A Fibre Model for Push-Over Analysis of Under Designed Reinforced Concrete Frames

8/6/2019 A Fibre Model for Push-Over Analysis of Under Designed Reinforced Concrete Frames

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A fibre model for push-over analysis of underdesignedreinforced concrete frames

Edoardo Cosenza, Gaetano Manfredi *, Gerardo M. Verderame

Department of Structural Analysis and Design, University of Naples Federico II, Via Claudio, 21, 80125 Napoli, Italy

Received 10 January 2005; accepted 1 February 2006

Abstract

Most of the existing reinforced concrete buildings were designed according to early seismic provisions or, sometimes, without applyingany seismic provision. Some problems of strength and ductility, like insufficient shear strength, pull-out of rebars, local mechanisms, etc.,could characterize their structural behaviour. The above mentioned topics lead to a number of problems in the evaluation of the seismicbehaviour of reinforced concrete (RC) frames. Therefore the assessment of existing RC structures requires advanced tools. A refinedmodel and numerical procedure for the non-linear analysis of reinforced concrete frames is presented. The current version of the modelproposed is capable of describing the non-linear behaviour of underdesigned reinforced concrete frames including brittle modes of fail-ure. Selected results of an experimentaltheoretical comparison are presented to show the capabilities of this model. The results show thecapacity of the model of describing both the global behaviour and the local deformation at service and ultimate state. 2006 Elsevier Ltd. All rights reserved.

Keywords: Non-linear; RC frames; Beam; Column; Bond; Hook

1. Modelling of reinforced concrete frames

Many models for the non-linear analysis of RC framesare proposed in the literature. They can be classifieddepending on the level of discretization [1] in point by pointmodel, member by member models and global models. Thechoice of the most suitable model depends on the goals ofthe analysis and by the structural properties. Structurescharacterized by brittle mechanisms require a non-linearanalysis and then the use of highly discretized models; on

the other hand, for structures with flexural collapse mech-anisms, member by member or globalmodels can be used toobtain reliable predictions. A good balance between com-putational effort and level of reliability of the results shouldbe achieved in choosing the model by taking into accountthe amount of basic information that each model requires.In the last few years, fibre models have become more and

more popular. They still keep the basic hypothesis of sub-dividing the structure in mono-dimensional elements, eventhough they could be defined as a hybrid between point bypoint and member by member models. The constitutivelaws of concrete and steel are introduced [2]; in recent ver-sions the hypothesis of perfect bond is removed [3] andshear collapse is taken into account [4]. Some authorsintroduced a joint element accounts for inelastic sheardeformation and bar bond slip in program DRAIN-2DX[5,6].

Manfredi and Pecce [7] proposed for beams and col-umns a fibre element that introduces explicitly the bondlaw ss. Limkatanyu and Spacone [8] showed that theaccurate representation of the bondslip behaviour is cru-cial in predicting the response of RC frames subjected toboth static and dynamic loadings.

In the assessment of the seismic capacity of existingunderdesigned RC structures all the brittle failure modesare potentially active, and this occurrence requires thedevelopment of a reliable numerical model in terms ofbehaviour and material properties [9]. In this paper a

0045-7949/$ - see front matter 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compstruc.2006.02.003

* Corresponding author. Tel./fax: +39 081 7683491.E-mail address: [email protected] (G. Manfredi).

www.elsevier.com/locate/compstruc

Computers and Structures 84 (2006) 904916
mailto:[email protected]:[email protected]


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numerical model for reinforced concrete frames is pre-sented, which is an extension of the model proposed inManfredi and Pecce [7]. The model is able to predict themain mechanisms influencing the non-linear behaviour ofreinforced concrete frames (Fig. 1). In particular, the pro-posed model considers an explicit introduction of advancedconstitutive bondslip relationships that allow to describ-

ing the structural behaviour in the large post yielding fieldfor elements under bending and axial forces and to intro-duce refined models for beamcolumn joints [13].

2. Element formulation

The beamcolumn element is characterized by a spreadof plasticity and distributed cracking: it belongs to the fibremodel family. The mechanical properties of the cross-sec-tion are evaluated by considering the constitutive laws ofthe materials.

The classic hypothesis of perfect bond between concreteand steel is removed and a stressslip bond constitutive law

is introduced [10]. Such an aspect allows for a more reliableassessment of the tension stiffening effect, for both elasticand plastic field, and avoids the approximations due tothe assumption of the plastic hinge length.

For the column, it is possible to consider the variation ofaxial forces due to lateral loads, and the related effects interms of overall strength and deformation capacity. Also,considering the axial deformation allows for a detailed sim-ulation of the interactions between columns and infill walls.

In the beamcolumn joints, that plays a significant influ-ence on the structural response, both in terms of strength(i.e., shear failure of the panel or pull-out of the rebars)

and deformation (due to the cracking of the concrete and

slippage of rebars), the rotation at the beamcolumn inter-face is computed taking into account either the bondbetween concrete and steel or the constitutive law of thehooks.

