-
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.
-
1356 K. ElMandooh Galal, A. Ghobarah / Engineering Structures 25
(2003) 13531367
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.
-
1357K. ElMandooh Galal, A. Ghobarah / Engineering Structures 25
(2003) 13531367
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.
-
1358 K. ElMandooh Galal, A. Ghobarah / Engineering Structures 25
(2003) 13531367
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
-
1359K. ElMandooh Galal, A. Ghobarah / Engineering Structures 25
(2003) 13531367
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
-
1360 K. ElMandooh Galal, A. Ghobarah / Engineering Structures 25
(2003) 13531367
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
-
1361K. ElMandooh Galal, A. Ghobarah / Engineering Structures 25
(2003) 13531367
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
-
1362 K. ElMandooh Galal, A. Ghobarah / Engineering Structures 25
(2003) 13531367
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].
-
1363K. ElMandooh Galal, A. Ghobarah / Engineering Structures 25
(2003) 13531367
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
-
1364 K. ElMandooh Galal, A. Ghobarah / Engineering Structures 25
(2003) 13531367
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-
-
1365K. ElMandooh Galal, A. Ghobarah / Engineering Structures 25
(2003) 13531367
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)
-
1366 K. ElMandooh Galal, A. Ghobarah / Engineering Structures 25
(2003) 13531367
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
-
1367K. ElMandooh Galal, A. Ghobarah / Engineering Structures 25
(2003) 13531367
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.
References
[1] Abrahamson NA, Litehiser JJ. Attenuation of vertical peak
accel-eration. Bulletin of the Seismological Society of
America1989;79(3):54980.
[2] Ghobarah A, Elnashai AS. Contribution of vertical ground
motionto the damage of RC buildings. In: Proceedings of the 11th
Euro-pean Conference on Earthquake Engineering. Rotterdam:
Balk-ema; 1998.
[3] Saatcioglu M, Derecho AT, Corley WG. Modelling
hystereticbehaviour of coupled walls for dynamic analysis.
EarthquakeEngineering and Structural Dynamics 1983;11:71126.
[4] Keshavarzian M, Schnobrich WC. Computed nonlinear
seismicresponse of R/C wall-frame structures. In: Civil
EngineeringStudies. Structural Research Series No. 515. Urbana:
Universityof Illinois; 1984.
[5] Takayanagi T, Schnobrich W. Computed behaviour of
reinforcedconcrete coupled shear walls. Civil engineering studies.
StructuralResearch Series No. 434. Urbana: University of Illinois,
1976.
[6] Lai SS, Will GT, Otani S. Model for inelastic biaxial
bendingof concrete members. Journal of Structural Engineering,
ASCE1984;110(11):256384.
[7] Lai SS, Will GT. R/C space frames with column axial force
andbiaxial bending moment interactions. Journal of Structural
Engin-eering, ASCE 1986;112(7):155372.
[8] Poston RW, Diaz M, Breen JE, Roesset JM. Design of
slender,nonpresmatic, and hollow concrete bridge piers. Research
Report254-2F, Center for Transportation Research, University of
Texas,Austin, Texas, 1983, p. 414.
[9] Mari AR. Nonlinear geometric, material and time
dependentanalysis of three dimensional reinforced and prestressed
concreteframes. UCB/SESM Report No. 1984/12, Department of
CivilEngineering, University of California, Berkeley, California,
1984,p. 152.
[10] Perdomo ME, Ramirez A, Florez-Lopez J. Simulation of
damagein RC frames with variable axial forces. Earthquake
Engineeringand Structural Dynamics 1999;28(3):31128.
[11] Takizawa H, Aoyama H. Biaxial effects in modelling
earthquakeresponse of R/C structures. Earthquake Engineering and
Struc-tural Dynamics 1976;4:52352.
[12] Chen PF, Powell GH. Generalized plastic hinge concepts for
3Dbeam-column elements. EERC Report No. 82-20,
EarthquakeEngineering Research Center, University of California,
Berkeley,California, 1982, p. 274.
[13] Mroz Z. An attempt to describe the behavior of metals
undercyclic loads using more general work-hardening model.
ActaMechanica 1969;7:199212.
[14] Ghobarah A, Biddah A, Mahgoub M. Seismic retrofit of RC
col-umns using steel jackets. European Earthquake
Engineering1997;15:105266.
[15] Vecchio F, Collins M. The modified compression-field theory
forreinforced concrete elements subjected to shear. ACI
StructuralJournal 1986;83(1):21931.
[16] Ricles JM, Yang YS, Priestley MJN. Modeling non-ductile
R/Ccolumns for seismic analysis of bridges. Journal of
StructuralEngineering, ASCE 1998;124(4):41525.
[17] Park R, Priestley MJN, Gill WD. Ductility of
square-confinedconcrete columns. Journal of Structural Division,
ASCE1982;108(ST4):92950.
[18] Alsiwat JM, Saatcioglu M. Reinforced anchorage slip under
mon-otonic loading. Journal of Structural Engineering,
ASCE1992;118:242138.
[19] Park R, Paulay NY. Reinforced concrete structures. New
York,NY: John Wiley and Sons Inc., 1975.
[20] Priestley MJN, Verma R, Xiao Y. Seismic shear strength
ofreinforced concrete columns. Journal of Structural
Engineering1994;120(8):231029.
[21] Aschheim M. Towards improved models of shear strength
degra-dation in reinforced concrete members. Structural
Engineeringand Mechanics 2000;9(6):60113.
[22] Abrams DP. Influence of axial force variations on flexural
behav-iour of reinforced concrete columns. ACI Structural
Journal1987;84(3):24654.
[23] Ramirez H, Jirsa JO. Effect of axial load on shear
behaviour ofshort RC columns under cyclic lateral deformations.
PMFSELReport 80-1, University of Texas, Austin, 1980, p. 162.
[24] Saatcioglu M, Ozcebe G. Response of reinforced concrete
col-umns to simulated seismic loading. ACI Structural
Journal1989;86(1):312.
[25] Bousias NS, Verzelletti G, Fardis MN, Gutierrez E.
Load-patheffects in column biaxial bending with axial force.
Journal ofEngineering Mechanics, ASCE 1995;121(5):596605.
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