Nonlinear Inelastic Analysis of Concrete-Filled Steel Tubular Slender Beam-Columns by Vipulkumar Ishvarbhai Patel, ME Thesis submitted in fulfillment of the requirement for the degree of Doctor of Philosophy College of Engineering and Science, Victoria University, Australia January 2013
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Nonlinear Inelastic Analysis of
Concrete-Filled Steel Tubular
Slender Beam-Columns
by
Vipulkumar Ishvarbhai Patel, ME
Thesis submitted in fulfillment of the requirement for the degree of
Doctor of Philosophy
College of Engineering and Science, Victoria University, Australia
January 2013
V.I. Patel: Nonlinear Inelastic Analysis of Concrete-Filled Steel Tubular Slender Beam-Columns ii
ABSTRACT
High strength thin-walled concrete-filled steel tubular (CFST) slender beam-columns
may undergo local and global buckling when subjected to biaxial loads, preloads or
cyclic loading. The local buckling effects of steel tube walls under stress gradients have
not been considered in existing numerical models for CFST slender beam-columns. This
thesis presents a systematic development of new numerical models for the nonlinear
inelastic analysis of thin-walled rectangular and circular CFST slender beam-columns
incorporating the effects of local buckling, concrete confinement, geometric
imperfections, preloads, high strength materials, second order and cyclic behavior.
In the proposed numerical models, the inelastic behavior of column cross-sections is
simulated using the accurate fiber element method. Accurate constitutive laws for
confined concrete are implemented in the models. The effects of progressive local
buckling are taken into account in the models by using effective width formulas. Axial
load-moment-curvature relationships computed from the fiber analysis of sections are
used in the column stability analysis to determine equilibrium states. Deflections caused
by preloads on the steel tubes arising from the construction of upper floors are included
in the analysis of CFST slender columns. Efficient computational algorithms based on
the Müller’s method are developed to obtain nonlinear solutions. Analysis procedures
are proposed for predicting load-deflection and axial load-moment interaction curves for
CFST slender columns under axial load and uniaxial bending, biaxial loads, preloads or
axial load and cyclic lateral loading.
V.I. Patel: Nonlinear Inelastic Analysis of Concrete-Filled Steel Tubular Slender Beam-Columns iii
The numerical models developed are verified by comparisons of computer solutions
with existing experimental results and then utilized to undertake extensive parametric
studies on the fundamental behavior of CFST slender columns covering a wide range of
parameters. The numerical models are shown to be efficient computer simulation tools
for designing safe and economical thin-walled CFST slender beam-columns with any
steel and concrete strength grades. The thesis presents benchmark numerical results on
the behavior of high strength thin-walled CFST slender beam-columns accounting for
progressive local buckling effects. These results provide a better understanding of the
fundamental behavior of CFST columns and are valuable to structural designers and
composite code writers.
V.I. Patel: Nonlinear Inelastic Analysis of Concrete-Filled Steel Tubular Slender Beam-Columns iv
DECLARATION
“I, Vipulkumar Ishvarbhai Patel, declare that the PhD thesis entitled ‘Nonlinear inelastic
analysis of concrete-filled steel tubular slender beam-columns’ is no more than 100,000
words in length including quotes and exclusive of tables, figures, appendices,
bibliography, references and footnotes. This thesis contains no material that has been
submitted previously, in whole or in part, for the award of any other academic degree or
diploma. Except where otherwise indicated, this thesis is my own work”.
Signature Date:
V.I. Patel: Nonlinear Inelastic Analysis of Concrete-Filled Steel Tubular Slender Beam-Columns v
ACKNOWLEDGMENTS
The author would like to thank his principal supervisor A/Prof. Qing Quan Liang for his
close supervision, productive discussions and invaluable suggestions throughout the
period of the research project. The author would also like to thank his associate
supervisor A/Prof. Muhammad N.S. Hadi of the University of Wollongong for his
discussions, invaluable suggestions, encouragement and support.
The research presented in this thesis was carried out in the College of Engineering and
Science at Victoria University (2010-2013). The author was supported by an Australian
Postgraduate Award and a Faculty of Health, Engineering and Science scholarship. The
financial support is gratefully acknowledged.
The author would also like to thank all staff in the College of Engineering and Science
and Postgraduate Research Office for their kind assistance. I am grateful to the staff of
the Victoria University library for their strong support of my enthusiasm to find the
research articles and reports.
The author wishes to express his appreciation to his friends and colleagues for their help
and support.
Last, but certainly not the least, the author would like to thank his family for their
support and understanding during three years study.
V.I. Patel: Nonlinear Inelastic Analysis of Concrete-Filled Steel Tubular Slender Beam-Columns vi
LIST OF PUBLICATIONS
Based on this research work, the candidate has written the following papers, which have
been published or submitted for publication in international journals and conference
proceedings.
Journal Articles
1. Patel, V. I., Liang, Q. Q. and Hadi, M. N. S., “High strength thin-walled
rectangular concrete-filled steel tubular slender beam-columns, Part I: Modeling,”
Journal of Constructional Steel Research, 2012, 70, 377-384.
2. Patel, V. I., Liang, Q. Q. and Hadi, M. N. S., “High strength thin-walled
rectangular concrete-filled steel tubular slender beam-columns, Part II: Behavior,”
Journal of Constructional Steel Research, 2012, 70, 368-376.
3. Patel, V. I., Liang, Q. Q. and Hadi, M. N. S., “Inelastic stability analysis of high
Interaction and Multiscale Mechanics, An International Journal, 2012, 5(2), 91-
104.
4. Liang, Q. Q., Patel, V. I. and Hadi, M. N. S., “Biaxially loaded high-strength
concrete-filled steel tubular slender beam-columns, Part I: Multiscale simulation,”
Journal of Constructional Steel Research, 2012, 75, 64-71.
5. Patel, V. I., Liang, Q. Q. and Hadi, M. N. S., “Biaxially loaded high-strength
concrete-filled steel tubular slender beam-columns, Part II: Parametric study,”
Journal of Constructional Steel Research (currently under review).
V.I. Patel: Nonlinear Inelastic Analysis of Concrete-Filled Steel Tubular Slender Beam-Columns vii
6. Patel, V. I., Liang, Q. Q. and Hadi, M. N. S., “Numerical analysis of circular
concrete-filled steel tubular slender beam-columns with preload effects,”
International Journal of Structural Stability and Dynamics, 2013, 13(3), 1250065
(23 pages).
7. Patel, V. I., Liang, Q. Q. and Hadi, M. N. S., “Nonlinear analysis of rectangular
concrete-filled steel tubular slender beam-columns with preload effects,” Journal
of Constructional Steel Research (currently under review).
8. Patel, V. I., Liang, Q. Q. and Hadi, M. N. S., “Numerical analysis of high-strength
concrete-filled steel tubular slender beam-columns under cyclic loading,” Journal
of Constructional Steel Research (currently under review).
Refereed Conference Papers
9. Liang, Q. Q., Patel, V. I. and Hadi, M. N. S., “Nonlinear analysis of biaxially
loaded high strength rectangular concrete-filled steel tubular slender beam-
columns, Part I: Theory,” Proceedings of the 10th International Conference on
Advances in Steel Concrete Composite and Hybrid Structures, Singapore, July
2012, 403-410.
10. Patel, V. I., Liang, Q. Q. and Hadi, M. N. S., “Nonlinear analysis of biaxially
loaded high strength rectangular concrete-filled steel tubular slender beam-
columns, Part II: Applications,” Proceedings of the 10th International Conference
on Advances in Steel Concrete Composite and Hybrid Structures, Singapore, July
2012, 411-418.
V.I. Patel: Nonlinear Inelastic Analysis of Concrete-Filled Steel Tubular Slender Beam-Columns viii
11. Patel, V. I., Liang, Q. Q. and Hadi, M. N. S., “Nonlinear inelastic behavior of
circular concrete-filled steel tubular slender beam-columns with preload effects,”
Proceedings of the 10th International Conference on Advances in Steel Concrete
Composite and Hybrid Structures, Singapore, July 2012, 395-402.
V.I. Patel: Nonlinear Inelastic Analysis of Concrete-Filled Steel Tubular Slender Beam-Columns ix
PART A:
DETAILS OF INCLUDED PAPERS: THESIS BY PUBLICATION
Please list details of each Paper included in the thesis submission. Copies of published papers and submitted and/or final draft Paper manuscripts should also be included in the thesis submission.
Item/
Chapter
No.
Paper Title
Publication
Status(e.g. published, accepted for publication, to be revised and resubmitted, currently under review, unsubmitted but proposed to be submitted)
Publication Title and
Details (e.g. date published, impact factor etc.)
3
High strength thin-walled rectangular concrete-filled steel tubular slender beam-columns, Part I: Modeling
Published.
Journal of Constructional Steel Research, 2012, 70, 377-384. ERA Rank: A*. Impact Factor: 1.251.
High strength thin-walled rectangular concrete-filled steel tubular slender beam-columns, Part II: Behavior
Published.
Journal of Constructional Steel Research, 2012, 70, 368-376. ERA Rank: A*. Impact Factor: 1.251.
Inelastic stability analysis of high strength rectangular concrete-filled steel tubular slender beam-columns
Published.
Interaction and Multiscale Mechanics, An International Journal, 2012, 5(2), 91-104.
Interaction and Multiscale Mechanics, An International Journal, 2012, 5(2), 91-
104.
67
68
69
70
71
72
Journal of Constructional Steel Research 70 (2012) 377–384
Contents lists available at SciVerse ScienceDirect
Journal of Constructional Steel Research
High strength thin-walled rectangular concrete-filled steel tubular slenderbeam-columns, Part I: Modeling
Vipulkumar Ishvarbhai Patel a, Qing Quan Liang a,⁎, Muhammad N.S. Hadi b
a School of Engineering and Science, Victoria University, PO Box 14428, Melbourne, VIC 8001, Australiab School of Civil, Mining and Environmental Engineering, University of Wollongong, Wollongong, NSW 2522, Australia
High strength thin-walled rectangular concrete-filled steel tubular (CFST) slender beam-columns under eccentricloading may undergo local and overall buckling. The modeling of the interaction between local and overallbuckling is highly complicated. There is relatively little numerical study on the interaction buckling of highstrength thin-walled rectangular CFST slender beam-columns. This paper presents a new numerical modelfor simulating the nonlinear inelastic behavior of uniaxially loaded high strength thin-walled rectangularCFST slender beam-columns with local buckling effects. The cross-section strengths of CFST beam-columns aremodeled using the fiber element method. The progressive local and post-local buckling of thin steel tube wallsunder stress gradients is simulated by gradually redistributing normal stresses within the steel tube walls.New efficient Müller's method algorithms are developed to iterate the neutral axis depth in the cross-sectionalanalysis and to adjust the curvature at the columns ends in the axial load–moment interaction strength analysisof a slender beam-column to satisfy equilibrium conditions. Analysis procedures for determining the load–deflection and axial load–moment interaction curves for high strength thin-walled rectangular CFST slenderbeam-columns incorporating progressive local bucking and initial geometric imperfections are presented. Thenew numerical model developed is shown to be efficient for predicting axial load–deflection and axialload–moment interaction curves for high strength thin-walled rectangular CFST slender beam-columns.The verification of the numerical model and parametric studies is given in a companion paper.
The local and overall instability problem is encountered in eccentri-cally loaded high strength thin-walled rectangular concrete-filled steeltubular (CFST) slender beam-columns with large depth-to-thicknessratios. The inelastic modeling of thin-walled CFST slender beam-columns under axial load and bending is complicated because it mustaccount for not onlymaterial and geometric nonlinearities aswell as as-sociated second order effects but also the interaction between progres-sive local and overall buckling. Although these composite beam-columns are increasingly used in high rise composite buildings, theirstructural performance cannot be accurately predictedwithout an accu-rate and efficient computer modeling tool. Currently, there is a lack ofsuch a modeling technique for thin-walled CFST slender beam-columns. This paper describes the important development of anew numerical model that simulates the behavior of high strengththin-walled CFST slender beam-columns incorporating the localbuckling effects of the steel tube walls under stress gradients.
+61 3 9919 4139.
l rights reserved.
Experiments have been undertaken by many researchers to studythe behavior of normal strength CFST beam-columns. Early experimentson slender steel tubular columns filled with normal strength concreteincluded those conducted by Bridge [1], Shakir-Khalil and Zeghiche [2]and Matsui et al. [3]. More recently, Schneider [4] performed tests tostudy the effects of the steel tube shape and the wall thickness on theultimate strength and ductility of CFST short columns. He reportedthat circular CFST columns possessed higher strength and ductilitythan square and rectangular ones. Experimental results presented byHan [5] illustrated that the ultimate axial strength andductility of axiallyloaded CFST columns increased with increasing the constraining factorbut decreased with an increase in the depth-to-thickness ratio. Ellobodyet al. [6] studied the behavior of normal and high strength circular CFSTshort columns under axial loading.
The local buckling behavior of thin-walled rectangular CFST shortcolumns under axial compressionwas studied experimentally byGe andUsami [7], Bridge and O'Shea [8] and Uy [9]. Tests results demonstratedthat the concrete core delayed the occurring of the steel tube localbuckling and forced the steel tube walls to buckle outward. In addition,it was found that local buckling remarkably reduced the ultimatestrengths of thin-walled CFST columns. Liang and Uy [10] and Lianget al. [11] employed the finite element method to investigate the localand post-local buckling behavior of steel plates in thin-walled CFST
Fig. 1. Stress–strain curve for concrete in rectangular CFST columns.
378 V.I. Patel et al. / Journal of Constructional Steel Research 70 (2012) 377–384
columns under axial load and biaxial bending. They proposed a set offormulas for determining the initial local buckling stresses and post-local buckling strengths of steel plates under stress gradients.
Recently, experimental research has focused on high strength CFSTbeam-columns as they are increasingly used in high rise compositebuildings. An experimental study on CFST beam-columns under eccen-tric loading was carried out by Chung et al. [12]. These columns weremade of steel tubes with a yield strength of 445 MPa filled with88 MPa high strength concrete. In addition, Liu [13] undertook testson eccentrically loaded high strength rectangular CFST beam-columnswith steel yield strength of 495 MPa and high strength concrete of60 MPa. His test results demonstrated that the ultimate loads of CFSTslender beam-columns were significantly reduced by increasing theload eccentricity ratio. Moreover, Lue et al. [14] reported experimentalresults on rectangular CFST slender beam-columns with concretecompressive strengths varying between 29 and 84 MPa.
Nonlinear analysis techniques are efficient and cost-effective per-formance simulation and design tools for CFST columns [2]. Lakshmiand Shanmugam [15] proposed a semi-analytical model for deter-mining the behavior of CFST slender beam-columns under biaxialbending. The limitation of their model is that it did not account forthe effects of local buckling, concrete confinement and concrete ten-sile strength. The load–deflection analysis procedure presented byVrcelj and Uy [16] could be used to analyze axially loaded high strengthsquare CFST slender beam-columns, but it has not considered the pro-gressive local buckling of the steel tube walls under stress gradients.Liang [17,18] developed a performance-based analysis (PBA) tech-nique for predicting the ultimate strength and ductility of thin-walled CFST short beam-columns under axial load and biaxialbending, incorporating effective width formulas proposed byLiang et al. [11] to account for the effects of progressive local buck-ling. Moreover, an efficient numerical model was created by Liang[19,20] that simulates the load–deflection responses and strengthenvelopes of high strength circular CFST slender beam-columns.
This paper extends the numericalmodels developed by Liang [17,19]to the nonlinear analysis of high strength thin-walled rectangular CFSTslender beam-columns under axial load and uniaxial bending. Materialconstitutive models for concrete in CFST columns and for structuralsteels are presented. Themodeling of cross-sectional strengths account-ing for the local buckling effects of the steel tube walls under stressgradients is formulated by the fiber element method. New efficientcomputational algorithms based on theMüller's method are developedto obtain nonlinear solutions. Computational procedures for simulatingthe load–deflection and axial load–moment interaction curves for highstrength CFST slender beam-columns are described in detail. Theverification of the numerical model developed and its applicationsare given in a companion paper [21].
2. Material stress–strain relationships
2.1. Stress–strain relationships for concrete
The rectangular steel tube provides confinement to the four cornersof the concrete core in a rectangular CFST column. This confinementdoes not have a significant effect on the compressive strength of theconcrete core so that it can be ignored in the analysis and design ofrectangular CFST columns. However, the ductility of the concretecore in rectangular CFST columns is improved and is included in theconcretemodel. The stress–strain relationship for concrete in rectangu-lar CFST columns is shown in Fig. 1. The concrete stress from O to A iscalculated based on the equations given by Mander et al. [22] as:
σ c ¼f ′ccλ
εcε′cc
� �λ−1þ εc
ε′cc
� �λ ð1Þ
λ ¼ EcEc− f ′cc
ε′cc
� � ð2Þ
where σc stands for the compressive concrete stress, fcc′ is the effectivecompressive strength of concrete, εcis the compressive concrete strain,εcc′is the strain at fcc′ and Ec is the Young's modulus of concrete whichis given by ACI [23] as
Ec ¼ 3320ffiffiffiffiffiffif ′cc
qþ 6900 MPað Þ: ð3Þ
The effective compressive strengths of concrete (fcc′) is taken asγcfc′, where γc is the strength reduction factor proposed by Liang[17] to account for the column size effect and is expressed as
γc ¼ 1:85D−0:135c 0:85≤γc≤1:0ð Þ ð4Þ
where Dcis taken as the larger of (B−2t) and (D−2t) for a rectangularcross section, B is the width of the cross-section, D is the depth of thecross-section, and t is the thickness of the steel tube wall.
High strength concrete becomes more brittle after reaching themaximum compressive strength. The strain corresponding to themaximum compressive strength of high strength concrete is greaterthan that of the normal strength concrete. In the numerical model,the strain εcc′ corresponding to fcc′ is taken as 0.002 for the effectivecompressive strength less than or equal to 28 MPa and 0.003forfcc′>82 MPa. For the effective compressive strength between 28and 82 MPa, the strain εcc′is determined by linear interpolation.
The parts AB, BC and CD of the stress–strain curve for concrete inCFST columns as shown in Fig. 1 are defined by the following equationsgiven by Liang [17]:
σ c ¼f ′cc for ε′cc bεc ≤0:005βc f
′cc þ 100 0:015−εcð Þ f
0
cc−βc f0
cc
� �for 0:005 bεc ≤0:015
βc f′‘cc for εc > 0:015
8><>: ð5Þ
where βc was proposed by Liang [17] based on experimental resultspresented by Tomii and Sakino [24] and is given by
βc ¼1:0 for
Bs
t≤ 24
1:5− 148
Bs
tfor24 b
Bs
t≤ 48
0:5 forBs
t> 48
8>>>>><>>>>>:
ð6Þ
where Bs is taken as the larger of B andD for a rectangular cross-section.The tensile strength of concrete is taken as 0:6
ffiffiffiffiffiffif ′cc
q, which is much
lower than its ultimate compressive strength. The stress–strainrelationship for concrete in tension is shown in Fig. 1. The concrete
74
B
ε
φ
y
y
D
Pn
x
y
n
e,i
t
i
n,i
Concrete fibers Steel fibers
N.A.
d
d
Load,
Fig. 3. Fiber element discretization of CFST beam-column section.
379V.I. Patel et al. / Journal of Constructional Steel Research 70 (2012) 377–384
tensile stress is considered zero at the ultimate tensile strain. Theconcrete tensile stress is directly proportional to the tensile strainof concrete up to concrete cracking. After concrete carking, the tensilestress of concrete is inversely proportional to the tensile strain ofconcrete up to the ultimate tensile strain due to concrete cracking.The ultimate tensile strain is taken as 10 times of the strain at cracking.
