evaluation of the numericalintegral based on the Gauss–Legendre quadrature, whereGreen’s theorem is used.� Non-linear concrete behavior: Popovics [24] modified by Collins

et al. [9] for uniaxial compression. The method suggested byVismann and Zilch (CEB-229 [33]) has been used for uniaxialtension.� Non-linear steel behavior: elastic–plastic stress–strain relation-

ship (Model Code-90 [12]).� Geometric nonlinearity: large displacements from the geomet-

ric matrix and large deformations from taking into consider-ation the equilibrium in the deformed shape of the structureby means of updating the node coordinate.� Time-dependent effects: creep and shrinkage (CEB ([11] and

[13])).

A detailed description of the numerical model can be found inBonet et al. [7] and Bonet [5].

Collins et al. [9] and Attard and Stewart [2] considered that thecolumn concrete strength differs from cylinder concrete strengthdue to a scale factor, vibration during casting, and the loading rate.This effect has been considered by a factor k3. This factor is alsolinked to the premature spalling of the concrete cover (Collinset al. [9]), as shown in Leite et al. [17] for HSC columns (Fig. 3). How-ever, these authors reduce the concrete strength by using the factork3 in the whole cross section, without taking into account that thespalling of the concrete cover only happens in the cover.

In this paper, the use of the k3 factor is only applied along thecover of the cross section to take into account the effect of thespalling-off, as well as a reduction of the concrete strength in thisarea. The area included in the middle axis of the lateral ties (Fig. 4)keeps the original concrete strength.

In this paper the k3 factor has been calibrated through a statis-tical analysis of the results from the following experimental tests:Series A1 in Leite et al. [17] – 12 tests – for an average strength of88 MPa; Series C in Rangan [25] – 12 tests – for an average strengthof 60 MPa; test specimens from C7 to C12 in Sarker and Rangan[27] – 6 tests – for an average strength of 62 MPa; and Series A2in Leite et al. [17] – 8 tests – for an average strength of 31 MPa.Thus, the selection includes the results of 38 experimental testsof pinned–pinned columns subjected to uniaxial bending withunequal eccentricities at the ends and the following characteris-tics: e1/e2 = -1 to 1; e2/h = 0.10 to 0.40; Le/h = 8.56 to 30; ql = 2%to 3%; fy = 454 to 538 MPa; fc = 28.40 to 91.40 MPa; and cnom/h = 0.10 to 0.20. The proposed k3 factor is the following, for mem-bers without lateral confinement:

k3 ¼ 0:05þ 55fcm� 1 ð6Þ

Fig. 4. Section with the use of different k3 factors.

L.C. Leite et al. / Engineering Structures 71 (2014) 73–87 77

The accuracy (nNS) was assessed as the ratio of the maximum axialload obtained in the experimental test (Ntest) and the maximumaxial load computed by the numerical model (NNS). Where nNS (Eq.(7)) reaches a value larger than one, the numerical model providesa conservative result.

nNS ¼Ntest

NNSð7Þ

The proposed k3 factor (Eq (6)) has been checked with 68 experi-mental tests on pinned slender columns performed by Leite et al.[16][17] (Fig. 5) subjected both to compression and uniaxial bend-ing with unequal eccentricities at the ends (Table 2), and underunequal eccentricities and skew angle (Table 3). As can be seen,the proposal is safer for biaxial bending. Nevertheless, althoughthe k3 factor depends on the skew angle and other parameters suchas the relative cover of the concrete (cnom/h), a simple expression forthe k3 factor is proposed, since the precision of the numerical modelis considered to be acceptable.

Finally, the numerical method has been checked with 70 exper-imental tests from the literature for pinned columns subjected tocompressive stress with unequal eccentricities at the ends, andwith 117 experimental tests from the literature subjected to bothuniaxial bending and biaxial bending, with equal eccentricities atthe ends. Such columns have been cast with both NSC and HSC.

The length of the columns and the size of the cross-sections inthe numerical model are the same as in the experimental tests.Moreover, it was verified that with a length of the finite element

Fig. 5. Cross-section of the column

equal to the height of the section, the obtained results showed rea-sonable accuracy.

Table 4 shows the parameters and the top and minimum valuesfor the studied parameters.

Table 5 shows the accuracy of the numerical method (Eq. (7))for several authors using the proposed k3 factor. 255 experimentaltests available in the literature were used to check the accuracy ofthe numerical method.

The spread of results seems to be consistent with the variationfound in laboratory specimens, and one may conclude that goodaccuracy is found between the results obtained using the numeri-cal method with the new k3 factor proposed and the experimentaltests.

3. Numerical simulation

3.1. Application to columns subjected to combined axial loads anduniaxial bending

The Cm-factor is numerically gathered for every axial load level-j(mj) applied to the column (Fig. 6) as the ratio between the relativemaximum first-order moment (lb=1), in the case of equal eccentric-ities at the ends, and the relative maximum first-order momentdue to the load with different eccentricities at the ends (lb–1).

Cm ¼lb¼1

lb–1ð8Þ

Table 6 shows the analyzed parameters and the range of variationthat has been considered. Ten levels of axial load have been consid-ered, from zero to the critical axial load of the column (Ncr) for eachcase (Fig. 6). The size and shape of the cross-section (square of120 � 120 mm) and the arrangement of the longitudinal reinforce-ment (1 rebar at each corner) were all considered as constant val-ues. The combination of the parameters approximately results in2700 numerical simulations.

