1 INTRODUCTION. Akbar Haghinejad Mahdi Nematzadeh *

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1 916 Three-Dimensional Finite Element Analysis of Compressive Behavior of Cirular Steel Tube-Confined Conrete Stub Columns by New Confinement Relationships Abstrat This paper presents a nonlinear analysis of axially loaded steel tube-onfined onrete (STCC) stub olumns with new onfinement relationships. For this aim, a 3-D finite element model of STCC olumns using ABAQUS program is developed and validated against the experimental data. Proper material onstitutive models are proposed and the onfinement parameters of onfined onrete are determined by mathing the numerial results via trial and error. The parameters onsidered for quantitative verifiation of the FE model inlude five different fators indiating the behavior of STCC olumns: ompressive strength orresponding to steel yielding point, initial peak strength and ultimate strength as well as longitudinal to irumferential stress ratio of steel tube at steel yielding point and initial peak point. For the qualitative verifiation, the axial and lateral stress strain relationships of STCC olumns are taken into aount. The omparison results indiate that the model an aurately predit the ompressive behavior of STCC stub olumns. Finally, a parametri study is also performed to evaluate the effet of tube diameter-to-wall thikness ratio (D/t), onrete ompressive strength (f) and steel yield strength (fy) on the ompressive behavior of STCC olumns. Aording to the results of the parametri study, the interfae shear stress and lateral onfining pressure are not affeted by f while signifiantly inrease with dereasing D/t. Akbar Haghinejad Mahdi Nematzadeh * Department of Civil Engineering, University of Mazandaran, , Babolsar, Iran * Corresponding author m.nematzadeh@umz.a.ir, Tel ; fax: Reeived In revised form Aepted Available online Keywords Confined onrete; Finite element analysis; Stub olumn; Confinement relationships; Compressive strength; Axial ompression. 1 INTRODUCTION Conrete onfinement reates a triaxial stress, whih inreases onrete strength and dutility (Bahrami et al. 2013; Han et al. 2005; Bahrami et al. 2014). In reent years, the use of steel tube-

2 A. Haghinejad and M. Nematzadeh / Three-Dimensional Finite Element Analysis of Compressive Behavior of Cirular Steel Tube-Confined onfined onrete (STCC) olumns in modern strutures has been the field of interest for many strutural engineers. In these olumns, the load is only applied on the onrete ore. Hene, the role of steel tube in onrete onfinement is more effetive. Moreover, beause of small ompressive axial load arried by the onfining tube in STCC olumns, the possibility of loal bukling of the steel tube dereases signifiantly (Wang et al. 2011; Aboutaha and Mahado 1998). In addition, the steel tube in STCC olumns is used as permanent formworks, leading to no need for onrete shuttering and uring and onsequently redution in the onstrution time and ost. So far, muh experimental researh has been arried out to investigate the performane of STCC olumns. It seems that Tomii et al. (1985) onduted the first studies on the behavior of STCC olumns. They improved the dutility of reinfored onrete stub olumns through onfining them by steel tubes and used it as a method to prevent shear failures. Experimental investigation of Han et al. (2005) illustrated that STCC olumns exhibit high levels of dutility and energy dissipation, espeially when they are under high axial loads. Aboutaha and Mahado (1998) studied the behavior of STCC olumns and ompared the obtained results with those of onrete filled-steel tube (CFST) olumns. They found that effetive onfinement of onrete ore inreases and the possibility of steel tube bukling dereases in STCC olumns more than CFST ones. One of the latest experimental researhes on STCC olumns has been performed by Nematzadeh (2012) in whih portion of steel tube onfinement is separated from its axial load arrying portion. Their results showed that the onfinement effetiveness oeffiient of STCCs is about half that of the experimental results of Rihart et al. (1928). There is little researh available in the literature on developing an exat model for simulating the onrete onfinement in STCC olumns. Using the ABAQUS program, Shneider (1998) presented a 3-D nonlinear finite element model for CFST olumns. To define the properties of onrete material in this model, the stress-strain urve of unonfined onrete was used in the absene of the strain hardening for steel material. His results showed that the finite element model an predit the elasti and inelasti behavior of the olumns preisely. Hu et al. (2003) used a onfined onrete stress-strain urve to define the properties of onrete material and a bilinear stress-strain urve for steel material in ABAQUS program. They verified the results obtained from the model with the experimental results of Shneider (1998) and Huang et al. (2002). From the results of numerial simulations, they also presented empirial equations of the lateral onfining pressure applied to the onrete ore. Ellobody et al. (2006) used a onfined onrete stress-strain and a multi linear stressstrain urve for onrete and steel material, respetively, and studied the effet of onrete strength and ross-setion geometries on the ompressive behavior of irular CFST olumns. Wang et al. (2011) investigated the numerial behavior of STCC olumns in their researh. Their results indiated that in STCC olumns due to small ompressive axial load arried by the steel tube, the eftiveness of the tube in onfining the onrete ore is high and its loal bukling possibility is low. Hene, thinner steel tube an be utilized in STCC olumns ompared with the CFST ones. Gupta and Singh (2014) investigated the numerial behavior of short onrete filled steel tubular (CFST) olumns by providing a 3-D finite element program and verifying the proposed model through omparison with the orresponding experimental speimens. It is observed that the value of radial onfining pressure in the area adjaent to top and bottom platens is markedly higher than the value inbetween due to the end restraint provided by the mahine platens. Yu et al. (2010) by using the

