Nonlinear analysis of axially loaded circular concrete-filled stainless steel tubular short columns

Transcription

1 Nonlinear analysis of axially loaded irular onrete-filled stainless steel tubular short olumns Vipulkumar Ishvarbhai Patel a, Qing Quan Liang b, *, Muhammad N. S. Hadi a Institute of Materials Engineering, Australian Nulear Siene and Tehnology Organisation, Loked Bag 1, Kirrawee, NSW 3, Australia b College of Engineering and Siene, Vitoria University, PO Box 1448, Melbourne, VIC 81, Australia Shool of Civil, Mining and Environmental Engineering, University of Wollongong, Wollongong, NSW 5, Australia ABSTRACT Experiments show that the ultimate ompressive strength of stainless steel is muh higher than its tensile strength. The full-range two-stage onstitutive model for stainless steels assumes that stainless steels follow the same stress strain behavior in ompression and tension, whih may underestimate the ompressive strength of stainless steel tubes. This paper presents a fiber element model inorporating the reently developed full-range three-stage stress strain relationships based on experimentally observed behavior for stainless steels for the nonlinear analysis of irular onrete-filled stainless steel tubular (CFSST) short olumns under axial ompression. The fiber element model aounts for the onrete onfinement effets provided by the stainless steel tube. Comparisons of omputer solutions with experimental results published in the literature are made to examine the auray of the fiber element model and material onstitutive models for stainless steels. Parametri studies are onduted to study the effets of various parameters on the behavior of irular CFSST short olumns. A design model based on Liang and Fragomeni s design formula is proposed for * Corresponding author. Tel.: address: Qing.Liang@vu.edu.au (Q. Q. Liang) 1

2 irular CFSST short olumns and validated against results obtained by experiments, fiber element analyses, ACI-318 odes and Euroode 4. The fiber element model inorporating the three-stage stress strain relationships for stainless steels is shown to simulate well the axial load strain behavior of irular CFSST short olumns. The proposed design model gives good preditions of the experimental and numerial ultimate axial loads of CFSST olumns. It appears that ACI-318 odes and Euroode 4 signifiantly underestimate the ultimate axial strengths of CFSST short olumns. Keywords: Conrete-filled steel tubes; Fiber element analysis; Stainless steel; Strength 1. Introdution The material onstitutive models used in the nonlinear inelasti analysis of irular onretefilled stainless steel tubular (CFSST) short olumns ould have a ruial influene on the auray of the predited behavior. The two-stage stress strain model for stainless steels proposed by Rasmussen [1] assumes that the stainless steel follows the same stress strain urve in tension and ompression. This model developed from tension oupon tests may underestimate the ultimate axial strengths of CFSST olumns. The three-stage onstitutive model for stainless steels proposed by Quah et al. [] aounts for different behavior of stainless steels in ompression and in tension based on experimental observations. The threestage material model is believed to be a more aurate formulation than the two-stage one. The inversion of the three-stage stress strain relationships for stainless steels given by Abdella et al. [3] expresses the stress as a funtion of strain. This gives a onvenient implementation of the material laws in numerial models. The three-stage stress strain model

3 for stainless steels has not yet been inorporated in numerial tehniques for the nonlinear analysis of CFSST olumns. Experimental studies on the behavior of axially loaded irular onrete-filled steel tubular (CFST) short olumns have extensively been onduted by researhers [4-7]. However, researh studies on irular CFSST short olumns under axial loading have been relatively limited. Young and Ellobody [8] onduted tests on the axial strengths of old-formed high strength square and retangular CFSST short olumns. Their results indiated that the implementation of the material properties of stainless steel obtained from tension oupon tests underestimated the ultimate axial strengths of CFSST olumns under axial ompression. This is beause the strain hardening of stainless steels in ompression is muh higher than that of stainless steels in tension. Tests on irular CFSST short olumns under axial ompression were arried out by Lam and Gardner [9]. They investigated the effets of the tube thikness, onrete ompressive strength and proof stress on the behavior of CFSST olumns under axial loading. Design formulas based on the Continuous Strength Method were proposed for determining the ultimate axial strengths of CFSST short olumns. Uy et al. [1] tested irular CFSST olumns under axial ompression. Both square and irular olumn setions were tested to study the effets of the tube shape, diameter-to-thikness ratio and onrete ompressive strength on the behavior of CFSST olumns. They ompared various design odes for irular CFSST olumns. They reported that existing design odes for omposite olumns provide onservative preditions of the ultimate strengths of CFSST olumns. Numerial models have been developed to study the behavior of irular CFST short olumns under axial loading [11-18]. However, there have been relatively limited numerial studies on the behavior of axially loaded irular CFSST short olumns. Nonlinear analysis methods for 3

