Flex ans Shear of col with varying axial ld Eng Str 2003.pdf

Engineering Structures 25 (2003) 13531367www.elsevier.com/locate/engstruct

Flexural and shear hysteretic behaviour of reinforced concretecolumns with variable axial load

K. ElMandooh Galal, A. Ghobarah Department of Civil Engineering, McMaster University, 1280 Main Street West, Hamilton, ON, Canada

Received 28 January 2002; received in revised form 7 April 2003; accepted 22 April 2003

Abstract

The importance of non-linear biaxial models that are applicable to the analysis of reinforced concrete members under cyclic anddynamic loads has been recognized. Variations in the axially applied force can influence strength, stiffness and deformation capacityof such members. In this study, an inelastic biaxial model based on plasticity theory, is proposed. This quadri-linear degradingmodel takes into account the effect of axial load variation on lateral deformation. The model predictions are examined againstavailable experimental results. Using the developed model, the effect of various axial loading patterns on the lateral deformationof reinforced concrete columns is investigated. 2003 Elsevier Ltd. All rights reserved.

Keywords: Reinforced concrete; Columns; Post-yield; Hysteretic behaviour; Global models; Variable axial load; Three-dimensional; Flexure; Shear

1. Introduction

Reinforced concrete (RC) members with non-ductilereinforcement detailing experienced both flexural andshear failures during recent earthquakes. Under strongearthquake ground motion, structures are subjected tolateral loads, which impose biaxial flexural and shearforces on the columns. In the analysis of columns,accounting for the effect of biaxial loading on yielding,moment resisting capacity, inelastic deformation, anddegradation of strength and stiffness of the member, isimportant. Considering the effect of these parameters isnecessary to achieve realistic predictions of the seismicresponse of frame structures.

There is a wealth of earthquake records that exhibit avertical component with peak ground acceleration wellin excess of the corresponding horizontal value [1,2].From these studies, it can be concluded that the engin-eering practice of assuming vertical/horizontal peakground acceleration ratio in the range of 1/22/3 mayinvolve significant underestimation of the effect of the

Corresponding author. Tel.: +1-905-525-9140; fax: +1-905-529-9688.

E-mail address: [email protected] (A. Ghobarah).

0141-0296/03/$ - see front matter 2003 Elsevier Ltd. All rights reserved.doi:10.1016/S0141-0296(03)00111-1

vertical component. Variable axial column forces mayalso result from the horizontal earthquake ground motioncomponent. The effect of high dynamic axial force onthe lateral hysteretic response cannot be neglected forRC structures because of the significant change in thehysteretic momentcurvature relationship, as well as theoverall structural behaviour.

Modeling RC elements subjected to biaxial flexuraland shear loads with a variable axial load has receivedrelatively little attention. Some analytical models for thenon-linear analysis of RC frame structures have beenproposed [310]. These range from simplified globalmodels [38] to the refined and complex local (finiteelement) models [9,10]. A technique called shifting ofthe primary curve has been used [35] to take intoaccount axial force variations with bending moments instructural coupled walls. A triaxial spring model that cansimulate varying axial force and stiffness degradationwas proposed [6,7]. The model discretizes the elementcross section into four effective steel springs and fiveeffective concrete springs. The non-linear response ofthe element was shown to be sensitive to the non-linearproperties of its nine constitutive springs. A fibreelement to treat solid and hollow cross sections and toinclude both prismatic and non-prismatic profiles wasadopted [8]. The model was used to design slender con-

1354 K. ElMandooh Galal, A. Ghobarah / Engineering Structures 25 (2003) 13531367

crete bridge piers and bents. Each member is dividedinto series of cross sections and each cross section intofibres. The flexibility matrix is formed by integrating thestress along the cross section, and then by integratingalong the segments. A more refined fibre element wasthen developed and used to model the three-dimensionaldynamic response of the RC frames [9]. A finite frameelement that accounts for variation in axial force wasdeveloped [10]. The element required a significantincrease in computational effort, which is a disadvantagein the analysis of large structures.

The objective of this study is to develop a lumpedplasticity non-linear analytical model for RC members.The model developed in this study is based on the basicbiaxial flexural model [11,12] with improvements to thestiffness degradation criteria, inclusion of shear defor-mation and the implementation of the effect of variationof the axial load. The element is intended to modelinelastic effects in RC columns and beams under a gen-eral dynamic load. Particular emphasis is placed onaccounting for axial forcebiaxial moment and shearinteraction along with degradation in stiffness andstrength of the element.