3 918 A. Haghinejad and M. Nematzadeh / Three-Dimensional Finite Element Analysis of Compressive Behavior of Cirular Steel Tube-Confined... experimental results of Han et al. (2005) developed a 3-D finite element model and evaluated the behavior of STCC olumns under axial ompression load. In order to define the onrete material in the model, the stress-strain urve similar to that of CFST olumns was used. Their results showed that load-arrying apaity of irular STCC olumns is more than the CFST ones. Despite that STCC olumns are widely used in engineering strutures and exhibit a good performane, there is not enough information available in the literature regarding their analysis. This paper presents a nonlinear analysis on the behavior of irular STCC stub olumns under axial ompression load along with onfinement effet. For this purpose, an appropriate 3-D finite element model is developed using ABAQUS (ABAQUS 6.12, 2012) program to be verified against the experimental results of STCC olumns onduted by Nematzadeh (2012). Appropriate equations are presented for the onrete onfinement parameters, by mathing the finite element (FE) results with the experimental data via trial and error. The parameters onsidered for quantitative verifiation of the FE model inlude the ompressive strength orresponding to the yield, initial peak and ultimate point of steel and ratio of longitudinal to hoop stress of steel tube at the yield and initial peak point of steel. Furthermore, both the axial and lateral stress strain relationships of STCC olumns are onsidered for quality validation. In order to define the onrete behavior in finite element model, the equivalent uniaxial stress-strain urve of onfined onrete is used, in whih the onfinement effetiveness oeffiient obtained from the experimental results of Nematzadeh (2012) is applied. Multi linear stress-strain urve is used for steel material. A parametri study is also performed and the effet of various parameters inluding tube diameter-to-wall thikness ratio, onrete ompressive strength and steel yield stress on ompressive behavior of STCC olumns is evaluated. 2 FINITE ELEMENT MODELING Modeling the behavior of steel tube-onfined onrete (STCC) olumns is performed in three main parts inluding onfined onrete, onfining steel tube and interation between onrete ore and steel tube. In addition, in finite element analysis of these olumns, seletion of element type and mesh size should be appropriate for aurate simulation of the behavior of the olumns in a reasonable omputational time. Sine STCC olumns under axial load are ompletely symmetrial, just 18of the speimen is needed to model whih leads to a signifiant redution in omputational time. 2.1 Element Type and Mesh Various elements were used for simulation of STCC olumns behavior and finally aording to the results, steel tube and onrete ore were modeled using 3-D solid element (C3D8R) available in ABAQUS program library. This element has eight nodes and eah one has three translational degrees of freedom. Also, strutured mesh is applied to simulate the onfined olumns. A pattern of the model in program is showed in Fig. 1. As shown in the figure, there is a height differene between top surfaes of onrete ore and steel tube whih is due to mathing with experimental onditions of STCC speimens for applying the load on the onrete ore only.

4 A. Haghinejad and M. Nematzadeh / Three-Dimensional Finite Element Analysis of Compressive Behavior of Cirular Steel Tube-Confined Figure 1: Finite element mesh of steel tube-onfined onrete. 2.2 Boundary Condition and Load Appliation To simulate the boundary ondition of STCC speimens in the model, top surfae of onrete ore is restrained in all degrees of freedom exept in loading diretion while upper surfae of steel tube is ompletely free. Also, bottom surfae of the omposite olumn inluding onrete and steel is free in all diretion exept in the loading diretion. Lateral displaement of ut surfaes (XZ and YZ surfaes) in diretion perpendiular to the surfae is prevented beause of the symmetry so that the enter line has no movement in both X and Y diretions. In the laboratory, solid steel plates were used for applying the axial load on end surfaes of onrete ore (Nematzadeh, 2012), but in modeling, the results of this ondition are similar to the ase without the loading plates. The axial load is applied on top surfae of the onrete ore equally in all surfae nodes through strategy of displaement ontrol in appropriate intervals via *STATIC GENERAL METHOD available in ABAQUS program library. 2.3 Modeling of Steel Tube Material Aording to Fig. 2, a trilinear stress-strain urve is used to define the steel material behavior in ABAQUS program whih inludes three stages: elasti, yield and strain hardening. Main parameters for defining the stress-strain urve of steel inlude yield stress ( f y ), ultimate stress ( f su ), yield strain ( y ), strain at the beginning of strain hardening ( p ) and ultimate strain ( su ), the values of whih are presented in Table 1, based on the experimental results of (Nematzadeh, 2012). First part of steel stress-strain urve represents the elasti behavior of steel material whih an be defined using *ELASTIC option in ABAQUS program library where Young s modulus and Poisson s ratio are onsidered equal to 210 GPa and 0.28, respetively. Also, *PLASTIC option in the program library is used to introdue the inelasti behavior of steel material inluding yield and strain hardening stages.