When applying unequal eccentricities at the ends, the maxi-mum total bending force may be placed at one of the ends of thecolumn, which behaves as the ultimate strength of the cross-sec-

s (Leite et al. ([16] and [17])).

Table 2Characteristics of the experimental tests and comparison with the numerical model in combined axial loads and uniaxial bending (Leite et al. [17]).

No column Cross section (mm �mm) ql (%) kg b e2/h et (m) eb (m) fc (MPa) Nmax (kN) NNS (kN) nNS

S01-A1 100 � 200 2.26 30 �0.5 0.1 �0.0050 0.0100 91.4 616.92 569.08 1.08S02-A1 100 � 200 2.26 30 �0.5 0.2 �0.0100 0.0200 87.8 590.63 473.96 1.25S03-A1 100 � 200 2.26 30 �0.5 0.4 �0.0200 0.0400 85.4 287.77 309.18 0.93S04-A1 100 � 200 2.26 30 0 0.1 0.0000 0.0100 86.8 497.21 496.15 1.00S05-A1 150 � 200 2.26 20 0 0.1 0.0000 0.0150 90.4 1382.59 1386.37 1.00S06-A1 150 � 200 2.26 20 0 0.2 0.0000 0.0300 87.4 1031.21 1050.67 0.98S07-A1 100 � 200 2.26 30 0 0.2 0.0000 0.0200 88.5 374.72 375.08 1.00S08-A1 100 � 200 2.26 30 0 0.4 0.0000 0.0400 90.2 210.11 227.93 0.92S09-A1 100 � 200 2.26 30 0.5 0.2 0.0100 0.0200 88.3 281.05 302.58 0.93S10-A1 100 � 200 2.26 30 0.5 0.4 0.0200 0.0400 87.3 162.85 172.67 0.94S11-A1 150 � 200 2.26 20 0.5 0.2 0.0150 0.0300 88.1 851.10 848.98 1.00S12-A1 150 � 200 2.26 20 0.5 0.4 0.0300 0.0600 88.2 487.96 466.62 1.05S01-A2 100 � 200 2.26 30 0 0.1 0.0000 0.0100 30.1 334.32 347.07 0.96S02-A2 100 � 200 2.26 30 0 0.2 0.0000 0.0200 28.4 265.78 264.04 1.01S03-A2 100 � 200 2.26 30 0 0.4 0.0000 0.0400 30.0 168.25 177.22 0.95S04-A2 100 � 200 2.26 30 �0.5 0.1 �0.0050 0.0100 30.5 430.31 390.67 1.10S05-A2 100 � 200 2.26 30 �0.5 0.2 �0.0100 0.0200 32.5 329.65 345.97 0.95S06-A2 100 � 200 2.26 30 �0.5 0.4 �0.0200 0.0400 31.5 251.12 234.22 1.07S07-A2 150 � 200 2.26 20 0 0.1 0.0000 0.0150 32.6 751.87 803.48 0.94S08-A2 150 � 200 2.26 20 0 0.2 0.0000 0.0300 32.2 615.68 644.10 0.96S01-A3 100 � 200 3.39 30 0 0.1 0.0000 0.0100 85.7 542.55 536.86 1.01S02-A3 100 � 200 3.39 30 0 0.2 0.0000 0.0200 92.3 370.33 412.81 0.90S03-A3 100 � 200 3.39 30 0 0.4 0.0000 0.0400 93.1 238.23 268.66 0.96S04-A3 100 � 200 3.39 30 �0.5 0.1 �0.0050 0.0100 93.3 657.32 613.85 1.07S05-A3 100 � 200 3.39 30 �0.5 0.2 �0.0100 0.0200 92.5 453.04 516.27 0.88S06-A3 100 � 200 3.39 30 �0.5 0.4 �0.0200 0.0400 93.2 334.32 357.23 0.94S01-A4 100 � 200 3.39 30 0 0.1 0.0000 0.0100 34.3 382.57 412.30 0.93S02-A4 100 � 200 3.39 30 0 0.2 0.0000 0.0200 31.3 318.23 311.35 1.02S03-A4 100 � 200 3.39 30 0 0.4 0.0000 0.0400 30.7 192.64 213.42 0.90S04-A4 100 � 200 3.39 30 �0.5 0.1 �0.0050 0.0100 30.0 464.61 426.79 1.09S05-A4 100 � 200 3.39 30 �0.5 0.2 �0.0100 0.0200 28.2 352.67 353.62 1.00S06-A4 100 � 200 3.39 30 �0.5 0.4 �0.0200 0.0400 29.5 258.19 264.11 0.98

Average 0.99CV 0.08Min 0.88Percentile of 5% 0.90Percentile of 95% 1.09Max 1.25

Table 3Characteristics of experimental tests and comparison with the numerical model in combined axial loads and biaxial bending (Leite et al. [16]).