The influence of shear forces on the behaviour of thebeams was modelled by Priestley et al. [11]; such a modelis based on a reduction of the shear strength depending

on the local ductility, as expressed in terms of linear varia-tion of the curvature. The model introduced here repre-sents an improvement of Priestleys since it enables thesectional ductility at any step of the analysis to be directlydetermined and then evaluate the shear strength of thosesections located in the plastic regions. Therefore, alongwith predicting ductile (i.e., flexural) and brittle (i.e., shear)failures, this method allows also failures characterized bylow ductility due to the bendingshear interaction to bedetermined.

The model for infill walls is based on the shear model byFardis and Panagiotakos [12]. It takes into account thestrength reduction due to the cracking of the panels and

the post-strength degradation. It is based on four differentsteps: initial shear behaviour of the uncracked panel,behaviour of the cracked panel as equivalent strut, its insta-bility after the maximum strength and final stage after com-plete failure characterized by constant residual strength.

2.1. The flexural model

The column is considered as a mono-dimensional ele-ment, by introducing a simplified deformation model forthe cross-section, as shown in Fig. 2. As mentioned beforethe hypothesis of perfect bond between steel and concrete is

removed. Thus, calculations in the generic cross-section

footingzone

exteriorjoint

beam

interiorjoint

column

panel zoneflexural shear

interaction

priestley

T

N

N

Mi

Mi+1

manfredi - pecce

bending moment

with axial force

fardis-panagiotakos

strut model

infills

fixed-endrotation

anchorage

hooked rebar pull-out

hooked rebar pull-out

panel zone

T

fardis-panagiotakos

strut model

-out

Fig. 1. The main mechanisms influencing non-linear behaviour of reinforced concrete frames.

E. Cosenza et al. / Computers and Structures 84 (2006) 904916 905


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comprised between two subsequent cracks are performedconsidering a linear strain diagram (i.e., concrete in com-pression and steel in tension).

Another significant hypothesis is based on pre-definingthe distance, Dl, between cracks (sub-element) which occur

in those sections where the cracking moment value is over-come. However, this is not a basic hypothesis in the pro-posed model since the sections where cracks open couldalso be determined as the analysis proceeds by computingwhere the tensile stress of the concrete reaches the limit ten-sile strength. Using the strain method, the problem is gov-erned by subsequent parameters: the maximumcompressive strain in the concrete ec(x,yg), the tensile strainin the steel es(x), the tensile strain in the concrete ect(x) andthe slip between steel rebar and the surrounding concrete intension s(x).

The material constitutive laws, concrete in compression

rc = rc(ec), concrete in tension rct = rct(ect), steel in ten-sion/compression rs = rs(es) and the steel/concrete bondslip relationship s = s(s, x), should be added to the aboveparameters.

The set of equations is composed by the force equilib-rium equation of the cross-section, the moment equilibriumequation around its geometric axis, the force equilibriumequation of the rebar and a compatibility equation betweensteel rebar and tensile concrete. The problem is formallysolved by the subsequent system of equations:

force equilibrium of the cross-section:

ZAc

rcx;yby dyZAct

rctxby dy Asrsx Nx

1

moment equilibrium around the geometric axis of thecross-section:ZAc

rcx;yybydy

ZAct

rctxybydyAsrsxhyg Mx

2

translational equilibrium of the steel bar:

drs

dx

4

Usx 0 3

compatibility equation for the steel bar and the concretein tension:

ds

dx esx ectx 4

where Ac, Act and As are the area of the concrete in com-

pression, of concrete in tension, of the steel bars and U isthe bar diameter, respectively.

Considering the three functions ec(x), es(x), ect(x) as gov-erning unknowns, the solution can be achieved by resolvingtwo non-linear algebraic equations and two differentialequations of the first order, linear and non-linear respec-tively. Since strains have been adopted as unknowns, thesolution in terms of stress is unique in any case, eventhough the constitutive laws show a descending branch.At any integration interval the boundary conditions haveto be associated to the differential equations system. In par-ticular for x = x* and for x = x* + Dl, that means in the

section corresponding to the crack formation, the stress(or strain) is immediately obtainable. In these sectionsrct = ect = 0.

Is then possible to separate the first two equations fromthe others to obtain the value of ec(x) and es(x) in the endsection of the sub-element.

It is to be noted that Eqs. (1) and (2) are coupled to thedifferential equations (3) and (4) by means of the twoparameters rct(x) and ect(x) describing the behaviour ofthe concrete in tension. By neglecting the ect(x) term inEq. (4), usually smaller if compared to es(x) particularlywhen the steel is yielded, the differential equations (3)

and (4) form a system in the es(x) and s(x) terms, separatefrom the global equilibrium of the section. In such anhypothesis it is then necessary to use Eqs. (1) and (2),related to the global equilibrium of the section at the pointwhere x = x* and x = x* + Dl, in order to obtain bound-ary conditions, while inside the element the problemdepends on the differential equations. That simplifies thecomputational effort, and gives a solution in terms ofs(x) and es(x) which is very close to the effective value.The numerical solution of the differential set of equationsis quite difficult due to the non-linear constitutive laws andthe fact that the bond relationship depends on the distanceof the section from the crack. The problem can be solvedonly by following a numerical approach; in particular, adiscretization is carried out using the finite differencemethod, by dividing the region included by two cracksinto (n 1) parts having lengths equal to Dx, as shownin Fig. 3.