Predictions of the developed model are compared totest data to validate the reliability of the element. Havingthe model verified, the capability of the element to modelthe influence of axial force variations on the hystereticbehaviour of RC columns is demonstrated by investigat-ing the effect of various axial load patterns on the lateralinelastic response of a slender RC column.

2. Biaxial quadri-linear degrading model withvarying axial load

The model development includes the formulation ofthe element and the determination of the yield surfacesand stiffness degradation behaviour of the flexure andshear subhinges. The effects of the axial load on theflexure and shear subhinges are accounted for. The finalpart of this section discusses how the model parametersare determined.

2.1. Element formulation

The three-dimensional beamcolumn element is for-mulated to model inelastic hysteretic behaviour of RCbeams and columns. The element is capable of rep-resenting the biaxial moment and shearaxial force inter-action, stiffness and strength degradation, and variationof axial load. The element consists of an elastic elementwith lumped plastic hinge at each end. In the three-dimensional space, the element may be arbitrarily ori-ented in a global XYZ coordinate system as shown inFig. 1a.

For the elastic element, initial elastic flexural, tor-

sional, axial, and shear stiffnesses need to be specified.The plastic hinge at each end of the element consists ofthree flexure plastic subhinges and one shear subhinge,as shown in Fig. 1b. Each of the three flexural plasticsubhinges is to represent a stage of the non-linear behav-iour of RC member (i.e. concrete cracking, steel yieldingor ultimate conditions), while the shear subhinge rep-resents a shear failure. The force and moment matrix ateach end SI or J consists of two parts; flexural forces andmoments SI or Jm , and shear forces SI or Js such that:(SIm)T [MIy MIz Mx Fx]; (1a)(SJm)T [MJy MJz Mx Fx]; and (1b)(Ss)T [Vy Vz] (1c)where Vy = (MIz + MJz) /L; and Vz = (MIy + MJy) /L as illus-trated in Fig. 2.

The interaction between the forces and moments ateach subhinge is represented by a yield surface. Theyield surfaces are YS1, YS2, and YS3 for the threeflexure subhinges, and YSS for the shear subhinge. Aquadratic function to describe the yield surface for eachsubhinge was used. For clarity, a 2D rather than 4Drelationship is shown in Fig. 3 for each flexure subhingei. The quadratic yield surface function can be written inthe form:

mi(Sm,ai) MyaiMyMiy 2

MzaiMzMiz 2

(2a)

MxaiMxMix 2

FxaiFxFix 2

1

in which ai is the current position of the yield surfacefor Fx, Mx, My and Mz at hinge i. For the shear subhinge,the failure surface is shown in Fig. 4.

The quadratic failure surface for the shear subhingecan be written as:

s(Ss,Fx) VyVfy2

VzVfz2

FxFavFultFav2

1 (2b)

where Fult is the elements axial compressive capacity,Tult is the tensile capacity, and Fav = (|Fult| + |Tult|) / 2 isthe average axial force. The lateral shear capacities in yand z directions at an average axial force are Vfy andVfz, respectively.

All the plastic subhinges are assumed to be initiallyrigid, thus the initial stiffness will be the stiffness of theelastic element. Under the action of flexural and shearforces, the subhinges experience some flexibility, there-fore a reduction in the element stiffness occurs. Theflexibility of the flexure subhinges upon load reversal isdivided into elastic, fIsem (recoverable) and plastic, fIspm(non-recoverable) flexibilities. During shear failure, anisotropic contraction of the shear yield surface towardsa residual shear surface that follows the shear subhinge

1355K. ElMandooh Galal, A. Ghobarah / Engineering Structures 25 (2003) 13531367

Fig. 1. (a) Element idealization; (b) Plastic hinge idealization.

Fig. 2. Element end forces and moments.

flexibility fspm, is assumed. The residual shear surface isrepresented by the shear capacity of transverse reinforce-ment.

The flexibility matrix for the entire element, Ft, canbe obtained by the appropriate addition of the elasticelement, fo, and the end hinges tangent flexibility matr-ices fIp and fJp at nodes I and J, respectively.

Ft fo fIp fJp 1Kt (3)

Fig. 3. Flexural hinge force-displacement and momentrotation relationships.

Fig. 4. Failure surface for shear subhinge.