Column Cross section (mm �mm) ql (%) kg exb (m) eyb (m) ext (m) eyt (m) fc (MPa) Ntest (kN) NNS (kN) nNS

S01_B1 125 � 125 2.01 24 0.0115 0.0048 0.0000 0.0000 85.3 514.88 502.49 1.02S02_B1 125 � 125 2.01 24 0.0231 0.0096 0.0000 0.0000 83.0 387.08 375.21 1.03S03_B1 125 � 125 2.01 24 0.0462 0.0191 0.0000 0.0000 85.9 198.76 217.89 0.91S04_B1 125 � 125 2.01 24 0.0231 0.0096 0.0115 0.0048 87.4 279.41 299.24 0.93S05_B1 125 � 125 2.01 24 0.0462 0.0191 0.0231 0.0096 90.3 175.55 163.27 1.08S06_B1 125 � 125 2.01 24 0.0115 0.0048 �0.0058 �0.0024 89.2 610.77 577.70 1.06S07_B1 125 � 125 2.01 24 0.0231 0.0096 �0.0115 �0.0048 84.9 492.10 469.94 1.05S08_B1 125 � 125 2.01 24 0.0115 0.0048 0.0058 0.0024 86.4 505.43 439.60 1.15S02_B2 125 � 125 2.01 24 0.0231 0.0096 0.0000 0.0000 37.4 274.55 295.24 0.93S03_B2 125 � 125 2.01 24 0.0462 0.0191 0.0000 0.0000 35.1 174.97 175.93 0.99S04_B2 125 � 125 2.01 24 0.0462 0.0191 0.0231 0.0096 28.2 134.58 124.56 1.08S05_B2 125 � 125 2.01 24 0.0231 0.0096 0.0115 0.0048 31.7 234.54 221.95 1.06S01_C1 125 � 125 2.01 24 0.0231 0.0096 0.0125 0.0000 89.5 306.92 304.82 1.01S02_C1 125 � 125 2.01 24 0.0462 0.0191 0.0250 0.0000 93.6 178.92 164.66 1.09S03_C1 125 � 125 2.01 24 0.0115 0.0048 �0.0063 0.0000 95.3 648.09 557.33 1.16S04_C1 125 � 125 2.01 24 0.0231 0.0096 �0.0125 0.0000 92.3 502.00 446.99 1.12S02_C2 125 � 125 2.01 24 0.0354 0.0354 0.0250 0.0000 91.5 182.97 173.08 1.06S03_C2 125 � 125 2.01 24 0.0088 0.0088 0.0000 0.0000 91.4 587.14 491.41 1.19S04_C2 125 � 125 2.01 24 0.0177 0.0177 0.0000 0.0000 99.1 408.97 359.34 1.14S05_C2 125 � 125 2.01 24 0.0354 0.0354 0.0000 0.0000 95.3 245.08 215.80 1.14S06_C2 125 � 125 2.01 24 0.0088 0.0088 �0.0063 0.0000 98.3 640.83 538.46 1.19S07_C2 125 � 125 2.01 24 0.0088 0.0088 0.0063 0.0000 89.5 527.44 444.60 1.19S01_C3 125 � 125 2.01 24 0.0177 0.0177 0.0115 0.0048 88.1 340.06 296.46 1.15S02_C3 125 � 125 2.01 24 0.0354 0.0354 0.0231 0.0096 88.0 200.63 167.50 1.20S03_C3 125 � 125 2.01 24 0.0088 0.0088 �0.0058 �0.0024 86.5 694.91 564.51 1.23S04_C3 125 � 125 2.01 24 0.0088 0.0088 0.0058 0.0024 86.3 512.11 434.75 1.18S01_C4 125 � 125 2.01 24 0.0115 0.0048 0.0063 0.0000 27.1 291.92 279.03 1.05S02_C4 125 � 125 2.01 24 0.0231 0.0096 0.0125 0.0000 29.9 232.91 219.20 1.06

78 L.C. Leite et al. / Engineering Structures 71 (2014) 73–87

Table 4Variation range of the parameters studied in the experimental tests.

Parameters Variation range

Concrete strength (fc in MPa) 18.53–101Steel strength (fy in MPa) 280–521Mechanical reinforcement ratio (x) (%) 0.08–0.62Geometric reinforcement ratio (ql) (%) 0.5–4.00Sides ratio (h/b) 0.5–2.0Geometric slenderness (kg) 3–40Relative axial load [m = Nmax/(Ac � fc)] 0.04–0.94End eccentricities ratio (b = e1/e2) �1.00–1.00Relative concrete cover (cnom/h) 0.05–0.21

Table 3 (continued)

Column Cross section (mm �mm) ql (%) kg exb (m) eyb (m) ext (m) eyt (m) fc (MPa) Ntest (kN) NNS (kN) nNS

S01_C5 125 � 125 2.01 24 0.0088 0.0088 0.0000 0.0000 27.6 348.81 305.05 1.14S02_C5 125 � 125 2.01 24 0.0177 0.0177 0.0000 0.0000 29.1 244.45 244.67 1.00S03_C5 125 � 125 2.01 24 0.0354 0.0354 0.0000 0.0000 29.1 172.90 155.32 1.11S04_C5 125 � 125 2.01 24 0.0088 0.0088 �0.0063 0.0000 31.3 389.76 357.62 1.09S05_C5 125 � 125 2.01 24 0.0088 0.0088 0.0063 0.0000 25.9 318.27 269.73 1.18S01_C6 125 � 125 2.01 24 0.0088 0.0088 0.0058 0.0024 33.3 313.74 313.97 1.00S02_C6 125 � 125 2.01 24 0.0177 0.0177 0.0115 0.0048 28.3 226.73 203.80 1.11S03_C6 125 � 125 2.01 24 0.0177 0.0177 �0.0115 �0.0048 27.0 299.16 270.99 1.10

Average 1.09CV 0.07Min 0.91Percentile of 5% 1.00Percentile of 95% 1.20Max 1.23

L.C. Leite et al. / Engineering Structures 71 (2014) 73–87 79

tion. As a result, these load cases have been excluded fromthe analysis. In short, 1311 numerical simulations have beenconsidered in the analysis. In theses numerical simulations, the

Table 5Accuracy of the numerical method (nNS = Ntest/NNS) according to experimental results.