The mechanical properties of the cross-section are eval-uated in a detailed way, using a fibre approach for materialproperties and introducing the appropriate constitutivelaws. The latter aspect enables the model to account alsofor specific issues such as the instability of steel rebars incompression or confinement effects. The pre-cracking stage(i.e., moment M< Mcr) is determined based on the hypoth-

esis of linear elastic behaviour.

H h

c

ygdc

ds/dxct(x)

s(x)

c(x)

dt

y

z

b(y)

As

H h

c

ygdc

ds/dx ct(x)

s(x)

c(x)

dt

y

z

b(y)

As

Fig. 2. Deformation model for the cross-section.

906 E. Cosenza et al. / Computers and Structures 84 (2006) 904916


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A reliable evaluation of the average steel strain es,mwithin the sub-element is possible as

es;m

1

DlZDl

0 esx dx 5

where es(x) is the distribution of steel strain along the entiresub-element and Dl is the length of the sub-element.

Generally, for a portion subjected at the ends by themoments M1 and Mn (see Fig. 3) a relationship averagemoment Mm average curvature /m of the sub-element,can be uniquely defined; in particular, the moment Mmcan be defined, assuming its linear variation between twoends, as the average of moments in cracked sections, whilethe average curvature /m can be obtained solving the sub-element as

/m

es;m

h dc;m 6

where dc,m represents the average neutral axis depth, con-sidering its linear variation between the values taken atthe ends of the sub-element, es,m is the average steel strainand h is the effective depth of the cross-section,respectively.

Defining Mim and /im as the moment and the average

curvature at the ith step, while Mi1m and /i1m the corre-

sponding quantities at the i+ 1th step, the moment Mmrelative to an average curvature within /im and /

i1m can

be determined as their linear interpolation, that is

Mm Mim M

i1

m M

i

m/i1m /

im

/m /im 7

and similarly the curvature /m relative to a given momentMm within two known values Mi and Mi+1, is determinedas

/m /i

m /

i1m /

im

Mi1m Mim

Mm Mim 8

In particular, the incremental ratio D/m/DMm in theprevious expression can be interpreted as the average flex-ural tangential deformability of the sub-element; actually,as DMm tends to 0, the ratio D/m/DMm becomes the deriv-

ative of the curve / = /(M), that is

limDMm!0

D/mDMm

d/mdMm

9

2.2. The axial model

During the loading process, the non-linear behaviour ofthe concrete, and the cracking and the plastic deformationsof the section result in a change in the axial deformability.The proposed fibre model allows the assessment of theactual axial behaviour of the member by defining boththe axial deformability of the section (cracked anduncracked) and the sub-element defined by two consecutivecracks.

In general, in the case of bending combined with axialload, if the strain at the geometric axis level of the element

is assumed as a reference, and for symmetric cross-sections,the axial strain can be expressed as

eo es / h H

2

10

where eo is the axial strain, es is the strain in steel, / repre-sent the curvature, and h and Hare the effective depth andgeometric height of the cross-section, respectively.

Considering two consecutive cracks, it is possible tocompute the axial deformability of cracked sections. Asthe analysis moves away from the cracked section, the ten-sile stress is progressively transferred from the steel rebarto the surrounding concrete and then the steel strain

decreases; this results also in a decreased curvature of thegeneric section belonging to each sub-element.

The remarks on the tension stiffening effect outlined inthe previous section allow to be observed the increasedstiffness of the sub-element lowers the axial strain of thegeneric section compared to the axial strain in the crackedsection.

If, for the seek of simplicity, the stress of the tensile con-crete, ect is neglected, an approach similar to the abovediscussed allows to define the axial strain of the generic sec-tion belonging to the sub-element described by two consec-utive cracks. The average axial strain of the sub-element

can be obtained by averaging the axial strains of different

dc,nMn

N N

s,n

l

M M+ M

N N

T+ TT

x

y

x=x* x=x*+ l

dc,1

x=x* x=x*+ l

M1

s,1

li=1

s,ns,1

i+1i

i

i=n

i+1

s,m

s,i s,i+1

x

dc,nMn

N N

s,n

l

M M+M

N N

T+TT

x

y

x=x* x=x*+l

dc,1

x=x* x=x*+l

M1

s,1

li=1

s,ns,1

i+1i

i

i=n

i+1

s,m

s,i s,i+1

x

Fig. 3. Beam/column element and cracked sub-element subdivision.