2.2. Stress–strain relationships for steels
Three types of the structural steels such as high strength structuralsteels, cold-formed steels and mild structural steels are considered inthe numerical model. The steel generally follows the same stress–strainrelationship under the tension and compression. The stress–strainrelationship for steel under uniaxial compression is shown in Fig. 2.The mild structural steels have a linear stress–strain relationship up tothe yield stress, however, it is assumed that high strength steels andcold-formed steels have a linear stress–strain relationship up to0.9fsy, where fsy is the steel yield strength. The rounded curve of thecold-formed structural steel can be defined by the equation proposedby Liang [17]. The hardening strain εst is assumed to be 0.005 for highstrength and cold-formed steels and 10εsy for mild structural steelsin the numerical model. The ultimate strain of steels is assumed tobe 0.2.
3. Modeling of cross-sectional strengths
3.1. Strain calculations
In thefiber elementmethod, the rectangular cross-section of a slenderbeam-column is divided into small concrete fiber elements and steelfiber elements for the purpose of the fiber element integration. Thetypical cross-section discretization is shown in Fig. 3. The strain ofeach fiber element is calculated by multiplication of the curvatureand the orthogonal distance of each fiber element from the neutralaxis. In the numerical model, the compressive strain is taken as positiveand the tensile strain is taken as negative. The strain εt at the top fiber ofthe section in the composite cross-section can be determined bymulti-plication of the curvature ϕ and the neutral axis depth dn.
For bending about the x-axis, the strains in concrete and steel fiberscan be calculated by the following equations given by Liang [17]:
yn;i ¼D2−dn ð7Þ
de;i ¼ yi−yn;i�� �� ð8Þ
Fig. 2. Stress–strain curves for structural steels.
εi ¼ϕde:i for yi≥yn;i
−ϕde;i for yibyn;i
�ð9Þ
wheredn is the neutral axis depth, de, iis the orthogonal distance from thecentroid of each fiber element to the neutral axis, yi is the coordinates ofthe fiber i and εi is the strain at the ith fiber element and yn, i is thedistance from the centroid of each fiber element to the neutral axis.
For bending about the y-axis, the strain in concrete and steel fiberscan be calculated by the following equations given by Liang [17]
xn:i ¼B2−dn ð10Þ
de:i ¼ xi−xn;i�� �� ð11Þ
εi ¼ϕde;i for xi ≥ xn;i−ϕde;i for xi b xn;i
�ð12Þ
where xi is coordinates of the fiber i and xn, i is the distance from thecentroid of each fiber element to the neutral axis.
3.2. Initial local buckling
Steel plates in thin-walled CFST columns with a large width-to-thickness ratiomay buckle locally outward, which reduces the strengthand ductility of the beam-columns. The initial local buckling stresses ofsteel plates depend on the width-to-thickness ratio, residual stresses,geometric imperfections, the yield strength of steel plates, and theapplied edge stress gradients. Liang and Uy [10] reported that localbuckling of a thin steel plate may occur when its width-to-thickness (b/t) ratio is greater than 30. Steel tube walls of a thin-walledCFST column under axial load and bending are subjected to uniform ornon-uniform stresses. Therefore, steel tube walls under both uniformand non-uniform compressive stresses must be taken into account inthe interaction buckling analysis of a thin-walled CFST beam-columnunder axial load and uniaxial bending. Liang et al. [11] proposed formulasfor determining the initial local buckling stresses of thin steel plates understress gradients. These formulas are incorporated in the numerical modelto account for initial local buckling effects on the behavior of highstrength thin-walled rectangular CFST slender beam-columns.
75
380 V.I. Patel et al. / Journal of Constructional Steel Research 70 (2012) 377–384
3.3. Post-local buckling
The post-local buckling strengths of thin steel plates can be deter-mined using the effective strength and width concept. Liang et al. [11]proposed effective strength and width formulas for determining thepost-local buckling strengths of steel plates in thin-walled CFSTbeam-columns under axial load and biaxial bending. Their formulasare incorporated in the numerical model to determine the ultimatestrengths of thin steel plates under stress gradients. Fig. 4 shows theeffective and ineffective areas of a rectangular thin-walled CFSTbeam-column cross-section under axial compression and uniaxialbending. The effective widths be1 and be2 shown in Fig. 4 are givenby Liang et al. [11] as
be1b
¼0:2777þ 0:01019
bt
� �−1:972� 10−4 b
t
� �2þ 9:605� 10−7 b
t
� �3forαs > 0:0
0:4186−0:002047bt
� �þ 5:355� 10−5 b
t
� �2−4:685� 10−7 b
t
� �3forαs ¼ 0:0
8>>><>>>:
ð13Þ
be2b
¼ 2−αsð Þ be1b
ð14Þ
where b is the clear width of a steel flange or web in a CFST columnsection, and αs=σ2/σ1, where σ2 is the minimum edge stress appliedto the plate and σ1 is the maximum edge stress applied to the plate.
3.4. Simulation of progressive post-local buckling
The behavior of a thin steel plate under increasing loads is character-ized by the progressive post-local buckling. In the post-local bucklingregime, stresses in the steel plate are gradually redistributed from theheavily buckled region to the unloaded edges. This implies that theheavily buckled central region in the plate carries lower stresses whilethe unloaded edges withstand higher stresses. The effective widthconcept assumes that a steel plate attains its ultimate strengthwhen the maximum stress in the plate reaches its yield strength. Theineffective width of a thin steel plate increases from zero to maximumvalue when the applied load is increased from the initial local bucklingload to the ultimate load. The maximum ineffective width of each thinsteel plate in a CFST beam-column at the ultimate load is determinedby
bne;max ¼ b− be1 þ be2ð Þ: ð15Þ
Ineffective steel areaEffective steel area
σ1
σ2
b e1
be2
D x
y
N.A.
B
t
Fig. 4. Effective area of steel tubular cross-section under axial load and bending.
As suggested by Liang [17], the ineffective width between zeroand bne,max can be approximately calculated using linear interpolationbased on the maximum stress level in the steel plate, and it is givenby
bne ¼σ1−σ1c
f sy−σ1c
!bne;max ð16Þ
where σ1c is the maximum edge stress in a steel tube wall when initiallocal buckling occurs.
If the maximum edge stress σ1 in a steel plate with a b/t ratiogreater than 30 is larger than the initial local buckling stress σ1c, itis assumed that steel fiber elements within the ineffective area donot carry any loads. Therefore, the normal stresses in those fibersare assigned to zero. However, if the maximum edge stress σ1 isgreater than the yield strength of the steel plate, stresses within thesteel tube wall need to be reduced by a factor of σ1/fsy to make surethat the effectivewidth concept is valid. The effective strength formulasproposed by Liang et al. [11] are employed to determine the ultimatestrength of the thin steel plate, when the total effective width of theplate (be1+be2) is greater than its width (b). If the maximum edgestress of the thin steel plate is greater than the ultimate edge stressσ1u of the steel plate, the stresses within the steel plate are reducedby a factor of σ1/σ1u.
3.5. Stress resultants
The axial force and bending moments carried by a rectangularCFST beam-column cross-section are determined as stress resultantsin the cross-section as follows:
P ¼Xnsi¼1
σ s;iAs;i þXncj¼1
σ c; jAc; j ð17Þ
Mx ¼Xnsi¼1
σ s;iAs;iyi þXncj¼1
σ c; jAc; jyj ð18Þ
My ¼Xnsi¼1
σ s;iAs;ixi þXncj¼1
σ c;jAc;jxj ð19Þ
where P is the axial force, σs, i is the stress of steel fiber i, As, i is the areaof steel fiber i, σc, j is the stress of concrete fiber j, Ac, j is the area ofconcrete fiber j, xi and yi are the coordinates of steel element i, xjand yj are the coordinates of concrete element j, ns is the total numberof steel fiber elements and nc is the total number of concrete fiberelements.
4. Müller's method algorithms
4.1. Determining the neutral axis depth
Computational algorithms based on the Müller's method [25] aredeveloped to iterate the neutral axis depth in a composite cross-sectionwith local bucking effects. The neutral axis depth dn of the compositecross-section is iteratively adjusted to maintain the force equilibriumin order to determine the internal moment carried by the section. TheMüller's method algorithm requires three initial values of the neutralaxis depth dn, 1, dn, 2 and dn, 3 to start the iterative process. The neutralaxis depth dn is adjusted by using the following proposed equations:
381V.I. Patel et al. / Journal of Constructional Steel Research 70 (2012) 377–384
a1 ¼dn;2−dn;3� �
rpu;1−rpu;3� �
− dn;1−dn;3� �
rpu;2−rpu;3� �
dn;1−dn;2� �
dn;1−dn;3� �
dn;2−dn;3� � ð21Þ
b1 ¼dn;1−dn;3� �2
rpu;2−rpu;3� �
− dn;2−dn;3� �2
rpu;1−rpu;3� �
dn;1−dn;2� �
dn;1−dn;3� �
dn;2−dn;3� � ð22Þ
c1 ¼ rpu;3 ð23Þ
where rpu is the residual moment at the mid-height of the column that isgiven by
rpu ¼ P eþ um þ uoð Þ−Mmi ð24Þ
where uo is the initial geometric imperfection at the mid-height of thebeam-column, um is the deflection at the mid-height of the beam-column, e is the eccentricity of the applied load as shown in Fig. 5and Mmi is the internal moment carried by the section.
The sign (+ or −) of the square root term in the denominator ofEq. (20) is chosen to be the same as the sign of b1 to keep dn, 4 close todn, 3. In order to obtain converged solutions, the values of dn, 1, dn, 2 anddn, 3 and corresponding residual moments rpu, 1, rpu, 2 and rpu, 3 need tobe exchanged. The values of dn, 1, dn, 2 and dn, 3 are temporarily storedin dn, 1
T , dn, 2T and dn, 3T respectively while the values of rpu, 1, rpu, 2 and
rpu, 3 are temporarily stored in rpu, 1T , rpu, 2T and rpu, 3
T respectively. Thefollowing computer codes are executed:
Eq. (20) and the above codes are executed repetitively until theconvergence criterion of |rpu|bεk is satisfied,where εk is the convergencetolerance that is taken as 10−4.
4.2. Determining the curvature at column ends
The maximum moment Me max at the column ends for a givenaxial load Pn needs to be determined in order to generate the axialload–moment interaction diagram for the slender beam-column.The curvature at the mid-height ϕm of the beam-column under axial
Fig. 5. Pin-ended beam-column model.
load and uniaxial bending is initialized and gradually increased. Foreach curvature increment at the mid-height of the beam-column,the curvature at the column ends ϕe is iteratively adjusted by theMüller's method algorithm to maintain equilibrium between theinternal momentMmi and the external moment Mme at the mid-heightof the beam-column. The Müller's method algorithm requires threeinitial values of the curvature at the column ends ϕe, 1, ϕe, 2 and ϕe, 3
to start the iterative process. The curvature ϕe at the column endsis adjusted by using the following proposed equations:
where rpm is the residual moment at the mid-height of the columnwhich is given by
rpm ¼ Me þ Pn um þ uoð Þ−Mmi: ð29Þ
The sign of the square root term in the denominator of Eq. (25) istaken as positive in order to obtain converged solutions. Similar to thedetermination of dn, the values of ϕe, 1, ϕe, 2 and ϕe, 3 and correspondingrpm, 1, rpm, 2 and rpm, 3 need to be exchanged in order to obtain the trueϕe.The values of ϕe, 1, ϕe, 2 and ϕe, 3 are temporarily stored in ϕe, 1
T , ϕe, 2T and
ϕe, 3T respectively and the values of rpm, 1, rpm, 2 and rpm, 3 are temporarily
stored in rpm, 1T , rpm, 2
T and rpm, 3T respectively. The following computer
codes are executed:
IF Abs ϕe;4−ϕTe;2
� �bAbs ϕe;4−ϕT
e;1
� �; ϕe;1 ¼ ϕT
e;2;ϕe;2 ¼ ϕTe;1; rpm;1 ¼ rTpm;2; rpm;2 ¼ rTpm;1;
h i;
IF Abs ϕe;4−ϕTe;3
� �bAbs ϕe;4−ϕT
e;2
� �; ϕe;2 ¼ ϕT
e;3;ϕe;3 ¼ ϕTe;2; rpm;2 ¼ rTpm;3; rpm;3 ¼ rTpm;2;
h i;
ϕe;3 ¼ ϕe;4; rpm;3 ¼ rpm;4:
The curvature at the column ends is iteratively adjusted untilconvergence criterion of |rpm|bεk is satisfied.
5. Modeling of load–deflection responses
5.1. Formulation
A new efficient numerical model has been developed to generateaxial load–deflection curves for high strength thin-walled rectangularCFST slender beam-columns under axial load and uniaxial bendingwith local buckling effects. The deflected shape of the column is as-sumed to be part of a sine wave as suggested by Shakir–Khalil andZeghiche [2]. The deflection at any point (y, z) along the columnlength is given by
u ¼ um sinπzL
� �ð30Þ
where L is the effective length of the beam-column.
77
Fig. 6. Flowchart for determining the axial load–deflection curve for rectangular thin-walled CFST slender columns.
382 V.I. Patel et al. / Journal of Constructional Steel Research 70 (2012) 377–384
The curvature (ϕ) along the length of the beam-column can bederived from Eq. (30) as
ϕ ¼ ∂2u∂z2
¼ πL
� �2um sin
πzL
� �: ð31Þ
The curvature at the mid-height of the beam-column is given by
ϕm ¼ πL
� �2um ð32Þ
The external bending moment at the mid-height of the beam-column with an initial geometric imperfection uo and under eccentricloading can be calculated by
Mme ¼ P eþ um þ uoð Þ ð33Þ
5.2. Analysis procedure
The axial load–deflection analysis procedure is started by first as-suming a small value of the mid-height deflection um in the slenderbeam-column under axial load with an eccentricity. The curvatureϕm at the mid-height of the slender beam-column is calculated forthe given mid-height deflection using Eq. (32). The proposed Müller'smethod algorithms are used to adjust the neutral axis depth in thecomposite cross-section tomaintain the force equilibrium. The internalbending moment Mmi is determined using the moment-curvatureresponse of the composite cross-section with local buckling effects.The mid-height deflection of the slender beam-column is graduallyincreased and the process is repeated. A pair of applied axial loadand deflection is used to plot the axial load–deflection diagram.The flowchart of the axial load–deflection analysis procedure isshown in Fig. 6. The key steps of the axial load–deflection diagramare given as follows:
(1) Input the dimensions of the beam-column, material propertiesof the steel and concrete, the loading eccentricity e and initialgeometric imperfection uo.
(2) Divide the concrete core and steel tube into fiber elements.(3) Initialize the mid-height deflection: um=Δum.(4) Compute the curvature ϕm at the mid-height of the beam-
column from the given mid-height deflection um by usingEq. (32).
(5) Initialize three values of the neutral axis depth of the compositecross-section: dn, 1=D/4, dn, 2=D/2 and dn, 3=D.
(6) Compute the stresses of steel and concrete using the stress–strainrelationships.
(7) Check local buckling and redistribute stresses in steel fibers iflocal buckling occurs.
(8) Calculate the internal bending moment Mmi and the externalbending moment Mme corresponding to dn, 1=D/4, dn, 2=D/2and dn, 3=D respectively.
(9) Compute the residualmoments rpu, 1, rpu, 2 and rpu, 3 correspondingtodn, 1=D/4, dn, 2=D/2 and dn, 3=D respectively.
(10) Calculate a1, b1 and c1 and adjust the neutral axis depth dnusing Eq. (20).
(11) Compute fiber element stresses and redistribute normal stressesin steel fibers if local buckling occurs.
(12) Calculate the internal bendingmomentMmi and external bendingmoment Mme corresponding to the neutral axis depth dn.
(13) Compute rpu using Eq. (24) and repeat Steps (10)–(12) untilthe convergence condition |rpu|bεk is satisfied.
(14) Increase the deflection um at the mid-height of the beam-column and repeat steps (4)–(13) until the ultimate load Pnis obtained or the deflection limit is reached.
(15) Plot the load–deflection curve.
The convergence tolerance εk is set to 10−4 in the analysis. Thecomputational procedure proposed can predict the complete axialload–deflection responses of uniaxially loaded rectangular CFST slenderbeam-columns with local buckling effects.
6. Modeling of axial load–moment interaction diagrams
6.1. Formulation
The ultimate bending moment Mn is calculated as the maximumapplied moment Me max at the column ends for a given axial load Pnto generate the axial load–moment interaction diagram. The maxi-mum moment Me max is obtained when the external moment Mme at-tains the ultimate bending strength of the beam-column cross-section for the given axial load Pn. The curvature at the column endsis adjusted using the Müller's method algorithms and the correspond-ing moment Me at the column ends is calculated. The external
Compute Poa using P-u analysis procedure including local buckling effects
Initialize
Initialize
Calculate um using Eq.(35)
Calculate Mmi using P-M- relationship accounting for local buckling effects
Initialize Compute
Adjust using Eq.(25)
Calculate Mmi and Me using P-M-relationship considering local buckling
effects
Is Pn < Poa?
Plot Pn-Mn diagram
Is Mn obtained?
Yes
No
No
NoYes
Yes
mm φφ Δ=
0=nP
φ
φ
eφ
mmm φφφ Δ+=
nnn PPP Δ+=
?kpmr ε<
,61,101 3,1, −=−= EE ee φφ3,2,1, ,, pmpmpm rrr( ) 23,1,2, eee φφφ +=
Fig. 7. Flowchart for determining the axial load–moment interaction diagram forrectangular thin-walled CFST slender beam-columns.
383V.I. Patel et al. / Journal of Constructional Steel Research 70 (2012) 377–384
moment at the mid-height of the slender beam-column is calculatedby
Mme ¼ Me þ Pn um þ uoð Þ: ð34Þ
The curvature at the mid-height of the slender beam-column isgradually increased to calculate the maximum bending moment Me max
for a given axial load. For each curvature increment at the mid-height of the beam-column, the mid-height deflection is calculatedby the following equation:
um ¼ Lπ
� �2ϕm: ð35Þ
6.2. Analysis procedure
The axial load–moment interaction diagram for high strengththin-walled rectangular CFST slender beam-column is determinedby the numerical model developed. The axial load–deflection analysisprocedure incorporating local buckling effects is used to determinethe ultimate axial load Poa of the slender column under axial loadalone. The applied axial load Pn is gradually increased. The curvatureϕm at the mid-height is initialized and gradually increased. Theload–moment–curvature relationship incorporating local bucklingeffects is used to determine the corresponding internal momentMmi at the mid-height of the beam-column. The proposed Müller'smethod algorithms are used to adjust the curvature ϕe at the columnends to produce themomentMe that satisfies equilibrium at themid-height of the beam-column. The maximum moment Me max at thecolumn ends is obtained for each increment of the applied axialload Pn. A pair of the maximum moment Me max at the column endsand the given applied axial load Pn is used to plot the axial load–momentinteraction diagram. Fig. 7 shows the flowchart for determining axialload–moment interaction diagrams for rectangular CFST slenderbeam-column. The main steps of the analysis procedure are givenas follows:
(1) Input the geometry of the beam-column, material properties ofthe steel and concrete and the initial geometric imperfectionuo.
(2) Divide composite section into concrete and steel fiber elements.(3) Compute the ultimate axial load Poa of the axially loaded slender
beam-column using the load–deflection analysis procedureaccounting for local buckling effects.
(4) Initialize the applied axial load: Pn=0.(5) Initialize the curvature at the mid-height of the beam-column:
ϕm=Δϕm.(6) Calculate the mid-height deflection um using Eq. (35) from the
mid-height curvature ϕm.(7) Compute the internal moment Mmi for the given axial load Pn
using the P−M−ϕ relationship accounting for local bucklingeffects.
(8) Initialize three values of the curvature at the column endsϕe, 1=10−10, ϕe, 3=10−6, ϕe, 2=(ϕe, 1+ϕe, 3)/2 and calculatethe corresponding rpm, 1, rpm, 2 and rpm, 3.