Authors No Average C

Experimental test under unequal eccentricitiesBaumann [4] 3 1.01 0Martin and Olivieri [21] 6 1.04 0MacGregor and Barter [18] 4 1.07 0Van Stekelenburg [32] 12 1.10 0Mehmel et al. [22] 2 0.94 0Robinson et al. [26] 16 0.93 0Rangan [25] 12 1.02 0Sarker and Rangan [27] 15 1.17 0Leite et al. [17] 32 0.99 0Leite et al. [16] 36 1.09 0Uniaxial bending 102 1.03 0Biaxial bending 36 1.09 0NSC 71 1.02 0HSC 67 1.07 0All 138 1.05 0

Experimental test under equal eccentricitiesPallarés et al. [34] 56 1.18 0Sarker and Rangan [27] 3 1.07 0Kim and Yang [35] 30 1.06 0Wang and Hsu [37] 8 1.08 0Wu and Huggins [36] 11 0.98 0Mavichak and Furlong [38] 9 1.04 0Uniaxial bending 56 1.12 0Biaxial bending 61 1.12 0NSC 38 1.03 0HSC 79 1.16 0

All 117 1.11 0

All 255 1.08 0

CV = coefficient of variation; P05 = percentile of 5%; P95 = percentile of 95%.

maximum moment along the column length exceeds the largestof the two end moments.

3.2. Application to columns subjected to combined axial loads andbiaxial bending

The numerical model has been used to analyze the degree of theaccuracy for the generalized application of the Cm factor to biaxialbending (Eqs. (2) and (3)).

Table 7 shows the analyzed parameters and their variationinterval corresponding to the numerical simulation which has beencarried out. For the analysis purposes b is defined as the ratio et/eb

between the moduli of the eccentricity located at the top end (et)and at the bottom end (eb) of the column. In the numerical simula-tions, the maximum eccentricity modulus of the column e2 isplaced at the bottom end of the column (e2 = eb); as a consequence,the b value is within the range between �1 and 1. The application

V Min P05 P95 Max

.13 0.87 0.89 1.12 1.13

.14 0.81 0.84 1.19 1.19

.13 0.94 0.94 1.21 1.23

.14 0.87 0.89 1.27 1.31

.02 0.93 0.93 0.95 0.95

.11 0.82 0.82 1.14 1.26

.19 0.81 0.82 1.32 1.38

.12 0.96 0.99 1.40 1.49

.08 0.88 0.90 1.09 1.25

.07 0.91 1.00 1.20 1.23

.14 0.81 0.85 1.27 1.49

.07 0.91 1.00 1.20 1.23

.12 0.81 0.85 1.22 1.31

.13 0.81 0.86 1.29 1.31

.13 0.81 0.85 1.26 1.49

.08 0.95 1.02 1.30 1.38

.12 0.99 0.99 1.20 1.22

.10 0.88 0.91 1.24 1.27

.07 0.84 0.86 1.03 1.03

.08 0.84 0.86 1.04 1.04

.08 0.88 0.92 1.15 1.16

.10 0.88 0.94 1.27 1.38

.10 0.84 0.95 1.27 1.32

.09 0.84 0.88 1.18 1.25

.09 0.90 0.99 1.27 1.38

.10 0.84 0.94 1.27 1.38

.12 0.81 0.85 1.27 1.49

Fig. 6. Equivalent moment factor Cm method.

Table 6Parameters analyzed in uniaxial bending. Numerical simulation.

Parameters Values Considered

cmec/h 0.10; 0.15 and 0.20Concrete strength (fc) (MPa) 30; 50; 60; 70; 80; 90 and 100Mechanical reinforcement ratio (x) 0.06; 0.25 and 0.50End eccentricities ratio (b) �0.75; �0.50; �0.25; 0; 0.25; 0.50; 0.75Geometric slenderness (kg = Le/h) 15; 20; 25; 30Axial load 10 values belonging to uniform steps, starting from zero to the critical axial load Ncr of the column

Table 7Parameters analyzed in biaxial bending. Numerical simulation.

Parameters Values considered

Cross section (mm) square: 120 � 120 and rectangular: 120 � 240Longitudinal reinforcement

Mechanical reinforcement ratio (x) 0.06; 0.25 and 0.50Concrete strength (fc) (MPa) 30 and 90End eccentricities ratio (b = e1/e2) �1; �0.50; �0.25; 0; 0.25; 0.50; 1Skew angle at the upper end of the column (atop) 0�; 22.5�; 26.57�; 45�; 63.43�; 90�Skew angle at the lower end of the column (abottom) 0�; 5.04�; 10.31�; 16.1�; 22.5�; 26.57�; 40.89�; 45�; 53.95�; 63.43�; 70.57�; 90�Geometric slenderness (kg = Le/h) 20 and 30Axial load 6 values belonging to uniform steps, starting from zero to the critical axial load Ncr of the column

80 L.C. Leite et al. / Engineering Structures 71 (2014) 73–87

point of the load at the top end is placed in the first quadrant. Inthe case of simple curvature (b > 0), the application point of theload at the bottom end is placed in the first quadrant (Fig. 7a); inthe opposite case (b < 0), it is placed in the third quadrant(Fig. 7b). The skew angles both at the top end (atop) and at the bot-tom end (abottom) are measured with respect to the x-axis and are