4 omposite olumns and strutures have been reviewed by Ellobody [19]. Ellobody and Young [] utilized the finite element analysis program ABAQUS to study the behavior of axially loaded retangular CFSST short olumns. They inorporated the measured tensile material properties in the finite element model and assumed the same material properties of stainless steel in ompression. Nonlinear finite element analyses of square CFSST short olumns using ABAQUS have been undertaken by Tao et al. [1]. The two-stage stress strain model for stainless steels proposed by Rasmussen [1] was employed in their study. The true stress strain urves were used in the finite element model. Reently, Hassanein et al. [] employed the finite element program ABAQUS to study the inelasti behavior of axially loaded irular lean duplex CFSST short olumns. The finite element results were verified against test results presented by Uy et al. [1]. In this paper, the inversion of the three-stage stress strain relationships for stainless steels [, 3] is inorporated in the fiber element model for simulating the nonlinear inelasti behavior of CFSST short olumns under axial ompression. The fiber element model aounts for the effets of onrete onfinement and high strength onrete. The fiber element analyses are performed to examine the auray of different onstitutive models for stainless steels. The effets of diameter-to-thikness ratio, onrete ompressive strength and stainless steel proof stress on the behavior of irular CFSST short olumns are investigated. A design model based on Liang and Fragomeni s formula [16] is proposed for the design of irular CFSST olumns and ompared with test results and design odes.. Nonlinear analysis.1. Assumptions 4

5 The nonlinear analysis of CFSST olumns under axial ompression is based on the fiber element method. The following assumptions are made in the fiber element formulation: The bond between the stainless steel tube and the onrete ore is perfet. The passive onfinement provided by the stainless steel tube inreases the ompressive strength and dutility of the onrete ore. The stress and strain of fibers are uniformly distributed on the ross-setion. Strain hardening of stainless steels in ompression is onsidered. Failure ours when the onrete fiber strain reahes the maximum axial strain. Loal bukling of the stainless steel tube is not onsidered. The effets of onrete reep and shrinkage are not onsidered... The fiber element method The fiber element method is an aurate numerial tehnique for determining the rosssetion behavior of steel-onrete omposite olumns [14, 15, 3-5]. In the fiber element method, a irular CFSST olumn ross-setion is disretized into fine fiber elements as depited in Fig. 1. Eah fiber element represents a fiber of material running longitudinally along the olumn and an be assigned either stainless steel or onrete material properties. The fiber stresses are alulated from fiber strains using the material uniaxial stress-strain relationships. Although the disretization of a CFSST olumn under axial ompression is not required, it is a prerequisite for the nonlinear analysis of CFSST short olumns under eentri loading or CFSST slender olumns [6, 7]. The present study is part of a researh program on the nonlinear analysis of CFSST slender olumns so that the omposite setion is 5

6 disretized using the fiber element method. However, the size of fiber elements does not affet the ultimate axial strength and behavior of CFSST short olumns..3. Material model for stainless steels The inversion of the full-range three-stage stress strain relationships for stainless steels presented by Abdella et al. [3] is based on the equations proposed by Quah et al. [], whih is implemented in the present fiber element model. The three-stage stress strain urve for stainless steels in ompression is shown in Fig.. In the first stage s of the stress strain urve, the stress is expressed by C s E s 1 C1 s (1) C4 C s s 1 C3 C1 where s is the stress in a steel fiber, s is the strain in the steel fiber, E is the Young s modulus of stainless steel,. is the strain at. and. is the % proof stress. The strain is alulated by the equation given by Ramberg and Osgood [8] as:. () E In Eq. (1), the positive onstants C 1, C, C 3 and C 4 are given by 6

7 C 1 (3) C 1 1 C 1 B (4) C C 3 G 1 C1 (5) (6) 4 G 1 in whih 1 1 4B1 (7) B G E n G 1 1 (8) E.E G (9) E n 1 G 1 (1) E E. (11) n 1. e where E. is the tangent modulus of the stress strain urve at the % proof stress, n is the nonlinearity index and e is the non-dimensional proof stress given as: e E (1) 7

8 In the seond stage Abdella et al. [3]: s., the stress is expressed by the equation presented by C6 s E s 1 C5 1. s (13) C8 C6 s s 1 C7 C where 1. is the strain at 1. and 1. is the 1.% proof stress. The 1.% proof stress under ompression an be alulated by the following equation given by Quah et al. []: (14) n The strain 1. is alulated by []: E E E 1.. (15) In Eq. (13), C 5, C 6, C 7 and C 8 are the positive onstants, C 1 5 C 1 (16) 6 8

9 1 C6 C8 ln 1 ln. H ln 1. H A (17) C 7 H 1 C5 (18) C8 1 H 1 (19) where A n 1 H H 1 n H 1 n H () E E E H (1) H n 1H () 1 nh E. 1. H (3). In Eq. (17),. is the strain at. and. is the.% proof stress. In Eq. (), n is given by Quah et al. [] as E 1. n (4) E The.% proof stress an be alulated by the following equation given by Quah et al. []: 9