5 920 A. Haghinejad and M. Nematzadeh / Three-Dimensional Finite Element Analysis of Compressive Behavior of Cirular Steel Tube-Confined... Sine the steel tubes used in experiments are seamless hot-rolled (Nematzadeh, 2012), they have no residual stresses aused by the welding proess. von Mises riterion, known as the maximum distortion energy riterion or otahedral shear stress theory, is used in this study to estimate the steel yield stress. Although the Tresa riterion known as the maximum shear stress riterion is generally easier to apply in omparison with the von Mises riterion, the latter is in better agreement with the atual response of most metals. The Tresa riterion usually gives onservative results in predition of the system strength (Lekie and Bello 2009). f y (MPa) f su (MPa) y p su Table 1: Mehanial properties of steel tube (Nematzadeh, 2012). f su f y ɛ e ɛ p ɛ su Figure 2: Equivalent stress-strain urve of steel. 2.4 Modeling of Confined Conrete Material Conrete Stress-Strain Curve Sine ABAQUS program is able to onsider the onfinement effet in improving the behavior of the onfined onrete by using models suh as Druker-Prager, it seems that unonfined onrete behavior should be used to define the onrete material in ABAQUS. In this field, Shneider (2003) onduted an analytial study on ompressive behavior of CFST olumns, in whih the unonfined uniaxial stress-strain urve is used to define the onrete material in the ABAQUS. Nevertheless, many researhers have applied the onfined onrete model in the program to define the ompressive behavior of onrete (Hu et al. 2003; Ellobody et al. 2006; Yu et al. 2010). One of the main reasons for this fat is the lak of real modeling of onfinement and the inreased strength aused by it in ABAQUS program espeially in high onfinement values. This is beause there are various parameters in the program to define the inelasti stage of onrete material, based on whih the empirial uniaxial stress-strain urve of onrete is introdued. In this ase, Hu et al. (2003) presented empirial equations for the onfinement levels of onfined onrete for the definition of onrete material in the program using the trial and error method and mathing the analytial results obtained from the ABAQUS with the experimental results of CFST olumns.

6 A. Haghinejad and M. Nematzadeh / Three-Dimensional Finite Element Analysis of Compressive Behavior of Cirular Steel Tube-Confined In the present study, the results obtained from the analytial model of STCC speimens by using the unonfined behavior of onrete material, exhibit a signifiant differene with the experimental results so that this differene inreases with inreasing the onfinement level. Hene, the onfined uniaxial stress-strain urve is applied to define the onrete material in STCC model, the harateristis of whih are a funtion of onfinement level. Using various onfinement levels for definition of onrete behavior in ABAQUS program and mathing the obtained results with the experimental ones, empirial equations are ahieved for the onrete onfinement value in STCC olumns, whih are presented later. It should be noted that in general, the onfinement level defined for the onrete material in STCC model is muh smaller than the real onfinement provided by the steel tube. The equivalent uniaxial stress-strain urve presented in this study to define onrete behavior in ABAQUS program is shown in Fig. 3. The urve onsists of three stages: linear stage, nonlinear stage for pre-peak and bilinear stage for post-peak. The first stage of the urve is linear with slope of E known as modulus of elastiity of onfined onrete. This stage of the urve ontinues to half the ompressive strength of onfined onrete ( 0.5 f ) (Hu et al. 2003). It should be noted that a disontinuity is reated in the stress-strain urve at the intersetion between the linear stage and the nonlinear stage after that. The number of data points for definition of the onrete stress-strain urve must be enough so that no disontinuity would be reated with negative slope at intersetion of the first and seond stage of the urve. In this ondition, a disontinuity with positive slope is reated in urve, whih has a negligible effet on the FE results. 0.55(1 K) f Kf Confined onrete f f 0.5 f Unonfined onrete Figure 3: Equivalent uniaxial stress-strain urves of onfined and unonfined onrete. Modulus of elastiity of onfined onrete is alulated from the relationship proposed by ACI 318 (2008) as follows E 4730 f (1) where E and f are modulus of elastiity and ompressive strength of onfined onrete in MPa, respetively. The value of f an be obtained from Eq. (2) proposed by Rihart et al. (1928) who

7 922 A. Haghinejad and M. Nematzadeh / Three-Dimensional Finite Element Analysis of Compressive Behavior of Cirular Steel Tube-Confined... onduted the tests on the onrete onfined with fluid pressure. In Eq. (2), k is onfinement effetiveness oeffiient and f and f l are ompressive strength of the unonfined onrete and lateral onfining pressure, respetively. Strain orresponding to f, an be determine by Eq. (3) proposed by Mander et al. (1988), based on the experimental results of Rihart et al. f f k fl (2) fl 15k f (3) where is strain of the unonfined onrete orresponding to f and is onsidered Aording to the equations proposed by Nematzadeh (2012) for STCC olumns, the parameter k is onsidered as a onstant value equal to However, a larger value is obtained by Rihart et al., whih is equal to 4.1. Due to the existene of the longitudinal ompressive stress of steel tube in STCC olumns, the onfinement effet and thus the onfinement effetiveness oeffiient is redued ompared with the ase without the longitudinal stress. The parameter f l an be alulated by the trial and error method and mathing the FE results with the experimental data, as follows fl f y D t (4) l y The above equation indiates that the parameter f l is independent of the onrete ompressive strength. The relationship between f / f obtained from the mathing and D / t (external diameter-to-wall thikness ratio of steel tube) is illustrated in Fig. 4. Sine in Eqs. (2) and (3), the parameters k and f l are multiplied together, it is possible to inrease the value of k to 4.1 (proposed by Rihart et al.) and in ontrast, f l is redued by a linear redution fator of Figure 4: f / f obtained from mathing results versus D/ t of STCC olumns. l y The seond stage of the stress-strain urve of onfined onrete is nonlinear and is determined from Eq. (5) proposed by Saenz (1964), whih starts at the end of linear stage and ontinues to f.

