FIBER KONA Č NI ELEMENT U NELINEARNOJ ANALIZI KVADRATNIH SPREGNUTIH CFT STUBOVA FIBER FINITE ELEMENT IN NONLINEAR ANALYSIS OF SQUARE CFT COLUMNS

Čelične cevi ispunjene betonom, to jest CFT (Concrete Filled Tubes CFT) stubovi predstavljaju jedan od tipova spregnutih stubova. Pri opisivanju njihovog ponašanja, neophodno je imati u vidu različite nelinearne uticaje koji ih karakterišu. U ovom radu predložen je numerički model za nelinearnu analizu kvadratnih CFT stubova, baziran na primeni fiber konačnog elementa raspodeljene plastičnosti. Prema dosadašnjim istraživanjima, ovaj konačni element pokazao se kao dosta pouzdan – kako prilikom modeliranja čeličnih i betonskih, tako i u slučaju spregnutih konstrukcija [1], [2]. Ovde su analizirani njegova primena i izbor parametara modela prilikom opisivanja nelinearnog ponašanja uzoraka izloženih delovanju statičkog opterećenja, pri različitoj vitkosti uzoraka, različitom odnosu D/t (odnos ukupne dimenzije čeličnog profila [D] i debljini čeličnog profila [t]), kao i uzoraka izloženih različitim tipovima statičkog naprezanja. Tačnost modela proverena je upoređivanjem sa eksperimentalnim podacima dostupnim u literaturi.


INTRODUCTION
Steel tubes filled with concrete, also known as CFT (Concrete Filled Tubes -CFT) columns represent one type of composite columns.To model their behaviour appropriately, it is necessary to take into account different nonlinear effects.In this work, a numerical model based on distributed plasticity fiber element is presented.According to previous studies, fiber element has shown great reliability in modelling steel, concrete and composite structures [1], [2].The application of the previously defined numerical model and determination the model parameters is presented.The ability to model nonlinear behaviour of samples exposed to static load with different slenderness, D/t ratio (where D is the total dimension of a cross section and t is the thickness of steel tube) and different loading conditions has been analyzed.Accuracy of the model has been verified by comparing numerical results with experiments found in literature.

DESCRIPTIONS OF CFT COLUMNS
In recent engineering practice, the use of CFT columns is increasing due to numerous advantages relative to steel or reinforced concrete columns [3], [4].Since steel tube forms the exterior of the cross section (Figure 1), it provides CFT columns with large moment of inertia, leading to high stiffness and flexural capacity.The concrete, that can also be reinforced, fills the core of the cross section increasing compressive strength of the CFT column and also delaying local buckling of the steel tube by forcing all buckling modes outward.The use of CFT columns is of great interests in seismically active areas due to their high ductility [5].

BEHAVIOUR OF CFT COLUMNS
Previous studies [5]- [7] have confirmed that the behaviour of CFT columns is nonlinear due to numerous effects such as: cracking of the concrete, confining effect in concrete, viscous deformations (creep and shrinkage), yielding of the steel tube, residual stresses, buckling of the steel tube, etc.
CFT columns are predominantly exposed to compressive loading.The confining effects take place at an axial strain of approximately 1‰ when micro cracking in concrete start to occur and lateral expansion rate of concrete increases approaching lateral expansion rate of steel [8].At that point, the steel tube acts as stirrups and confines concrete resulting in an increase in compressive strength due to triaxial stresses in concrete.Circular steel tubes provide higher degree of confinement than flat sides of rectangular tubes, hence, the increase of compressive strength due to confining effect is more evident in circular than in rectangular cross sections [8].However, experiments have shown that confining effect is negligible under certain conditions.In slender CFT columns, when L/D ratio is higher than 15, loss of stability can appear even before axial strain in concrete achieves value when confining effects occur.In these cases increase in ductility and compressive strength of concrete due to confinement pokazano je da je povećanje čvrstoće betona usled efekta utezanja zanemarljivo i kod ekscentrično opterećenih kružnih i pravougaonih stubova ukoliko je effect should be neglected [9].In addition, when square or circular CFT columns are loaded eccentrically, it is shown that for e/D (e being force eccentricity) ratio odnos e/D (eekscentricitet sile) veći od 0.125 [7].
In this work, nonlinear material models are used to take into account nonlinear behaviour of steel and concrete, parts of a CFT column cross-section.Confinement effect is taken into account using modified constitutive models for concrete.Viscous deformations of concrete, buckling of steel tube and residual stresses in steel are neglected.In CFT columns, the effect of viscous deformations has much smaller influence than in reinforced concrete columns.The steel tube serves as an enclosed environment, so conditions remain ideally humid minimizing the effects of creep and shrinkage [10].Also, due to concrete core, local buckling of steel tube is postponed.Local buckling should be taken into account in CFT columns with small thickness of the steel tube.In rectangular cross sections, if D/t ratio is smaller than local buckling will not occur before steel yields [10].All CFT columns considered in this paper have D/t ratios smaller than , and the effect of local buckling was not included in the numerical model.Residual stresses in steel tube are neglected, and are a subject of the future study.