(9) Calculate a2, b2 and c2 and adjust the curvature at the columnsend φe using Eq. (25).
(10) Compute the moment Me at the column ends and the internalbendingmomentMmi using the P−M−ϕ relationship accountingfor local buckling effects.
(11) Calculate rpm using Eq. (29) and repeat steps (9)–(10) until|rpm|bεk.
(12) Increase the curvature at the mid-height of the beam-columnby ϕm=ϕm+Δϕm.
(13) Repeat steps (6)–(12) until the ultimate bending strengthMn(=Me max) at the column ends is obtained.
(14) Increase the axial load by Pn=Pn+ΔPn, where ΔPn=Poa/10.(15) Repeat steps (5)–(14) until the maximum load increment is
reached.(16) Plot the axial load–moment interaction diagram.
Numerical analysis shows that the proposed Muller algorithms arevery efficient for obtaining converged solutions. The ultimate pureaxial load Poa is calculated by specifying the eccentricity of the appliedto zero in the axial load–deflection analysis. Similarly, the ultimate purebending strength of a slender beam-column is obtained by specifyingthe axial load to zero in the axial load–moment interaction strengthanalysis.
7. Conclusions
This paper has presented a new numerical model for simulatingthe behavior of high strength thin-walled rectangular CFST slenderbeam-columns under axial load and uniaxial bending. The effects of
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local buckling, column slenderness, eccentricity of loading, highstrength materials, initial geometric imperfections and material andgeometric nonlinearities are considered in the numerical modeldeveloped. The numerical model for pin-ended CFST slender beam-columns with equal end eccentricities and single curvature bendingwas developed based on fiber element formulations. New Müller'smethod algorithms were developed to iterate the neutral axis depthin the composite cross-section and to adjust the curvature at thecolumn ends in the slender beam-column to satisfy equilibriumconditions. The new numerical model is shown to be efficient forpredicting the complete axial load–deflection and axial load–momentinteraction curves for high strength thin-walled rectangular CFSTslender beam-columns with local buckling effects. The comparisonwith corresponding experimental results and parametric studies isgiven in a companion paper.
References
[1] Bridge RQ. Concrete filled steel tubular columns. School of Civil Engineering,University of Sydney, Sydney, Australia, Research Report No. R283; 1976.
[2] Shakir-KhalilH, Zeghiche J. Experimental behaviour of concrete-filled rolled rectangularhollow-section columns. The Structural Engineer 1989;67(19):346–53.
[3] Matsui C, Tsuda K, Ishibashi Y. Slender concrete filled steel tubular columns undercombined compression and bending. The 4th Pacific Structural Steel ConferenceSingapore, Pergamon 3(10): 1995. p. 29–36.
[5] Han LH. Tests on stub columns of concrete-filled RHS sections. J Constr Steel Res2002;58(3):353–72.
[6] Ellobody E, Young B, Lam D. Behaviour of normal and high strength concrete-filledcompact steel tube circular stub columns. J Constr Steel Res 2006;62(7):706–15.
[7] Ge H, Usami T. Strength of concrete-filled thin-walled steel box columns: experiment.J Struct Eng ASCE 1992;118(11):3036–54.
[8] Bridge RQ, O'Shea MD. Behaviour of thin-walled steel box sections with or withoutinternal restraint. J Constr Steel Res 1998;47(1–2):73–91.
[9] Uy B. Strength of concrete filled steel box columns incorporating local buckling.J Struct Eng ASCE 2000;126(3):341–52.
[10] Liang QQ, Uy B. Theoretical study on the post-local buckling of steel platesin concrete-filled box columns. Computers and Structures 2000;75(5):479–90.
[11] Liang QQ, Uy B, Liew JYR. Local buckling of steel plates in concrete-filled thin-walled steel tubular beam-columns. J Constr Steel Res 2007;63(3):396–405.
[12] Chung J, Tsuda K, Matsui C. High-strength concrete filled square tube columnssubjected to axial loading. The Seventh East Asia-Pacific Conference on StructuralEngineering & Construction, Kochi, Japan, vol. 2; 1999. p. 955–60.
[13] Liu D. Behaviour of eccentrically loaded high-strength rectangular concrete-filledsteel tubular columns. J Constr Steel Res 2006;62(8):839–46.
[14] Lue DM, Liu JL, Yen T. Experimental study on rectangular CFT columns with high-strength concrete. J Constr Steel Res 2007;63(1):37–44.
[15] Lakshmi B, Shanmugam NE. Nonlinear analysis of in-filled steel-concrete compositecolumns. J Struct Eng ASCE 2002;128(7):922–33.
[16] Vrcelj Z, Uy B. Strength of slender concrete-filled steel box columns incorporatinglocal buckling. J Constr Steel Res 2002;58(2):275–300.
[17] Liang QQ. Performance-based analysis of concrete-filled steel tubular beam-columns, Part I: theory and algorithms. J Constr Steel Res 2009;65(2):363–72.
[18] Liang QQ. Strength and ductility of high strength concrete-filled steel tubularbeam-columns. J Constr Steel Res 2009;65(3):687–98.
[19] Liang QQ. High strength circular concrete-filled steel tubular slender beam-columns, Part I: numerical analysis. J Constr Steel Res 2011;67(2):164–71.
[20] Liang QQ. High strength circular concrete-filled steel tubular slender beam-columns, Part II: fundamental behavior. J Constr Steel Res 2011;67(2):172–80.
[21] Patel VI, Liang QQ, Hadi MNS. High strength thin-walled rectangular concrete-filled steel tubular slender beam-columns, Part II: behavior. Journal of Construc-tional Steel Research 2012;70(C):368–76.
[22] Mander JB, PriestlyMNJ, Park R. Theoretical stress–strainmodel for confined concrete.J Struct Eng ASCE 1988;114(8):1804–26.
[23] ACI-318. Building code requirements for reinforced concrete. Detroit, MI: ACI;2002.
[24] Tomii M, Sakino K. Elastic–plastic behavior of concrete filled square steel tubularbeam-columns. Trans Architec Inst Jpn 1979;280:111–20.
[25] Müller DE. A method for solving algebraic equations using an automatic computer.MTAC 1956;10:208–15.
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Contents lists available at SciVerse ScienceDirect
Journal of Constructional Steel Research
High strength thin-walled rectangular concrete-filled steel tubular slenderbeam-columns, Part II: Behavior
Vipulkumar Ishvarbhai Patel a, Qing Quan Liang a,⁎, Muhammad N.S. Hadi b
a School of Engineering and Science, Victoria University, PO Box 14428, Melbourne, VIC 8001, Australiab School of Civil, Mining and Environmental Engineering, University of Wollongong, Wollongong, NSW 2522, Australia
Article history:Received 5 April 2011Accepted 18 October 2011Available online 13 November 2011
Keywords:Concrete-filled steel tubesHigh strength materialsLocal and post-local bucklingNonlinear analysisSlender beam-columns
Experimental and numerical research on full-scale high strength thin-walled rectangular steel slender tubesfilled with high strength concrete has not been reported in the literature. In a companion paper, a new nu-merical model was presented that simulates the nonlinear inelastic behavior of uniaxially loaded highstrength thin-walled rectangular concrete-filled steel tubular (CFST) slender beam-columns with local buck-ling effects. The progressive local and post-local buckling of thin steel tube walls under stress gradients wasincorporated in the numerical model. This paper presents the verification of the numerical model developedand its applications to the investigation into the fundamental behavior of high strength thin-walled CFSTslender beam-columns. Experimental ultimate strengths and load-deflection responses of CFST slenderbeam-columns tested by independent researchers are used to verify the accuracy of the numerical model.The verified numerical model is then utilized to investigate the effects of local buckling, column slendernessratio, depth-to-thickness ratio, loading eccentricity ratio, concrete compressive strengths and steel yieldstrengths on the behavior of high strength thin-walled CFST slender beam-columns. It is demonstratedthat the numerical model is accurate and efficient for determining the behavior of high strength thin-walled CFST slender beam-columns with local buckling effects. Numerical results presented in this studyare useful for the development of composite design codes for high strength thin-walled rectangular CFSTslender beam-columns.
The behavior of high strength thin-walled rectangular concrete-filledsteel tubular (CFST) slender beam-columns under axial load and bendingis influenced by many parameters such as the depth-to-thickness ratio,loading eccentricity, column slenderness, concrete compressive strength,steel yield strength, initial geometric imperfections and second order ef-fects. Thin-walled CFST slender beam-columns under eccentric loadingmay fail by local and overall buckling. Researches on the interactionlocal and overall buckling behavior of high strength CFST slender beam-columns have been comparatively very limited. No design specificationson the design of high strength thin-walled CFST slender beam-columnsare given in current design codes such as Eurocode 4 [1], LRFD [2] andACI 318-05 [3]. Apparently, the fundamental behavior of high strengththin-walled CFST slender beam-columns with large depth-to-thicknessratios has not been fully understood and effective design methods havenot been developed and incorporated in current composite designcodes. In this paper, a numerical model developed is validated and
+61 3 9919 4139.
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employed to investigate the behavior of high strength thin-walled CFSTslender beam-columns with local buckling effects.
The local instability problem of CFST slender beam-columns wasnot identified in early experimental researches undertaken by Bridge[4] and Shakir-Khalil and Zeghiche [5]. Their test results indicatedthat all tested normal strength CFST slender columns with a depth-to-thickness ratio (D/t) of 20 or 24 failed by overall buckling withoutthe occurring of local buckling. Local buckling of the steel tubes wasexpected in CFST columns with a D/t ratio greater than 35 as thosetested by Matsui et al. [6]. Chung et al. [7] and Zhang et al. [8] testedhigh strength CFST slender beam-columns with D/t ratios rangingfrom 31 to 52 and compressive concrete cube strength of 94.1 MPa.Han [9] carried out tests on the flexural behavior of normal strengthCFST members with D/t ratios ranging from 20 to 51. Further testson CFST columns were conducted by Vrcelj and Uy [10], Varma etal. [11] and Giakoumelis and Lam [12]. However, the behavior ofhigh strength rectangular steel tubular slender beam-columns filledwith high strength concrete has not been investigated.
In design practice, the axial load–moment interaction diagramsneed to be developed and are used to check for the strength adequacyof CFST slender beam-columns under axial load and bending. Most ofthe experiments conducted were to measure the load–deflection re-sponses of CFST slender columns. It is highly expensive and time
369V.I. Patel et al. / Journal of Constructional Steel Research 70 (2012) 368–376
consuming to obtain complete axial load–moment interaction curvesfor CFST slender beam-columns by experimental methods. Althoughthe load–deflection analysis procedures such as those presented byVrcelj and Uy [10] and Lakshmi and Shanmugam [13] can be usedto develop strength envelopes by varying the loading eccentricity,they are not efficient when the loading eccentricity is large. Despiteof this, the axial load–moment interaction diagrams of high strengthrectangular CFST slender beam-columns have not been studied byVrcelj and Uy [10] and Lakshmi and Shanmugam [13]. Ellobody andYoung [14] used the finite element program ABAQUS to study theaxial load–strain behavior of CFST short columns. Liang [15,16] uti-lized his numerical models to investigate the effects of various impor-tant parameters on the axial load–moment interaction curves forthin-walled CFST short beam-columns under axial load and biaxialbending. Moreover, Liang [17,18] studied concrete confinement ef-fects on column curves and axial load–moment interaction curvesfor high strength circular CFST slender beam-columns.
Presented in this paper is the verification of the numerical modeldeveloped in a companion paper [19] and the behavior of highstrength thin-walled CFST slender beam-columns with local bucklingeffects. Firstly, the numerical model is verified against existing exper-imental results. An extensive parametric study is then undertaken toinvestigate the effects of various important parameters on the ulti-mate strengths, load–deflection responses and axial load–momentinteraction diagrams of high strength thin-walled CFST slenderbeam-columns. Benchmark numerical results obtained from the para-metric study are presented and discussed.
2. Verification of the numerical model
The efficiency and accuracy of the developed numerical model aredemonstrated through comparisons between the numerical andexisting experimental results for a large number of slender beam-columns with different parameters. The ultimate axial and bendingstrengths and load–deflection curves of CFST slender beam-columnsare considered in the verification of the numerical model developed.
2.1. Ultimate axial strengths of normal strength CFST columns
The geometry, material properties and experimental results ofnormal strength CFST slender columns tested by various researchersare given in Table 1. Specimens SCH-1, SCH-2 and SCH-7 were testedby Bridge [4]. The width (B) of the steel tubes varied between
Table 1Ultimate axial strengths of normal strength CFST slender beam-columns.
Specimens B×D× t (mm) D/t L (mm) ex (mm) ey (mm) uo (mm) f
152.5 mm and 204.0 mm while the depth (D) of the steel tubes ran-ged from 152.3 to 203.9 mm. The depth-to-thickness ratios (D/t) ran-ged from 20 to 24 while the length of the columns was either2130 mm or 3050 mm. Specimen SCH-2 was tested under a concen-tric axial load. Initial geometric imperfections (uo) at the mid-heightof the specimens were measured as shown in Table 1. The axial loadwas applied at an eccentricity (e) of either 38 mm or 64 mm. Theslender beam-columns were constructed by steel tubes filled withnormal strength concrete of either 29.9 or 31.1 MPa. The steel yieldstrengths (fsy) varied between 254 MPa and 291 MPa. The ultimatetensile strength (fsu) of steel tubes was assumed to be 410 MPa. TheYoung's modulus (Es) of the steel material was 205 GPa.
Specimens R1, R2 and R5 shown in Table 1 were tested by Shakir-Khalil and Zeghiche [5]. These slender beam-columns with a cross-section of 80×120 mm were constructed by cold-formed steel tubesfilled with normal strength concrete. The yield strength of the steeltubes ranged from 343.3 MPa to 386.3 MPa while the compressivestrength of concrete varied between 34 MPa and 37.4 MPa. Speci-mens were fabricated from 5 mm thick steel tubes so that theirdepth-to-thickness ratio (D/t) was 24. The effective buckling lengthsof the beam-columns were 3210 mm and 2940 mm about the majorand minor axes respectively. The ultimate tensile strength of thesesteel tubes was assumed to be 430 MPa. Specimen R1 was testedunder a concentric axial load while other specimens were testedunder eccentric loads about either the major or minor axis. The load-ing eccentricity ratios (e/D) were in the range of 0.0 to 0.5. The initialgeometric imperfections of specimens were not measured. Shakir-Khalil and Zeghiche [5] realized that the tested columns had initialgeometric imperfections that should be recorded before testing.Therefore, the initial geometric imperfections of L/1000 at the mid-height of the columns were taken into account in the present study.
Slender columns tested byMatsui et al. [6]were also considered in thisstudy. Test parameters of specimens designated by series S1 to S12 arepresented in Table 1. The columns had a square cross-section of149.8 mm wide and 4.27 mm thick. Steel tubes with yield strength of445 MPa were filled with normal strength concrete of 31.9 MPa. The ec-centricity of the applied axial load varied from zero to a maximum of125 mm. The length of the columns varied from 600mm to 4500 mmcovering a wide range of column slenderness. However, only slender col-umnswere considered in this paper. Specimens S1, S5 and S9were testedunder concentric axial loads. Experimental and analytical studies con-ducted by Matsui et al. [6] indicated that these specimens had significantgeometric imperfections that were not recorded. The initial geometric
′c (MP) fsy (MPa) fsu (MPa) Es (GPa) Pu. exp (kN) Pu. num (kN) Pu:numPu:exp
370 V.I. Patel et al. / Journal of Constructional Steel Research 70 (2012) 368–376
imperfections of L/1000 at the mid-height of the tested specimens wereconsidered in the present numerical analysis of these specimens.
The load–deflection analysis procedure developed by Patel et al.[19] was used to predict the ultimate strengths of these tested speci-mens. Test results indicated that specimen R1 did not attain its ulti-mate strength so that the measured maximum compressiveconcrete strain was used in the numerical analysis to determine themaximum load. The computational and experimental ultimate axialstrengths of normal strength CFST slender beam-columns are givenin Table 1, where Pu. exp represents the experimental ultimate axialstrength and Pu. num denotes the ultimate axial strength predicted bythe numerical model. It can be seen from Table 1 that the predictedultimate axial strengths of tested specimens are in good agreementwith experimental results. The mean ultimate axial strength pre-dicted by the numerical model is 1.03 times the experimental valuewith a standard deviation of 0.03 and a coefficient of variation of 0.03.
2.2. Ultimate axial strengths of high strength CFST columns
Table 2 presents the geometries, material properties and experi-mental results of high strength CFST slender columns tested by inde-pendent researchers. Chung et al. [7] tested a number of high strengthCFST slender beam-columns subjected to axial load and uniaxialbending. The specimens were designated by series C18, C24 andC30. The lengths of specimens C18, C24 and C30 were 2250, 3000and 3750 mm respectively. All specimens had a square cross-sectionof 125×125 mm. The steel tubes with yield strength of 450 MPawere filled with high strength concrete of 94.1 MPa. The tensilestrength of the steel tubes was measured as 528 MPa. The depth-to-thickness ratio (D/t) of these specimens was 39. Specimens C30-0 was tested under a concentric axial load while other columnswere subjected to axial loads at an eccentricity of either 20.5 mm or61.5 mm.
Specimens CCH1, CCH2, CCM1 and CCM2 shown in Table 2 weretested by Vrcelj and Uy [10]. These columns were made of hot-rolled steel tubes which were cut into 1770 mm length. In order tosimulate pinned-end support conditions, the specimen was providedwith special knife-edge supports secured at their ends. The loadingeccentricity ratio (e/D) was approximately equal to 0.05. The concretecompressive strength was either 52 MPa or 79 MPa while the steelyield strength was either 400 MPa or 450 MPa.
The load–deflection analysis procedure [19] was used to deter-mine the ultimate axial strengths of these high strength specimenstested. The initial geometric imperfections of these high strengthCFST slender beam-columns were not measured. However, real CFSTslender beam-columns particularly small-scale ones usually have ini-tial geometric imperfections. Therefore, the initial geometric imper-fections of L/1000 at the mid-height of the columns were taken intoaccount in the present numerical analysis of these columns. The
Table 2Ultimate axial strengths of high strength CFST slender beam-columns.
Specimens B×D× t (mm) D/t L (mm) ex (mm) ey (mm) f′c (MP)
ultimate axial strengths of these specimens obtained from the numer-ical analysis and experiments are given in Table 2. It is seen that thenumerical model predicts very well the ultimate axial strengths ofhigh strength CFST slender beam-columns. The ratio of the mean ulti-mate axial strength computed by the numerical model to the experi-mental value is 1.01. The standard deviation of Pu. num/Pu. exp is 0.07and the coefficient of variation is 0.07.
2.3. Ultimate bending strengths
Experimental results presented by Bridge [4], Shakir-Khalil andZeghiche [5], Matsui et al. [6] and Chung et al. [7] were used toexamine the accuracy of the axial–load moment interaction strengthanalysis program [19]. The initial geometric imperfections of L/1000at the mid-height of the columns were taken into account in the anal-ysis of the specimens tested by Shakir-Khalil and Zeghiche [5], Matsuiet al. [6] and Chung et al. [7]. The ultimate axial loads obtained fromexperiments were applied in the numerical analysis to determinethe ultimate bending strengths. The experimental and numericalultimate bending strengths are presented in Table 3. Theexperimental ultimate bending strength Mn. exp was calculated asMn. exp=Pn. exp×e. It appears from Table 3 that the numerical modelyields accurate predictions of the ultimate bending strengths of highstrength CFST slender beam-columns. The ratio of the mean ultimatebending strength computed by the numerical model to the experi-mental value is 1.03 with the standard deviation of 0.05 and the coef-ficient of variation of 0.05.