8 A. Haghinejad and M. Nematzadeh / Three-Dimensional Finite Element Analysis of Compressive Behavior of Cirular Steel Tube-Confined f E 1 ( R RE 2) (2R1) R 2 3 (5) where f and are uniaxial stress and strain of onfined onrete, respetively. Also, R E and R are modular ratio and ratio relation, respetively, and are obtained by the following equations. R E E f (6) R RE ( R 1) 1 2 ( R 1) R (7) Constants R and R are strain ratio and stress ratio, respetively, whih determine the stressstrain urve shape of onrete after peak point (desending branh). They are taken as 4.0 as reommended by Hu and Shnobrih (1989). Sine these two parameters influene the asending branh of the onrete stress-strain urve, they are an be used to define the seond stage of the onfined onrete stress-strain urve. The third stage of the onfined onrete stress-strain urve is bilinear. This stage starts at and ended at strain equal to 11. The final stress is equal to K f where, K is the produt of the two parameters K and K s whih are related to the ompressive strength of unonfined onrete and the tube outer diameter-to-wall thikness ratio, respetively. By mathing the FE results to the experimental stress-strain urve of STCC speimens, parameter K is obtained as Eqs. (8a) and (8b), the value of whih an be higher or lower than 1. It an be found from these equations that the value of K and onsequently K dereases with inreasing the onrete ompressive strength beause of the redution in onrete dutility. Also, the inrease in D / t leads to the derease in the onfining pressure and as a result the redution in dutility of the onfined onrete and thus redution of K s and K. Fig. 5 shows the urves of K s and K versus D / t and f, respetively. D K KK s ( f) t D , 16.7 f 52.6 t D , 16.7 f 52.6 t D K KK s ( f) t D , 16.7 f 52.6 t (8a) (8b) Stress and strain at the intersetion between two lines of the third stage is onsidered 1.1 times higher than the average value of stress and strain between the endpoint of the seond stage and the

9 924 A. Haghinejad and M. Nematzadeh / Three-Dimensional Finite Element Analysis of Compressive Behavior of Cirular Steel Tube-Confined... endpoint of the third stage. Therefore, the stress and strain in the middle of the third stage are obtained equal to 0.55( K 1) f and 6.6, respetively. As mentioned for the steel stress-strain urve, it should be noted that the end point of the third stage in the onfined onrete stress-strain urve does not imply the failure point, but the ABAQUS program onsiders a onstant stress ondition after endpoint. Figure 5: Relationship of K s and K versus D / t and f for STCC olumns. Sine the behavior of the onfined onrete is used in the modeling, the plasti strain must be defined in the program as follows p f fl 2 E E (9) where and f are uniaxial strain and stress of the onfined onrete, respetively, obtained from p Fig. 3. Also, is plasti strain and is Poisson s ratio of the onfined onrete whih is onsidered equal to 0.2. In Eq. (9) and other equations of this researh, the sign of ompressive stress and strain is defined to be positive Yield Surfae of Conrete Sine the onrete ore in STCC olumns is under triaxial ompression, ompressive yield surfae of onrete inreases with inreasing the hydrostati pressure. Druker-Prager model (Chen and Saleeb 1994) an be used as one of the yield riteria of onrete, in whih the onrete shear strength is expressed based on the hydrostati pressure. Also, by onsidering a linear relationship between the shear strength and the onfining pressure of onrete, the linear Druker-Prager an be used for the onrete yield surfae as Eq. (10). F q ptan d 0 (10) where is the frition angle of the material and d is the ohesion of the material. Also, p and q are equivalent pressure stress and Mises equivalent stress, respetively, whih are defined in terms of the first stress invariant ( I 1 ) and the seond deviatori stress invariant ( J 2 ) as follows

10 A. Haghinejad and M. Nematzadeh / Three-Dimensional Finite Element Analysis of Compressive Behavior of Cirular Steel Tube-Confined I (11) 3 1 p, I q 3 J2, J2 [( 1 2) ( 3 2) ( 3 1) ] (12) 6 where 1, 2 and 3 are prinipal stresses. In ABAQUS, a modified yield riterion is used for onrete material whih is a D-P type plastiity model, referred to the extended Druker-Prager model. In this model, an additional parameter known as the flow stress ratio is adopted. This parameter ontrols the dependene of the yield surfae on the value of the intermediate prinipal stress and is physially defined as the ratio of the yield stress in triaxial tension to that in triaxial ompression, equal to the shear strength ratio of onrete under equal biaxial ompression to that under triaxial ompression. Figure 6: Extended linear Draker-Prager yield surfae.

11 926 A. Haghinejad and M. Nematzadeh / Three-Dimensional Finite Element Analysis of Compressive Behavior of Cirular Steel Tube-Confined... Fig. 6a illustrates the extended Druker-Prager yield surfae whih is ompared with von Mises yield surfae. Aording to this figure, the yield surfae funtion (F) of the extended Druker-Prager model is expressed as Eq. (13) in whih t is replaed by q in Eq. (10). F t ptan d 0 (13) r where t q1 1 is shear strength, in whih K is shear strength ratio and r 2 K K q is defined in terms of the third invariant of deviatori stress ( J 3 ), as following equation: r J J p p p (14) 33 3 /2, 3 ( 1 )( 2 )( 3 ) Fig. 6b shows the shape of the yield surfae in meridional plane for a ertain value of K. This figure indiates that two elements with the same material and the same hydrostati pressure ( p ) an exhibit different yield strengths ( q ). It an also be seen that for r / q 1 orresponding to the triaxial pressure ondition, the equation t q is obtained that is the same as Eq. (10) for the onventional Druker-Prager. Also, for r / q 1 whih orresponds to the equal biaxial pressure ondition, the equation t q / K is ahieved and the yield surfae an be written as Eq. (15) where the frition angle and the ohesion of the material are K times those in the triaxial pressure ondition. F t ptan K Kd 0 (15) In the ase of onrete material whih has the hardening and softening behavior, the yield surfae is not onstant and hanges parallel to the initial yield surfae so that only the intereption point ( d ) is different. Hene, the initial and final failure surfaes orrespond to the beginning of the onrete nonlinear stage and the onrete failure, respetively. Fig. 6 indiates the yield surfae in the deviatori plane for the two ases of K 1 and K 0.8. It an be seen from the figure that in the ase of the former, the shape of yield surfae is irular and yield stress for various r / q is the same and equal to q (similar to the onventional Druker-Prager model). In the ase of the later, the shape of yield surfae is not irular and yield stress for various r / q is different (similar to the extended Druker-Prager model). In order to make sure that the yield surfae remains onvex, it is neessary to apply the ondition of K 1.0. In irular STCC olumns, the lateral stresses applied on the onrete ore in different diretions are the same, as shown in Fig. 7, and thus the stress parameters of Druker-Prager model in the onfined onrete an be summarized as Eqs. (16)-(18).