NUMERICAL MODEL
Contrary to numerous experimental research of CFT column, relatively few numerical models that can model their behaviour have been developed so far.For detailed modelling of connections between CFT columns and beam, 3D solid models can be used.However, when modelling whole structures, this model is unacceptable due to its low numerical efficiency.Therefore, when modelling larger structures, frame finite elements are still primarily used.
In this work, modelling of square CFT columns is done using distributed plasticity force based fiber beam/column elements [2], that were proven as very efficient and reliable when used for modelling steel, reinforced concrete and composite structures [1], [2].
In this element, a certain number of cross sections are being monitored along the element axis.Also, cross section discretization is done for each of the monitored cross-sections; they are discretized into a number of integration points (fibers).A corresponding uniaxial material model is assigned to each of the integration points (fibers) (Figure 2).The section response (the section stiffness matrix and the section resisting forces) is obtained through integration over the cross section (1), (2). (2) gde je Ksmatrica krutosti poprečnog preseka, a ssile u preseku.
where Ks -stiffness matrix of the cross section, and ssection forces.
Za modeliranje čeličnog dela preseka korišćeni su sledeći modeli materijala: bilinearni model s kinematičkim ojačanjem i Giuffré-Menegotto-Pinto model sa izotropnim ojačanjem.Za modeliranje betonskog dela preseka, korišćeni su Kent-Scott-Park model, Popov model i Chang-Mander-ov model sa uzimanjem u obzir čvrstoće betona na zatezanje i bez toga.Na osnovu detaljne parametarske analize [14], zaključeno je da se najbolje poklapanje sa eksperimentima dobija u modelima sa Giuffré-Menegotto-Pinto modelom materijala sa izotropnim ojačanjem (Slika 3) za čelični deo preseka i Chang-Mander modelom materijala (Slika 4) za betonski deo preseka.Pored toga, parametarska analiza pokazala je i to da je -u slučaju monotonog Force based formulated fiber beam/column elements are used in this work.The main advantage of the force formulation is that nonlinear behaviour of CFT column can be approximated by less number of finite elements (one or two elements per length of the column) with few integration points along the axis of the finite element.In addition, distributed element loading can easily be included into the formulation [11].
Nonlinear geometry is taken into account using the corotational beam formulation [12].
Stresses in fiber elements are obtained from strains, using previously defined constitutive relations.All stressstrain models are uniaxial, while space stress conditions are approximated indirectly, using the modified uniaxial stress-strain relations.The computer program "OpenSees" [13] is used for all numerical simulations and stress-strain material models available in this program are used for numerical models.
Furthermore, it is shown that when samples are exposed to monotonic static load, tensile strength of concrete has little influence on numerical results.In all the samples, difference between ultimate forces, when tensile strength is included in the model and when it is neglected, was smaller than 0.25%.Therefore, in the following evaluation study, tensile strength of concrete is neglected.In the case of cyclic or dynamic loading, this should be further analyzed, although most researchers agree that tensile strength can be neglected when calculating ultimate forces for CFT columns [15], [16] irrespective of loading conditions.
Parameters for defining Giuffré-Menegotto-Pinto model (Figure 3) are yield strength f y , modulus of elasticity E s and strain-hardening ratio b.Transition curve from elastic to plastic branch and hysteretic behaviour of the model is defined using parameters R 0 , cR 1 , cR 2 .Values of these parameters are obtained through a parametric analysis.Vrednosti navedenih parametara sračunate su na sledeći način.Čvrstoća utegnutog betona na pritisak usvojena je prema preporukama autora eksperimenata [6], [7].Dilatacija pri maksimalnoj čvrstoći betona sračunata je prema izrazu [17]: Parameters for defining Chang-Mander's model (Figure 4) are: confined compressive strength f' c , concrete strain at maximum compressive strength ε' co , initial elastic modulus E c , tensile strength f t , tensile strain at maximum tensile strength ε ct , non-dimensional parameter that defines the strain at which the straight line descent begins in compression x n , non-dimensional parameter that defines the strain at which the straight line descent begins in tension x p and parameter that controls the nonlinear descending branch of the concrete model r.