2.4. Load–deflection curves
The load–deflection curves for CFST slender beam-columns pre-dicted by the numerical model are compared with experimental re-sults provided by Bridge [4], Shakir-Khalil and Zeghiche [5] andMatsui et al. [6]. Fig. 1 shows the load–deflection curves for specimenSCH2 predicted by the numerical model developed by Patel et al. [19]and obtained from the experiments conducted by Bridge [4]. It can beobserved that the load–deflection curve predicted by the numericalmodel agrees reasonably well with the experimental results. The ini-tial stiffness of the load–deflection curve predicted by the model isslightly higher than that of the experimental one. This is likely attrib-uted to the uncertainty of the actual concrete stiffness and strength asthe average concrete compressive strength was used in the numericalanalysis. However, it can be seen from Fig. 1 that the numerical load–deflection curve is in excellent agreement with experimental resultsin the ultimate load range. The predicted and experimental axialload–deflection curves for specimen R5 tested by Shakir-Khalil andZeghiche [5] are depicted in Fig. 2. The figure shows that the initialstiffness of the specimen predicted by the numerical model is almostthe same as that of the experimental one up to loading level about
fsy (MPa) fsu (MPa) Es (GPa) Pu. exp (kN) Pu. num (kN) Pu:numPu:exp
371V.I. Patel et al. / Journal of Constructional Steel Research 70 (2012) 368–376
100 kN. After this loading, the stiffness of the specimen obtained fromtests slightly differs from numerical predictions. This is likely attribut-ed to the uncertainty of the concrete stiffness and strength. The nu-merical model was able to predict the post-peak behavior of thetested specimen R5.
Fig. 3 presents a comparison between numerical and experimentalaxial load–deflection curves for specimen S3. It is seen that the stiff-ness of the column predicted by the numerical model is in excellentagreement with experimental data for loading up to the ultimateload. In the post-peak range, the numerical model also predicts wellthe stiffness of the tested specimen but the predicted load capacityis lower than the test data. The verification shows that the numericalmodel yields good predictions of the ultimate axial and bendingstrengths and axial load–deflection curves for normal and highstrength CFST slender beam-columns.
3. Parametric study
An extensive parametric study was performed to investigate theinfluences of local buckling of the steel tube, column slendernessratio, depth-to-thickness ratio, eccentricity ratio, concrete compres-sive strengths and steel yield strengths on the fundamental behaviorof full-scale high strength thin-walled rectangular CFST slenderbeam-columns under axial load and uniaxial bending. Only one vari-able was considered at a time to assess its individual effect.
3.1. Influences of local buckling
The numerical model was employed to study the effects of localbuckling on the behavior of high strength rectangular CFST slender
0
500
1000
1500
2000
2500
3000
3500
0 1 2 3 4
Axi
al lo
ad (
kN)
Mid-height deflection um (mm)
Experiment
Numerical analysis
Fig. 1. Comparison of predicted and experimental axial load–deflection curves for spec-imen SCH-2.
beam-columns. A thin-walled CFST beam-column with a cross-section of 700×700 mm was considered. The thickness of the steeltube was 7 mm so that its depth-to-thickness ratio (D/t) was 100.The yield and tensile strengths of the steel tube were 690 MPa and790 MPa respectively while its Young's modulus was 200 GPa. Highstrength concrete with compressive strength of 70 MPa was filledinto the steel tube. The column slenderness ratio (L/r) was 40. Theaxial load was applied at an eccentricity 70 mm so that the loadingeccentricity ratio (e/D) was 0.1. The initial geometric imperfectionat the mid-height of the beam-column was assumed to be L/1500.The slender beam-column was analyzed by considering and ignoringlocal buckling effects respectively.
Fig. 4 shows the influences of local buckling on the axial load–deflection curves for thin-walledCFST slender beam-columns andhigh-lights the importance of considering local buckling in the analysis. It canbe seen from Fig. 4 that local buckling remarkably reduces the stiffnessand ultimate axial strength of the slender beam-column under eccentricloading. When the applied axial load is higher than about 10,000 kN, thestiffness of the slender beam-column is found to gradually reduce by pro-gressive post-local buckling as illustrated in Fig. 4. The steel tube wallswith aD/t ratio of 100 undergone elastic local and post-local buckling be-fore yielding. Because the beam-column was subjected to axial load andbending, the two webs of the section were subjected to bending stressesunderwhich local bucklingwas not considered in the effectivewidth for-mulas given by Liang et al. [20]. After steel yielded, strain hardening oc-curred so that the two webs and the tension flange of the steel tubecould carry the higher load than the yield load. This resulted in an in-crease in the load carrying capacity as shown on the load–deflectioncurve depicted in Fig. 4. However, thin-walled CFST slender beam-columns with a D/t ratio less than 80 exhibit smooth load–deflection
Mid-height deflection um (mm)
0
50
100
150
200
250
0 20 40 60 80 100
Axi
al lo
ad (
kN)
Experiment
Numerical analysis
Fig. 2. Comparison of predicted and experimental axial load–deflection curves for spec-imen R5.
84
Mid-height deflection um (mm)
0
100
200
300
400
500
600
0 20 40 60 80 100 120
Axi
al lo
ad (
kN)
Experiment
Numerical analysis
Fig. 3. Comparison of predicted and experimental axial load–deflection curves for specimenS3.
Moment Mn/Mo
Axi
al lo
ad P
n/P o
a
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
Local buckling ignoredLocal buckling considered
Fig. 5. Influence of local buckling on the axial load–moment interaction diagrams forthin-walled CFST slender beam-columns.
372 V.I. Patel et al. / Journal of Constructional Steel Research 70 (2012) 368–376
curves as shown in Fig. 9. The ultimate axial load of the slender beam-column is overestimated by 8.32% if local buckling was not considered.
Fig. 5 demonstrates the influences of local buckling on the axialload–moment interaction diagram for the CFST slender beam-column.As shown in Fig. 5, the ultimate axial strengthwas normalized by the ul-timate axial load (Poa) of the axially loaded slender column includinglocal buckling effects, while the ultimate moment was normalized bythe ultimate pure bending moment (Mo) of the slender beam-columnwith local buckling effects. It can be observed fromFig. 5 that local buck-ling considerably reduces the ultimate strength of the slender beam-column. If local buckling was not considered in the numerical analysis,the ultimate axial strength of the slender beam-column is overesti-mated by 5.02% while the maximum ultimate bending strength of theslender beam-column is overestimated by 7.49%.
3.2. Influences of column slenderness ratio
The numerical model developed was used to examine the effectsof the column slenderness ratio (L/r) which is an important parame-ter that influences the behavior of CFST beam-columns. High strengththin-walled CFST slender beam-columns with a cross-section of600×600 mm were considered. The depth-to-thickness ratio of the
Mid-height deflection um (mm)
0
5000
10000
15000
20000
25000
30000
0 10 20 30 40 50 60
Axi
al lo
ad (
kN)
Local buckling ignored
Local buckling considered
Fig. 4. Influence of local buckling on the axial load–deflection curves for thin-walledCFST slender beam-columns.
section was 50. Column slenderness ratios (L/r) of 22, 50, 65 and 80were considered in the parametric study. The loading eccentricityratio (e/D) of 0.1 was specified in the analysis of slender beam-columns with various L/r ratios. The initial geometric imperfectionof L/1500 at the mid-height of the beam-columns was incorporatedin the model. The beam-columns were made of high strength steeltubes with the yield and tensile strengths of 690 MPa and 790 MParespectively. The Young's modulus of the steel tubes was 200 GPa.The steel tubes were filled with high strength concrete of 80 MPa.
Fig. 6 presents the axial load–deflection curves for the beam-columns with various slenderness ratios. It appears that the ultimateaxial load is strongly affected by the column slenderness ratio. In-creasing the column slenderness ratio significantly reduces its ulti-mate axial load. Columns with a smaller slenderness ratio areshown to be less ductile than the ones with a larger slendernessratio. The mid-height deflection at the ultimate axial strength of thebeam-columns increases with increasing the column slendernessratio. Fig. 7 shows the ratio of the ultimate axial load (Pn) to the ulti-mate axial strength (Poa) of the column's section under eccentricloading as a function of the column slenderness ratio. As
Mid-height deflection um (mm)
L/r=22
L/r=50
L/r=65L/r=80
0
5000
10000
15000
20000
25000
30000
35000
0 50 100 150 200 250 300 350
Axi
al lo
ad (
kN)
Fig. 6. Influence of column slenderness ratio on the axial load–deflection curves forthin-walled CFST slender beam-columns.
85
Axi
al lo
ad P
n/P o
Moment Mn/Mo
L/r=0
L/r=22
L/r=35
L/r=500
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Fig. 8. Influence of column slenderness ratio on the axial load–moment interaction di-agrams for thin-walled CFST slender beam-columns.
40000
373V.I. Patel et al. / Journal of Constructional Steel Research 70 (2012) 368–376
demonstrated in Fig. 7, the ultimate axial strength of the beam-column with a L/r ratio of 22 is 99% of the section's strength.
Fig. 8 demonstrates the effects of the column slenderness ratio onthe axial load–moment strength interaction curves for CFST beam-columns. In Fig. 8, the ultimate axial strength (Pn) was normalizedto the ultimate axial load (Po) of the axially loaded beam-column sec-tion while the ultimate moment (Mn) was normalized to the ultimatepure bending moment (Mo) of the beam-column. The column slen-derness ratio L/r=0 was used in the axial load–moment interactionanalysis to determine the ultimate strengths of the composite section.It can be seen from Fig. 8 that reducing the column L/r ratio enlargesthe axial load–moment interaction diagram. The ultimate bendingstrength of the beam-column is shown to decrease as the columnslenderness ratio increases.
3.3. Influences of depth-to-thickness ratio
Local buckling of a thin-walled steel tubes depends on its depth-to-thickness ratio (D/t). The numerical model was utilized to examinethe effects of D/t ratio on the load–deflection and axial load–momentinteraction curves for high strength CFST slender beam-columns. Asquare cross-section of 650×650 mm was considered in the analysis.The D/t ratios of the column sections were calculated as 40, 60, 80 and100 by changing the thickness of the steel tubes. The column slender-ness ratio (L/r) was 35. The loading eccentricity ratio (e/D) was takenas 0.1 in the analysis. The initial geometric imperfection of the beam-columns at mid-height was specified as L/1500. The yield and tensilestrengths of the steel tubes were 690 MPa and 790 MPa respectivelywhile the Young's modulus of the steel tubes was 200 GPa. The com-pressive strength of the in-filled concrete was 60 MPa.
The influences of D/t ratio on the axial load–deflection curves forhigh strength thin-walled CFST slender beam-columns are illustratedin Fig. 9. The figure shows that increasing the D/t ratio of the slenderbeam-columns slightly reduces their initial stiffness. However, in-creasing the D/t ratio significantly reduces the ultimate axial strengthof eccentrically loaded CFST slender beam-columns. This is attributedto the fact that a column section with a larger D/t ratio has a lessersteel area and it may undergone local buckling which reduces the ul-timate strength of the column.
Fig. 10 demonstrates the influence of D/t ratio on the axial load–moment interaction diagrams for high strength thin-walled CFSTslender beam-columns. In each interaction curve, the axial load (Pn)was divided by the ultimate axial strength (Poa) of the axially loadedslender beam-column, while the moment (Mn) was divided by the ul-timate pure bending moment (Mo) of the beam-column. It can be
Ulti
mat
e ax
ial l
oad
P n/P
oa
Slenderness ratio L/r
0
0.2
0.4
0.6
0.8
1
1.2
0 20 40 60 80 100 120
Fig. 7. Influence of column slenderness ratio on the ultimate axial loads of thin-walledCFST slender beam-columns.
seen from Fig. 10 that decreasing the D/t ratio of the slender beam-column narrows the axial load–moment interaction diagram for theslender beam-column. In addition, the flexural strength at the maxi-mum moment point is shown to increase remarkably with increasingthe D/t ratio.
3.4. Influences of loading eccentricity ratio
The fundamental behavior of a rectangular CFST slender beam-column under axial load and uniaxial bending is influenced by theloading eccentricity ratio (e/D). To investigate this effect, a beam-columnwith a cross-section of 600×700 mm and loading eccentricityranged from 0.1 to 0.2, 0.4 and 0.6 were analyzed using the numericalmodel. The depth-to-thickness ratio of the section was 60. The slen-derness ratio (L/r) of the beam-column was 30. The initial geometricimperfection of the beam-column at the mid-height was assumed tobe L/1500. The yield and tensile strengths of the steel tubes were690 MPa and 790 MPa respectively. The beam-column was filled
D/t=40
D/t=60D/t=80
D/t=100
0
5000
10000
15000
20000
25000
30000
35000
0 20 40 60 80 100 120 140 160 180
Axi
al lo
ad (
kN)
Mid-height deflection um (mm)
Fig. 9. Influence of D/t ratio on the axial load–deflection curves for thin-walled CFSTslender beam-columns.
86
Moment Mn/Mo
D/t=40
D/t=60
D/t=80
D/t=100
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Axi
al lo
ad P
n/P o
a
Fig. 10. Influence of D/t ratio on the axial load–moment interaction diagrams for thin-walled CFST slender beam-columns.
374 V.I. Patel et al. / Journal of Constructional Steel Research 70 (2012) 368–376
with high strength concrete of 90 MPa. The Young's modulus of200 GPa was specified for the steel tubes in the analysis.
The computed axial load–deflection curves for high strength rect-angular CFST slender beam-columns with different loading eccentric-ity ratios are presented in Fig. 11. It can be seen that the initialstiffness of the slender beam-columns decreases with an increase inthe e/D ratio. In addition, increasing the e/D ratio significantly reducesthe ultimate loads of the beam-columns. This is attributed to the factthat increasing the e/D ratio increases the bending moments at thecolumn ends which significantly reduce the ultimate axial load ofthe slender beam-columns. Furthermore, the mid-height deflectionsat the maximum axial load of the slender beam-column increasewith an increase in the eccentricity ratio. The displacement ductilityof a slender beam-column is shown to be improved when increasingthe eccentricity ratio. Fig. 12 shows that the ultimate axial load is afunction of the loading eccentric ratio. The figure clearly shows thatthe effect of the loading eccentricity ratio on the ultimate axialstrengths of CFST slender bean-columns is significant.
Mid-height deflection um (mm)
Axi
al lo
ad (
kN)
e/D=0.1
e/D=0.2
e/D=0.4
e/D=0.6
0
5000
10000
15000
20000
25000
30000
35000
0 50 100 150 200 250
Fig. 11. Influence of loading eccentricity ratio on the axial load–deflection curves forthin-walled CFST slender beam-columns.
3.5. Influences of concrete compressive strengths
The effects of concrete compressive strengths on the strengths andbehavior of high strength CFST slender beam-columns with localbuckling effects were studied by the numerical model. In the para-metric study, the concrete compressive strength was varied from60 MPa to 100 MPa. The width and depth of the steel tube were600 mm and 800 mm respectively. The steel tube wall was 10 mmthick so that its depth-to-thickness ratio (D/t) was 80. The slendernessratio (L/r) of the beam-columnwas 32. The initial geometric imperfec-tion of the beam-column at the mid-height was taken as L/1500. Theloading eccentricity ratio (e/D) was 0.1. The steel yield and ultimatetensile strengths were 690 MPa and 790 MPa respectively and theYoung's modulus of the steel tubes was 200 GPa.
The axial load–deflection curves for high strength rectangularCFST slender beam-columns with different concrete strengths aredepicted in Fig. 13. It would appear from Fig. 13 that the ultimateaxial strengths of rectangular CFST slender beam-columns increasesignificantly with an increase in the concrete compressive strength.Increasing the concrete compressive strength from 60 MPa to80 MPa and 100 MPa increases the ultimate axial strength by18.13%and 35.84%, respectively. It can be seen from Fig. 13 that increasingthe concrete compressive strength results in a slight increase in theinitial stiffness of the slender beam-columns.
The normalized axial load–moment interaction diagrams for rect-angular CFST slender beam-columns are given in Fig. 14. It can be ob-served that increasing the concrete compressive strength enlarges theP–M interaction curves. The ultimate bending moment of the slenderbeam-columns is found to increase considerably with increasing theconcrete compressive strength. By increasing the concrete compres-sive strength from 60 MPa to 80 MPa and 100 MPa, the ultimatebending moment of the column is increased by 7.23% and 13.87%, re-spectively. The ultimate pure bending strengths of rectangular CFSTbeam-columns are also found to increase with an increase in the com-pressive strength of the in-filled concrete.
3.6. Influences of steel yield strengths
Rectangular thin-walled CFST slender beam-columns with differ-ent steel yield strengths and a cross-section of 700×800 mm wereanalyzed using the numerical model. The depth-to-thickness (D/t)ratio of the section was 70. The yield strengths of the steel tubeswere 500 MPa, 600 MPa and 690 MPa and the corresponding tensilestrengths were 590 MPa, 690 MPa and 790 MPa, respectively. The
0
0.2
0.4
0.6
0.8
1
1.2
0 0.5 1 1.5 2
Eccentricity ratio e/D
Axi
al lo
ad P
n/P o
a
Fig. 12. Influence of loading eccentricity ratio on the ultimate axial loads of thin-walledCFST slender beam-columns.
87
Mid-height deflection um (mm)
Axi
al lo
ad (
kN)
fc'=60 MPa
fc'=80 MPa
fc'=100 MPa
0
5000
10000
15000
20000
25000
30000
35000
40000
45000
0 20 40 60 80 100 120 140 160
Fig. 13. Influence of concrete compressive strengths on the axial load–deflection curvesfor thin-walled CFST slender beam-columns.
Mid-height deflection um (mm)
fsy=500MPafsy=600MPafsy=690MPa
0
5000
10000
15000
20000
25000
30000
35000
40000
45000
0 30 60 90 120 150 180 210 240 270 300
Axi
al lo
ad (
kN)
Fig. 15. Influence of steel yield strengths on the axial load–deflection curves for thin-walled CFST slender beam-columns.
375V.I. Patel et al. / Journal of Constructional Steel Research 70 (2012) 368–376
column slenderness ratio (L/r) of 45 was used in the parametric studywith a loading eccentricity ratio (e/D) of 0.1. The initial geometric im-perfection of the beam-column at themid-height was taken as L/1500.The Young's modulus of steel was 200 GPa. The steel tubes were filledwith 100 MPa concrete.
Fig. 15 illustrates the influences of steel yield strengths on theaxial load–deflection curves for high strength rectangular CFST slen-der beam-columns. It can be observed from Fig. 15 that the steelyield strength does not have an effect on the initial stiffness of thebeam-columns. However, the ultimate axial strength of slenderbeam-columns is found to increase significantly with an increase inthe steel yield strength. By increasing the steel yield strength from500 MPa to 600 MPa and 690 MPa, the ultimate axial load of the slen-der beam-column is found to increase by 4.92% and 8.95%respectively.
The normalized axial load–moment interaction diagrams for rect-angular CFST beam-columns made of different strength steel tubesare presented in Fig. 16. The figure shows that the normalized axialload–moment interaction curve is enlarged by reducing the steelyield strength. In addition, the ultimate pure bending strength ofthe slender beam-column increases by 13.61% and 28.07%, when
Moment Mn/Mo
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Axi
al lo
ad P
n/P o
a
fc'=60 MPa
fc'=80 MPa
fc'=100 MPa
Fig. 14. Influence of concrete compressive strengths on the axial load–moment interac-tion diagrams for thin-walled CFST slender beam-columns.
increasing the steel yield strength from 500 MPa to 600 MPa and690 MPa, respectively. The ultimate pure bending strength of theslender beam-columns increases significantly with increasing thesteel yield strength. When increasing the steel yield strength from500 MPa to 600 MPa and 690 MPa, the maximum ultimate bendingmoment of the slender beam-columns is increased by 13.15% and28.35% respectively.