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sections of the sub-element. Supposing to adopt theapproximate approach, the axial strain for the generic sec-tion i belonging to the sub-element can be defined as

eo;i es;i es;i

h dc;m h

H

2

11

where dc,m represents the average neutral axis depth.The sub-element average axial strain can be then

obtained averaging the axial strains of the different sec-tions, as follows:

eo;m es;m es;m

h dc;m h

H

2

es;m /m h H

2

12

Therefore, as already stated for the flexural model, if theaxial load and the axial strain at the ith step are defined as

N

i

and ei

o;m, while N

i+1

and ei1

o;m are the axial load and theaxial strain at next step, the average axial tangential defor-mability of the sub-element can be defined as

fNm ei1o;m e

io;m

Ni1 Ni13

2.3. Stiffness matrix elements

In order to evaluate the stiffness matrix of the element,the flexural and axial tangential deformability of the sub-element within two subsequent cracks have been defined;in particular, average flexural and axial deformability (or

stiffness) are the following:

fMm d/mdMm

kMm dMmd/m

14

fNm deo;mdNm

kNm dNmdeo;m

15

Within the considered uncracked element intervals, theaverage flexural and axial stiffness are defined by the corre-sponding quantities evaluated assuming the concrete grosssection.

Each term belonging to the stiffness matrix can be eval-uated starting by the flexibility coefficients of the corre-

sponding auxiliary beam. The generic flexibilitycoefficient can be evaluated applying the principle of vir-tual work; in particular, for the rotational coefficient itresults:

aij

ZL

MiMj

kMmdx 16

where L is the elements length, Mi and Mjare the momentsdiagrams evaluated on the auxiliary element with ith andjth unitary action, while kMm is the actual average flexuraltangential stiffness of the sub-element, as previously stated.

The compatibility conditions provide the following stiff-

ness matrix rotational terms:

K33 ajj

aii ajj a2ij17

K66 aii

aii ajj a2ij18

K36 aij

aii ajj a2ij19

Via equilibrium the following mixed terms can be deduced:

K32 K35 K33 K66=L ajj aij

aii ajj a2ij L20

K62 K65 K36 K66=L aii aij

aii ajj a2ij L21

Finally, from equilibrium and compatibility, the followingtranslational stiffness are obtained:

K22 K55 K25 K23 K26=L

aii ajj 2aij

aii ajj a2

ij L2

22

To these stiffness coefficients, those relative to extensionalterms must be added. In a similar way, applying the prin-ciple of virtual work, the extensional strain coefficients ofthe element are calculated and the stiffness coefficientsafterwards.

3. Special elements and detailing formulation

3.1. The fixed-end rotation model

The proposed joint model is governed by the slippage

between the anchored rebar and concrete, while the stiff-ness of the panel is assumed as infinite and, therefore, doesnot affect the joint deformability. The slip of steel rebars atthe jointelement (i.e., beam or column) interface is evalu-ated by a procedure similar to that above described for theflexural model; a force equilibrium equation (3) for therebar and a compatibility equation (4) between steel rebarand concrete in tension control the problem. The strain ofthe element can be obtained if the boundary conditions interms of steel strains are provided in the two end sections ofrebar.

In the section at the interface between joint and beamelement, x = Lh the boundary condition is obtained usingthe equilibrium equations (1) and (2) and considering thesection s part of the element; in the section, x = 0the boundary conditions depends on the behaviour of theanchorage detailing (see Fig. 4). From a theoretical pointof view, in this section, two limit boundary conditionscan be identified:

if the anchorage is not present, straight rebar is charac-terized by a free end, thus a unrestricted slippage occurs,so that es = 0;

if anchorage is rigid, slip at the inner end of the rebar isequal to zero s = 0, and a steel stress develops on the

anchoring device.



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Commonly, anchorage devices have a stressslipresponse that lies between the above boundaries, so thatboth slip and steel strain are not zero and are dependentupon the response of the end anchorage. In particular,when the rebar is terminated with an end hook, the steelelement can be treated as composed by the hook plus thestraight portion [13].

The steelconcrete interaction in the straight region isdescribed by the ss bond law, while the hook can be mod-elled as a translational non-linear spring whose behaviouris governed by the stressslip, rs,hsh relationship [14] com-puted in the common section between the hook and thestraight part, as depicted in Fig. 4.

Therefore, for the generic flexural moment in the inter-face section, it is possible to calculate the end rotation hrelated to slip sj as

h sj

h dc23

where dc represents the neutral axis depth, sj is the slip ofsteel rebars and h is the effective depth of the cross-section,respectively, in the jointelement interface section.

Once the Mh function is known, the joint deformabilitycan be computed. This can be done by considering thebeamcolumn intersection as a rotational non-linear springwhose behaviour is represented by the Mh relationship.

The model proposed for the joint can also be used for theinterfacial section between columns and footing; in this case,the longitudinal rebars of the columns are terminated intothe footing similarly to exterior joints of a RC plane frame.