Fig. 7. Location of the points of application of the load in the column in biaxialbending.

within the range between 0� and 90� (Fig. 7a and b). A geometricalslenderness (kg) lower than 20 has not been considered, with theaim of ensuring that the critical cross-section of the column (theone with the highest total bending moment, including the secondorder effects) is placed between the ends of the column. The yield-ing stress of the steel has been considered as fixed (fy = 500 MPa) aswell as the mechanical cover of the longitudinal reinforcement(15% of the height). Through the numerical model, different seriesof numerical simulations have been obtained in which all theparameters, including the skew angle at the top end atop, have beenfixed, and the skew angle at the bottom end abottom is varied(Fig. 7). The combination of the different values adopted for theabove mentioned parameters has resulted in approximately 2611numerical simulations.

4. New proposed expression for determining the equivalentfactor Cm

Pallarés et al. [23] found from the theoretical analysis that theCm factor linearly depends on the ratio N/Ncr (where N is theapplied axial load and Ncr is the critical axial load of the column).Furthermore, these authors also found from a numerical simula-

0.000.100.200.300.400.500.600.700.800.901.00

0 100 200 300 400 500 600 700

ν.λg2

λg = 15λg = 20λg = 25λg = 30

β = -0.75

Cm proposal

Austin

0.000.100.200.300.400.500.600.700.800.901.00

ν.λg2

λg = 15λg = 20λg = 25λg = 30

β = -0.50

Cm proposal

Austin

0.000.100.200.300.400.500.600.700.800.901.00

ν.λg2

λg = 15λg = 20λg = 25λg = 30

β = -0.25Cm proposal

Austin

0.000.100.200.300.400.500.600.700.800.901.00

ν.λg2

λg = 15λg = 20λg = 25λg = 30 β = 0

Cm proposal

Austin

0.000.100.200.300.400.500.600.700.800.901.00

ν.λg2

λg = 15λg = 20λg = 25λg = 30 β = 0.25

Cm proposal

Austin

0.000.100.200.300.400.500.600.700.800.901.00

ν.λg2

λg = 15λg = 20λg = 25λg = 30 β = 0.50

Cm proposal

Austin

0.000.100.200.300.400.500.600.700.800.901.00

ν.λg2

Cm

λg = 15λg = 20λg = 25λg = 30 β = 0.75

Cm proposalAustin

(a) (b)

(c) (d)

(g)

(f)

n 19= n 2 04=

n 541=

n 561=

n 161= 331=n

178=n

0 100 200 300 400 500 600 700

0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700

0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700

0 100 200 300 400 500 600 700

Cm

Cm

Cm Cm

Cm

Cm

(e)

Fig. 8. Proposal Cm factor.

L.C. Leite et al. / Engineering Structures 71 (2014) 73–87 81

tion that the Cm factor depends on the ratio N/Ncr and, since thisratio is proportional to the factor m � kg

2, the authors suggested ana-lyzing the Cm factor as a function of m � kg

2.In this work, the following expression was adopted for the Cm

factor as all-enveloping in order to ensure that all the cases becomeconservative. Fig. 8 shows the proposed adjustments.

Cm ¼ 1�m � k2

g

600� Cmin ð9Þ

where Cmin = 0.60 + 0.40 � b P 0.40; m is the relative axial load [N/(Ac � fc)]; N is the applied axial load; Ac is the gross area of thecross-section; fc is the concrete strength; kg is the geometric slen-

Table 8Cm-factor accuracy in columns under unequal eccentricities and skew angle loads at the ends. Numerical simulation.

No Average CV Min P05 P95 Max

Cm-Austin Uniaxial bending 1311 1.04 0.16 0.42 0.75 1.32 1.74Biaxial bending 2611 1.07 0.06 0.70 0.99 1.19 1.50All 3922 1.06 0.11 0.42 0.90 1.23 1.74

Cm-proposal Uniaxial bending 1311 1.16 0.11 0.96 1.02 1.38 1.90Biaxial bending 2611 1.14 0.12 0.90 1.01 1.43 1.92All 3922 1.14 0.12 0.90 1.01 1.40 1.92

CV = coefficient of variation; P05 = percentile of 5%; P95 percentile of 95%;

0.0

0.5

1.0

1.5

2.0

0 100 200 300 400 500 600 700

ν·λg2

ξ ( μ

e/ μβ=

1)

Cm - Austin

0.0

0.5

1.0

1.5

2.0

-1.0

0

-0.7

5

-0.5

0

-0.2

5

0.00

0.25

0.50

0.75

1.00

β

ξ ( μ

e/ μβ=

1)

Cm - Austin

0.0

0.5

1.0

1.5

2.0

ν·λg2

ξ ( μ

e/ μβ=

1)

Cm - proposal(a) (b)

(c) (d)0.0

0.5

1.0

1.5

2.0-1

.00

-0.7

5

-0.5

0

-0.2

5

0.00

0.25

0.50

0.75

1.00

β

ξ ( μ

e/ μβ=

1)

Cm - proposal

n 1131=

n 1131=n 1131=

n 1131=

0 100 200 300 400 500 600 700

Fig. 9. Analysis of Cm-factor accuracy using numerical simulation. Uniaxial bending.

82 L.C. Leite et al. / Engineering Structures 71 (2014) 73–87

derness [Le/h]; Le is the effective length of column; h is the overallthickness of cross section perpendicular to the axis bending; b isthe ratio between end eccentricities.