10 B n A E E E A n n (5) where E E E B A (6) 1.18 E E E B (7) The strain. is expressed by the following equation given by Quah et al. []: n E E E (8) In the third stage s su., the stress is expressed as a funtion of strain as [3]: s s s B A (9) where

11 A. B (3) 1 1 su su.. B 3 (31) su. where su is the ultimate strain and su is the ultimate stress. The ultimate strain and stress are proposed by Quah et al. [] as follows: su ut (3) 1 ut su 1 1 (33) 1 ut where ut and ut are the ultimate tensile strength and ultimate tensile strain respetively and are given by 1.375n 5 ut (34) 185e (35). ut 1 ut.4. Material model for onfined onrete The onfinement provided by the steel tube on the onrete ore in irular onrete-filled stainless steel or steel tubular olumns is passive. The onrete expansion auses the elongation of the steel tube that indues ompressive onfining stresses on the onrete ore. The onfining pressure inreases with inreasing the axial strain. Madas and Elnashai [9] 11

12 proposed a passive onfinement model for onfined onrete in whih the onrete onfinement varies with the axial strain. They reported that the onfinement model given by Mander et al. [3] overestimates the onfining pressures on the onrete. Liang and Fragomeni [16] proposed a onfining pressure model for onrete in irular CFST olumns. The general stress strain urve for onfined onrete in irular CFST olumns suggested by Liang and Fragomeni [16] is shown in Fig. 3. The onrete stress from O to A on the stress strain urve is alulated based on the equations given by Mander et al. [3] as f / (36) 1 / / E (37) E f E 33 f 69 MPa (38) in whih denotes the ompressive onrete stress, f is the effetive ompressive strength of onfined onrete, f is the ompressive strength of onrete ylinder, is the ompressive onrete strain, is the strain at f, E is the Young s modulus of onrete given by ACI [31], and is the strength redution fator proposed by Liang [14] to aount for the olumn setion size effet, expressed by D (39) 1

13 where D is the diameter of the onrete ore. The equation proposed by Mander et al. [3] for determining the ompressive strength of onfined onrete was modified by Liang and Fragomeni [16] using the strength redution fator as follows: f f k1 f (4) rp f rp 1 k (41) f where f rp is the lateral onfining pressure on the onrete and k 1 and k are taken as 4.1 and.5 respetively. The strain is the strain at f of the unonfined onrete, given by. for f 8 MPa f 8. for 8 f 8MPa (4) 54.3 for f 8MPa Based on the work of Hu et al. [1] and Tang et al. [3], Liang and Fragomeni [16] proposed an aurate onfining pressure model for normal and high strength onrete in irular CFST olumns, whih is adopted in the present numerial model as follows: f rp t.7 ve vs D t D t D for 47 t D for t (43) 13

14 in whih D is the outer diameter and t is the thikness of the steel tube and v e and v s are Poisson s ratio of the steel tube with and without onrete infill, respetively. Tang et al. [3] suggested that Poisson s ratio v s is taken as.5 at the maximum strength point and v e is given by f f f v e ve v e (44) 3 6 D 4 D D v e (45) t t t It is noted that the onfining pressure model proposed by Liang and Fragomeni [16] an be used for onrete onfined by high strength steel tubes with signifiant strain hardening and has been verified by experimental results [15, 7]. In addition, it has been used to simulate the behavior of onfined onrete in irular CFSST olumns with aeptable auray []. The parts AB and BC of the stress strain urve shown in Fig. 3 an be desribed by f f u u f f for for u u (46) where u is taken as. as suggested by Liang and Fragomeni [16] based on the experimental results, and is a fator aounting for the onfinement effet by the stainless steel tube on the post-peak strength and dutility of the onfined onrete, whih is given by Hu et al. [1] as 14

15 D 1. for 4 t (47) D D D for 4 15 t t t.5. Analysis proedure In the nonlinear analysis, the axial load-strain urve for a CFSST short olumn is determined by gradually inreasing the axial strain and omputing the orresponding axial fore (P), whih is alulated as the stress resultant in the omposite ross-setion. When the axial load drops below a speified perentage of the maximum axial load ( P max ) suh as.5pmax or when the axial strain in onrete exeeds the speified ultimate strain u, the iterative analysis proess an be stopped as disussed by Liang [14, 33]. The ultimate axial strength of a CFSST short olumn under axial ompression is taken as the maximum axial load from its omplete axial load-strain urve [33]. The analysis proedure for determining the axial load-strain urve for a CFSST short olumn under axial ompression is given as follows [33]: 1. Input dada.. Disretize the olumn ross-setion into fiber elements. 3. Initialize fiber axial strains. 4. Calulate fiber stresses using the material uniaxial stress-strain relationships. 5. Compute the axial fore P as the stress resultant in the ross-setion. 6. Inrease the fiber axial strain by. 7. Repeat Steps 4 to 6 until P.5Pmax or u. 15