3.2. Shearflexural interaction model

As regards the behaviour of bending elements carryinghigh shear forces, refined models must be introduced toevaluate the shear strength and take into account the ele-ments decreasing flexural ductility due to shear interaction.The shear strength of beam/column elements basically

depends on their ductility level. In plastic areas, decreasing

shear strength of the element is noted, due to a decreasingor absent shear resistant mechanisms in the concrete.

The influence of shear forces on the behaviour of thebeams was modelled by Priestley et al. [11]; such model isbased on a reduction of the shear strength depending onthe local ductility, expressed in terms of curvature varyingwith a linear trend. The introduced model represents animprovement of Priestleys because it enables the sectionalductility at any step of the analysis to be directly deter-mined and then the shear strength of those sections locatedin the plastic regions to be evaluated. Therefore, along withpredicting ductile (i.e., flexural) and brittle (i.e., shear)failures, this method allows to determine also failures char-

acterized by low ductility due to the bendingshear interac-tion (see Fig. 5).

Usually, the shear capacity of a frame element is givenby three terms regarded as independent: a concrete compo-nent Vc which is function of the section ductility level, acompression strength component Vp and finally a Vscomponent, whose extent is function of the transversereinforcement quantity. So

V Vc Vp Vs 24

The concrete resistance component Vc, valid both for circu-lar and rectangular elements, decreases as ductility in-creases, according to following equation:

Shear Capacity

Curvature ductility,

=1

ShearForce,

V

Flexural Response

Flexural Response

Flexural Response

Shear Capacity

Curvature ductility,

=1

ShearForce,

V

Flexural Response

Flexural Response

Flexural Response

,

=1

Fig. 5. Bendingshear interaction: failures mode.

Fig. 4. Fixed-end rotation model.



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Vc cffiffiffiffif0c

pAe 25

where c depends on section curvature ductility level, l/,following a linear relationship, while Ae is the actual shearstrength area considered as 80% of the sections geometricarea only.

It is assumed that the presence of the axial load, Ndeter-

mines a shear strength mechanism due to the formation ofa sloped compressed strut. Therefore, the shear strength Vpis given by the horizontal component of the diagonal com-pression strength as

Vp N H dc

2L26

where H is the geometric section height, L is the length ofcolumn from the critical section to the point of contraflex-ure, and dc is the neutral axis depth. Vp does not decreasewith increasing ductility.

The transverse reinforcement, contribution, Asw, on

shear strength is based on a mechanism that does notdecrease with increasing ductility. The shear strength isthen given by

Vs Aswfyhh

0

scot b 27

where h0 is the distance between the outside edge of the stir-rups, s is the stirrups spacing and b is the slope of the com-pressed diagonal strut with respect to the centroid of theelement.

3.3. Infill model

Infill walls can represent in some cases a crucial sourceof strength for the structure. Their presence and high stiff-ness can play an important role on the global response ofthe building and on the local performance of RC elements.

The masonry panels are implemented introducing themodel proposed by Fardis and Panagiotakos [12] thatcorrelate the shear force V on the infill wall, with the hori-zontal displacement D (see Fig. 6). In the resultant diagramfour different segments can be analyzed which representrespectively: (a) the initial shear behaviour of theuncracked panel; (b) the equivalent strut behaviour of thecracked panel; (c) the instability of the panel over its max-

imum strength; (d) the final stage of the panel when thefailure is achieved and the residual strength remains con-stant. The main parameters of the model are

initial stiffness of the uncracked wall, K1, obtained onaverage as Gwtwlw/hw, where Gw is the shear modulus

obtained by the diagonal compressive test; secant stiffness, K2, equal to the equivalent strut stiffness

computed with the elastic modulus, Ew, equal to themodulus of the panel in diagonal direction and withthe strut cross-section dimensions according to Main-stone [15];

cracking load, Vcr that can be computed as the productof the shear strength, fws (obtained by the diagonal com-pressive test) with the plan dimensions of the panel,fwstwlw;

maximum load, Vmax equal to 1.3Vcr.

The model is implemented by introducing a diagonal

compressive strut.

4. Global stiffness matrix

The models of different elements, constituting the beamelement and the column element, have been previously dis-cussed one by one; in particular, the flexural and nodal ele-ments are serially jointed to form the beam or columnelement. In this viewpoint, a flexibility matrix of thebeam/column element can be defined as the sum of thesingle sub-elements flexibility matrix, i.e.

Felem Fflex Fjnt 28

where Fflex is the flexibility matrix of the flexural element,while Fjnt is the flexibility matrix of the nodal element.

It is important to point out that the above stated matrixvaries within every loading step, for the non-linearitiesassociated with the moment-curvature, or the moment-rotation, and involves some variations in elements charac-teristics. Therefore, the matrix represents the tangentialflexibility matrix of the beam/column element.