In the case where the applied axial load is zero, the second ordereffects are zero and the maximum bending force is located at oneof the column ends. In this case, lb=1 = lb–1 and therefore, Cm = 1(Eq. (8)). This condition has been taken into account in theproposed Cm factor formula (Eq. (9)).

Fig. 10. Cm-factor accurac

In this paper, a Cm is proposed which is independent from thecritical axial load Ncr ð¼ p2 � EI=L2

e Þ of the column as a simpleexpression and, at the same time, on the safe side. The critical axialload of the column Ncr depends on the effective flexural stiffness EIof the column. This stiffness EI in turn depends on the loadsapplied, as well as on the mechanical and geometric characteristicsof the column (Bonet et al. [7]) including the effect of the spallingof the concrete. In this paper it is proposed that those effects are

y in biaxial bending.

0.0

0.5

1.0

1.5

2.0

-1.0

0

-0.7

5

-0.5

0

-0.2

5

0.00

0.25

0.50

0.75

1.00

β

ξ (| μ

e|/| μ

β=1

|)

Cm - Austin

0.0

0.5

1.0

1.5

2.0

ν·λg2

ξ (| μ

e|/| μ

β=1

|)

Cm - proposal

0.0

0.5

1.0

1.5

2.0

-1.0

0

-0.7

5

-0.5

0

-0.2

5

0.00

0.25

0.50

0.75

1.00

β

ξ (| μ

e|/| μ

β=1

|)Cm - proposal

0.0

0.5

1.0

1.5

2.0

0 100 200 300 400 500 600 700

ν·λg2

ξ (| μ

e|/| μ

β=1

|)

Cm - Austin

(a) (b)

(c) (d)

n 1162= n 1162=

n 1162=n 1162=

0 100 200 300 400 500 600 700

Fig. 11. Analysis of Cm-factor accuracy using numerical simulation. Biaxial bending Fig. 12. Accuracy degree (nmethod) using the simplified approaches proposed by: EC-2 [12].

Table 9Cm-factor accuracy in columns under unequal eccentricities and skew angle loads at the ends. Simplified methods.

Cm-Austin Cm-proposal

EC-2 ACI-318 EC-2 ACI-318

Uniaxial Bending Average 1.53 1.43 1.91 1.69CV 0.32 0.30 0.35 0.25Min 0.64 0.59 1.02 1.01P05 0.87 0.77 1.20 1.13P95 2.41 2.09 3.01 2.46Max 3.45 2.58 4.22 2.90

Biaxial Bending Average 1.96 1.70 2.20 1.81CV 0.24 0.24 0.31 0.24Min 1.34 1.05 1.34 1.12P05 1.40 1.06 1.46 1.18P95 2.72 2.29 3.38 2.48Max 2.88 2.57 3.50 2.80

All Average 1.64 1.50 1.98 1.72CV 0.32 0.29 0.34 0.24Min 0.64 0.59 1.02 1.01P05 0.89 0.80 1.21 1.13P95 2.67 2.22 3.38 2.47Max 3.45 2.58 4.22 2.90

CV = coefficient of variation; P05 = percentile of 5%; P95 percentile of 95%;

L.C. Leite et al. / Engineering Structures 71 (2014) 73–87 83

taken into account in the expression to calculate the moment mag-nifier factor dns( = 1/(1 � N/Ncr)), in the same way as it is currentlyapplied in the design codes (ACI-318 [1] and EC-2 [15]).

5. Validation of the proposed Cm factor for uniaxial bending

Fig. 8 plots the Cm factor obtained through the numerical simu-lation using the Cm factor proposed in this paper (Eq. (9)) comparedto the results using the proposals by Austin [3].

The accuracy of the proposals was assessed by the followingfactor:

n ¼ le

lb¼1¼

Cm;prop � lb:¼1

lb¼1¼ Cm;prop

Cm;NSð10Þ

where le is the reduced equivalent moment obtained as a productof the equivalent moment factor proposed by each autor Cm,prop

and the maximum reduced moment lb–1 at the ends; and lb=1 isthe reduced first-order bending moment corresponding to the caseof equal eccentricities at the ends (Fig. 6). Both lb–1 and lb=1 areobtained for the same level of applied axial load through thenumerical simulation. If the Eq. (8) is known, the degree of precisiónn (Eq. (10) can be expressed as the quotient between the equivalent

Leite et al. (2013) -Uniaxial bendingLeite et al. (2012) -Biaxial bendingOther Authors

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Cm - proposal

EC2

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

-1.0

0

-0.7

5

-0.5

0

-0.2

5

0.00

0.25

0.50

0.75

1.00

Cm - proposal

EC2

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 100 200 300 400 500 600 700 800

Cm - Austin

EC-2

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

-1.0

0

-0.7

5

-0.5

0

-0.2

5

0.00

0.25

0.50

0.75

1.00

Cm - Austin

EC-2

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 20 40 60 80 100 120

fc (MPa)

Cm - Austin

EC-2

(a) (b)

(c) (d)

(e) (f)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Cm - proposal

EC2

0 100 200 300 400 500 600 700 800

0 20 40 60 80 100 120fc (MPa)

Fig. 12. Accuracy degree (nmethod) using the simplified approaches proposed by: EC-2 [12].

84 L.C. Leite et al. / Engineering Structures 71 (2014) 73–87

moment factor proposed by each autor Cm,prop and the equivalentmoment factor obtained in the numerical simulation Cm,NS (Eq.(8)). If the ratio n is higher than one, it is on the safe side.