16 The axial fore ating on the omposite setion is determined as the stress resultant, whih is expressed by P ns n s, i As, i, j i1 j1 A (48), j where P is the axial load, s, i is the longitudinal stress at the entroid of steel fiber i, A s, i is the area of steel fiber i, is the longitudinal stress at the entroid of onrete fiber j, j, j is the area of onrete fiber j, ns is the total number of steel fiber elements and n is the total number of onrete fiber elements. A, 3. Validation of the fiber element model 3.1. Ultimate axial strengths The ultimate axial strengths of irular CFSST short olumns predited by the fiber element model are ompared with experimental results presented by Lam and Gardner [9] and Uy et al. [1]. Geometry and material properties of speimens are given in Table 1. The predited and experimental ultimate axial strengths of irular CFSST olumns under axial ompression are also given in Table 1, where P is the experimental ultimate axial load and P u. fib is u. exp denoted as the ultimate axial load predited by the fiber element model at the measured maximum axial strain max. It an be seen from Table 1 that the fiber element model generally gives good preditions of the ultimate axial strengths of irular CFSST short olumns. The mean ultimate axial strength omputed by the fiber element model is 97% of the experimental value. The standard deviation of P P is.8 while its oeffiient of u. fib u. exp 16

17 variation is.9. The disrepany between the predited and experimental results is attributed to the unertainty of the atual onrete ompressive strength beause the average onrete ompressive strength was used in the fiber element analysis. It is noted that speimens shown in Table 1 inlude normal strength stainless steel tubes filled with normal strength onrete and normal strength stainless steel tubes filled with high strength onrete. Therefore, the proposed fiber element model an be used for the design and analysis of axially loaded CFSST olumns made of normal and high strength onrete. 3.. Axial load strain urves Fig. 4 shows the predited and experimental axial load strain urves for Speimen CHS C3 tested by Lam and Gardner [9]. It appears that the fiber element model generally predits well the experimental axial load strain urve for the speimen. The predited initial axial stiffness of the olumn is slightly higher than the experimental one. This is likely due to the unertainty of the atual onrete stiffness and strength as the average onrete ompressive strength was used in the fiber element analysis. However, it an be seen from Fig. 4 that the fiber element model aurately predits the strain-hardening behavior of the speimen. The results obtained for Speimen CHS C6 tested by Lam and Gardner [9] are presented in Fig. 5. The figure shows that the fiber element results are in good agreement with experimentally observed behavior. The initial axial stiffness of the olumn is well predited, but the experimental axial load strain urve differs from the omputational one at the axial load higher than 11 kn. The predited axial load strain urve for Speimen C3-5 1.A is ompared with experimental one provided by Uy et al. [1] in Fig. 6. It an be observed that the fiber element 17

18 model losely predits the initial axial stiffness of the speimen before attaining a loading level about 8 kn. After this loading, the axial stiffness of the speimen obtained from tests slightly differs from fiber element preditions. This is likely attributed to the unertainty of the onrete stiffness and strength. Further omparison of axial load strain urves for Speimen C-1 1.6B tested by Uy et al. [1] is shown in Fig. 7. The figure demonstrates that these two urves are almost idential up to the loading level of about 35 kn. After that the experimental urve departs from omputational one. The disrepany is most likely due to the unertainty of the atual onrete ompressive strength and stiffness as the average onrete ompressive strength was used in the analysis. The omparative studies demonstrate that the fiber element model yields good preditions of the behavior of irular CFSST short olumns under axial ompression. 4. Parametri study The fiber element model developed was employed to investigate the effets of the diameterto-thikness ratio, onrete ompressive strength and stainless steel proof stress on the behavior of irular CFSST short olumns under axial loading. Uy et al. [1] reported that CFSST short olumns undergone large plasti deformations with signifiant strain hardening. The tests of CFSST short olumns were stopped before failure ourred owing to the large plasti deformation. Tests indiated that CFSST short olumns exhibited very good dutility and the axial strain of CFSST short olumns under axial ompression ould be up to. Therefore, the ultimate axial strain ( u ) was taken as in the following parametri study. For stress-strain urves without desending part, the ultimate axial strengths of CFSST short olumns were taken as the axial load orresponding to the ultimate axial strain of. The Young s modulus of stainless steel was GPa. 18