According to the process shown so far, the stiffnessmatrix of each beam or column element can be deduced;in particular, for what it may concern columns, the geomet-

K1

K2

K3=(0.5%-10%)K

1

V

Vmax

Vcr

Vres=(5%-10%)Vmax

V

lw

hwK

1

K2

K3=(0.5%-10%)K

1

V

Vmax

Vcr

Vres=(5%-10%)Vmax

V

lw

VV

lw

hw

Fig. 6. Infill model: shear force versus horizontal displacement.



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ric stiffness matrix is considered as well. The whole struc-ture stiffness matrix can be formally drawn with respectto the stiffness method as follows:

Kstruct Xelem

Kbeam Kcol 29

where Kstruct is the whole structure matrix, Kbeam represents

the beam element matrix while Kcol represents the columnelement matrix.

5. Solution algorithm

In a finite element that is based on the stiffness methodof analysis the section deformations are obtained directlyfrom the elements end deformations by the deformationinterpolation function; the corresponding section resistingforces are determined from the section forcedeformationrelationship. The integral of the section resisting forcesover the element length yields the element resisting forces.

In a finite element that is based on the flexibility method,the first step is the determination of the element forces fromthe current element deformation using the stiffness matrix.Thus, the force interpolation function yields the sectionforces. The first problem is the determination of the sectiondeformation from the given section force, since the non-lin-ear section forcedeformation is commonly expressed as anexplicit function of section deformation; another problemarises from the fact that changes in the section stiffness pro-duce a new element stiffness matrix which change the ele-ment forces for the given deformation. These problemsare solved in Taucer et al. [16] by a special non-linear solu-tion method.

At the ith NewtonRaphson iteration it is necessary todetermine the element resisting forces for the current ele-ment deformation:

qi qi1 Dqi 30

To this end an iterative process denoted by j is introducedinside the NewtonRaphson iteration. With the initialelement tangent stiffness matrix Kj=0 = Ki1 and the givenelement deformation increments Dqi the corresponding ele-ment force increments are

DQj1 Kj0el Dqj1 31

The section force increments DDj=1 can be determinedfrom the force interpolation function. With the section flex-ibility matrix fj=0 = fi1 the linearization of the sectionforcedeformation relationship (moment-curvature) yieldsthe section increments:

Ddj1 fj0DDj1 32

The section deformations are updated dj = dj1 + Ddj.According to the section forcedeformation relationshipsection deformations dj correspond to resisting forcesDj1R and a new tangent flexibility matrix f

j=1 .The section unbalanced forces Dj1u D

j1 Dj1R aredetermined and they are then transformed to residual

section deformations rj1 fj1

Dj

1

u .

The residual section deformations are the errors made inthe linearization of the section forcedeformation relation-ship. The residual section deformation are integrated alongthe element to obtain the residual element deformationssj=1. The residual section deformation and the residual ele-ment deformation are determined but the corresponding

deformation vectors are not updated. The presence ofresidual element deformation violates the compatibility ofthe element. In order to restore the element compatibilitycorrective forces DQj2 Kj1el s

j1 must be applied atthe ends of the element, where Kj1el is the updated elementtangent stiffness matrix. A corresponding section forceincrement DDj2 bK

j1el s

j1 is determined inducing sec-tion deformation incrementfj1bK

j1el s

j1.Thus, in the next iteration j the state of the element

change as Qj=2 = Qj=1 + DQj=2 and the section forcesDj=2 = Dj=1 + DDj=2 and deformations dj=2 = dj=1 + Ddj=2

are updated, where Ddj2 rj1 fj1bKj1elemsj1.

Convergence is achieved when the selected convergence

criterion is satisfied. The presented non-linear analysismethod offers several advantages. Equilibrium along theelement is always strictly satisfied, since the section forcesare derived from the element force by the force interpola-tion functions. While equilibrium and compatibility aresatisfied along the element the section forcedeformationis only satisfied within a specified tolerance when the con-vergence is achieved. When all elements have convergedthe ith NewtonRaphson iteration is complete. The ele-ment force vectors are assembled to form the updatedstructure resisting forces. The new structure stiffness matrixis formed by assembling the current element stiffness

matrix. The structure resisting forces are compared withthe total applied load. If the differences is not within thespecified tolerance, a new NewtonRaphson iterationbegins.

6. A numericalexperimental comparison

In order to give an example of the capabilities of theproposed model a numericalexperimental comparison ispresented; the experimental reference is a full-scale RCbuilding, designed according to criteria and constructionmethods used in the last 40 years [17] in a large part ofSouth Europe and tested in the laboratory of the JointResearch Center of Ispra.

In particular, pseudo-dynamic tests on two full-scale RCframes (having the same structure, materials and geometrybut one with infill walls and one without) have been per-formed (Fig. 7). Numericalexperimental comparisons onboth of them allowed assessing the feasibility of the pro-posed model with reference to the non-linear behaviourof RC frames and to the interaction problems betweenthe RC elements and the masonry panels.