12 A. Haghinejad and M. Nematzadeh / Three-Dimensional Finite Element Analysis of Compressive Behavior of Cirular Steel Tube-Confined Figure 7: Stress ondition of onfined onrete in irular STC. 1 p ( f 2 fl ) (16) 3 q f f (17) l r f f (18) l It should be noted that in this study Eqs. (16) (18) are true at the mid-height of the STCC speimens beause the frition at the interfae between onrete ore and steel tube makes a shear stress in the longitudinal diretion, whih hanges the prinipal stresses ondition indiated in Fig. 7. Using Eqs. (16) (18), shear strength ratio (K) is obtained as (see Appendix A), whih in this researh is onsidered to be 0.8 for the modeling of STCC speimens in ABAQUS. The results obtained from the FE model of STCC olumns indiate that the ompressive behavior of these olumns exhibits little sensitivity to the K parameter. This fat an be learly observed in Fig. 8. The reason for this is that aording to Eqs. (16) (18), r is equal to q ( r / q 1) and hene in the shear strength equation, t will be equal to q and independent of the parameter K. Figure 8: Parameter study of flow stress ratio (axial load-displaement urves of STCC for different values of K ).

13 928 A. Haghinejad and M. Nematzadeh / Three-Dimensional Finite Element Analysis of Compressive Behavior of Cirular Steel Tube-Confined... The frition angle is defined for shear failure and it is the slope of the linear Draker-Prager yield surfae in the meridional plane geometrially. To apply the yield surfae for the ase of uniaxial ompression, the ondition of 71.5 must be established. This parameter an be alulated based on the triaxial and uniaxial ompressive tests. In the present study, using the onfinement relationship proposed in the previous artile (Nematzadeh, 2012) for STCC olumns, the frition angle of onrete material is obtained to be 40 (see Appendix B). It should be noted that the improved behavior of onrete material is used for the modeling in ABAQUS program so that its ompressive strength and stress-strain urve are not in aordane with the experimental results of unonfined onrete, but are a funtion of the lateral onfining pressure. For this reason, the frition angle ahieved by the alulations leads to a poor agreement between the analytial and the experimental results. Hene, by mathing the FE results via trial and error method for all the onfined speimens, the frition angle is obtained to be 20. The ohesion parameter d is geometrially the interept of the linear yield surfae, and is related to the yield stress of the uniaxial ompression as: d tan (1 ) f (19) 3 For the STCC speimens, the parameter d is obtained to be To determine the plasti deformation of the material, the flow potential (G) is defined for the linear Druker-Prager model as follows G t ptan (20) where is the volumetri dilation angle whih is a major parameter affeting the behavior of the material that governs the D-P flow rule and is used as a material parameter in ABAQUS. Physially, the dilation angle is defined as the ratio of plasti volume hange to plasti shear strain and geometrially, it is the slope of the potential funtion in the p t plane, as shown in Fig. 9. Aording this figure, If 0 the material dilates; if 0 the material ontrats and if 0 the inelasti deformation is inompressible. To apply the flow potential for the ase of uniaxial ompression, the ondition of 71.5 must be established that is likely for real material. Also, assoiated flow potential is referred to the ase of and non-assoiated one is referred to the ase of in whih the flow potential is different from the yield funtion. The flow rule determines the diretion p of plasti deformation and relates G to the inremental plasti strain ( d ), defined as: p G dij (21) ij where is a salar hardening parameter whih may vary throughout the straining proess. Also, ij is the omponents of prinipal stress inluding 1, 2 and 3 in radial, irumferential and longitudinal diretions, respetively. In this researh, a non-assoiated flow rule is used to define the diretion of the plasti flow. The dilation angle of onfined onrete is obtained to be about 35 whih is higher than the frition

14 A. Haghinejad and M. Nematzadeh / Three-Dimensional Finite Element Analysis of Compressive Behavior of Cirular Steel Tube-Confined angle while the dilation angle is usually lower than the frition angle. The results of this study indiate that for the ase of, although the axial stress-strain relationship obtained from the FE analysis is in a good agreement with the experimental results, there is a signifiant differene between their longitudinal to irumferential stress ratio of the steel tube in the plasti strains. The inrease of up to the values higher than, although insignifiantly affets the ompressive strength and the shape of stress-strain urve of STCC olumns, as shown in Fig. 10, onsiderably dereases the longitudinal to irumferential stress ratio of the steel tube at the steel yielding point and partiularly at the strain hardening point. A parametri study of the dilation angle onduted in this study on the ompressive behavior of the onfined onrete indiates that the best agreement with the experimental results is reahed for a dilation angle between 20 and 40. It should be noted that low values of the dilation angle produe a brittle behavior while higher values make more dutile behavior. Moreover, a higher level of onfinement is resulted in a more dutile behavior and onsequently a higher value of the dilation angle. Hene, due to the high dutile of the onfined onrete in STCC olumns, a high value of the dilation angle is preditable. The flow potential eentriity is onsidered 0.1 in ABAQUS. Figure 9: Plasti flow and volumetri behavior. Figure 10: Parameter study of dilation angle with frition angle of 10 (axial stress-strain urves of STCC for different values of ).