4. Conclusions
The verification and applications of a numerical model developedfor the nonlinear analysis of high strength thin-walled rectangularCFST slender beam-columns with local buckling effects have beenpresented in this paper. The numerical model was verified by com-parisons of computational solutions with experimental results of nor-mal and high strength CFST slender beam-columns presented byindependent researchers. The numerical model can accurately predictthe axial load–deflection responses and strength envelopes of thin-walled CFST slender beam-columns. This paper has provided newbenchmark numerical results on the behavior of full-scale highstrength thin-walled steel slender tubes filled with high strength con-crete with various parameters including the effects of local buckling,
Moment Mn/Mo
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Axi
al lo
ad P
n/P o
a
fsy=500MPa
fsy=600MPa
fsy=690MPa
Fig. 16. Influence of steel yield strengths on the axial load–moment interaction dia-grams for thin-walled CFST slender beam-columns.
88
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column slenderness ratio, depth-to-thickness ratio, loading eccentric-ity ratio, concrete compressive strengths and steel yield strengths.The numerical results presented can be used to validate other non-linear analysis techniques and to develop composite design codesfor high strength thin-walled rectangular CFST slender beam-columns. The numerical model can be used to analyze and designthin-walled CFST slender columns in practice.
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Qing Quan Liang a,⁎, Vipulkumar Ishvarbhai Patel a, Muhammad N.S. Hadi b
a School of Engineering and Science, Victoria University, PO Box 14428, Melbourne, VIC 8001, Australiab School of Civil, Mining and Environmental Engineering, University of Wollongong, Wollongong, NSW 2522, Australia
The steel tube walls of a biaxially loaded thin-walled rectangular concrete-filled steel tubular (CFST) slenderbeam-column may be subjected to compressive stress gradients. Local buckling of the steel tube walls understress gradients, which significantly reduces the stiffness and strength of a CFST beam-column, needs to beconsidered in the inelastic analysis of the slender beam-column. Existing numerical models that do notconsider local buckling effects may overestimate the ultimate strengths of thin-walled CFST slender beam-columns under biaxial loads. This paper presents a newmultiscale numerical model for simulating the structuralperformance of biaxially loaded high-strength rectangular CFST slender beam-columns accounting for progressivelocal buckling, initial geometric imperfections, high strength materials and second order effects. The inelasticbehavior of column cross-sections is modeled at the mesoscale level using the accurate fiber element method.Macroscale models are developed to simulate the load-deflection responses and strength envelopes of thin-walled CFST slender beam-columns. New computational algorithms based on the Müller's method are developedto iteratively adjust the depth and orientation of the neutral axis and the curvature at the column's ends to obtainnonlinear solutions. Steel and concrete contribution ratios and strength reduction factor are proposed for evaluat-ing the performance of CFST slender beam-columns. Computational algorithms developed are shown to be anaccurate and efficient computer simulation and design tool for biaxially loaded high-strength thin-walled CFSTslender beam-columns. The verification of the multiscale numerical model and parametric study are presentedin a companion paper.
High strength thin-walled rectangular concrete-filled steel tubular(CFST) slender beam-columns in composite frames may be subjectedto axial load and biaxial bending. Biaxially loaded thin-walled CFSTslender beam-columnswith large depth-to-thickness ratios are vulner-able to local and global buckling. No numericalmodels have been devel-oped for the multiscale inelastic stability analysis of biaxially loadedhigh strength thin-walled CFST slender beam-columns accounting forthe effects of progressive local buckling of the steel tube walls understress gradients. The difficulty is caused by the interaction betweenlocal and global buckling and biaxial bending. However, it is importantto accurately predict the ultimate strength of a thin-walled CFST slenderbeam-column under biaxial loads because this strength is needed in thepractical design. This paper addresses the important issue of multiscalesimulation of high strength thin-walled rectangular CFST slender beam-columns under combined axial load and biaxial bending.
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Extensive experimental investigations have been undertaken to de-termine the ultimate strengths of short and slender CFST columnsunder axial load or combined axial load and uniaxial bending [1–9].Test results indicated that the confinement provided by the rectangularsteel tube had little effect on the compressive strength of the concretecore but considerably improved its ductility. In addition, local bucklingof the steel tubes was found to remarkably reduce the ultimate strengthand stiffness of thin-walled CFST short columns as reported by Ge andUsami [10], Bridge and O'Shea [11], Uy [12] and Han [13]. As a result,the ultimate strengths of rectangular CFST short columns can be deter-mined by summation of the capacities of the steel tube and concretecore, providing that local buckling effects are taken into account asshown by Liang et al. [14]. Moreover, experimental results demonstratedthat the confinement effect significantly increased the compressivestrength and ductility of the concrete core in circular CFST short columns.However, this confinement effect was found to reduce with increasingthe column slenderness as illustrated by Knowles and Park [2] andLiang [15]. In comparisons with researches on CFST columns underaxial load and uniaxial bending, experimental investigations on biaxiallyloaded rectangular thin-walled CFST slender beam-columns have re-ceived little attention [16–18].
Fig. 1. Fiber element discretization and strain distribution of CFST beam-columnsection.
65Q.Q. Liang et al. / Journal of Constructional Steel Research 75 (2012) 64–71
Although the performance of CFST columns could be determinedby experiments, they are highly expensive and time consuming. Toovercome this limitation, nonlinear analysis techniques have beendeveloped by researchers for composite columns under axial load orcombined axial load and uniaxial bending [19–23]. However, only afew numerical models have been developed to predict the nonlinearinelastic behavior of slender composite columns under biaxial bend-ing. El-Tawil et al. [24] and El-Tawil and Deierlein [25] proposed afiber element model for determining the inelastic moment-curvatureresponses and strength envelopes of concrete-encased composite col-umns under biaxial bending. The fiber model, which accounted for con-crete conferment effects and initial stresses caused by preloads, was usedto investigate the strength and ductility of concrete-encased compositecolumns. A fiber element model was also developed by Muñoz andHsu [26] that was capable of simulating the behavior of biaxially loadedconcrete-encased slender composite columns. The relationship be-tween the curvature and deflection was established by using the finitedifferent method. The incremental deflection approach was employedto capture the post-peak behavior of slender concrete-encased compos-ite columns.
Lakshmi and Shanmugam [27] presented a semi-analytical modelfor predicting the ultimate strengths of CFST slender beam-columnsunder biaxial bending. An incremental-iterative numerical schemebased on the generalized displacement control methodwas employedin themodel to solve nonlinear equilibrium equations. Extensive com-parisons of computer solutions with test results were made to exam-ine the accuracy of the semi-analytical model. However, the effectsof local buckling and concrete tensile strength were not taken intoaccount in the semi-analytical model that may overestimate the ulti-mate strengths of thin-walled rectangular CFST columns with largedepth-to-thickness ratios. Recently, Liang [28,29] developed a numer-ical model based on the fiber element method for simulating the in-elastic load-strain and moment-curvature responses and strengthenvelopes of thin-walled CFST short beam-columns under axial loadand biaxial bending. The effects of local bucklingwere taken into accountin the numerical model by using effective width formulas proposed byLiang et al. [14]. Secant method algorithms were developed to obtainnonlinear solutions. Liang [29] reported that the numerical model wasshown to be an accurate and efficient computer simulation tool for biax-ially loaded thin-walled normal and high strength CFST short columnswith large depth-to-thickness ratios.
This paper extends the numerical models developed by Liang[21,28] and Patel et al. [22,23] to biaxially loaded high-strength rectan-gular CFST slender beam-columns with large depth-to-thickness ratios.The mesoscale model is described that determines the inelastic behaviorof column cross-sections incorporating progressive local buckling. Mac-roscale models are established for simulating the load-deflection re-sponses and strength envelopes of slender beam-columns under biaxialbending. New computational algorithms based on the Müller's methodare developed to obtain nonlinear solutions. Steel and concrete contribu-tion ratios and strength reduction factor are proposed for CFST slenderbeam-columns. The verification of the numerical model developed andits applications are given in a companion paper [30].
2. Mesoscale simulation
2.1. Fiber element model
The mesoscale model is developed by utilizing the accurate fiberelement method [28] to simulate the inelastic behavior of compositecross-sections under combined axial load and biaxial bending. Therectangular CFST beam-column section is discretized into fine fiberelements as depicted in Fig. 1. Each fiber element can be assigned eithersteel or concrete material properties. Fiber stresses are calculated fromfiber strains using the material uniaxial stress–strain relationships.
2.2. Fiber strains in biaxial bending
It is assumed that plane section remains plane under deformation.This results in a linear strain distribution throughout the depth of thesection. In the numerical model, the compressive strain is taken aspositive while the tensile strain is taken as negative. Fiber strains inbiaxial bending depend on the depth (dn) and orientation (θ) of theneutral axis of the section as illustrated in Fig. 1. For 0°≤θb90°, con-crete and steel fiber strains can be calculated by the following equationsproposed by Laing [28]:
yn;i ¼ xi−B2
�������� tanθþ D
2− dn
cosθ
� �ð1Þ
εi ¼ϕ yi−yn;i�� �� cosθ for yi ≥ yn;i
−ϕ yi−yn;i�� �� cosθ for yi b yn;i
�ð2Þ
in which B and D are the width and depth of the rectangular columnsection respectively, xi and yi are the coordinates of fiber i and εi is thestrain at the ith fiber element and yn, i is the distance from the centroidof each fiber to the neutral axis.
When θ=90°, the beam-column is subjected to uniaxial bendingand fiber strains can be calculated by the following equations givenby Liang [28]:
εi ¼ϕ xi−
B2−dn
� ��������� for xi ≥ xn;i
−ϕ xi−B2−dn
� ��������� for xi b xn;i
8>><>>: ð3Þ
where xn, i is the distance from the centroid of each fiber element tothe neutral axis.
2.3. Stresses in concrete fibers
Stresses in concrete fibers are calculated from the uniaxial stress–strain relationship of concrete. A general stress–strain curve for con-crete in rectangular CFST columns is shown in Fig. 2. The stress–straincurve accounts for the effect of confinement provided by the steeltube, which improves the ductility of the concrete core in a rectangular
112
C D
A B
εc0.0150.005εccεtc
'
fct
ε tu
fcc'
βc f
cc
'
c
o
σ
Fig. 2. Stress–strain curve for confined concrete in rectangular CFST columns.
Fig. 3. Stress–strain curves for structural steels.
66 Q.Q. Liang et al. / Journal of Constructional Steel Research 75 (2012) 64–71
CFST column. The concrete stress from O to A in the stress–strain curveis calculated based on the equations given by Mander et al. [31] as:
σ c ¼f′ccλ
εcε ′cc
� �λ−1þ εc
ε′cc
� �λ ð4Þ
λ ¼ EcEc− f′cc
ε′cc
� � ð5Þ
Ec ¼ 3320ffiffiffiffiffif′cc
qþ 6900 MPað Þ ð6Þ
in which σc stands for the compressive concrete stress, f′cc representsthe effective compressive strength of concrete, εc denotes the compres-sive concrete strain, ε′cc is the strain at f′cc and is between0.002 and 0.003depending on the effective compressive strength of concrete [28]. TheYoung's modulus of concrete Ec was given by ACI [32]. The effectivecompressive strength of concrete f′cc is taken as γcf′c, where γc is thestrength reduction factor proposed by Liang [28] to account for the col-umn size effect and is expressed by
γc ¼ 1:85D−0:135c 0:85≤ γc ≤ 1:0ð Þ ð7Þ
where Dc is taken as the larger of (B−2t) and (D−2t) for a rectangularcross-section, and t is the thickness of the steel tube wall as shown inFig. 1.
The parts AB, BC and CD of the stress–strain curve for concreteshown in Fig. 2 are defined by the following equations proposed byLiang [28]:
σ c ¼f′cc for ε ′cc b εc ≤ 0:005
βcf′cc þ 100 0:015−εcð Þ f′cc−βcf′ccð Þ for 0:005 b εc ≤ 0:015βcf′cc for εc > 0:015
8<: ð8Þ
where βc was proposed by Liang [28] based on experimental resultsprovided by Tomii and Sakino [33] to account for confinement effectson the post-peak behavior and is given by
βc ¼1:0 for
Bs
t≤ 24
1:5− 148
Bs
tfor24 b
Bs
t≤ 48
0:5 forBs
t> 48
8>>>>><>>>>>:
ð9Þ
where Bs is taken as the larger of B and D for a rectangular cross-section.
The stress–strain curve for concrete in tension is shown in Fig. 2.The constitutive model assumes that the concrete tensile stress in-creases linearly with the tensile strain up to concrete cracking. After
concrete cracking, the tensile stress of concrete decreases linearly tozero as the concrete softens. The concrete tensile stress is consideredto be zero at the ultimate tensile strain which is taken as 10 times ofthe strain at concrete cracking. The tensile strength of concrete istaken as 0:6
ffiffiffiffiffif′cc
p.
2.4. Stresses in steel fibers
Stresses in steel fibers are calculated from uniaxial stress–strainrelationship of steel material. Steel tubes used in CFST cross-sectionsare normally made from three types of structural steels such as highstrength structural steels, cold-formed steels and mild structuralsteels, which are considered in the numerical model. Fig. 3 showsthe stress–strain relationship for three types of steels. The steel materialgenerally follows the same stress–strain relationship under the com-pression and tension. The rounded part of the stress–strain curve canbe defined by the equation proposed by Liang [28]. The hardening strainεst is assumed to be 0.005 for high strength and cold-formed steels and10εsy for mild structure steels in the numerical model. The ultimatestrain εsu is taken as 0.2 for steels.
2.5. Initial local buckling
Local buckling significantly reduces the strength and stiffness ofthin-walled CFST beam-columns with large depth-to-thickness ratios.Therefore, it is important to account for local buckling effects in theinelastic analysis of high strength CFST slender beam-columns. How-ever, most of existing numerical models for thin-walled CFST beam-columns have not considered local buckling effects. Thismay be attrib-uted to the complexity of the local instability problem as addressed byLiang et al. [14]. The steel tube walls of a CFST column under axial loadand biaxial bending may be subjected to compressive stress gradientsas depicted in Fig. 4. Due to the presence of initial geometric imperfec-tions, no bifurcation point can be observed on the load-deflectioncurves for real thin steel plates. The classical elastic local buckling the-ory [34] cannot be used to determine the initial local buckling stress ofreal steel plates with imperfections. Liang et al. [14] proposed formulasfor estimating the initial local buckling stresses of thin steel plates understress gradients by considering the effects of geometric imperfectionsand residual stresses. Their formulas are incorporated in the numericalmodel to account for initial local buckling of biaxially loaded CFSTbeam-columns with large depth-to-thickness ratios.
2.6. Post-local buckling
The effective width concept is commonly used to describe thepost-local buckling behavior of a thin steel plate as illustrated in
113
D
N.A.
x
y
b
b
Pn
e,1
e,2
1
2
t
θ
B
Fig. 4. Effective and ineffective areas of steel tubular cross-section under axial load andbiaxial bending.
67Q.Q. Liang et al. / Journal of Constructional Steel Research 75 (2012) 64–71
Fig. 4. Liang et al. [14] proposed effective width and strength formulasfor determining the post-local buckling strengths of the steel tubewalls of thin-walled CFST beam-columns under axial load and biaxialbending. Their formulas are incorporated in the numerical model toaccount for the post-local buckling effects of the steel tube wallsunder compressive stress gradients. The effective widths be1 and be2of a steel plate under stress gradients as shown in Fig. 4 are givenby Liang et al. [14] as
be1b
¼0:2777þ 0:01019
bt
� �−1:972� 10−4 b
t
� �2
þ 9:605� 10−7 bt
� �3
forαs > 0:0
0:4186−0:002047bt
� �þ 5:355� 10−5 b
t
� �2−4:685� 10−7 b
t
� �3forαs ¼ 0:0
8>>><>>>:
ð10Þ
be2b
¼ 2−αsð Þ be1b
ð11Þ
in which b is the clear width of a steel flange or web of a CFST columnsection, and the stress gradient coefficient αs=σ2/σ1, where σ2 is theminimum edge stress acting on the plate and σ1 is the maximum edgestress acting on the plate.
Liang et al. [14] suggested that the effective width of a steel platein the nonlinear analysis can be calculated based on the maximumstress level within the steel plate using the linear interpolationmethod.The effectivewidth concept implies that a steel plate attains its ultimatestrength when the maximum edge stress acting on the plate reaches itsyield strength. Stresses in steel fiber elements within the ineffectiveareas as shown in Fig. 4 are taken as zero after themaximumedge stressσ1 reaches the initial local buckling stress σ1c for a steel plate with a b/tratio greater than 30. If the total effective width of a plate (be1+be2) isgreater than its width (b), the effective strength formulas proposed byLiang et al. [14] are employed in the numerical model to determinethe ultimate strength of the tube walls.
2.7. Stress resultants
The internal axial force and bending moments acting on a CFSTbeam-column section under axial load and biaxial bending are deter-mined as stress resultants in the section as follows:
P ¼Xnsi¼1
σ s;iAs;i þXncj¼1
σ c;jAc;j ð12Þ
Mx ¼Xnsi¼1
σ s;iAs;iyi þXncj¼1
σ c;jAc;jyj ð13Þ
My ¼Xnsi¼1
σ s;iAs;ixi þXncj¼1
σ c;jAc;jxj ð14Þ
in which P stands for the axial force, Mx and My are the bending mo-ments about the x and y axes, σs, i denotes the stress of steel fiber i, As, i
represents the area of steel fiber i, σc, j is the stress of concrete fiber j,Ac, j is the area of concrete fiber j, xi and yi are the coordinates of steelelement i, xj and yj stand for the coordinates of concrete element j, nsis the total number of steel fiber elements and nc is the total numberof concrete fiber elements.
2.8. Inelastic moment-curvature responses
The inelastic moment-curvature responses of a CFST beam-columnsection can be obtained by incrementally increasing the curvature andsolving for the corresponding moment value for a given axial load (Pn)applied at a fixed load angle (α). For each curvature increment, thedepth of the neutral axis is iteratively adjusted for an initial orientationof the neutral axis (θ) until the force equilibrium condition is satisfied.The moments ofMx andMy are then computed and the equilibrium con-dition of tan α=My/Mx is checked. If this condition is not satisfied, theorientation of the neutral axis is adjusted and the above process is repeat-ed until both equilibrium conditions aremet. The effects of local bucklingare taken into account in the calculation of the stress resultants. Thedepth and orientation of the neutral axis of the section can be adjustedby using the secant method algorithms developed by Liang [28] or theMüller's method [35] algorithms which are discussed in Section 4. A de-tailed computational procedure for predicting the inelastic moment-curvature responses of composite sections was given by Liang [28].
3. Macroscale simulation
3.1. Macroscale model for simulating load-deflection responses
The pin-ended beam-column model is schematically depicted inFig. 5. It is assumed that the deflected shape of the slender beam-column is part of a sine wave. The lateral deflection of the beam-column can be described by the following displacement function:
u ¼ um sinπzL
� �ð15Þ
where L stands for the effective length of the beam-column and um isthe lateral deflection at the mid-height of the beam-column.
The curvature at themid-height of the beam-column can be obtainedas
ϕm ¼ πL
� �2um ð16Þ
For a beam-column subjected to an axial load at an eccentricity ofe as depicted in Fig. 5 and an initial geometric imperfection uo at the
Fig. 6. Computer flowchart for predicting the axial load-deflection responses of thin-walled CFST slender beam-columns under biaxial loads.
68 Q.Q. Liang et al. / Journal of Constructional Steel Research 75 (2012) 64–71
mid-height of the beam-column, the external moment at the mid-height of the beam-column can be calculated by
Mme ¼ P eþ um þ uoð Þ ð17Þ
To capture the complete load-deflection curve for a CFST slenderbeam-column under biaxial loads, the deflection control method isused in the numerical model. In the analysis, the deflection at themid-height um of the slender beam-column is gradually increased.The curvature ϕm at the mid-height of the beam-column can be calcu-lated from the deflection um. For this curvature, the neutral axis depthand orientation are adjusted to achieve the moment equilibrium atthe mid-height of the beam-column. The equilibrium state for biaxialbending requires that the following equations must be satisfied:
P eþ um þ uoð Þ−Mmi ¼ 0 ð18Þ
tanα−My
Mx¼ 0 ð19Þ
in which Mmi is the resultant internal moment which is calculated as
The macroscale model incorporating the mesoscale model isimplemented by a computational procedure. A computer flowchartis shown in Fig. 6 to implicitly demonstrate the computational proce-dure for load-deflection responses. The main steps of the computa-tional procedure are described as follows:
(1) Input data.(2) Discretize the composite section into fine fiber elements.(3) Initialize themid-height deflection of the beam-column um=Δum.(4) Calculate the curvatureϕm at themid-height of the beam-column.(5) Adjust the depth of the neutral axis (dn) using the Müller's
method.(6) Compute stress resultants P andMmi considering local buckling.