Table 8 shows some statistical values of the accuracy degree (n)for the proposal of the Cm factor (Eq. (9)) and for the expressionsuggested by Austin [3] in columns subjected to uniaxial bending.Fig. 9 shows the accuracy degree with respect to the variables onwhich Eq. (9) (m � kg

2; b) depends, for both proposals.The formula suggested by Austin [3] presents a more centered

solution with an average ratio of 1.04, a coefficient of variationequal to 0.16, and a percentile of 5% of 0.75. However, as shownin Fig. 9, this proposal becomes unsafe for low and medium valuesof (m � kg

2), lower than 400, and double curvature bending (b < 0).By using the new proposal (Eq. (9)), an average ratio of 1.16 hasbeen obtained, as well as a variation coefficient of 0.11 and a per-centile of 5% of 1.02, standing on the conservative side for most ofthe cases. With regard to the end eccentricities ratio (b) and the

values of (m � kg2), the proposed formula improves both the safety

and the accuracy (lower coefficient of variation).

6. Verification of the generalized application of the Cm factor tobiaxial bending

The accuracy degree (n) of the application of the Cm factor tobiaxial bending is assessed through the calculation of the followingexpression.

n ¼le

�� ��lb¼1

��� ��� ð11Þ

where |le| is the modulus of the reduced equivalent momentobtained from the generalized application of the equivalentmoment factor proposed by each author Cm,prop to biaxial bending(Eqs. (2) and (3)) and |lb=1| is the modulus of the first-order reduced

Leite et al. (2013) -Uniaxial bendingLeite et al. (2012) -Biaxial bendingOther Authors

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0ACI-318

Cm - proposal

(a) (b)

(c) (d)

(e) (f)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

-1.0

0

-0.7

5

-0.5

0

-0.2

5

0.00

0.25

0.50

0.75

1.00

Cm - Austin

ACI-318

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 100 200 300 400 500 600 700 800

Cm - Austin

ACI-318

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 20 40 60 80 100 120fc (MPa)

Cm - Austin

ACI-318

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Cm - proposal

ACI-318

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

-1.0

0

-0.7

5

-0.5

0

-0.2

5

0.00

0.25

0.50

0.75

1.00

ACI-318

Cm - proposal

0 100 200 300 400 500 600 700 800

0 20 40 60 80 100 120fc (MPa)

Fig. 13. Accuracy degree (nmethod) using the simplified approaches proposed by: ACI-318 [1].

L.C. Leite et al. / Engineering Structures 71 (2014) 73–87 85

moment corresponding to the case of equal eccentricities at theends in biaxial bending, see Fig. 10. Both |le| and |lb=1| are obtainedfor the same level of applied axial load and reduced skew angle aer

through the numerical simulation.Table 8 shows some statistical values of the accuracy degree n

(Eq. (11)) for the proposal of the Cm factor (Eq. (9)) and for theexpression suggested by Austin [3] in columns subjected to biaxialbending. Fig. 11 shows the accuracy degree with regard to the vari-ables on which Eq. (9) (m � kg

2; b) depends, for both proposals.In both proposals, the mean value of the ratio n for all the con-

sidered numerical simulations is on the safe side, being the varia-tion coefficient lower than 0.12 and the percentile of 5% around thevalue of one. The proposal of the Cm factor (Eq. (9)) is shown to bemore conservative than that of Austin [3], being its variation coef-ficient slightly higher.

The application of the proposal made by Austin [3] to biaxialbending can be on the unsafe side for columns with double curva-ture (b < 0) and low and medium values of m � kg

2 (lower than 400).In Fig. 11 it can be noted that the proposal of the Cm factor (Eq. (9))improves the safety level in these cases.

With the application of the Cm factor (Eq. (9)) to the calculationof the equivalent moment factor Cm in columns subjected to biaxialbending with unequal eccentricities and skew angles at the ends, amean value of the accuracy degree n of 1.14 and a variation coeffi-cient (CV) of 0.12 was obtained, thus being on the safe side fornearly all of the considered cases.

Finally, the application of the proposal made by Austin [3], bothfor uniaxial and biaxial bending, results in situations on the unsafeside for columns with double curvature (b < 0) and for low andmedium values of m � kg

2 (lower than 400). Nevertheless, with the

86 L.C. Leite et al. / Engineering Structures 71 (2014) 73–87

proposal of the Cm factor (Eq. (9)), the safety level is improved,being on the safe side for nearly all of the considered cases. Thevalue of the percentile of 5% for Austin [3] is 0.90 on the unsafeside, whereas for the proposal (Eq. (9)) it is 1.01 on the safe side.

7. Implementation of the new equation proposed to computethe equivalent moment factor Cm

The purpose of this section is to analyze the accuracy of simpli-fied methods proposed by the design codes (EC-2 [15] and ACI-318[1]) for experimental tests in columns subjected to unequal eccen-tricities at both ends, taking the following ratio (nmethod) as a refer-ence to measure accuracy:

nmethod ¼Ntest

Nmethodð12Þ

where Ntest is the maximum axial load from the experimental test;Nmethod is the maximum axial load applying the simplified method;the simplified approaches have been tested with the same experi-mental tests found in the literature (Tables 2, 3 and 5) which wereused to verify the numerical model. Table 4 shows the variationrange of the parameters studied in the different experimental tests.