15 930 A. Haghinejad and M. Nematzadeh / Three-Dimensional Finite Element Analysis of Compressive Behavior of Cirular Steel Tube-Confined Conrete-Steel Tube Interfae Modeling The ontat between the steel tube and onrete ore is modeled by the interfae element whih mathes the faes of steel and onrete elements. Using normal behavior and hoosing the hard ontat interfae available in the ABAQUS material library, the two ontat elements are not allowed to penetrate eah other. Also, the interfae element allows the ontat surfaes to separate under a tensile fore. The frition at interfae maintains until two faes remain in ontat. By mathing the FE results by trial and error, the frition oeffiient is obtained to be Also, by onduting a parametri study, it is found that with inreasing the frition oeffiient, the longitudinal to hoop stress ratio of the steel tube inreases at the yield and strain hardening points. 2.6 Failure Criterion Sine, the failure of all speimens is due to the rupture of steel tube at mid-height, the shear damage is seleted in ABAQUS to ut the ompressive stress-strain urve of STCC olumns. The Shear damage riterion is a model for prediting the onset of damage due to shear band loalization and is used for dutile metal. The shear riterion an be used in onjuntion with the Mises. The model assumes that the equivalent plasti strain at the onset of damage is a funtion of the shear stress ratio and strain rate. The damage parameters in this model are frature strain whih is equivalent frature strain at damage initiation and shear stress ratio ( s ) whih is defined as follows where max q ksp / max s (22) is the maximum shear stress and k s is material parameter. 3 MODEL VERIFICATION In order to verify the proposed model, the STCC speimens tested by Nematzadeh (2012) are modeled in ABAQUS program and the results are ompared with the experimental ones. Charateristis of STCC speimens along with their ID are presented in Table 2, aording to the experimental researh (Nematzadeh, 2012). Also, the analytial results obtained from the modeling along with the experimental results are given in Table 3. These results inlude the ompressive strength at the steel yield stress, the initial peak strength (strength at the steel strain hardening) and the ultimate strength as well as the longitudinal to hoop stress ratio of the steel tube at the yield stress and the strain hardening. It an be seen from the table that the experimental and analytial results of the ompressive strengths, espeially the initial peak strength, are very lose to eah other, indiating a good agreement for the olumns strength. Fig. 11 illustrates the experimental results of the ompressive strength versus the analytial ones. As an be seen from the figure, the data points are properly distributed lose to the bisetor line. Aording to the FE results, the mean values of longitudinal to hoop stress ratio of the steel tube at the yield stress and the strain hardening are equal to 1.3 and 2.1, respetively. These values exhibit a little differene with the experimental values, whih are reported equal to 1 and 2 (Nematzadeh, 2012), respetively. Also, the axial and lateral stress-strain relationships of STCC speimens obtained from FEA are ompared with the experimental results and a good agreement is

16 A. Haghinejad and M. Nematzadeh / Three-Dimensional Finite Element Analysis of Compressive Behavior of Cirular Steel Tube-Confined ahieved, as shown in Figs. 12 and 13. A omparison between the analytially predited failure mode and the experimentally observed one is illustrated in Fig. 14 for a typial STCC speimen (N50P ). Aording to the figure, the FE deformed shape at the failure point is in a good agreement with that obtained from the experimental test. By omparing the results of finite element analysis with the experimental results for the various parameters, it an be found that the FE proposed model is able to properly predit the ompressive behavior of STCC olumns and alibrate the parameters required for modeling in ABAQUS program. Speimen ID N25P N25P N25P N35P N15P N25P N50P N15P N35P N25P N25P N25P N45P N47P Steel tube wall thikness (mm) Outer diameter-to-wall thikness ratio Speimen length (mm) *The inner diameter of steel tube is onstant and equal to 55.5 mm in all speimens. 39 Conrete ompressive strength (MPa) Yield and ultimate strength of steel (MPa) 339, , , , , , , , , , , , , ,480 Table 2: Properties of STCC speimens (Nematzadeh, 2012). Figure 11: Relationship between experimental results of ompressive strength and analytial results of STCC at the points of steel yielding, initial peak and failure.

17 932 A. Haghinejad and M. Nematzadeh / Three-Dimensional Finite Element Analysis of Compressive Behavior of Cirular Steel Tube-Confined... Speimen ID Strength orresponding to steel yielding (MPa) EXP/FEA Initial peak strength (MPa) EXP/FEA Failure strength (MPa) EXP FEA EXP FEA EXP FEA EXP/FEA Longitudinal to hoop stress ratio of steel tube Yielding point Strain hardening point N25P N25P N25P N35P N15P N25P N50P N15P N35P N25P N25P N45P N25P N47P Table 3: Finite element analysis results.

18 A. Haghinejad and M. Nematzadeh / Three-Dimensional Finite Element Analysis of Compressive Behavior of Cirular Steel Tube-Confined Figure 12: Axial stress-strain urves of STCC speimens.

19 934 A. Haghinejad and M. Nematzadeh / Three-Dimensional Finite Element Analysis of Compressive Behavior of Cirular Steel Tube-Confined... Figure 13: Lateral stress-strain urves of STCC speimens.