(7) Compute the residual moment rma =Mme−Mmi.(8) Repeat Steps (5)–(7) until |rma |bεk.(9) Compute bending moments Mx and My.
(10) Adjust the orientation of the neutral axis (θ) using the Müller'smethod.
(11) Calculate the residual moment rbm ¼ tanα−My
Mx.
(12) Repeat Steps (5)–(11) until |rmb |bεk.
115
mm φφ Δ=
0=n
oa
P
mmm φφφ Δ+=
nnn PPP Δ+=
?kc
mr ε<
?kb
mr ε<
?ka
mr ε<
eφ
n
m
d
cmr
amr
eM miM
miM
bmr
θ
xM yM
n
n oa
n
n
Fig. 7. Computer flowchart for simulating the strength envelops of thin-walled CFSTslender beam-columns under biaxial loads.
69Q.Q. Liang et al. / Journal of Constructional Steel Research 75 (2012) 64–71
(13) Increase the deflection at the mid-height of the beam-columnby um=um+Δum.
(14) Repeat Steps (4)–(13) until the ultimate axial load Pn isobtained or the deflection limit is reached.
(15) Plot the load-deflection curve.
In the above procedure, εk is the convergence tolerance and takenas 10−4 in the numerical analysis.
3.2. Macroscale model for simulating strength envelopes
In design practice, it is required to check for the design capacitiesof CFST slender beam-columns under design actions such as the de-sign axial force and bending moments, which have been determinedfrom structural analysis. For this design purpose, the axial load-momentstrength interaction curves (strength envelopes) need to be developedfor the beam-columns. For a given axial load applied (Pn) at a fixed loadangle (α), the ultimate bending strength of a slender beam-column is de-termined as the maximum moment that can be applied to the columnends. The moment equilibrium is maintained at the mid-height of thebeam-column. The external moment at the mid-height of the slenderbeam-column is given by
Mme ¼ Me þ Pn um þ uoð Þ ð20Þ
inwhichMe is themoment at the column ends. The deflection at themid-height of the slender beam-column can be calculated from the curvatureas
um ¼ Lπ
� �2ϕm ð21Þ
To generate the strength envelope, the curvature (ϕm) at the mid-height of the beam-column is gradually increased. For each curvatureincrement, the corresponding internal moment capacity (Mmi) is com-puted by the inelastic moment-curvature responses discussed inSection 2.8. The curvature at the column ends (ϕe) is adjusted andthe corresponding moment at the column ends is calculated untilthe maximum moment at the column ends is obtained. The axialload is increased and the strength envelope can be generated by re-peating the above process. For a CFST slender beam-column undercombined axial load and bending, the following equilibrium equationsmust be satisfied:
Pn−P ¼ 0 ð22Þ
tan α−My
Mx¼ 0 ð23Þ
Me þ Pn um þ uoð Þ−Mmi ¼ 0 ð24Þ
Fig. 7 shows a computer flowchart that implicitly illustrates thecomputational procedure for developing the strength envelope. Themain steps of the computational procedure are described as follows:
(1) Input data.(2) Discretize the composite section into fine fiber elements.(3) The load-deflection analysis procedure is used to compute the
ultimate axial load Poa of the axially loaded slender beam-column with local buckling effects.
(4) Initialize the applied axial load Pn=0.(5) Initialize the curvature at the mid-height of the beam-column
ϕm=Δϕm.(6) Compute the mid-height deflection um from the curvature ϕm.(7) Adjust the depth of the neutral axis (dn) using the Müller's
method.(8) Calculate resultant force P considering local buckling.(9) Compute the residual force rm
c =Pn−P.
(10) Repeat Steps (7)–(9) until |rmc |bεk.(11) Compute bending moment Mx and My.(12) Adjust the orientation of the neutral axis (θ) using the Müller's
method.(13) Calculate the residual moment rbm ¼ tanα−My
Mx.
(14) Repeat Steps (7)–(13) until |rmb |bεk.(15) Compute the internal resultant moment Mmi.
116
70 Q.Q. Liang et al. / Journal of Constructional Steel Research 75 (2012) 64–71
(16) Adjust the curvature at the column end ϕe using the Müller'smethod.
(17) Compute the moment Me at the column ends accounting forlocal buckling effects.
(18) Compute rma =Mme−Mmi.
(19) Repeat Steps (16)–(18) until |rma |bεk.(20) Increase the curvature at the mid-height of the beam-column
by ϕm=ϕm+Δϕm.(21) Repeat Steps (6)–(20) until the ultimate bending strength
Mn(=Me max) at the column ends is obtained.(22) Increase the axial load by Pn=Pn+ΔPn, where ΔPn=Poa/10.(23) Repeat Steps (5)–(22) until the maximum load increment is
reached.(24) Plot the axial load-moment interaction diagram.
4. Numerical solution scheme
4.1. General
As discussed in the preceding sections, the depth and orientationof the neutral axis and the curvature at the column ends need to beiteratively adjusted to satisfy the force and moment equilibrium condi-tions in the inelastic analysis of a slender beam-column. For this pur-pose, computational algorithms based on the secant method havebeen developed by Liang [21,28]. Although the secant method algo-rithms are shown to be efficient and reliable for obtaining converged so-lutions, computational algorithms based on the Müller's method [35],which is a generalization of the secant method, are developed in thepresent study to determine the true depth and orientation of the neutralaxis and the curvature at the column ends.
4.2. The Müller's method
In general, the depth (dn) and orientation (θ) of the neutral axis andthe curvature (ϕe) at the column ends of a slender beam-column aredesign variables which are denoted herein by ω. The Müller's methodrequires three starting values of the design variables ω1, ω2, and ω3.The corresponding force or moment functions rm, 1, rm, 2 and rm, 3 arecalculated based on the three initial design variables. The new designvariableω4 that approaches the true value is determined by the follow-ing equations:
When adjusting the neutral axis depth and orientation, the sign ofthe square root term in the denominator of Eq. (25) is taken to be thesame as that of bm. However, this sign is taken as positive when adjust-ing the curvature at the column ends. In order to obtain converged so-lutions, the values of ω1, ω2 and ω3 and corresponding residual forcesor moments rm, 1, rm, 2 and rm, 3 need to be exchanged as discussed byPatel et al. [22]. Eq. (25) and the exchange of design variables andforce or moment functions are executed iteratively until the conver-gence criterion of |rm|bεk is satisfied.
In the numerical model, three initial values of the neutral axis depthdn, 1, dn, 3 and dn, 2 are taken as D/4, D and (dn, 1+dn, 3)/2 respectively;
the orientations of the neutral axis θ1, θ3 and θ2 are initialized to α/4,α and (θ1+θ3)/2 respectively; and the curvature at the column endsϕe, 1, ϕe, 3 and ϕe, 2 are initialized to 10−10, 10−6 and (ϕe, 1+ϕe, 3)/2respectively.
5. Performance indices for CFST slender beam-columns
Performance indices are used to evaluate the contributions of the con-crete and steel components to the ultimate strengths of CFST slenderbeam-columns and to quantify the strength reduction caused by the sec-tion and column slenderness, loading eccentricity and initial geometricimperfections. These performance indices can be used to investigate thecost effective designs of CFST slender beam-columns under biaxial loads.
5.1. Steel contribution ratio (ξs)
The steel contribution ratio is used to determine the contribution ofthe hollow steel tubular beam-column to the ultimate strength of theCFST slender beam-column under axial load and biaxial bending, whichis given by
ξs ¼Ps
Pnð29Þ
where Pn is the ultimate axial strength of the CFST slender beam-columnand Ps is the ultimate axial strength of the hollow steel tubular beam-column,which is calculated by setting the concrete compressive strengthf′c to zero in the numerical analysis while other conditions of the hollowsteel tubular beam-column remain the same as those of the CFST beam-column. The effects of local buckling are taken into account in the deter-mination of both Pn and Ps.
5.2. Concrete contribution ratio (ξc)
The concrete contribution ratio quantifies the contribution of theconcrete component to the ultimate axial strength of a CFST slenderbeam-column. The slender concrete core beam-column without rein-forcement carries very low loading and does not represents the concretecore in a CFST slender beam-column. Portolés et al. [9] used the capacityof the hollow steel tubular beam-column to define the concrete contri-bution ratio (CCR), which is given by
CCR ¼ Pn
Psð30Þ
Eq. (30) is an inverse of the steel contribution ratio andmay not accu-rately quantify the concrete contribution. To evaluate the contribution ofthe concrete component to the ultimate axial strength of a CFST slenderbeam-column, a new concrete contribution ratio is proposed as
ξc ¼Pn−Ps
Pnð31Þ
It can be seen from Eq. (31) that the concrete contribution to theultimate axial strength of a CFST slender beam-column is the differencebetween the ultimate axial strength of the CFST column and that of thehollow steel column.
5.3. Strength reduction factor (αc)
The ultimate axial strength of a CFST short column under axialloading is reduced by increasing the section and column slenderness,loading eccentricity, and initial geometric imperfections. To reflect onthese effects, the strength reduction factor is defined as
αc ¼Pn
Poð32Þ
117
71Q.Q. Liang et al. / Journal of Constructional Steel Research 75 (2012) 64–71
where Po is the ultimate axial strength of the column cross-sectionunder axial compression. The ultimate axial strengths of Pn and Poare determined by considering the effects of local buckling of thesteel tubes.
6. Conclusions
This paper has presented a new multiscale numerical model forthe nonlinear inelastic analysis of high strength thin-walled rectangularCFST slender beam-columns under combined axial load and biaxialbending. At the mesoscale level, the inelastic axial load-strain andmoment-curvature responses of column cross-sections subjected to bi-axial loads aremodeled using the accurate fiber elementmethod, whichaccounts for the effects of progressive local buckling of the steel tubewalls under stress gradients. Macroscale models together with compu-tational procedures have been described that simulate the axial load-deflection responses and strength envelopes of CFST slender beam-columns under biaxial bending. Initial geometric imperfections andsecond order effects between axial load and deformations are takeninto account in the macroscale models. New solution algorithms basedon the Müller's method have been developed and implemented in thenumerical model to obtain converged solutions.
The computer program that implements the multiscale numericalmodel developed is an efficient and powerful computer simulationand design tool that can be used to determine the structural performanceof biaxially loaded high strength rectangular CFST slender beam-columnsmade of compact, non-compact or slender steel sections. This overcomesthe limitations of experiments which are extremely expensive and timeconsuming. Moreover, the multiscale numerical model can be imple-mented in frame analysis programs for the nonlinear analysis of compos-ite frames. Steel and concrete contribution ratios and strength reductionfactor proposed can beused to study the optimal designs of high strengthCFST beam-columns. The verification of the numerical model and para-metric study are given in a companion paper [30].
References
[1] Furlong RW. Strength of steel-encased concrete beam-columns. J Struct Div ASCE1967;93(5):113–24.
[2] Knowles RB, Park R. Strength of concrete-filled steel tubular columns. J Struct DivASCE 1969;95(12):2565–87.
[4] Varma AH, Ricles JM, Sause R, Lu LW. Seismic behavior andmodeling of high-strengthcomposite concrete-filled steel tube (CFT) beam-columns. J Constr Steel Res 2002;58:725–58.
[5] Sakino K, Nakahara H, Morino S, Nishiyama I. Behavior of centrally loadedconcrete-filled steel-tube short columns. J Struct Eng ASCE 2004;130(2):180–8.
[6] Fujimoto T, Mukai A, Nishiyama I, Sakino K. Behavior of eccentrically loadedconcrete-filled steel tubular columns. J Struct Eng ASCE 2004;130(2):203–12.
[7] Ellobody E, Young B, Lam D. Behaviour of normal and high strength concrete-filledcompact steel tube circular stub columns. J Constr Steel Res 2006;62(7):706–15.
[8] Liu D. Behaviour of eccentrically loaded high-strength rectangular concrete-filledsteel tubular columns. J Constr Steel Res 2006;62(8):839–46.
[9] Portolés JM, Romero ML, Bonet JL, Filippou FC. Experimental study on highstrength concrete-filled circular tubular columns under eccentric loading. J ConstrSteel Res 2011;67(4):623–33.
[10] Ge HB, Usami T. Strength of concrete-filled thin-walled steel box columns: exper-iment. J Struct Eng ASCE 1992;118(11):3036–54.
[11] Bridge RQ, O'Shea MD. Behaviour of thin-walled steel box sections with or with-out internal restraint. J Constr Steel Res 1998;47(1–2):73–91.
[12] Uy B. Strength of concrete filled steel box columns incorporating local buckling. JStruct Eng ASCE 2000;126(3):341–52.
[13] Han LH. Tests on stub columns of concrete-filled RHS sections. J Constr Steel Res2002;58(3):353–72.
[14] Liang QQ, Uy B, Liew JYR. Local buckling of steel plates in concrete-filled thin-walled steel tubular beam-columns. J Constr Steel Res 2007;63(3):396–405.
[15] Liang QQ. High strength circular concrete-filled steel tubular slender beam-columns, part II: fundamental behavior. J Constr Steel Res 2011;67(2):172–80.
[16] Bridge RQ. Concrete filled steel tubular columns. School of Civil Engineering, TheUniversity of Sydney, Sydney, Australia, Research Report No. R283; 1976.
[17] Shakir-Khalil H, Zeghiche J. Experimental behaviour of concrete-filled rolledrectangular hollow-section columns. Struct Eng 1989;67(19):346–53.
[18] Shakir-Khalil H, Mouli M. Further tests on concrete-filled rectangular hollow-section columns. Struct Eng 1990;68(20):405–13.
[19] Vrcelj Z, Uy B. Strength of slender concrete-filled steel box columns incorporatinglocal buckling. J Constr Steel Res 2002;58(2):275–300.
[20] Hu HT, Huang CS, Wu MH, Wu YM. Nonlinear analysis of axially loaded concrete-filled tube columns with confinement effect. J Struct Eng ASCE 2003;129(10):1322–9.
[21] Liang QQ. High strength circular concrete-filled steel tubular slender beam-columns, part I: numerical analysis. J Constr Steel Res 2011;67(2):164–71.
[22] Patel VI, Liang QQ, Hadi MNS. High strength thin-walled rectangular concrete-filledsteel tubular slender beam-columns, part I: modeling. J Constr Steel Res 2012;70:377–84.
[23] Patel VI, Liang QQ, Hadi MNS. High strength thin-walled rectangular concrete-filled steel tubular slender beam-columns, part II: behavior. J Constr Steel Res2012;70:368–76.
[24] El-Tawil S, Sanz-Picon CF, Deierlein GG. Evaluation of ACI 318 and AISC (LRFD)strength provisions for composite beam-columns. J Constr Steel Res 1995;34(1):103–23.
[25] El-Tawil S, Deierlein GG. Strength and ductility of concrete encased compositecolumn. J Struct Eng 1999;125(9):1009–19.
[26] Muñoz PR, Hsu CTT. Behavior of biaxially loaded concrete-encased compositecolumns. J Struct Eng ASCE 1997;123(9):1163–71.
[27] Lakshmi B, Shanmugam NE. Nonlinear analysis of in-filled steel-concrete compos-ite columns. J Struct Eng ASCE 2002;128(7):922–33.
[28] Liang QQ. Performance-based analysis of concrete-filled steel tubular beam-columns, part I: theory and algorithms. J Constr Steel Res 2009;65(2):363–72.
[29] Liang QQ. Performance-based analysis of concrete-filled steel tubular beam-columns, part II: verification and applications. J Constr Steel Res 2009;65(2):351–62.
[30] Patel VI, Liang QQ, Hadi MNS. Biaxially loaded high-strength concrete-filled steeltubular slender beam-columns, part II: parametric study. J Constr Steel Ressubmitted for publication.
[31] Mander JB, Priestley MJN, Park R. Theoretical stress–strain model for confinedconcrete. J Struct Eng ASCE 1988;114(8):1804–26.
[32] ACI-318. Building code requirements for reinforced concrete. Detroit, MI: ACI;2002.
[33] Tomii M, Sakino K. Elastic–plastic behavior of concrete filled square steel tubularbeam-columns. Trans Archit Inst Japn 1979;280:111–20.
[34] Bulson PS. The stability of flat plates. London: Chatto and Windus; 1970.[35] Müller DE. A method for solving algebraic equations using an automatic computer.
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118
Chapter 4: Rectangular CFST Slender Beam-columns under Axial Load and Biaxial Bending
V. I. Patel: Nonlinear Inelastic Analysis of Concrete-Filled Steel Tubular Slender Beam-Columns 119
Tubular Slender Beam-Columns, Part II: Parametric Study
Vipulkumar Ishvarbhai Patela, Qing Quan Liang
a,*, Muhammad N. S. Hadi
b
aSchool of Engineering and Science, Victoria University, PO Box 14428, Melbourne, VIC 8001, Australia
bSchool of Civil, Mining and Environmental Engineering, University of Wollongong,
Wollongong, NSW 2522, Australia
Corresponding author:
Dr. Qing Quan Liang School of Engineering and Science Victoria University PO Box 14428 Melbourne VIC 8001 Australia Phone: +61 3 9919 4134 Fax: +61 3 9919 4139 E-mail: [email protected]
Chapter 4: Rectangular CFST Slender Beam-columns under Axial Load and Biaxial Bending
V. I. Patel: Nonlinear Inelastic Analysis of Concrete-Filled Steel Tubular Slender Beam-Columns 120
Biaxially loaded high-strength concrete-filled steel tubular slender beam-columns, Part II: Parametric study
Vipulkumar Ishvarbhai Patela, Qing Quan Lianga,, Muhammad N. S. Hadib
a School of Engineering and Science, Victoria University, PO Box 14428, Melbourne,
VIC 8001, Australia b School of Civil, Mining and Environmental Engineering, University of Wollongong,
Wollongong, NSW 2522, Australia
ABSTRACT
Biaxially loaded high strength rectangular concrete-filled steel tubular (CFST) slender
beam-columns with large depth-to-thickness ratios, which may undergo local and global
interaction buckling, have received very little attention. This paper presents the
verification of a multiscale numerical model described in a companion paper and an
extensive parametric study on the performance of high strength thin-walled rectangular
CFST slender beam-columns under biaxial loads. Comparisons of computer solutions
with existing experimental results are made to examine the accuracy of the multiscale
numerical model developed. The effects of the concrete compressive strength, loading
eccentricity, depth-to-thickness ratio and columns slenderness on the ultimate axial
strength, steel contribution ratio, concrete contribution ratio and strength reduction
factor of CFST slender beam-columns under biaxial bending are investigated by using
the numerical model. Comparative results demonstrate that the multiscale numerical
model is capable of accurately predicting the ultimate strength and deflection behavior
School of Civil, Mining and Environmental Engineering
University of Wollongong
Wollongong NSW 2522, Australia
Received 20 February 2012
Accepted 27 May 2012
Published 20 March 2013
This paper presents a new numerical model for the nonlinear analysis of circular concrete-¯lled
steel tubular (CFST) slender beam-columns with preload e®ects, in which the initial geometric
imperfections, de°ections caused by preloads, concrete con¯nement and second order e®ects areincorporated. Computational algorithms are developed to solve the nonlinear equilibrium
equations. Comparative studies are undertaken to validate the accuracy of computational
algorithms developed. Also included is a parametric study for examining the e®ects of the
preloads, column slenderness, diameter-to-thickness ratio, loading eccentricity, steel yield stressand concrete con¯nement on the behavior of circular CFST slender beam-columns under
eccentric loadings. The numerical model is demonstrated to be capable of predicting accurately
the behavior of circular CFST slender beam-columns with preloads. The preloads on the steel
tubes can a®ect signi¯cantly the behavior of CFST slender beam-columns and must be takeninto account in the design.