NUMERICAL MODEL EVALUATION
Accuracy of the previously described numerical model is evaluated by comparing numerical with experimental results.Tomii and Sakino experiments [6] and Bridge experiments [7] are used for validating the numerical model.Samples were exposed to non-proportional and proportional monotonically increasing static loading.
Beam was modelled using one fiber element with 5 integration points along the axis of the element.Considering that samples in this experiment are exposed to uniaxial bending, discretization has been done only in the y direction.Concrete core is discretized with 10 fibers along the y axis, while steel tube is discretized with 10 layers through the height of the concrete core and one layer per tube thickness (Figure 6).Further increase in the number of fibres has no influence on the results.Geometrical nonlinearities are taken into account using the corotational formulation [12].
For some of steel model parameters (Table 3) experimentally obtained values are used [6] (for E y , f y and b), while for other parameters values are obtained through a detailed parametric study: R 0 =15, cR 1 =0.925 and cR2=0.15.

Tabela 3. Parametri konstitutivnog modela čelika
Table 3 [7][8][9][10] show comparison between end moment (M) -end rotation (φ) relations obtained using previously defined numerical model and experimental results.For sample II-2, two diagrams are shown to illustrate the small effect of the modelling of the tensile strength of numerički rezultati modela sa uzimanjem u obzir čvrstoće betona na zatezanje i bez toga.Kod ostalih uzoraka, prikazani su samo numerički rezultati dobijeni bez uzimanja u obzir ove čvrstoće (f t =0).Vrednosti graničnih momenata nosivosti uzoraka dati su u Tabeli 4. Sračunata srednja vrednost i standardna devijacija za ovu grupu testova potvrđuju visok stepen tačnosti modela za sve nivoe aksijalne sile.concrete on ultimate capacity.One is when tensile strength is included in the concrete model (f t ≠0), and the other one is when it is neglected (f t =0).For other samples, numerical results presented below are obtained not taking into account tensile strength of concrete.Ultimate moment capacities of samples are shown in Table 4. Calculated average and standard deviation for this group of test show high level of accuracy of the numerical model.Also, it is evident that numerical model is very good at describing nonlinear behaviour of CFT columns exposed to constant axial force and monotonically increasing end moments for all P/P 0 ratios.