After this, Ft can be inverted to obtain the elementstotal tangent stiffness matrix, Kt. The tangent flexibilitymatrices fIp and fJp at each end are calculated by appropri-ate summation of the flexibility of their constitutingsubhinges, such that:

fIp 3i 1

(fIsem,i fIspm,i) fsps; and (4a)

fJp 3i 1

(fJsem,i fJspm,i) fsps. (4b)

2.1.1. Flexural subhinge stiffness degradationThe flexure subhinge flexibility is divided into two

parts; elastic and plastic flexibilities. Both flexibilitiesare initially zero. A certain event such as crack, yield orultimate condition triggers a yield surface. Continuousloading follows kinematic strain hardening and the yieldsurface translates in the force space till it reaches thenext yield surface [13]. Both yield surfaces move with-out change in size or shape till they reach the next yieldsurface, and so on. The process of occurrence of a cer-tain event (reaching a yield surface) results in a finiteplastic flexibility, fI or Jspm,i, of such subhinge. Upon loadreversal, a finite elastic flexibility, fI or Jsem,i, is assigned to atriggered flexure subhinge (in addition to the plasticflexibility). Including elastic and plastic flexure subhingeflexibilities represents elastic stiffness degradation withreduced overall element stiffness (and consequentlystrength) when the element is subjected to reversed load-ing at the same displacement level.

The plastic flexibility matrix fspm,i of a yielded flexuresubhinge was derived by Chen and Powell [12] and wasshown to be equal to:

fspm,i nnT

nTKspm,in(5)

where n is the outward normal vector to the yield surfaceat the action point; and Kspm,i is the diagonal plastic stiff-ness matrix from the individual flexural actiondefor-mation relationships for each force component, definedas:

Kspm,i diag[KpMy,i KpMy,i KpMy,i KpMy,i] (6a)in which the plastic stiffness after yield for each forcecomponent is given by:

Kpi KiKi+1

KiKi+1(6b)

in which the stiffnesses Ki are the initial input data rep-resenting the elastic beam stiffness.

In a typical forcedeformation relationship obtainedfrom tests on concrete elements, it is observed that thelevel of degradation in stiffness increases as the ductility

Fig. 5. Typical forcedisplacement relationship (Ghobarah et al.[14]).

of the element increase or in case of repeated cycles atthe same level of ductility. An example of such forcedeformation relationship is that obtained from a testedcantilever column shown in Fig. 5 [14]. Stiffness degra-dation is introduced in the model when reversed loadingis applied. In the formulation of the 3D element, it isassumed that the stiffness degrades independently foreach moment or force component for each subhinge. Thedegraded stiffness is inversely proportional to the pre-vious hinge secant stiffness, Ks.

Thus the elastic subhinge flexibility after unloadingfor each force component is shown in Fig. 6 andgiven by:

fsem,i am dpi3i 1

dpi 1Ks (7)

Fig. 6. Degradation coefficients.


where Ks is the secant stiffness of the previous cycle andam is an arbitrary degradation coefficient that rangesfrom 0 to 1 where 0 indicates no stiffness degradation.A practical range for am was found to be from 0.03 to0.1. The strength degradation depends on the plasticdeformation of each subhinge in proportion to the totalplastic deformation of all subhinges during the previouscycle. It is possible to assign different degradation coef-ficients amy and amz in each loading direction. The modelcan be extended to have a different coefficient aim foreach subhinge; this means that the degradation level foreach subhinge can be different. This feature was not usedin the current applications in order not to increase thenumber of parameters that need to be defined. Theelement in its quadri-linear degrading formulationimplicitly includes the Bauschinger effect.

2.1.2. Shear subhinge stiffness degradationBased on the tests and the theoretical verifications of

Vecchio and Collins [15], a simplified ellipsoidal shearstrengthaxial strength interaction surface was used asthe shear subhinge yield surface as shown in Fig. 7. Theyield surface is defined by the element axial compressiveFult, and tensile Tult capacities, as well as the lateral shearcapacities Vfy and Vfz at an average axial force Fav in yand z directions, respectively.

When the shear cracks develop in a reinforced con-crete member, inelastic shear deformation commences.The shear transferred by aggregate interlock and dowelaction across the shear cracks will be reduced as thecrack widens, therefore the elements shear strengthdecreases. If large deformation is imposed, the shear willbe resisted primarily by transverse steel reinforcement.In addition, slippage of the bars in tension results in sig-nificant pinching of the hysteretic loops. Based on thediscussed physical behaviour, the shear forcedefor-mation relationship (V) for the element in each direc-tion as well as the stiffness degradation are assumed asshown in Fig. 8.

The shear subhinge flexibility matrix fsps is initiallynil till the shear yield function s(Ss,Fx) is equal to 1.The shear subhinge starts to contract and finite shearsubhinge flexibility fsps develops. An uncoupled shearforce flexibility matrix, similar to that proposed byRicles et al. [16], is used for the descending branch dur-ing shear failure:

fsps 1

Kvy0

01

Kvz (8)

where Kvy, Kvz represent plastic stiffness for shearsubhinge in the y and z axes, respectively. During shearfailure, which corresponds to contraction of shear

Fig. 7. Shear strengthaxial strength interaction diagram; (a) experi-mental [15]; (b) approximate model.