Keywords: Concrete-¯lled steel tubes; high strength; nonlinear analysis; preloads; slender
beam-columns.
1. Introduction
A common construction practice of high rise composite buildings is to erect the
hollow steel tubes and composite °oors several stories ahead of ¯lling the wet con-
crete. This construction method induces preloads on the steel tubes, which causes
*Corresponding author.
International Journal of Structural Stability and DynamicsVol. 13, No. 3 (2013) 1250065 (23 pages)
#.c World Scienti¯c Publishing Company
DOI: 10.1142/S0219455412500654
1250065-1
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initial stresses and deformations and may signi¯cantly reduce the sti®ness and
ultimate strength of concrete-¯lled steel tubular (CFST) slender beam-columns.
Extensive experimental studies have been undertaken to determine the behavior of
short and slender CFST columns without preload e®ects.1�9 However, research
studies on the performance of circular CFST slender beam-columns under preloads
have been relatively limited. As a result, design speci¯cations on high strength cir-
cular CFST slender beam-columns with preloads are not given in current design
codes, such as Eurocode 4,10 LRFD11 and ACI 318-11.12 This paper addresses an
important issue on the numerical analysis of circular CFST slender beam-columns
with the preload e®ects.
Only limited tests on the behavior of CFST slender beam-columns with preloads
have been conducted in the past. Zha13 performed tests on 23 eccentrically loaded
circular CFST beam-columns to study the preload e®ects on their ultimate axial
strengths. Test results indicated that the preload caused a reduction in the ultimate
axial strength of CFST beam-columns, but an increase in the deformations. The
behavior of eccentrically loaded circular CFST slender beam-columns with preloads
was investigated experimentally by Zhang et al.14 The variables considered in the
test program included the preload ratio, column slenderness and loading eccentricity.
Han and Yao15 conducted experiments to examine the e®ects of preloads on the load-
de°ection responses of square CFST beam-columns. It was found that the high
preload can result in about 20% reduction in the ultimate axial strength of CFST
slender columns under eccentric loading. Liew and Xiong16 studied the experimental
behavior of axially loaded circular CFST slender columns with preloads on the steel
tubes. Their study indicated that the preload has a pronounced e®ect on inter-
mediate and slender columns and little in°uence on short columns. The preload with
a ratio of 0.6 reduced the ultimate axial strength of circular CFST slender beam-
columns by more than 15%.
Finite element and numerical models have been developed by researchers to study
the behavior of circular CFST columns without preloads.17�24 Lakshmi and
Shanmugam25 developed a semi-analytical model for simulating the behavior of
circular and rectangular CFST slender beam-columns under biaxial bending. How-
ever, the semi-analytical model does not account for the e®ects of concrete con¯ne-
ment and preloads on the strength and ductility of circular CFST beam-columns.
Gayathri et al.26,27 presented analytical models for nonlinear analysis of CFST col-
umns under cyclic loadings. Liang28�30 proposed numerical models for predicting theinelastic behavior of CFST beam-columns without preloads. His models account for
the e®ects of local buckling on the behavior of rectangular CFST columns31�33 and ofconcrete con¯nement on the strength and ductility of circular CFST beam-columns.
Xiong and Zha34 used the ¯nite element programABAQUS to investigate the e®ects
of initial stresses on thebehavior of circularCFSTslanderbeam-columnsunder eccentric
loadings. The results obtained from their model were compared with experimental data
given by Zhang et al.14 In addition, the model was utilized to investigate the e®ects of
V. I. Patel, Q. Q. Liang & M. N. S. Hadi
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steel yield stress, concrete compressive strength, steel ratio and column slenderness ratio
on the strength reduction caused by preloads. Liew and Xiong16 also developed a ¯nite
element model using ABAQUS to examine the e®ects of preloads on the strength and
deformations of circular CFST columns under axial loading.
This paper extends the numerical models developed by Liang28,29 and Patel
et al.35,36 to the inelastic stability analysis of eccentrically loaded circular CFST
slender beam-columns with preloads. The con¯nement e®ect is incorporated in the
constitutive models for concrete and steels. Computational procedures and solution
method for determining the initial de°ections caused by preloads and the load-
de°ection curves of CFST slender beam-columns are described. Comparisons with
test results and an extensive parametric study are given.
2. Material Constitutive Models
2.1. Material model for con¯ned concrete
The con¯nement e®ect, which increases the compressive strength and ductility of
concrete in circular CFST columns, is incorporated in the constitutive model for
concrete. The general stress�strain curve for con¯ned concrete in circular CFST col-
umns suggested by Liang29 is shown in Fig. 1. The concrete stress from O to A on the
stress�strain curve is calculated based on the equations given by Mander et al.37 as
�c ¼f0cc�
"c"0cc
� ��� 1þ "c
" 0cc
� ��; ð1Þ
� ¼ Ec
Ec � f 0cc" 0cc
� � ; ð2Þ
Ec ¼ 3320ffiffiffiffiffiffiffiffiffiffi�cf
0c
pþ 6900 ðMPaÞ ð3Þ
C
A
B
εcεccεtc'
fct
εtu
fcc'
βc f
cc'
σc
o εcu
Fig. 1. Stress�strain curve for con¯ned concrete in circular CFST columns.
Numerical Analysis of Circular CFST Slender Beam-Columns with Preload E®ects
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in which �c denotes the compressive concrete stress, f0cc the e®ective compressive
strength of con¯ned concrete, f 0c the compressive strength of concrete cylinder, "c thecompressive concrete strain, " 0cc the strain at f 0cc, Ec Young's modulus of concrete
given by ACI 318-11,12 and �c is the strength reduction factor proposed by Liang28 to
account for the column section size e®ect, expressed by
�c ¼ 1:85D�0:135c ð0:85 � �c � 1:0Þ: ð4Þ
Here Dc is the diameter of the concrete core. The equations proposed by Mander
et al.37 for determining the compressive strength of con¯ned concrete were modi¯ed
by Liang and Fragomeni22 using the strength reduction factor �c as follows:
f 0cc ¼ �cf0c þ k1frp; ð5Þ
" 0cc ¼ " 0c 1þ k2frp�cf
0c
� �; ð6Þ
where frp is the lateral con¯ning pressure on the concrete and k1 and k2 are taken as 4.1
and 20.5 respectively.38 The strain " 0c is the strain at f 0c of the uncon¯ned concrete,28
" 0c ¼
0:002 for �cf0c � 28 ðMPaÞ
0:002þ �cf0c � 28
54000for 28 < �cf
0c � 82 ðMPaÞ
0:003 for < �cf0c � 82 ðMPaÞ
8>>><>>>:
: ð7Þ
Based on thework ofHu et al.17 andTang et al.,39 Liang andFragomeni22 proposed an
accurate con¯ning pressure model for normal and high strength concrete in circular
CFST columns, which is adopted in the present numerical model. Namely,
frp ¼0:7ð�e � �sÞ
2t
D� 2tfsy for
D
t� 47
0:006241� 0:0000357D
t
� �fsy for 47 <
D
t� 150
8>><>>: ð8Þ
in which D is the outer diameter and t is the thickness of the steel tube, fsy is the steel
yield strength and ve and vs are Poisson's ratios of the steel tube with and without
concrete in¯ll, respectively. Tang et al.39 suggested that Poisson's ratio vs is taken as
0.5 at the maximum strength point and ve is given by
ve ¼ 0:2312þ 0:3582v 0e � 0:1524f 0cfsy
� �þ 4:843v 0e
f 0cfsy
� �� 9:169
f 0cfsy
� �2
; ð9Þ
v 0e ¼ 0:881� 10�6D
t
� �3
� 2:58� 10�4D
t
� �2
þ 1:953� 10�2D
t
� �þ 0:4011: ð10Þ
The con¯ning pressuremodel for con¯ned concrete given inEq. (8) has been veri¯ed by
experimental results.22,23,29,30
V. I. Patel, Q. Q. Liang & M. N. S. Hadi
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The parts AB and BC of the stress�strain curve shown in Fig. 1 can be describedby
�c ¼�cf
0cc þ
"cu � "c"cu � " 0cc
� �ðf 0cc � �cf
0ccÞ for " 0cc < "c � "cu
�cf0cc for "c > "cu
8><>: ; ð11Þ
where "cu is taken as 0.02 as suggested by Liang and Fragomeni22 based on the
experimental results, and �c is a factor accounting for the con¯nement e®ect by the
steel tube on the post-peak strength and ductility of the con¯ned concrete, which is
given by Hu et al.17 as
�c ¼1:0 for
D
t� 40
0:0000339D
t
� �2
� 0:010085D
t
� �þ 1:3491 for 40 <
D
t� 150
8>><>>: : ð12Þ
The stress�strain curve for concrete in tension is shown in Fig. 1. It is assumed thatthe concrete tensile stress increases linearly with the increase in tensile strain up to
concrete cracking. After that, the tensile stress of concrete decreases linearly to zero
as the concrete softens. In the numerical model, the tensile strength of concrete is
taken as 0:6ffiffiffiffiffiffif 0cc
p, while the ultimate tensile strain is taken as 10 times the cracking
strain of concrete.
2.2. Material models for structural steels
The con¯nement e®ect induces a biaxial stress state in the steel tube of a circular
CFST beam-column. The hoop tension developed in the steel tube reduces the yield
stress in the longitudinal direction. This e®ect is considered in the constitutive
law represented by a linear-rounded-linear stress�strain curve shown in Fig. 2. Therounded part of the curve is replaced by a straight line for high strength steels.
The rounded part of the curve can be modeled by the following formula given by
εsε y0.9 εst
f
σs
o εsu
0.9 ff
sy
sy
su
Fig. 2. Stress�strain relationships for structural steels.
Numerical Analysis of Circular CFST Slender Beam-Columns with Preload E®ects
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Liang28 as
�s ¼ fsy"s � 0:9"sy"st � 0:9"sy
� � 145 ð0:9"y < "s � "stÞ; ð13Þ
where �s is the stress, "s the strain, "sy the yield strain of the steel ¯ber, and "st is the
steel strain at strain hardening as shown in Fig. 2, with "su indicating the ultimate
strain and fsu the tensile strength of steel material. The hardening strain "st of 0.005
is speci¯ed in the present numerical studies.
3. Ultimate Axial Strengths of Slender Hollow Steel Tubes
The numerical model developed by Liang29 can be used to determine the ultimate
axial strength of circular hollow steel tubes. For design purpose, however, a simple
design method given in Eurocode 410 is used to calculate the ultimate axial strength
of a slender hollow steel tube under axial compression, given as:
Ns;Ek ¼ �sNs;pl:Rk; ð14Þwhere Ns;pl;Rk is the cross-sectional plastic resistance of the steel tube, given by
Ns;pl:Rk ¼ Asfsy; ð15Þwhere As is the cross-sectional area of the steel tube. In Eq. (14), �s is the slenderness
where Ns;cr is the Euler buckling load of the pin-ended hollow steel tube,
Ns;cr ¼2EsIsL2
ð20Þ
with Es indicating Young's modulus of the steel tube, Is the second moment of area
of the hollow steel tubular section and L the e®ective length of the steel tube.
V. I. Patel, Q. Q. Liang & M. N. S. Hadi
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The preload ratio is de¯ned as
�a ¼Ppre
Pus
; ð21Þ
in which Ppre is the preload acting on the steel tubular column and Pus is the ultimate
axial strength of the hollow steel tube.
4. Initial De°ections Caused by Preloads
4.1. The ¯ber element method
The accurate ¯ber element method28 is used in the numerical model to simulate the
inelastic behavior of composite cross-sections under axial load and bending.
The column section is discretized into ¯ne ¯ber elements as depicted in Fig. 3.
Each ¯ber element can be assigned either steel or concrete material properties.
Fiber stresses are calculated from ¯ber strains using material uniaxial stress�strain relationships. It is assumed that the plane section remains plane
under deformations, which results in a linear strain distribution. The strains in
concrete and steel ¯bers can be calculated by the following equations given by
Liang28:
yn;i ¼D
2� dn; ð22Þ
"i ¼ yi � yn;i�� �� for yi � yn;i
� yi � yn;i�� �� for yi < yn;i
(; ð23Þ
where yi is the coordinate of the ¯ber element, dn is the neutral axis depth and
the curvature.
φD
d
ε t
εe,i
de,i
Steel fibers Concrete fibers
yi
yn,i
x
n
y
N.A.
Fig. 3. Stain distribution in circular CFST column section.
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The axial force and bending moments carried by a circular CFST beam-column
section are determined as stress resultants by integration as follows:
P ¼Xnsi¼1
�s;iAs;i þXncj¼1
�c;jAc;j; ð24Þ
Mx ¼Xnsi¼1
�s;iAs;iyi þXncj¼1
�c;jAc;jyj; ð25Þ
where P is the axial force, �s;i is the stress of steel ¯ber i,As;i is the area of steel ¯ber i,
�c;j is the stress of concrete ¯ber j, Ac;j is the area of concrete ¯ber j, yi is the coordi-
nate of steel element i, yj are the coordinate of concrete element j,ns is the total number
of steel ¯ber elements and nc is the total number of concrete ¯ber elements.
4.2. Load control approach
In the present study, the load control method is used in the load-de°ection analysis
to determine the mid-height de°ection induced by preloads. By employing the sine
displacement function, the de°ection at the mid-height of the pin-ended hollow steel
tubular beam-column caused by preload29 can be obtained from the curvature as
umo ¼L
� �2
mo; ð26Þ
where mo is the curvature at the mid-height of the hollow steel tube.
The external bending moment at the mid-height of the hollow steel tube with an
initial geometric imperfection uo and eccentricity of loading e can be calculated by
Mme ¼ Ppreðeþ umo þ uoÞ: ð27ÞThe curvature at the mid-height of the steel tube is adjusted to satisfy the force and
moment equilibriums at the mid-height of the beam-column as follows:
(11) Calculate the external and moments Mme and Mmi at the column
mid-height.
(12) Repeat Steps (10)–(12) until jrmj < "kð"k ¼ 10�4Þ.
5. Load-De°ection Analysis of CFST Slender Beam-Columns
5.1. Theoretical model
The computational procedure based on the load control method given in the pre-
ceding section can accurately determine the de°ections caused by a given preload.
However, it cannot predict the complete load-de°ection curves for columns with
softening behavior. To overcome this problem, the de°ection control method as
suggested by Liang29 is employed in the numerical model. The curvature at the mid-
height of the beam-column can be obtained as
m ¼
L
� �2um; ð30Þ
where um is the mid-height de°ection of the CFST beam-column.
The de°ection at the mid-height of the steel tube caused by the preload ðumoÞ istreated as initial geometric imperfection. The total initial geometric imperfection at
the mid-height of the CFST beam-column becomes ðuo þ umoÞ. The external momentat the mid-height of the CFST beam-column can be determined as
Mme ¼ P ðeþ uo þ umo þ umÞ: ð31Þ
In the analysis, the de°ection at the mid-height um of the CFST slender beam-
column is gradually increased until the ultimate axial load Pn is obtained. The
curvature m at the mid-height of the beam-column can be calculated from the
de°ection um. For this curvature, the neutral axis depth is adjusted to achieve
the moment equilibrium at the mid-height of the beam-column. The equilibrium
equation for an eccentrically loaded CFST slender beam-column with preload e®ects
is written as
P ðeþ uo þ umo þ umÞ �Mmi ¼ 0: ð32Þ
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5.2. Computational procedure
The main steps of the computational procedure for predicting the load-de°ection
responses of circular CFST slender beam-columns incorporating preload e®ects are
given as follows:
(1) Input data.
(2) Discretize the composite section into ¯ber elements.
(3) Calculate the mid-height de°ection umo of the hollow steel tube under preload
using the load control method.
(4) Set uo ¼ uo þ umo.
(5) Initialize the mid-height de°ection of the beam-column um ¼ �um.
(6) Calculate the curvature m at the mid-height of the beam-column.
(7) Adjust the neutral axis depth dn using the Müller's method.(8) Compute ¯ber stresses from ¯ber strains using material constitutive models.
(9) Calculate stress resultants P and moment Mmi.
(10) Compute the residual moment rm ¼Mme �Mmi.
(11) Repeat Steps (7)–(10) until jrmj < "k.
(12) Increase the de°ection at mid-height of the beam-column by um ¼ um þ�um.
(13) Repeat Steps (6)–(12) until the ultimate axial load Pn is obtained or the
de°ection limit is reached.
(14) Plot the load-de°ection curve.
The computational procedure proposed can predict the complete load-de°ection
responses of eccentrically loaded circular CFST slender beam-columns with preload
e®ects.
6. Solution Method
The Müller's method40 algorithms based on the secant method proposed by
Liang28,29 are developed to adjust the neutral axis depth and curvatures at the
mid-height of the beam-column to satisfy equilibrium conditions. The Müller'smethod requires three initial values of the neutral axis depth ðdnÞ and the cur-
vature ðmÞ to start the iterative process. These initial values are treated as designvariables !1, !2 and !3. The corresponding force or moment residuals rm;1, rm;2
and rm;3 are calculated based on the three initial values of the design variables.
The new design variable !4 that approaches the true value is determined by the
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loading eccentricity ratio ðe=DÞ was 0.2. The yield stress of the steel tube was
690MPa while the compressive strength of the concrete core was 40MPa.
Figure 9 demonstrates the e®ects of the preload ratio on the axial load-de°ection
curve for the CFST slender beam-column. It can be seen that increasing the preload
ratio signi¯cantly reduces the ultimate axial strength of the CFST slender beam-
column. In addition, the °exural sti®ness of the beam-column decreases with an
0
50
100
150
200
250
300
0 10 20 30 40 50
Axi
al lo
ad (
kN)
Mid-height deflection um (mm)
Experiment (A204)
Numerical analysis
Fig. 5. Comparison of computational and experimental axial load-de°ection curves for Specimen A204.
0
50
100
150
200
250
300
350
400
450
0 5 10 15 20 25 30 35
Axi
al lo
ad (
kN)
Mid-height deflection um (mm)
Experiment (A124)
Numerical analysis
Fig. 6. Comparison of computational and experimental axial load-de°ection curves for Specimen A124.
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increase in the preload ratio. Moreover, the presence of the preload on the steel tube
increases the mid-height de°ection at the ultimate loads. When increasing the pre-
load ratio from 0 to 0.4 and 0.8, the ultimate axial strength of the slender beam-
column reduces by 7.7% and 17.4% respectively.
0
1000
2000
3000
4000
5000
6000
0 5 10 15 20 25 30 35 40 45 50
Axi
al lo
ad (
kN)
Mid-height deflection um (mm)
Experiment (CFT-I-100-0P)
Numerical analysis
Fig. 7. Comparison of computational and experimental axial load-de°ection curves for Specimen
CFT-I-100-0P.
0 10 20 30 40 50 60 70 80 900
1000
500
1500
2000
2500
3000
3500
4000
Axi
al lo
ad (
kN)
Mid-height deflection um (mm)
Experiment (CFT-L-40-30P)
Numerical analysis
Fig. 8. Comparison of computational and experimental axial load-de°ection curves for Specimen
CFT-L-40-30P.
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8.2. E®ects of preloads on column strength curves
The e®ects of preloads on the column strength curves for eccentrically loaded high
strength steel tubes ¯lled with high strength concrete were examined. The diameter
of the circular column section was 500mm with a thickness of 10mm. The column
slenderness ratio varied from 0 to 100 while the dimension of the section was not
changed. The loading eccentricity ratio of 0.2 was considered. The yield stress and
tensile strength of steel tubes ¯lled with 60MPa concrete were 690MPa and 790MPa
respectively.