The frame was designed for gravity loads and lateralforces equal to about 8% of its mass. The structure has fourfloors, typical floor height of 2.70 m and 3 spans; two of

them are 5.00 m long, while the third has a length equal



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to 2.50 m. The beam layout is the same at each floor, withequal cross-section and reinforcement, with a slab thick-ness of 15 cm. The columns have rectangular cross-sec-tions, constant dimensions along the height and placedalong their weak axis.

The infill frame has wall panels in each span (Fig. 7). Inparticular, on each span the openings are located asfollows:

left span: windows openings with dimensions of1.20 1.10 m at each of the four levels;

central span: door opening with dimension of2.00 1.90 m and windows openings at the other levelswith dimensions of 2.00 1.00 m;

right span: no openings.

In order to verify the potentialities of the proposedmodel the experimental pseudo-dynamic tests are simu-lated assuming in approximate way that the push-overanalysis results are similar to the envelope of the cyclic test.The push-over analysis has been developed using the pro-posed numerical model and applying a pre-defined distri-bution of horizontal forces; in particular, the distributionhas been selected in order to model the first linear modeof vibration.

The gravity loads, applied on the frames during thetests, have been defined simulating dead and live load. InFig. 8 load values [18] are reported in order to performthe frame analysis; the uniform loads on the beams andthe point loads on the top of the columns represent a

scheme of the loads really applied on the tested frame.

6.1. The bare frame

The comparison with experimental results shows thatthe use of a bond slip model significantly improves theprediction capacity over the use of rigid bond model, but

several analyses [9] highlight that much of this differencederives from bondslip in the foundations and in the beamcolumn joints, respect to beam column elements, as demon-strated also by others authors [19].

In this direction a comparison between the resultsobtained by applying the proposed model and the experi-mental curves is proposed in Fig. 9, which depicts the storyshear versus the interstorey drift. Two different numericalanalyses have been carried out, graphically shown usingtwo different color curves: the grey curve represents thenumerical result obtained through a rigid joint hypothesisnot including the fixed-end rotation effect, while the blackone is obtained through a strained joint hypothesis includ-ing the fixed-end rotation effect.

If the joint is assumed to be stiff, the prediction of bothstrength and tangent stiffness overestimate the experimen-tal results, while when the deformability of the joint is con-sidered allows a very satisfactory agreement with test isachieved. The graph shows the numerical outcomes for aconcrete strain in compression up to ec = 0.5%.

This result highlights the large influence of the fixed-endrotation on the local and global behaviour of the structure.In particular, for low values of the story shear the twocurves are overlapped; in fact, the low flexural moment atthe end of beamcolumn elements supplies, in the hypoth-

esis of deformable joint (black curve), a low slip demand in

Fig. 7. Geometry of the Ispra frame: elevation view.



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the tensile reinforcement and therefore a low joint defor-mation. As the story shear increases, the black curve showsa lower stiffness, as consequence of an increasing jointdeformability, and it is in agreement with the experiments.In all cases, the numerical analyses appear to agree with the

lab evidence.

6.2. The infill frame

The interaction between infill walls and RC memberscan be seen to behave as something in between a shearpanel and a diagonal strut [20]. The boundary stress are

very different for those two cases: in the latter, the reaction

2.7

0m

2.7

0m

2.7

0m

2.7

0m

2.50 m5.00 m5.00 m

44.3 kN 60.2 kN76.1 kN 28.4 kN

56.4 kN 72.3 kN88.2 kN 40.5 kN

56.4 kN 72.3 kN88.2 kN 40.5 kN

56.4 kN 72.3 kN88.2 kN 40.5 kN

12.7 kN/m

15.1 kN/m

15.1 kN/m

15.1 kN/m

2.7

0m

2.7

0m

2.7

0m

2.7

0m

2.50 m5.00 m5.00 m

44.3 kN 60.2 kN76.1 kN 28.4 kN

56.4 kN 72.3 kN88.2 kN 40.5 kN

56.4 kN 72.3 kN88.2 kN 40.5 kN

56.4 kN 72.3 kN88.2 kN 40.5 kN

12.7 kN/m12.7 kN/m

15.1 kN/m15.1 kN/m

15.1 kN/m15.1 kN/m

15.1 kN/m15.1 kN/m

Fig. 8. Scheme of the frame gravity loads.

-300

-200

-100

0

100

200

300

-40 -30 -20 -10 0 10 20 30 40

drift [mm]

Shear, V [kN]Storey 1

with fixed end rotation

without fixed end rotation -300

-200

-100

0

100

200

300

-40 -30 -20 -10 0 10 20 30 40

drift [mm]



without fixed end rotation

-200

-150

-100-50

0

50

100

150

200

-80 -60 -40 -20 0 20 40 60 80

drift [mm]

Storey 3 Shear, V [kN]Storey 3 Shear, V [kN]


without fixed end rotationwithout fixed end rotation

-150

-100

-50

0

50

100

150

-40 -30 -20 -10 0 10 20 30 40

drift [mm]



Fig. 9. Numerical simulation of bare frame: effect of the fixed-end rotation.