19 4.1. Evaluation of material onstitutive models for stainless steels The auray of full-range two-stage stress strain urves proposed by Rasmussen [1] and three-stage stress strain relationship presented by Abdella et al. [3] have been investigated using the present fiber element analysis program developed. Table shows a omparison of the ultimate axial strengths of CFSST olumns determined using these two material models for stainless steels. The material properties of these speimens an be found in Table 1. It an be seen from Table that the two-stage stress strain relationships proposed by Rasmussen [1] generally underestimate the ultimate axial strengths of irular CFSST olumns. The mean ultimate axial strength omputed by the fiber element model with two-stage urves is 81% of the experimental value. The standard deviation of P P is.8 while its oeffiient of u. fib1 u. exp variation is.1. The mean ultimate axial strength predited using Abdella et al. [3] model is 97% of the experimental value. The standard deviation of P u. fib Pu. exp is.8 and its oeffiient of variation is.9. The evaluation demonstrates that the three-stage stress strain relationships given by Abdella et al. [3] provide reliable results for irular CFSST olumns under axial ompression. The auray of the onstitutive models for stainless steels an be assessed by omparisons of predited axial load strain urves using the two-stage and three-stage stress strain relationships. Fig. 8 presented a omparison of the axial load strain urves omputed by Rasmussen onstitutive model [1] and Abdella et al. material model [3]. Close agreement between these two material models is ahieved up to the proof stress of the stainless steel. However, there are signifiant differenes between the axial load strain urves obtained from both material models for the stress higher than the proof stress. 19

20 4.. Effets of D t ratio The D t ratio is one of the important variables affeting the behavior of irular CFSST short olumns under axial ompression. This is due to both the onfinement effets and steel area varies with the D t ratio. Column C7 given in Table 3 was analysed to examine the effets of D t ratios. The typial D t ratios of 3, 4 and 64 were onsidered by hanging the thikness of the stainless steel tube while maintaining the same ross-setion size. Fig. 9 depits the axial load strain urves for irular CFSST olumns with various D t ratios. It appears that the ultimate axial load of irular CFSST olumns dereases with inreasing the D t ratio. When inreasing the D t ratio from 3 to 4 and 64, the ultimate axial strength of the short olumn dereases by 17% and 44%, respetively. In addition, the irular CFSST olumn exhibits strain hardening behavior when the D t ratio is small. This is due to both the steel area and onfinement effet inreases with dereasing the D t ratio. More importantly, the dutility of irular CFSST short olumns inreases with dereasing the D t ratio. Similar onlusion was observed by Uy et al. [1] in their experimental researh on irular CFSST short olumns under axial loading. The dutility of a strutural member or setion is defined by Liang [14] as the ability to undergo large plasti deformation without signifiant strength degradation Effets of onrete ompressive strengths It is important to understand the effets of onrete ompressive strengths on the stiffness and dutility of irular CFSST short olumns under axial loading. Columns C13, C14 and C15

21 given in Table 3 were analysed again to investigate the effets of the onrete ompressive strengths on the behavior of irular CFSST olumns. Fig. 1 shows the axial load strain urves for the irular CFSST olumns filled with onrete of different ompressive strengths. The ultimate axial strength and stiffness of irular CFSST olumns are found to inrease with inreasing the onrete ompressive strength. Inreasing the onrete ompressive strength from 7 MPa to 9 MPa and 11 MPa inreases the ultimate axial strength by 1% and %, respetively. It an also be seen from Fig. 1 that inreasing the onrete ompressive strength results in a slight inrease in the initial stiffness of the olumn. Despite of the use of high strength onrete, these irular CFSST short olumns exhibit good dutility, whih is the ability to undergo large plasti deformation without signifiant strength degradation Effets of stainless steel strengths The numerial model developed was utilised to investigate the effets of the proof stress of stainless steel on the ompressive behavior of irular CFSST olumns. Column C36 given in Table 3 was analysed to examine the effets of proof stress on the behavior of CFSST olumns. Austeniti and duplex stainless steel tubes with proof stresses of 4 MPa and 53 MPa and non-linearity index of 7 and 5 were onsidered. The axial load strain urves for the irular CFSST olumns made of different strength stainless steel tubes are presented in Fig. 11. It an be observed from the figure that inreasing the proof stress of the stainless steel notieably inreases the ultimate axial loads of irular CFSST olumns. When inreasing the proof stress of the stainless steel from 4 MPa to 53 1

22 MPa, the ultimate axial strength of the olumn inreases by 31%. Computational results indiate that the proof stress of stainless steel does not have an effet on the initial axial stiffness of the olumns. 5. Design models 5.1. ACI-318 odes The design equation given in ACI-318 [31] odes does not aount for the onrete onfinement effets on the ultimate axial strength of omposite olumns, whih is expressed by P 85A f (49) u. ACI As fsy. where A is the ross-setional area of the onrete ore, A s is the ross-setional area of steel tube, f sy is the steel yield strength whih is taken as % proof stress for stainless steels. 5.. Euroode 4 The ultimate axial loads of irular CFSST olumns were ompared with the design strengths predited by the Euroode 4 [34]. The ode takes into aount the onrete onfinement effet by the irular steel tube. The EC4 equation for ultimate axial strength of a CFST olumn is given as