Bridge experiment[7]
Bridge experiment [7] is conducted on a simply supported beam exposed to eccentric axial force (Figure 11).Axial compressive force N is monotonically increasing, while the mid-span deflection is being monitored.In this group of tests, the effects of loading eccentricity, column slenderness and biaxial bending on nonlinear behaviour of CFT columns are studied.Geometrijski podaci o uzorcima su prikazani u Tabeli 5, sa oznakama veličina prikazanim na slici 12.
Beam was modelled with two fiber elements with three integration points along the element axis.Considering that beam is exposed to biaxial bending, the discretization was done along both y and z directions.Total number of cross section integration points (fibers) is 140 (100 fibers for concrete core and 40 fibers for steel tube) (Figure 13).
The increase of compressive strength due to confinement for these samples was neglected, because e/D ratio was higher than 0.125, as explained in section 3. Remaining material model parameters are calculated using expressions (3)(4)(5)(6)(7)(8) and their values are shown in Table 6.Parameter b has a value of 0.025 Tabela 6. Parametri konstitutivnih modela materijala Table 6.Parameters for models of materials [ Slike 14-17 prikazuju poređenje numerički i eksperimentalno dobijenih relacija aksijalna sila (N) -vertikalno pomeranje na sredini raspona (v).Granične vrednosti aksijalne sile su date u Tabeli 7. Kao i u prethodnoj grupi testova, srednja vrednost i standardna devijacija potvrđuju visoku tačnost predloženog numeričkog modela.Najveće odstupanje numerički i eksperimentalno dobijenih rezultata je kod uzoraka SHC-7 i SHC-8 koji imaju najveću vitkost.Odnos L/D kod ovih uzoraka jeste 20, odnosno, veći je od 15.Kao što je objašnjeno u uvodnom delu, u ovim slučajevima stabilnost nosača dominantno određuje njegovo ponašanje, što dati numerički model ne uzima u obzir i što će biti predmet naknadne analize.Kod ostalih uzoraka, granična sila određena je s greškom manjom od 7%.Figures 14-17 show numerically and experimentally obtained relations between axial force (N) -vertical midsection displacement (v).Numerically and experimentally obtained values of ultimate axial forces are shown in Table 7.As with previous group of test, average value and standard deviation confirm the ability of the proposed numerical model to approximate well the nonlinear behaviour of CFT columns exposed to eccentric axial force.Larger differences between numerical and experimental results are observed in samples with higher slenderness such as SHC-7 and SHC-8.Ratio L/D for these samples is 20, i.e. higher than 15.As was explained in introduction, in these cases stability of CFT column dominantly governs the behaviour of the sample.In proposed numerical model, this is not considered and will be a subject of future study.In other samples, ultimate axial force obtained numerically differs from experimental results by 7% at most.

CONCLUSION
In this work, a numerical model for nonlinear analysis of square CFT columns using distributed plasticity fiber finite element is proposed.Numerical model presented in this work takes into consideration: nonlinear behaviour of concrete and steel using nonlinear material models, confinement effect and geometrical nonlinearities.Local buckling of steel tube, residual stresses and viscous characteristics of concrete are not considered.
For modelling of steel and concrete parts of the cross section various material models are analysed.Parametric study is conducted and it is concluded that the best agreement between numerical and experimental results is achieved using Giuffré-Menegotto-Pinto model for steel and Chang-Mander model for concrete.Afterwards, validation of such numerical model is done by comparing numerical and experimental results of CFT columns exposed to proporizvršena je evaluacija tako definisanog numeričkog modela.U ovim testovima stubovi su bili izloženi tional and non-proportional monotonically increasing loading.Based on numerical results it can be concluded statičkom, monotono rastućem opterećenju.Na osnovu dobijenih rezultata, može se zaključiti da predloženi numerički model karakteriše visok stepen tačnosti u nelinearnoj analizi kvadratnih CFT stubova pri statičkom opterećenju.
that the proposed numerical model is convenient for use in nonlinear analysis of CFT columns, and shows high level of accuracy.

Nikola BLAGOJEVIC Svetlana M. KOSTIC Sasa STOSIC
The paper presents nonlinear analysis of square CFT columns using distributed plasticity fiber elements.Behaviour of CFT columns is nonlinear and it is necessary to include different nonlinear effects into the numerical model in order to simulate their behaviour properly.Model proposed in this work considers: nonlinear behaviour of concrete and steel using nonlinear stress-strain models, confinement effect and geometrical nonlinearities.Tests exposed to static loading with different slenderness, different D/t ratio (where D is the total dimension of a cross section and t is the thickness of steel tube) and different loading conditions are analyzed.Stress-strain models that best approximate the behaviour of CFT columns are determined from a detailed parametric study.The proposed numerical model is validated by comparing numerical with experimental results available in the literature.
Key words: fiber beam/column element, CFT column, nonlinear analysis

Figures
show comparison between end moment (M) -end rotation (φ) relations obtained using previously defined numerical model and experimental results.For sample II-2, two diagrams are shown to illustrate the small effect of the modelling of the tensile strength of

Table 1 .
Dimensions and material properties of samplesTestČelična cev (mm) D x B x t Steel tube (mm) D x B x t Slika 6. Diskretizacija poprečnog preseka u testovima Tomii i Sakina Figure . Steel model parameters t ), za uzorak II-2, prikazani su

Table 4 .
Numerical and experimental results

Table 5 .
Geometrical properties for Bridge experiment Test Čelična cev (mm) D x B x t Steel tube (mm) D x B x t

Table 7 .
Numerical and experimental results