20 A. Haghinejad and M. Nematzadeh / Three-Dimensional Finite Element Analysis of Compressive Behavior of Cirular Steel Tube-Confined Figure 14: A omparison between analytially predited and experimentally observed failure mode for a typial STCC speimen (N50P ). 4 PARAMETRIC STUDY 4.1 Speimens By alibrating the required parameters for modeling the STCC olumns in ABAQUS program, the effet of geometrial and mehanial properties of the omposite setion on its ompressive behavior an be investigated as a parametri study. These parameters inlude the onrete ompressive strength ( f ), tube outer diameter-to-wall thikness ratio ( D / t ) and steel yield stress ( f ). For y this purpose, 100 speimens with 5, 5 and 4 different values of f, D / t and f, respetively, are y seleted the properties of whih are given in Table 4. To investigate the effet of the strain hardening stage of steel on the ompressive behavior of STCC olumns, bilinear and trilinear urves are used to define the equivalent stress strain relationship of steel. f (MPa) t (mm) D / t f y (MPa) D (mm) L (mm) L s (mm) 20, 30, 40, 50, 60 1, 1.2, 1.5, 2.0, , 48.3, 39.0, 29.8, (bilinear), 339 (bilinear), 339 (trilinear), 400 (bilinear) Table 4: Speimen information for parametri study. Nomenlature of the speimens for identifiation is as Sf-t-f where the haraters after letter S, y represent the values of onrete ompressive strength, steel tube wall thikness and steel yield stress along with the type of steel stress-strain relationship as bilinear ( b ) or trilinear ( t ), respetively. For example, S b represents a STCC speimen with onrete ompressive strength of of 35 MPa, steel tube wall thikness of 2 mm and steel yield stress of 339 MPa with bilinear stress-strain

21 936 A. Haghinejad and M. Nematzadeh / Three-Dimensional Finite Element Analysis of Compressive Behavior of Cirular Steel Tube-Confined... urve. Also, to identify a group of the speimens with onstant value for one of the parameters, only the value of that parameter is replaed by the orresponding letter in speimens' nomenlature. For example, Sf-1-f represents all STCC speimens with steel tube wall thikness of 1 mm. y 4.2 Results and Disussion After modeling the STCC speimens under axial ompression load, the results obtained from finite element analysis are evaluated. These results inlude yield strength, initial peak strength, ultimate strength, axial stress-strain urve of omposite setion and its omponents and the interation between onrete and steel inluding interfae shear stress and onfining pressure. As mentioned before, the point orresponding to beginning of the steel strain hardening is onsidered as the initial peak point. Sine strain hardening in the steel with bilinear urve does not our, the infletion point in the omposite olumn stress-strain urve where the urve hanges from onave downward to onave upward, is onsidered as the initial peak point. The findings of this researh indiate that the differene in the stress-strain urve of STCC olumns between the speimens with bilinear and trilinear steel stress-strain relationship begins from the infletion point whih is orresponding to the strain hardening point of the trilinear one. The ompressive strength orresponding to the initial peak point is applied to evaluate the omposite setion strength under small deformations. It should be noted that the ultimate strength for STCC olumns with an appropriate onfinement ours at the failure point with a large axial strain, whih is between 0.37 and 0.56 for the speimens of this study Composite Setion Compressive Strength The relationship of the omposite setion ompressive strength at the steel yielding point ( initial peak point ( f ) and the ultimate point ( f u f y ), the ) versus the onrete ompressive strength for different values of the tube diameter-to-wall thikness ratio is illustrated in Fig. 15. The yield stress of steel tube is onstant for all the speimens and is equal to 339 MPa with trilinear urve ( S ). f -t-339t Figure 15: Relationship of ompressive strength of STCC olumns ( S f -t-339t of steel yielding, initial peak and failure for different values of D/ t. ) versus f at the points

22 A. Haghinejad and M. Nematzadeh / Three-Dimensional Finite Element Analysis of Compressive Behavior of Cirular Steel Tube-Confined Aording to the figure, with inreasing the onrete ompressive strength, the omposite ompressive strength inreases so that the rate of inrease in the ompressive strength at the steel yielding point and the initial peak point is approximately onstant and exhibits a dereasing trend at the ultimate point, whih is due to a faster failure in the speimens with a higher onrete ompressive strength. Hene, the effet of inreasing the onrete ompressive strength on the improvement of the omposite failure strength signifiantly dereases with the inrease in the onrete ompressive strength. Furthermore, inreasing the tube outer diameter-to-wall thikness ratio leads to a redution in the onfining pressure and thus a redution in the omposite ompressive strength. This trend is more evident for lower values of the tube diameter-to-wall thikness ratio. One of the parameters affeting the omposite ompressive strength is the steel yield stress, whih is investigated in Fig. 16. In the study of this parameter, the wall thikness of steel tube is onsidered onstant and equal to 1 mm ( ). Also, the stress-strain urve of steel is bilinear S f -1-fy with the yield stress of 300, 339 and 400 MPa in all the speimens. It an be seen from Fig. 16 that the relationship between the omposite ompressive strength (, f and f u ) and the steel yield stress is linear. Also, the parallel lines in Fig. 16 demonstrate that the trend of hanges in the omposite ompressive strength versus the steel yield stress for different values of the onrete ompressive strength is the same. In addition, it an be onluded from the equal distanes between the omposite ompressive strength urves at the steel yielding point and the initial peak point that there is a linear relationship between the omposite ompressive strength and the onrete ompressive strength, as demonstrated earlier. This trend for the omposite failure strength is assoiated with a redution in the distane between the urves as the onrete ompressive strength inreases, whih is aused by a faster failure in the speimens with a higher onrete ompressive strength beause of a higher brittleness. f y failure strength Figure 16: Relationship of ompressive strength of STCC olumns ( S f-1-f of steel yielding, initial peak and failure for different values of f. y ) versus f y at the points Interfae Shear Stress Shear stress distribution of STCC speimens in the onrete ore height due to the frition at the onrete steel interfae at the points of steel yielding, initial peak and failure is shown in Fig. 17. In