Fig. 8. Shear hingeshear deformation relationship envelope.


subhinge yield surface, the shear forces remain coupledwith the flexural forces through equilibrium. A minimumshear subhinge surface, as shown in Fig. 4 correspondsto the residual shear resistance of the reinforcing ties inthe concrete member in each direction Vry or z. The effectof variation in axial force during the contraction of theshear subhinge yield surface is discussed in Section2.2.2.

Upon reversing the load, the stiffness of the shearsubhinge reduces when the shear force reaches zero.This is to model the pinching behaviour in the hystereticresponse. The shear subhinge flexibility matrix duringpinching is defined as:

fsps apfs (9)where ap is an arbitrary coefficient (from 01) with apractical range from 0.01 to 0.02; and fs is the initialshear flexibility matrix

fs 1GAL1 11 1in which GA is the effective shear rigidity, and L is theelements length.

The pinching flexibility matrix is nil when the elementtotal deformation is zero.

After a complete cycle, shear stiffness degradation isintroduced. Reduced initial shear stiffness fs accordingto the following equation was used:

fs as 1Ks (10)

where as is an arbitrary coefficient (from 01) with apractical value of approximately 0.03 and Ks is thesecant stiffness as shown in Fig. 6.

2.2. Variation of axial load

There are two ways to incorporate the effect of axialload variation on the state of the yield surfaces of ahinge. The first approach is to consider the variation inaxial deformation (extension and compression), whilethe second approach is to consider the variation of axialforce as the control input. It is more convenient to usethe first approach when there are considerable differ-ences in the forceextension stiffnesses of eachsubhinge, thus keeping track of the appropriate axialforce corresponding to a certain level of axial defor-mation. Fig. 9 shows the axial loaddeformationrelationship for tied and spiral columns. From the figure,it can be postulated that the forceextension relationshipis almost linear with constant initial stiffness up to thefirst peak load level. On the basis of this assumption,the second approach that considers the variation of axialforce as the control input was adopted.

Fig. 9. Axial loaddeformation curves of tied and spiral columns(Park and Paulay [19]).

2.2.1. Effect on flexure subhingesSince the event to event solving technique rather

than the iterative technique is used on the elementlevel, the factor FACM which causes a certain flexureevent to occur such as reaching a yield surface andchange in stiffness, is calculated assuming a linearinterpolation along the axial load path, as shown in Fig.10. A curved path can be achieved if required, by subdi-viding the axial force variation increment into severallinear sub-increments.

2.2.2. Effect on shear subhingeThe post-shear failure forcedeformation response fol-

lows a softening branch as the failure surface contractsgradually. The rate of softening is specified by the plasticstiffness for the shear subhinge along the y and z axesKvy and Kvz. After contraction of the failure surface, it isrequired that the shear force state remains on the failuresurface. Considering a current state at the beginning ofa load step, the following must hold:

s(Vy,Vz,P,Vfy,Vfz) VyVfy2

VzVfz2

(11a)

FFavFultFav2

1

where Fav and Fult are defined in Fig. 7. While after load-ing:

s Vy FACSdVyVfacVfy 2

Vz FACSdVzVfacVfz 2

(11b)

F FACSdFFavFultFav 2

1

where FACS is the event factor for shear subhinge fail-ure, Vfac is the contraction factor for shear subhinge shearforce capacity (Fig. 11), Vy, Vz are the shear forces alongthe y and z axes, dVy, dVz are the incremental shear forcereductions along y and z axes, dF is the increment of


Fig. 10. Effect of variable axial load on response.

Fig. 11. Shear subhinge failure surface contraction: (a) case of constant axial load; (b) case of variable axial load.

variation in axial force, V fy, Vfz are the shear capacitiesalong y and z axes; and dV fy, dV fz are the increments ofshear capacity deterioration along y and z axes.

Fig. 11 shows the shear subhinge failure surface con-traction for the cases of constant and variable axial loads.In the figure, the element is assumed subjected to cycliclateral displacement from points 0 to 9 (loading 02,unloading 26, reloading 69). The shear subhingefailure event FACS is dependent on the contractionfactor Vfac. Fig. 11a shows the determination of the fac-tor Vfac in case of constant axial load, while Fig. 11bshows the change in the force state and consequentlyVfac, in case of variable axial force as shown in the samefigure where the axial force is varying through the points0 to 9 (increasing 02, decreasing 26, increasing69).