23 t f sy P u. EC 4 a As fsy A f 1 (5) D f a, , in whih is the relative where slenderness of a CFSST olumn, alulated as N pl. Rk (51) N r where N. is the ross-setion plasti resistane of the CFSST olumn, given by pl Rk N A A f (5) pl. Rk s In Eq. (51), N r is the Euler bukling load of the pin-ended CFSST olumn and expressed by N r EI eff (53) L eff where EI is the effetive flexural stiffness of a CFSST setion, given as EI EsIs. 6EmI eff (54) In Eq. (53), L is the effetive length of the CFSST olumn The proposed design model based on Liang and Fragomeni s formula 3

24 The ultimate axial strength of a CFSST short olumn under axial ompression is influened by onfinement of the onrete ore. A design model for determining the ultimate axial strength of a CFSST short olumn with onfinement effets was proposed by Liang and Fragomeni [16] as follows: f 4. frp A s As Pu. design 1. (51) where is laulated using Eq. (39), f rp is given by Eq. (43), and s is the strength fator for the stainless steel tube, whih is proposed as follows:.1 D D 3 for 8 t t s (5).1 D D for 8 t t 5.4. Comparisons of design models The fiber element model was utilized to analyze CFSST olumns in order to develope a design model. The geometri and material propeties used in the nonlinear regression analysis are given in Table 3. It should be noted that olumns are seleted with a wide range of diameter-to-thikness ratio ranging from to 1. The stainless steel tubes were filled with nornal and high strength onrete wth ompressiove strengths ranging from 3 MPa to 11 MPa. The ultimate axial strengths of irular CFSST olumns alulated using Eq. (51) are ompared with numerial results in Table 3, where P u. design is the ultimate axial strength laualed using Eq. (51). It an be seen from the table that the proposed design formula gives very good preditions of the ultimate axial strengths of 4

25 irular CFSST olumns exept for Columns C6, C41, C4, C46 and C51. The ultimate axial strengths obtained from the fiber element model for these olumns are governed by the strain hardening of stainless steel. The mean utlimate axial strength predited by Eq. (51) is 97% of the numerial value. The standard deviation of P u design Pu. num. is.5 while its oeffiient of variation is also.6. Therefore, it an be onluded that the proposed design formula an be used in the design of axially loaded irular CFSST olumns made of nornal and high strength onrete and stainless steel tubes with D t ratios ranging from to 1. The ultimate axial strengths for axially loaded CFSST olumns predited by the proposed design model and design odes are ompared with the experimental results in Table 4. It an be observed that the ultimate axial strengths alulated by the proposed design model agrees well with experimental results. However, existing design odes ACI-318 [31] and Euroode 4 [34] give onservative preditions of the ultimate axial strengths of CFSST olumns. The mean ultimate axial strength alulated using the proposed design model is 99% of the experimental value. The standard deviation of P P is.11 while its oeffiient of variation is.11. u. design u. exp The mean ultimate axial strength predited using ACI-318 odes is 54% of the experimental value. The standard deviation of P P is.9 and its oeffiient of variation is.17. u. ACI u. exp The mean ultimate axial strength predited using Euroode 4 is 78% of the experimental value. The standard deviation of P P is.11 and its oeffiient of variation is.14. u. EC 4 u. exp 6. Conlusions A fiber element model inorporating the full-range three-stage stress strain relationships for stainless steel has been presented in this paper for the nonlinear analysis of irular CFSST short olumns under axial ompression. The omparative studies performed show that the 5

26 three-stage material onstitutive model for stainless steels gives more aurate preditions of the experimentally observed behavior of CFSST short olumns than the two-stage material model. It is demonstrated that the developed fiber element model onsidering onrete onfinement effets predits well the load strain behavior and ultimate axial strengths of CFSST short olumns tested by independent researhers. The proposed design model for axially loaded CFSST short olumns is verified by experimental and numerial results. It is found that the design method given in ACI-318 odes [31] is highly onservative for estimating the ultimate axial strengths of irular CFSST short olumns beause the odes do not aount for the onrete onfinement effets and signifiant strain hardening of stainless steels in ompression. Euroode 4 [34] onsiders the effets of onrete onfinement in the alulation of the ultimate axial strength of irular onrete-filled steel olumns. However, it still provides onservative design strengths sine the signifiant strain hardening of stainless steel has not been taken into aount in Euroode 4. The onfining pressure model proposed by Liang and Fragomeni [16] gives onservative preditions of the onfinement in CFSST olumns. Further researh is needed to develop an aurate passive onfinement model for onrete in irular CFSST olumns to aount for the effets of signifiant strain hardening of stainless steel tubes on the onfinement. Referenes [1] Rasmussen KJR. Full-range stress strain urves for stainless steel alloys. J Constr Steel Res 3;59(1): [] Quah WM, Teng JG, Chung KF. Three-stage full-range stress strain model for stainless steels. J Strut Eng ASCE 8;134(9):