23 938 A. Haghinejad and M. Nematzadeh / Three-Dimensional Finite Element Analysis of Compressive Behavior of Cirular Steel Tube-Confined... order to study this parameter, the results of two groups of speimens inluding S and f t S are evaluated. It an be seen from Fig. 17 that the shear stress distribution in the height f t diretion is non-uniform so that its value is maximum at the two ends and is zero at the mid-height of the speimens due to the symmetry. It an also be found from Fig. 17 that the shear stress between the onrete ore and steel tube at the points of steel yielding and initial peak is not influened by the onrete ompressive strength while with inreasing the steel tube wall thikness, the interfae shear stress inreases signifiantly. This is due to the fat that with inreasing the tube wall thikness, the onfining pressure and onsequently the frition stress at the interfae inrease while inreasing the onrete ompressive strength has a negligible effet on it. At the failure point, the sign of shear stress hanges around both ends of the speimens. Sine there is a large deformation of STCC speimens at the failure point, the end parts of onrete ore tend to ontrat and due to the end restraints, an outward shear stress is reated at both ends. Figure 17: Interfae shear stress along onrete height at the points of steel yielding, initial peak and failure Lateral Confining Pressure Fig. 18 shows the onfining pressure distribution in the height of onrete ore at the points of steel yielding, initial peak and failure. In order to study this parameter, the results of two groups of the speimens inluding and S are evaluated. It an be found from the figure that in all Sf t f t the speimens, the maximum lateral pressure ours at the two ends of the speimens and the minimum one is reated at the mid-height. Sine the lateral deformation of the speimens at the ends is ompletely prevented, a signifiant pressure is applied on the onrete ore while the end effets are insignifiant in the middle of the speimens, and the lateral strains are only restrained by the steel tube. Therefore, the onfining pressure at the mid-height of the speimens is minimum. Aording to the results of this researh, longitudinal stress of the steel tube at the mid-height of the speimens is maximum. Hene, based on the von Mises yield riterion and isotropi hardening model, the irumferential tensile stress and onsequently the onfining pressure at the mid-height of the steel tube are lower than those at other points of the height.

24 A. Haghinejad and M. Nematzadeh / Three-Dimensional Finite Element Analysis of Compressive Behavior of Cirular Steel Tube-Confined It an also be seen from Fig. 18 that the inrease in the onrete ompressive strength has an insignifiant effet on the onfining pressure of the STCC speimens at the points of steel yielding, 2 f y t initial peak and failure. This fat an be proved using the equation fl derived from the D 2t lassial elastiity theory, in whih the lateral onfining pressure ( f l ) is independent of the onrete ompressive strength. Moreover, the lateral onfining pressure obtained in Eq. (4) is only a funtion of the geometrial and mehanial properties of steel tube. It an be also found from Fig. 18 that with inreasing the tube wall thikness, the lateral onfining pressure signifiantly inreases over the height of STCC speimens. Figure 18: Confining pressure along onrete height at the points of steel yielding, initial peak and failure Load -Carrying Capaity Load-arrying apaity of steel tube and onrete ore during the loading for S and f t S speimens is shown in Fig. 19, whih is obtained at the mid-height of the STCC speimens. f t It an be seen from the figure that with inreasing the onrete ompressive strength, load arrying portion of onrete ore inreases while that of steel tube is hanged insignifiantly. Also, with inreasing the axial load in large deformations, load arrying portion of onrete ore inreases while that of steel tube dereases. The reason is that the lateral pressure applied on the inner wall of the steel tube in large lateral deformations leads to a signifiant redution in the longitudinal ompressive stress of steel tube and its load-arrying portion and onsequently an inrease in the loadarrying portion of onrete ore. It should be noted that the vertial omponent of the onfining pressure is in the opposite diretion of the interfae shear stress vertial omponent.

25 940 A. Haghinejad and M. Nematzadeh / Three-Dimensional Finite Element Analysis of Compressive Behavior of Cirular Steel Tube-Confined... Figure 19: Load arrying portion of onrete ore and steel tube in STCC olumns for different values of f Volumetri Strain Volumetri strain of onrete ore ( v ) an be alulated from the sum of axial and lateral strains. Normalized axial stress relationship of the STCC speimens (ratio of axial stress to stress at steel yielding point) versus volumetri strain of onrete is presented in Fig. 20. The positive and negative volumetri strains indiate the ontration and expansion of the onrete ore, respetively. It an be observed from the figure that the onrete volumetri strain until the steel yielding point is independent of the onrete ompressive strength and all the urves in eah of the groups Sf t and Sf t are onsistent with eah other. Nonetheless, the onrete volumetri strain is signifiantly affeted by the steel tube wall thikness. It an be found from the figure that in all the speimens, the maximum ontration volumetri strain ours near the steel yielding point. Also, the zero volumetri strain, where the volumetri strain hanges from the ontration to the expansion ondition, happens shortly after the steel yielding point and before the initial peak point so that with inreasing the onrete ompressive strength and the steel tube wall thikness, the normalized stress orresponding to the zero volumetri strain inreases. Figure 20: Normalized axial stress of omposite olumn versus volumetri strain of onrete ore.