In addition to the influence of axial load variation on

the force state and the yield surfaces of the subhinges,it affects the flexibility matrix for the active flexuralsubhinges and consequently affects the elements totalflexibility and stiffness matrices as well. Eq. (5) showsthat the plastic flexibility matrix fspm,i of a yielded flexuresubhinge is dependant on the outward normal vector tothe yield surface, n, which in turn depends on the currentforce state.

2.3. Input data

The input data includes the properties of the yield sur-face for each subhinge, in terms of strength and stiffness.The data for the flexure subhinges can be determinedfrom the momentrotation analysis about each principalaxis. This analysis should take into consideration factors


that affect the lateral deformation behaviour of RCelements, such as bond slip of tensile reinforcement.

The rotation q and deflection of a reinforced con-crete member at any point along its length is due to thedistributed curvature of the member along its length andthe lumped rotation near the fixation point due to theslippage of steel bars at the tension side of the memberas well as the member shear deformation. The totalrotation and deflection at any point along the RC mem-ber is the algebraic sum of these three components asillustrated in Fig. 12.tot f s v (12a)qtot qf qs gv (12b)where tot is the total lateral tip displacement corre-sponding to the total rotation qtot; f and qf are the lateraldeflection and rotation due to flexure; s and qs are thelateral deflection and rotation due to reinforcement bondslip; v and gv are the lateral deflection and rotation dueto shear.

The rotation qf of a column is calculated by integratingthe distributed curvature, f, along the member, while thedeflection f is calculated by summing the static momentof the area under the column theoretical curvature dia-gram taken about the point of contraflexure. The theor-etical momentcurvature diagrams were obtainedassuming trilinear and parabolic stressstrain relation-ships for steel and concrete, respectively. Confined andunconfined concrete were modeled using the modifiedKent and Park model [17]. A plastic hinge zone withconstant curvature was assumed at the fixed end aftersteel yielding. In the current application, a simplifiedapproach was adopted where the plastic hinge length was

Fig. 12. Deformation components of an element.

assumed equal to half the section depth at yield andequal to the section depth at ultimate curvature. Fromthe momentrotation backbone relationship for the mem-ber, the points representing M1M3 and the slopes K1K4 of various segments are determined.

The lumped rotation at the fixation point, qs due toreinforcement bond slip is defined as:

qs ds

dc (13a)

The corresponding deflection at any point at distance xfrom support will be equal to qsx and the maximumtip deflection will be:

s qsL dsL

dc (13b)

The value of ds is determined from the anchorage slipanalysis of the embedded reinforcement bar under mono-tonic pull, utilizing the model developed by Alsiwat andSaatcioglu [18].

Shear deflections were calculated using the shear stiff-ness expression derived by Park and Paulay [19]. Forassumed 45 diagonal crack, the shear stiffness may beexpressed as:

Kv,45 rs

1 4nrsEsbd (14)

where 1/Kv,45 is the shear deflection in one unit lengthdue to one unit shear load; Es is the elastic modulus ofshear reinforcement; n is the modular ratio; and rs is theratio of transverse steel volume to volume of concretecore.

The shear subhinge Vfy or z, and Vry or z data are basedon the nominal shear capacity equation proposed byPriestley et al. [20].

The determination of the post-peak unloading stiffnessKvy and Kvz, in case of shear failure, was explored by anumber of researchers. Ricles et al. [16] proposed anunloading stiffness for a moderate ductility failure witha negative value of the elements initial elastic shearstiffness. In case of brittle shear failure, negativeunloading shear stiffness in the range of 2550% of theelastic shear strength was adopted. More recently,Aschheim [21] used the degrading MohrCoulomb fail-ure surface in an attempt to analytically model the shearstrength degradation in RC members. The tentativemodel was verified using seven specimens. Morerefinement to the model was recommended.

In the current analysis, a simple approach was adoptedfor the determination of the unloading post-peak shearstiffness. It is assumed that the column will undergopost-peakshear stiffness degradation, Kvy and Kvz, whichis equivalent to the loss of strength from Vfy or z toVry or z through a lateral displacement equivalent to twicethe yield displacement. This assumption is valid for a


moderate ductility type of response. On the other hand, alimited ductility response will be accompanied by brittleshear failure with a higher unloading stiffness, which isequivalent to the loss of strength from Vfy or z to Vry or zthrough a lateral displacement equivalent to the yielddisplacement. For y direction:

Kvy VfyVry

2yfor moderate ductility (15a)

Kvy VfyVryy

for limited ductility. (15b)

3. Element verification

The proposed 3D element was examined by compar-ing the analytical predictions to experimental measure-ments for the cases of uniaxial flexural behaviour of can-tilever RC columns under constant and variable axialloads tested by Abrams [22], and a case of biaxial shearfailure of biaxially loaded square fixedfixed RC columnwith variable axial load tested by Ramirez and Jirsa [23].