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of the wall panel affects the axial loads of both columnsand beams; in the former, the axial loads do not change.This means that the choice of an appropriate model repre-sents a crucial step toward a reliable assessment of theframe characterized by the interaction between bare struc-ture and infill walls. In order to validate this opinion, a the-oreticalexperimental comparison was carried out on theinfill frame tested at Ispra. The RC elements of the frame

was studied using the flexural model discussed in Section2.1 while the axial behaviour is modelled considering twodifferent hypotheses: using the model described in Section2.2 (non-linear model), that allows evaluating the non-lin-earities depending on section cracking and reinforcementyielding, and using a simple linear model (linear model)with a constant axial stiffness during the incrementalanalysis. The masonry panels were analyzed by a model

NM

M

N Axial deformation

Axial deformation

N

N

Linear model

Non-Linear model

STrut model

ST-NL

ST-L

NM

M

N Axial deformation

Axial deformation

N

N

Linear model

Non-Linear model

SHear model

SH-NL

SH-LV

V

NM

M

N Axial deformation

Axial deformation

N

N

Linear model

Non-Linear model

STrut model

ST-NL

ST-L

NM

M

N Axial deformation

Axial deformation

N

N

Linear model

Non-Linear model

SHear model

SH-NL

SH-LV

V

Fig. 10. Infill/RC column interaction models.

-800

-600

-400

-200

0

200

400

600

800

-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0

drift [mm]

Storey 1 Shear, V [kN]

ST-NL

ST-L

SH-NL

SH-L

-800

-600

-400

-200

0

200

400

600

800

-5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0

drift [mm]


-600

-400

-200

0

200

400

600

-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0

drift [mm]


-400

-300

-200

-100

0

100

200

300

400

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

drift [mm]


ST-NL

ST-L

SH-NL

SH-L

ST-NL

ST-L

SH-NL

SH-L

ST-NL

ST-L

SH-NL

SH-L

Fig. 11. Numerical simulation of infilled frame: effect of models on frame drift.



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implementing that proposed by Fardis and Panagiotakos[12]; in particular, both shear and strut models were usedfor the panels. The basic hypotheses are summarized inFig. 10.

Fig. 11 shows a comparison between the model out-comes and the experimental curves, in terms of story shear

versus the interstorey drift. It highlights how differentapproaches provided similar global results, while the differ-ence emerges at high levels, where the strut model gives lar-ger values than the shear one in terms of deformability.This is due to different column axial loads in the two mod-els, which is more evident at high floors where low gravityloads lower the axial stress.

All this is expressed in Fig. 12 where story shear versusaxial displacements are reported for the exterior columnsof the first two floors. It shows how the predictions ofthe shear model do not fit the experimental evidence; theadoption of a linear axial model rather than a non-linearone does not make a great difference. On the other hand,the strut model allow for a good agreement with experi-mental records; the choice of a non-linear axial modelresults in a excellent simulation which is able to reproducethe non-linear behaviour of the columns due to the interac-tion with infill walls and to the decrease of axial load underhorizontal actions.

7. Conclusions

The paper describes a fibre model proposed for the non-linear analysis of RC frames. This model directly intro-duces the ss bond relationship; such aspect makes it very

useful in the analysis of existing structures where smooth

bars change the bond mechanism. The axial deformabilityof elements influencing the interaction between frames andinfill walls is also taken into account. The joint modelallows considering the contribution of the fixed-end rota-tion when straight and/or hook ends are present.

Overall the model allows for the assessment of the seis-

mic capacity of underdesigned RC structures, wheresmooth bars, non-linearities for low levels of load, lowquality of constructive details and consequent potentialfor brittle failure modes require sophisticated tools forthe numerical simulation.

The previous numericalexperimental comparisons showthe models potentialities and how it is able to highlightstrains and strengths of frames subjected to extensivenon-linear behaviour of elements. As regards the influenceof joint deformability on global behaviour of structure, themodel superbly simulates the strain increasing due to thefixed-end rotation, as the ISPRA test points out.

The proposed model can be used in the push-over analy-sis under seismic loading of reinforced concrete buildingswhere the structure is subjected to an incremental staticanalysis under a pre-defined pattern of horizontal forces.

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[1] RC frames under earthquake loadingstate of the art report. CEBBull 1996;231(May).

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-800

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-1.00 -0.50 0.00 0.50 1.00

axial displacement [mm]

column 1 Shear, V [kN]

ST-NL

ST-L

SH-NL

SH-L -800

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-1.00 -0.50 0.00 0.50 1.00



ST-NL

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-600-400

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-1.00 -0.50 0.00 0.50 1.00



ST-NL

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SH-L -800

-600-400

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-1.00 -0.50 0.00 0.50 1.00



ST-NL

ST-L

SH-NL

SH-L

Fig. 12. Numerical simulation of infilled frame: effect of models on column response.



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A Fibre Model for Push-Over Analysis of Under Designed Reinforced Concrete Frames

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