The simplified parabola-rectangle diagram defined in the EC-2[15] and the equivalent rectangular concrete stress distributiondefined in the ACI-318 [1] have been used to compute the ultimatebending force of the cross-section in the column.

In order to take into account the second order effects, the mag-nified moment factor defined in the section 5.8.7.3 of the EC-2 [15]has been calculated, as well as the formulas 10.12 and 10.14 fromthe ACI-318 [1].

All safety factors have been taken as equal to one and the aver-age of the concrete strength fcm of the specimens as equal to theconcrete compressive strength fc.

Both EC-2 [15] and ACI-318 [1] adopted the formula proposedby Austin [3] to compute the Cm factor (0.6 + 0.4 � (M1/M2), whereM1 and M2 are the bending forces at the ends of the column, andM1/M2 is positive if the column is bending in single curvature,and negative if the member is bending in double curvature. TheEC-2 [15] defines a minimum value of Cm equal to 0.4.

This section also discusses the accuracy degree reached if onedesigns a member using the Cm factor proposed in this paper (Eq.(9)) instead of using the Cm factor proposed by Austin [3] in theapproaches EC-2 [15] and ACI-318 [1].

Table 9 shows the accuracy degree using the expression pro-posed by Austin [3] and the Cm factor proposed in this paper com-pared with the experimental results. For example, Figs. 12 and 13plot the results using the simplified approach proposed by EC-2[15] and ACI-318 [1]. A few non-conservative cases (nmethod < 1)applying the Cm factor proposed by Austin [3], may be observed,as well as the enhancement of safety for these cases when substi-tuting that formula with the Eq. (9) proposed in this paper.

In general, it may be seen (Table 9) that Austin’s proposal [3]and the new one (Eq. (9)) lead to conservative results with largecoefficients of variation.

Many cases using the formula proposed by Austin [3] becamenon-conservative, in particular, for low and medium values of geo-metric slenderness (kg), for HSC columns and values of m � kg

2 lowerthan 200. It should be noted that tests with HSC up to 100 MPawere carried out by Sarker and Rangan [27] and presented a geo-metric slenderness (kg) lower than 10. These authors suggestedthat the simplified approach proposed by the ACI-318 [1] usingAustin’s Cm [3] leads to non-conservative cases, as confirmed inthis analysis.

Using the new Cm factor (Eq. (9)), all cases become conservativeand safety is improved as compared with Austin’s proposal [3], for

both uniaxial bending and biaxial bending. In all cases, the EC-2[15] is more conservative, and its coefficient of variation is higher.

8. Conclusions

The following conclusions may be drawn from the analysis car-ried out in this paper:

� From the numerical model calibration, a k3 factor has been pro-posed to take into account the spalling of the concrete cover(Eq. (6)). That factor reduces the concrete strength only at theouter edge of the cross section, keeping the material propertiesin the concrete core, which is not affected by the spalling of theconcrete cover.� The formula proposed by Austin [3] for the Cm factor, and used

by most design codes, presents a more centered solution foruniaxial bending with an average ratio of 1.04, a variation coef-ficient of 0.16 and a percentile of 5% of 0.75. However, this pro-posal becomes unsafe for low and medium values of (m � kg

2),lower than 400, and double curvature bending (b < 0).� From the results of the numerical analysis, a new equation has

been proposed to compute the Cm –factor (Eq. (9)) for slendercolumns subjected to short term ultimate loads and unequaleccentricities at the ends, both for NSC and HSC. This proposed(Eq. (9)) enhances safety compared to the formula proposed byAustin [3], and presents for uniaxial bending an average ratio of1.16, a variation coefficient equal to 0.11 and a percentile of 5%of 1.02.� In general, the generalized application of the Cm factor to biaxial

bending is on the safe side. For the Cm proposal (Eq. (9)), themean value of the accuracy degree (Eq. (11)) is 1.14, the varia-tion coefficient is 0.12 and a percentile of 5% of 1.01. As a con-sequence, the application of the Cm factor in each bendingdirection in reinforced concrete columns whose material behav-ior is non-linear, is generally on the safe side. However, the for-mula proposed by Austin [3] becomes unsafe for low andmedium values of (m � kg

2), lower than 400, and double curva-ture bending (b < 0).� For all the cases carried out through the numerical simulation,

in both uniaxial and biaxial bending, with the proposal of Cm

(Eq. (9)) the results were a mean ratio of 1.14, a variation coef-ficient of 0.12 and a percentile of 5% of 1.01 on the safe side.� The use of the new Cm factor (Eq. (9)) significantly improves the

safety in the application of the simplified approaches proposedby the Design Codes (EC-2 [15] and ACI-318 [1]) compared withthe results obtained using the equation proposed by Austin [3],mainly for values of m � kg

2 lower than 200 and for HSC columns(fc P 50 MPa).� Finally, the proposed formula (Eq. (9)) allows slender reinforced

concrete columns under unequal eccentricities and skew angleloads at the ends to be both checked and designed with enoughaccuracy for practice engineering, and it is simple to apply.

Acknowledgements

The content of this article is part of a research work carried outby the Universitat Politècnica de València and the Institute of Con-crete Science and Technology (ICITECH). The authors of this workwish to thank the Research Bureau of the Spanish Ministry of Econ-omy and Competitivenes, the Plan E and the European RegionalDevelopment Fund for the co-funding of the research projectsBIA2006-06429 and BIA BIA-2012-32645.

L.C. Leite et al. / Engineering Structures 71 (2014) 73–87 87

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