3.1. Uniaxial flexural behaviour under constant andvariable axial loads

Two tests on RC cantilever columns, C1 and C8,under compressive axial loads and cyclic lateral dis-placements reported by Abrams [22] were considered.The columns cross section was 230 305 mm and thelength was 1.60 m. The vertical reinforcement is four #6bars (rv = 1.71%) with #3 ties at 64 mm spacing asshown in Fig. 13. The average concrete compressivestrength from cylinder tests is 44.1 MPa, and steel yieldstrength is 423 MPa. The axial load was kept constant

Fig. 13. Dimensions of test specimens by Abrams [22].

during testing specimen C1. For specimen C8, the axialload was varied following a proportional pattern withrespect to the lateral displacement. This includedincreasing the axial load for increased lateral push anddecreasing it for increased lateral pull up to twice theyield deflection, after which the axial load was held con-stant. Table 1 shows the input data used to model thetwo tested specimens. The degradation parameters, am,as and ap are necessary to model the post-peak behav-iour of reinforced concrete. They are selected within theranges described after Eqs. (7), (9) and (10). Variationof the degradation parameters within the recommendednarrow ranges has a reasonably small impact on theresults. The test measurements of the moment rotationrelationship are shown in Fig. 14. The figure also showsthe analytical results for both test cases using thedeveloped model.

Good agreement between the analytical predictionsand the experimental results is observed. For the case ofconstant axial load, the analytically predicted strengthand stiffness were fairly close to the experimental ones.For the case of variable axial load, the analytical yieldand ultimate strengths as well as the loading andunloading stiffnesses at both the push and pull sides cor-related well with the test results. The model was capableof representing the degradation in strength in the pullside due to the variation of axial load.

In the current analytical model, pinching is pertinentto shear failure. In the case of combined variable axialand flexural loads, the pinching zone in the momentrotation relationship was simulated using an equivalentaverage stiffness. Although pinching was not explicitlymodeled in the case of axial and flexural loads, the use ofequivalent average stiffness resulted in predicted energydissipation capacities of the analyzed and tested columnsto be close.

3.2. Biaxial shear failure with variable axial load

A squat square fixedfixed RC column, ATC-B, testedby Ramirez and Jirsa [23] under non-proportional biaxialcyclic loading in conjunction with an alternating com-pressive and tensile axial load was analyzed. The columnhad a cross section of 305 305 mm and 0.914 m length.The vertical reinforcement is eight #6 bars (rv =2.58%) with #2 ties at 64 mm. The average concretecompressive strength was 32.4 MPa, and the steel yieldstrength was 450 MPa. Fig. 15 shows the specimensdimensions and details of reinforcement as well as thedeflection history in NS and EW directions and thesequence of variation of axial load. Table 1 shows theinput data used to model the tested specimen.

The experimental lateral loaddisplacement hystereticresponse in the NS and EW directions is shown in Fig.16a,c. The specimen experienced initial flexural crackingfollowed by inelastic flexural deformations in both NS


Table 1Test data and calculated model parameters

Variable Symbol C1 [22] C8 [22] ATC-B [23]

Column height L (mm) 1600 1600 914Axial load F (kN) 310 45 to 575 +220 to 550Longitudinal bar diameter db (mm) 19.0 19.0 19.5Embedded length lbar (mm) 500 500 760Concrete compressive strength fc (MPa) 42.3 45.9 34.6Longitudinal steel yield strength fy (MPa) 423 423 448Longitudinal steel ultimate strength fu (MPa) 650 650 690Momentrotation stiffnesses K1 (kN.m/rad) 120,000 120,000 54,000

K2 (kN.m/rad) 50,000 50,000 22,780K3 (kN.m/rad) 5000 5000 9520K4 (kN.m/rad) 500 500 4,680

Momentrotation strengths M1 (kN.m) 60 60 45M2 (kN.m) 100 100 136M3 (kN.m) 115 115 176

Axial balanced load Fav (kN) 688 688 1335Axial compressive capacity Fult (kN) 1788 1788 3560Degradation factor am 0.022 0.022 0.035Initial total shear capacity Vf (kN) 351 351 366Residual shear capacity Vr (kN) 193 193 68Shear degradation factor as 0.030 0.030 0.029Shear pinching factor ap 0.01 0.01 0.01

Fig. 14. Comparison between experimental (left) and analytical (right) results of columns under axial load and cyclic lateral displacements: (a)constant axial load C1; (b) variable axial load C8 tested by Abrams [22].