27 [3] Abdella K, Thannon RA, Mehri AI, Alshaikh FA. Inversion of three-stage stress strain relation for stainless steel in tension and ompression. J Constr Steel Res 11;67(5):86-3. [4] Shneider SP. Axially loaded onrete-filled steel tubes. J Strut Eng ASCE 1998;14(1): [5] O Shea MD, Bridge RQ. Design of irular thin-walled onrete filled steel tubes. J Strut Eng ASCE ;16(11): [6] Giakoumelis G, Lam D. Axial apaity of irular onrete-filled tube olumns. J Constr Steel Res 4;6(7): [7] Sakino K, Nakashara H, Morino S, Nishiyama I. Behavior of entrally loaded onretefilled steel-tube short olumns. J Strut Eng ASCE 4;13(): [8] Young B, Ellobody E. Experimental investigation of onrete-filled old-formed high strength stainless steel tube olumns. J Constr Steel Res 6;6(5): [9] Lam D, Gardner L. Strutural design of stainless steel onrete filled olumns. J Constr Steel Res 8;64(11): [1] Uy B, Tao Z, Han LH. Behaviour of short and slender onrete-filled stainless steel tubular olumns. J Constr Steel Res 11;67(3): [11] Lakshmi B. Shanmugam NE. Nonlinear analysis of in-filled steel-onrete omposite olumns. J Strut Eng ASCE ;18(7):9-33. [1] Hu HT, Huang CS, Chen ZL. Finite element analysis of CFT olumns subjeted to an axial ompressive fore and bending moment in ombination. J Constr Steel Res 5;61(1): [13] Hatzigeorgiou GD. Numerial model for the behavior and apaity of irular CFT olumns, Part I: Theory. Eng Strut 8;3(6):

28 [14] Liang QQ. Performane-based analysis of onrete-filled steel tubular beam-olumns, Part I: Theory and algorithms. J Constr Steel Res 9;65(): [15] Liang QQ. Performane-based analysis of onrete-filled steel tubular beam-olumns, Part II: Verifiation and appliations. J Constr Steel Res 9;65(): [16] Liang QQ, Fragomeni S. Nonlinear analysis of irular onrete-filled steel tubular short olumns under axial loading. J Constr Steel Res 9;65(1): [17] Choi KK, Xiao Y. Analytial studies of onrete-filled irular steel tubes under axial ompression. J Strut Eng ASCE 1;136(5): [18] Tao Z, Wang ZB, Yu Q. Finite element modelling of onrete-filled steel stub olumns under axial ompression. J Constr Steel Res 13;89: [19] Ellobody E. A onsistent nonlinear approah for analysing steel, old-formed steel, stainless steel and omposite olumns at ambient and fire onditions. Thin Wall Strut 13;68:1-17. [] Ellobody E, Young, B. Design and behavior of onrete-filled old-formed stainless steel tube olumns. Eng Strut 6;8(5): [1] Tao Z, Uy B, Liao FY, Han LH, Nonlinear analysis of onrete-filled square stainless steel stub olumns under axial ompression. J Constr Steel Res 11;67(11): [] Hassanein MF, Kharoob OF, Liang QQ. Behaviour of irular onrete-filled lean duplex stainless steel tubular short olumns. Thin Wall Strut 13;68: [3] El-Tawil S, Sanz-pión CF, Deierlein GG. Evaluation of ACI 318 and AISC (LRFD) strength provisions for omposite beam-olumns. J Constr Steel Res 1995;34(1):13-6. [4] Hajjar JF, Gourley BC. Representation of onrete-filled steel tube ross-setion strength. J Strut Eng ASCE 1996;1(11): [5] Shanmugam NS, Lakshmi B, Uy B. An analytial model for thin-walled steel box olumns with onrete in-fill. Eng Strut ;4(6):