Fig. 15. Column tested by Ramirez and Jirsa [23]: (a) dimensions oftest specimens; (b) planned deflection and axial load paths.

and EW directions. Shear failure occurred when dis-placement first reached 15.2 mm in the NS directionleading to subsequent reduction in shear strength with ahighly pinched hysteretic behaviour, as seen in Fig.16a,c. A value of 22 kN/mm (126 kips/in.) was cal-culated for coefficients kvy and kvz to account for shearstrength degradation. However, these values did not havemuch effect on the analytical results since the degra-dation in stiffness upon load reversal combined withvariation of axial load dominated the shear strengthcapacity. Therefore, at each new cycle, the reduced stiff-ness led to a reduced shear strength capacity.

The predicted response based on the analysis of thetest specimen in the NS and EW directions is shownin Fig. 16b,d. The analytical and experimental responsesare shown to be in good agreement. The results of thisanalysis demonstrate that the element formulation is cap-able of modeling the hysteretic response of a reinforcedconcrete column with deterioration in shear capacity inthe plastic flexural hinge zone due to excessive flexuralductility demand.

4. Effect of different axial load patterns

Having the analytical model verified against experi-mental results, the next step was to study the effect ofdifferent axial load patterns on the response of laterallyloaded columns. Testing of columns under varying axialload patterns is difficult. Very few experimental resultsare available in the published literature [2225]. More-over, the axial load variations during severe earthquakeshave not been measured or accurately predicted by areliable analysis. Thus, an available analytical tool canachieve results and conclusions that have not yet beenverified by experimental work.

In this study, a slender RC cantilever column withgiven momentrotation properties, was subjected toeight different axial load paths as shown in Fig. 17. Theaxial load paths were selected to cover different possi-bilities of axial load variation with respect to the lat-eral deformation.

Path 1 had a constant axial load at 0.5 Pb (Pb is thebalanced compressive axial load). Path 2 had a variableaxial load between 0 and Pb (i.e. 0.5 Pb from theinitially applied load of 0.5Pb). In path 3, the axialload was varied from 0 to 2 Pb. In path 4, the axialload was varied between 0 and 0.5 Pb. In path 5, theaxial load was varied between 0.5 Pb (i.e. compressionand tension). Paths 25 were in phase loading cycles;such that the axial load was proportional to the lateralload with maximum axial load coinciding withmaximum lateral deflection. Path 6 was an out of phaseloading case, with a phase shift of quarter cycle betweenthe axial load and lateral deformation variations. In theout of phase loading of case 6, the maximum axial loadcoincided with zero lateral displacement. Paths 7 and 8represent an axial load that was varied at twice the num-ber of cycles with which the lateral load was varied. Inthe case of path 7, the axial load was varied such thatno axial load applies at maximum lateral displacementwhile the maximum axial load was applied when thelateral displacement was zero. In the case of path 8, theaxial load variation was such that the maximum axialload occurred at maximum lateral displacement while noaxial load was applied at zero lateral displacement.

The main purpose of the analysis of the column shownin Fig. 17 is to investigate the effect of the axial loadpath on the energy dissipation capacity of the column.For this reason, a slender column is selected where sheareffects are minimized.

The momentrotation relationships for the eightimposed variable axial load paths are shown in Fig. 18.It is recognized that there ought to be pinching of theloops under moment loading. The effect of pinching onthe energy dissipation capacity is reduced by using equi-valent average stiffness. In the analyzed problem, shearand pinching effects were selected to be of minor impacton the behaviour under investigation. The cumulative


Fig. 16. Comparison between experimental and analytical results of column under variable axial load and biaxial lateral cyclic displacementstested by Ramirez and Jirsa [22]: (a) experimental NS direction; (b) analytical NS direction; (c) experimental EW direction; (d) analytical EW direction.

dissipated energydisplacement ductility factor relation-ship for the eight different axial load paths are plottedin Fig. 19. From the two figures, the following behaviouris observed:

1. Comparing the cases of axial load paths 24 with path1, it is observed that the lateral moment capacity ofthe column subjected to a variable compressive axialload corresponds to the forcemoment interactionrelationship for the column section. In effect, themaximum moment capacity occurs when the axialload reaches the balanced compressive axial load.