29 [6] Liang QQ. High strength irular onrete-filled steel tubular slender beam-olumns, Part I: Numerial analysis. J Constr Steel Res, 11; 67(): [7] Liang QQ. High strength irular onrete-filled steel tubular slender beam-olumns, Part II: Fundamental behavior. J Constr Steel Res, 11; 67(): [8] Ramberg W, Osgood WR. Desription of stress strain relations from offset yield strength values. NACA Tehnial Note no. 97, [9] Madas P, Elnashai AS. A new passive onfinement model for the analysis of onrete strutures subjeted to yli and transient dynami loading. Earthquake Eng and Strut Dyn 199; 1(5): [3] Mander JB, Priestly MNJ, Park R. Theoretial stress strain model for onfined onrete. J Strut Eng ASCE 1988;114(8): [31] ACI Building ode requirements for strutural onrete and ommentary ACI Committee 318, Detroit (MI) 11. [3] Tang J, Hino S, Kuroda I, Ohta T. Modeling of stress strain relationships for steel and onrete in onrete filled irular steel tubular olumns. Steel Constr Eng JSSC 1996; 3(11): [33] Liang QQ. Analysis and design of steel and omposite strutures. London and New York: CRC Press, Taylor and Franis Group, 14. [34] Euroode 4 (1994), Design of omposite steel and onrete strutures, Part 1.1, General rules and rules for buildings, DD ENV , London, United Kingdom. 9

30 Figures and tables Table 1 Ultimate axial loads of irular CFSST short olumns. Speimens D (mm) t (mm) D/t L (mm). (MPa) E (GPa) n f (MPa) max P u. exp (kn) P u. fib (kn) Pu. fib Pu.exp Ref. CHS 14 -C CHS 14 -C CHS 14 -C CHS C [9] CHS C CHS C C-5 1.A C-5 1.B C3-5 1.A C-5 1.6A C-5 1.6B C A [1] C B C-1 1.6A C-1 1.6B C A C B Mean.97 Standard deviation (SD).8 Coeffiient of variation (COV).9 3

31 Table Comparison of ultimate axial strengths of irular CFSST short olumns. Rasmussen s model Abdella et al. model Speimens P u. exp (kn) P (kn) u. fib1 Pu. fib1 Pu.exp P u. (kn) fib Pu. fib Pu.exp CHS 14 -C CHS 14 -C CHS 14 -C CHS C CHS C CHS C C-5 1.A C-5 1.B C3-5 1.A C-5 1.6A C-5 1.6B C A C B C-1 1.6A C-1 1.6B C A C B Mean Standard deviation (SD).8.8 Coeffiient of variation (COV)

32 Table 3 Comparison of ultimate axial strengths of CFSST olumns determined by numerial model and design model. Speimens D (mm) t (mm) D/t (MPa) n f (MPa) P u. num (kn) P u. design (kn) P u. design P u. num C C C C C C C C C C C C C C C C C C C C C C C C C C C

33 C C C C C C C C C C C C C C C C C C C C C C C C C C C C Mean.97 Standard deviation (SD).5 Coeffiient of variation (COV).6 33

34 Table 4 Comparison of ultimate axial strengths of CFSST olumns determined by experiment, design odes and design model. Speimens D/t P u.exp (kn) P u. ACI (kn) ACI ode EC4 ode Design model Pu. ACI u.ec4 P u.exp P (kn) P u.exp u. EC4 P P u. design P u. design P u.exp CHS 14 -C CHS 14 -C CHS 14 -C CHS C CHS C CHS C C-5 1.A C-5 1.B C3-5 1.A C-5 1.6A C-5 1.6B C A C B C-1 1.6A C-1 1.6B C A C B Mean Standard deviation (SD) Coeffiient of variation (COV)

35 Conrete fibers y Stainless steel fibers x Fig. 1.Fiber element disretization of irular CFSST setion. Fig.. Stress strain urves for stainless steels in ompression. 35

36 f A f B C o u Fig. 3. Stress strain urve for onfined onrete in irular CFSST olumns Axial load (kn) Experiment [9] Numerial model Strain Fig. 4. Comparison of predited and experimental axial load strain urves for Speimen CHS C3. 36

37 Axial load (kn) Experiment [9] Numerial model Strain Fig. 5. Comparison of predited and experimental axial load strain urves for Speimen CHS C6. 5 Axial load (kn) Experiment [1] Numerial model Strain Fig. 6. Comparison of predited and experimental axial load strain urves for Speimen C3-5 1.A. 37

38 8 7 6 Axial load (kn) Experiment [1] Numerial model Strain Fig. 7. Comparison of predited and experimental axial load strain urves for Speimen C B. 6 5 Axial load (kn) Two-stage model [1] Three-stage model [3] Strain Fig. 8. Comparisons of two-stage and three-stage onstitutive model for Speimen C1. 38

39 Axial load (kn) D/t=3 D/t=4 D/t= Strain Fig. 9. Effets of D t ratio on the axial load strain behaviour of irular CFSST olumns f =11 MPa f =9 MPa f =7 MPa 35 Axial load (kn) Strain Fig. 1. Effets of onrete ompressive strengths on axial load strain urves for irular CFSST olumns. 39

40 1 1 σ =53 MPa Axial load (kn) σ =4 MPa Strain Fig. 11. Effets of stainless steel strengths on axial load strain urves for irular CFSST olumns. 4