2. Comparing the column behaviour when subjected toan axial load varying according to path 2 (0 to Pb)

to path 5 (0.5 Pb), it is observed that the axial loadwith reversing sign (i.e. compression and tension)causes approximately 25% decrease in the lateralmoment capacity of the column that is subjected toaxial load that remains compressive. The decrease inthe lateral moment capacity in the case of an axialload with reversing sign is accompanied by anincrease in the unloading stiffness which results inaccumulated energy dissipation capacity approxi-mately equal to the case of an axial load of the sameamplitude but remaining compressive.

3. Comparing cases of axial load path 6 with path 2, itis observed that out of phase loading (path 6) causesslight decrease in the moment capacity and approxi-


Fig. 17. Hypothetical cantilever column properties; (a) momentrotation relationship; (b) momentaxial force interaction diagram; (c) eight axialload paths.

mately 15% decrease in the energy dissipatingcapacity of the RC column as compared to the inphase loading. The in phase loading of path 2 refersto the case when the maximum axial load is appliedat the maximum lateral push while the minimum axialload is applied at the maximum lateral pull. The outof phase loading of path 6 represents the case whenthe maximum and minimum axial loads coincide withzero lateral displacement.

4. Comparing the column behaviour with an axial loadfollowing paths 7 and 8 with that of an axial loadfollowing paths 26, it is noted that applying twoaxial loading cycles for every one lateral load cyclewill decrease the energy dissipating capacity of thecolumns.

5. From the results of cases of axial load paths 7 and 8,it is observed that varying the compressive axial loadsuch that the maximum axial load is applied atmaximum lateral displacement and zero axial load isapplied at zero lateral displacement will decrease thelateral moment capacity, stiffness and energy dissipat-ing capacity of RC column as compared to the reverseload pattern (i.e. zero axial load at maximum lateral

displacement and maximum axial load at zero lateraldisplacement). This can be attributed to the fact thatincreasing the axial load while increasing the lateraldeformation will decrease the lateral stiffness of thecolumn.

5. Conclusions

A global element was developed to model the biaxialflexural and shear behaviours of RC columns subjectedto varying axial load. The model included stiffnessdegradation upon load reversals. The behaviour of theelement was verified using experimental results and wasshown to be fairly accurate in predicting the response ofaxially loaded columns with an applied variable axialload.

The effect of eight different axial load variation pat-terns on the response of laterally loaded RC columnswas studied. The patterns covered different load paths(in phase or out of phase) with different axial loadfrequency of application as compared to lateral load (oneor two axial load cycles for every cycle of lateral load)


Fig. 18. Momentrotation relationship for different axial load paths.

Fig. 19. Cumulative dissipated energydisplacement ductility factorrelationship for different axial load paths.

at different load levels (compressive 0 to 2 Pb or alter-nating compressive and tensile at 0.5 Pb).

It was concluded that the magnitude of the axial loaddid have a considerable effect on the lateral momentcapacity of RC columns in accordance with the momentaxial force interaction relationship. Thus, it is importantto accurately identify the possible expected levels ofaxial loads arising from different cases of loading dueto horizontal and vertical earthquake ground motioncomponents, as they will affect the section design.

A cyclic axial load that is reversing sign (compressionand tension) causes lower lateral moment capacity andenergy dissipation capacity of the RC column as com-pared to a cyclic axial load that remains compressive.

It was found that varying the compressive axial loadsuch that the maximum axial load is applied at maximum


lateral displacement and zero axial load is applied at zerolateral displacement will decrease the lateral momentcapacity, stiffness and energy dissipating capacity of thereinforced concrete column as compared to the reverseload pattern (i.e. zero axial load at maximum lateral dis-placement and maximum axial load at zero lateraldisplacement). This conclusion is relevant since it willhelp to focus the seismic analysis on the worst-casescenario of combined axial and lateral loading.

It was also concluded that increasing the frequency ofthe axial load cycles with respect to one cycle of lateraldisplacement (for example, two axial load cycles forevery lateral loading cycle) will considerably decreaselateral moment capacity, stiffness and energy dissipatingcapacity of the column. This conclusion is importantsince the frequency content of the vertical ground motioncomponent is usually higher than that of the horizontalcomponent.

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Flexural and shear hysteretic behaviour of reinforced concrete columns with variable axial loadIntroductionBiaxial quadri-linear degrading model with varying axial loadElement formulationFlexural subhinge stiffness degradationShear subhinge stiffness degradation

Variation of axial loadEffect on flexure subhingesEffect on shear subhinge

Input data

Element verificationUniaxial flexural behaviour under constant and variable axial loadsBiaxial shear failure with variable axial load

Effect of different axial load patternsConclusionsReferences

Flex ans Shear of col with varying axial ld Eng Str 2003.pdf

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