The column strength curves for circular CFST slender beam-columns with various
preload ratios are presented in Fig. 10, where Po the ultimate axial load of the
column section under axial compression. It is observed that increasing the column
slenderness ratio signi¯cantly reduces its ultimate axial strength regardless of the
preload ratio. In addition, the higher the preload ratio, the larger is the reduction in
the ultimate axial strength of the CFST beam-column. For the slender beam-column
with an L=r ratio of 100, the ultimate axial strength of CFST columns is reduced by
9% and 21.9% respectively when the preload ratios are 0.4 and 0.8 respectively.
However, for the short beam-column with an L=r ratio of 22, the preload with ratios
of 0.4 and 0.8 reduces the ultimate axial strength of the CFST column by only 0.98%
and 1.8% respectively. This implies that the preload e®ect on short columns can be
ignored in the design.
8.3. E®ects of preloads and diameter-to-thickness ratio
The e®ects of preloads on the ultimate axial strengths of normal strength steel tubes
¯lled with high strength concrete were investigated herein. The diameter of column
βa=0
βa=0.4
βa=0.8
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
0 100 200 300 400 500 600
Axi
al lo
ad (
kN)
Mid-height deflection um (mm)
Fig. 9. E®ects of preloads on the load-de°ection curves for circular CFST slender beam-columns.
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sections was 700mm. The diameter-to-thickness ratio was varied from 20 to 100. The
columns slenderness ratio was 80. The loading eccentricity ratio was taken as 0.2 in
the analysis. The yield stress and tensile strength of the steel tubes were 300MPa and
410MPa respectively. The compressive strength of the concrete was 70MPa.
Figure 11 presents the ultimate axial strength of circular CFST slender beam-
columns with various preload ratios as a function of the D=t ratio. The ¯gure shows
ββa=0
βa=0.4
βa=0.8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 30 60 90 120 150
Ulti
mat
e ax
ial l
oad
P n/P
o
Slenderness ratio L/r
Fig. 10. E®ects of preloads on the column strength curves.
βa=0
βa=0.4
βa=0.8
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
0 20 40 60 80 100 120 140
Ulti
mat
e ax
ial l
oad
(kN
)
Diameter-to-thickness ratio D/t
Fig. 11. E®ects of preloads and diameter-to-thickness ratio on the ultimate axial strengths.
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that increasing the D=t ratio remarkably reduces the ultimate axial loads of circular
CFST slender beam-columns regardless of the preload ratio. For a CFST slender
beam-column with a D=t ratio of 20, the preload with a ratio of 0.4 causes a 4.3%
reduction in its ultimate axial strength. However, when increasing the D=t ratio of
the column section to 100 and the preload ratio to 0.8, the strength reduction could
be as much as 14.7%.
8.4. E®ects of preloads and loading eccentricity ratio
The numerical model was utilized to examine the important e®ects of preloads and
loading eccentricity on the behavior of normal strength CFST slender beam-col-
umns. A circular CFST slender beam-column with a diameter of 550mm and
thickness of 10mm and with e=D ratios ranging from 0.0 to 2.0 was analyzed. The
column had a slenderness ratio of 80. The yield stress and tensile strength of the steel
tube were 300MPa and 410MPa respectively. The beam-column was ¯lled with
40MPa concrete.
Figure 12 shows that increasing the eccentricity ratio signi¯cantly reduces
the ultimate axial strength of the slender beam-column with preload e®ects. The
strength reduction caused by preloads is given in Fig. 13. The reduction in the
ultimate axial strength increases as the eccentricity ratio increases from 0.0 to 0.4
regardless of the preload ratio. When the e/D ratio is greater than 0.4, however, the
strength reduction tends to decrease with an increase in the e/D ratio. It is inter-
esting to note that the preload on the steel tube causes a maximum reduction in the
ultimate axial strength when the e/D ratio is equal to 0.4. For the axially loaded
CFST slender column with a preload ratio of 0.8, the strength reduction caused by
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
βa=0
βa=0.4
βa=0.8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Ulti
mat
e ax
ial l
oad
P n/P
o
Eccentricity ratio e/D
Fig. 12. E®ects of preloads and loading eccentricity ratio on the ultimate axial strengths.
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the preload is 4.9%. However, for the CFST beam-column with an e=D ratio of 0.4
and a preload ratio of 0.8, the strength reduction of 11.3% is expected.
8.5. E®ects of preloads and steel yield strengths
The e®ects of steel yield strengths on the behavior of preloaded circular steel tubes
¯lled with normal strength concrete of 50MPa were discussed herein. The diameter
of the circular CFST beam-column section was 600mm with a thickness of 12mm.
The column slenderness ratio of 80 and the loading eccentricity ratio e=D of 0.2 were
considered in the investigation. The yield stress of the steel tube varied from 250 to
690MPa.
The predicted ultimate axial strengths of CFST slender beam-columns are given in
Fig. 14. The ultimate axial strength of slender beam-columns tends to decrease when
increasing the yield strength of the steel tube regardless of the preload ratio. The
reason for this is that increasing the steel yield strength results in an increase in the
preload on the steel tube for the same preload ratio, which reduces the ultimate axial
load of the CFST column. The strength reduction is shown to be more pronounced
when the steel yield strength is greater than 350MPa. This means that the preload has
a more signi¯cant e®ect on high strength steel tubes than on normal strength steel
tubes. For the high strength CFST slender beam-column with yield steel strength of
690MPa and a preload ratio of 0.8, the preload causes a strength reduction of 17.3%.
8.6. E®ects of concrete con¯nement
The numerical model was used to examine the e®ects of concrete con¯nement on the
behavior of high strength circular CFST beam-columns with preloads on the steel
ββa=0.4
βa=0.8
0
2
4
6
8
10
12
0 0.4 0.8 1.2 1.6 2 2.4
Stre
ngth
red
uctio
n (%
)
Eccentricity ratio e/D
Fig. 13. Strength reduction caused by preloads in circular CFST beam-columns with various loadingeccentricity ratios.
Numerical Analysis of Circular CFST Slender Beam-Columns with Preload E®ects
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tubes. The beam-columns studied in the Sec. 8.2 with a preload ratio of 0.4 were
analyzed again by ignoring the concrete con¯nement e®ects. Liang30 suggested that
this can be done by setting the con¯ning pressure frp to zero in the analysis.
The e®ects of concrete con¯nement on the column strength curve are shown in
Fig. 15. It can be observed that the con¯nement has a considerable e®ect on the
ultimate axial strengths of cross-sections with zero length. However, the con¯nement
e®ect is shown to decrease with increasing the column slenderness. For the beam-
column with zero length, the con¯nement e®ect increases the ultimate axial strength
0
0.2
0.4
0.6
0.8
1
1.2
0 20 40 60 80 100 120
Ulti
mat
e ax
ial l
oad
P n/P
nc
Slenderness ratio L/r
Confinment ignored
Confinment considered
Fig. 15. E®ects of concrete con¯nment on column sterngth curves.
ββa=0
βa=0.4
βa=0.8
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 100 200 300 400 500 600 700 800 900
Ulti
mat
e ax
ial l
oad
P n/P
o
Steel yield strength fsy (MPa)
Fig. 14. E®ects of preloads and steel yield strengths on the ultimate axial strengths.
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by 10.5%. However, for the slender beam-column with an L=r ratio of 80, the
strength increases by only 2.4%. This means that the con¯nement e®ect can be
ignored in the design of very slender CFST beam-columns.
The normalized ultimate axial load of the column with an L=r ratio of 50 is shown
in Fig. 16, where Pnc is the ultimate axial strength of the column considering con-
¯nement e®ects. It can be seen that increasing the eccentricity ratio considerably
reduces the con¯nement e®ect. For the beam-column under axial loading, its ulti-
mate axial strength is found to increase by 7.1% due to the con¯nement e®ect.
However, for the slender beam-column with an e=D ratio of 2.0, the strength increase
owing to the con¯nement e®ect is only 1.9%. It can be concluded that the concrete
con¯nement e®ect on slender beam-columns with an e=D ratio greater than 0.2 can
be neglected in the design.
9. Conclusions
A new numerical model for predicting the load-de°ection responses of eccentrically
loaded circular CFST slender beam-columns incorporating preload e®ects has been
presented in this paper. Comparative studies conducted indicate that the numerical
model developed is highly e±cient and accurate for simulating the inelastic behavior
of eccentrically loaded normal and high strength circular CFST slender beam-col-
umns with preload e®ects. The parametric study performed demonstrates that for
slender beam-columns with an L=r ratio of 100 and preload ratio of 0.8, the preload
may reduce their ultimate axial strength by 21.9%. However, the preload e®ect can
be ignored in the design of short columns. The preload causes a maximum strength
reduction when the e=D ratio is 0.4. The con¯nement e®ect on the strength of
0
0.2
0.4
0.6
0.8
1
1.2
0 0.5 1 1.5 2 2.5
Ulti
mat
e ax
ial l
oad
P n/P
nc
Eccentricity ratio e/D
Confinment ignored
Confinment considered
Fig. 16. E®ects of concrete con¯nment on the ultimate axial strengths of circular CFST beam-columns
with various loading eccentricity ratios.
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circular CFST beam-columns is found to decrease with increasing the column
slenderness or loading eccentricity ratio.
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Numerical Analysis of Circular CFST Slender Beam-Columns with Preload E®ects
1250065-23
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Chapter 5: Circular and Rectangular CFST Slender Beam-columns with Preload Effects
V. I. Patel: Nonlinear Inelastic Analysis of Concrete-Filled Steel Tubular Slender Beam-Columns 181
Manuscript prepared for
Journal of Constructional Steel Research
August 2012
Nonlinear Analysis of Rectangular Concrete-Filled
Steel Tubular Slender Beam-Columns with Preload
Effects
Vipulkumar Ishvarbhai Patela, Qing Quan Liang
a,*, Muhammad N. S. Hadi
b
aSchool of Engineering and Science, Victoria University, PO Box 14428, Melbourne, VIC 8001, Australia
bSchool of Civil, Mining and Environmental Engineering, University of Wollongong,
Wollongong, NSW 2522, Australia
Corresponding author:
Dr. Qing Quan Liang School of Engineering and Science Victoria University PO Box 14428 Melbourne VIC 8001 Australia Phone: +61 3 9919 4134 Fax: +61 3 9919 4139 E-mail: [email protected]
Chapter 5: Circular and Rectangular CFST Slender Beam-columns with Preload Effects
V. I. Patel: Nonlinear Inelastic Analysis of Concrete-Filled Steel Tubular Slender Beam-Columns 215
Fig. 1. Fiber strain distribution in CFST beam-column section under axial load and biaxial bending.
Fig. 2. Stress-strain curve for concrete in rectangular CFST columns.
Concrete fibers
D
B
N.A.
Pn
x
y
dn
de,i
t
i
yi
yn,i
Steel fibers
t
C D
A B
c0.0150.005 cctc
'
fct
tu
fcc'
c fcc'
c
o
Chapter 5: Circular and Rectangular CFST Slender Beam-columns with Preload Effects
V. I. Patel: Nonlinear Inelastic Analysis of Concrete-Filled Steel Tubular Slender Beam-Columns 216
Fig. 3. Stress-strain curves for structural steels.
Fig. 4. Effective and ineffective areas of steel tubular cross-section under axial load and biaxial bending.
esu 0
fsu
fsy
0.9fsy
ss
0.9esy est es
D x
y
b
b
Pn
e1
e2
1
2
t
N.A.
Effective steel area Ineffective steel area
b
B
Chapter 5: Circular and Rectangular CFST Slender Beam-columns with Preload Effects
V. I. Patel: Nonlinear Inelastic Analysis of Concrete-Filled Steel Tubular Slender Beam-Columns 217
Fig. 5. Computer flowchart for predicting the axial load-deflection responses of CFST
beam-columns with prleoad effect
Start
Discretize the composite section
Initialize um=∆um
Calculate curvature
Adjust dn
Calculate fiber stresses and strains
Compute P, Mmi
Check local buckling and update steel stresses
Plot P-um diagram
um=um+∆ umIs Pn obtained?or um > limit?
End
Input data
Yes
Yes
No
No
m
Compute Mx and My
Adjust
No
Yes
Compute bmr
?kb
mr
Compute amr
?ka
mr
Compute umo using load control analysis procedure
Set uo = uo + umo
Chapter 5: Slender Beam-Columns with Preload Effects
V. I. Patel: Nonlinear Inelastic Analysis of Concrete-Filled Steel Tubular Slender Beam-Columns 218
Fig. 6. Comparison of computational and experimental axial load-deflection curves for specimen LP-1.
Fig. 7. Comparison of predicted and experimental axial load-deflection curves for specimen LP-6.
0
100
200
300
400
500
600
0 5 10 15 20 25 30 35
Axi
al lo
ad (k
N)
Mid-height deflection um (mm)
Experiment (LP-1)
Numerical analysis
0
50
100
150
200
250
300
350
400
450
0 5 10 15 20 25 30 35
Axi
al lo
ad (k
N)
Mid-height deflection um (mm)
Experiment (LP-6)
Numerical analysis
Chapter 5: Slender Beam-Columns with Preload Effects
V. I. Patel: Nonlinear Inelastic Analysis of Concrete-Filled Steel Tubular Slender Beam-Columns 219
Fig. 8. Effects of local buckling on the axial load-deflection curves for CFST slender beam-column 3.0a .
Fig. 9. Effects of local buckling on the strength envelopes for CFST slender beam-column 3.0a .
0
2000
4000
6000
8000
10000
12000
0 30 60 90 120 150
Axi
al lo
ad (k
N)
Mid-height deflection um (mm)
Local buckling ignored
Local buckling considered
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
Axi
al lo
ad P
n/Poa
Moment Mn/Mo
Local buckling ignored
Local buckling considered
Chapter 5: Slender Beam-Columns with Preload Effects
V. I. Patel: Nonlinear Inelastic Analysis of Concrete-Filled Steel Tubular Slender Beam-Columns 220
Fig. 10. Effects of preloads on the axial load-deflection curves for CFST beam-columns.
Fig. 11. Effects of preloads on the strength envelopes for CFST slender beam-column.
ba=0.8
ba=0ba=0.4
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
0 100 200 300 400 500
Axi
al lo
ad (k
N)
Mid-height deflection um (mm)
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
Axi
al lo
ad P
n/Pop
Moment Mn/Mo
Preload effect ignored
Preload effect considered
Chapter 5: Slender Beam-Columns with Preload Effects
V. I. Patel: Nonlinear Inelastic Analysis of Concrete-Filled Steel Tubular Slender Beam-Columns 221
Fig. 12. Effects of preloads and diameter-to-thickness ratio on the ultimate axial strengths.
Fig. 13. Strength reduction caused by preloads in CFST beam-column with various depth-to-thickness ratios.
ba=0
ba=0.3
ba=0.6
0
5000
10000
15000
20000
25000
30000
35000
40000
0 10 20 30 40 50 60
Axi
al lo
ad (k
N)
Depth-to-thickness ratio D/t
ba=0.3
ba=0.6
0
2
4
6
8
10
12
14
0 10 20 30 40 50 60
Stre
ngth
redu
ction
(%)
Depth-to-thickness ratio D/t
Chapter 5: Slender Beam-Columns with Preload Effects
V. I. Patel: Nonlinear Inelastic Analysis of Concrete-Filled Steel Tubular Slender Beam-Columns 222
Fig. 14. Effects of preloads on the column strength curves.
Fig. 15. Strength reduction caused by preloads in CFST beam-columns with various slenderness ratios.
ba=0
ba=0.3
ba=0.6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 20 40 60 80 100 120
Ulti
mat
e axi
al lo
ad P
n/P
o
Slenderness ratio L/r
ba=0.3
ba=0.6
0
2
4
6
8
10
12
14
16
18
20
0 20 40 60 80 100 120
Stre
ngt
h r
edu
ctio
n (%
)
Slenderness ratio L/r
Chapter 5: Slender Beam-Columns with Preload Effects
V. I. Patel: Nonlinear Inelastic Analysis of Concrete-Filled Steel Tubular Slender Beam-Columns 223
Fig. 16. Effects of preloads and loading eccentricity ratio on the ultimate axial strength.
Fig. 17. Strength reduction caused by preloads in CFST beam-columns with various loading eccentricity ratios.
ba=0
ba=0.3
ba=0.6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
Ulti
mat
e axi
al lo
ad P
n/P
o
Eccentricity ratio e/D
ba=0.3
ba=0.6
0
1
2
3
4
5
6
7
8
9
10
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
Stre
ngth
redu
ction
(%)
Eccentricity ratio e/D
Chapter 5: Slender Beam-Columns with Preload Effects
V. I. Patel: Nonlinear Inelastic Analysis of Concrete-Filled Steel Tubular Slender Beam-Columns 224
Fig. 18. Effects of preloads and steel yield strengths on the ultimate axial strength.
Fig. 19. Strength reduction caused by preloads in CFST beam-columns with various steel yield strengths.
ba=0ba=0.3ba=0.6
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 100 200 300 400 500 600 700 800 900
Ulti
mat
e axi
al lo
ad P
n/P
o
Steel yield strength fsy (MPa)
ba=0.3
ba=0.6
0
2
4
6
8
10
12
14
0 100 200 300 400 500 600 700 800
Stre
ngth
redu
ction
(%)
Steel yield strength fsy (MPa)
Chapter 5: Slender Beam-Columns with Preload Effects
V. I. Patel: Nonlinear Inelastic Analysis of Concrete-Filled Steel Tubular Slender Beam-Columns 225
5.5 CONCLUDING REMARKS
Numerical models for simulating the behavior of eccentrically loaded circular and
rectangular CFST slender beam-columns with preload effects have been developed in
this chapter. The effects of preloads, local buckling, concrete confinement, initial
geometric imperfections and second order have been considered in the numerical
models. The inelastic behavior of composite cross-sections with local buckling effects is
modeled using the accurate fiber element method. Deflections caused by preloads on the
steel tubes are incorporated in the inelastic stability analysis of CFST slender beam-
columns as additional geometric imperfections. Computational procedures have been
developed for simulating the load-deflection responses of CFST slender beam-columns
under axial load and bending. The accuracy of the numerical models developed has
been established by comparisons of computer solutions with existing experimental
results. The numerical models were used to investigate the effects of the preload ratio,
depth-to-thickness ratio, column slenderness and loading eccentricity on the behavior of
thin-walled CFST slender beam-column under axial load and bending.
Chapter 6: Rectangular CFST slender Beam-Columns under Cyclic Loading
V. I. Patel: Nonlinear Inelastic Analysis of Concrete-Filled Steel Tubular Slender Beam-Columns 226
Chapter 6
RECTANGULAR CFST SLENDER BEAM-
COLUMNS UNDER CYCLIC LOADING
6.1 INTRODUCTION
This chapter presents a numerical model for simulating the cyclic behavior of high
strength thin-walled rectangular CFST slender beam-columns with cyclic local buckling
effects. Uniaxial cyclic stress-strain relationships for the concrete core and structural
steels are incorporated in the numerical model. The effects of initial geometric
imperfections, high strength materials and second order are also taken into account in
the numerical model for CFST slender beam-columns under constant axial load and
cyclically varying lateral loading. Müller’s method algorithms are developed and
implemented in the numerical model to iterate the neutral axis depth in a thin-walled
CFST beam-column section. The ultimate lateral loads and cyclic load-deflection curves
for thin-walled CFST beam-columns predicted by the numerical model are verified by
Chapter 6: Rectangular CFST slender Beam-Columns under Cyclic Loading
V. I. Patel: Nonlinear Inelastic Analysis of Concrete-Filled Steel Tubular Slender Beam-Columns 227
experimental data. The numerical model is then utilized to investigate the effects of
cyclic local buckling, column slenderness ratio, depth-to-thickness ratio, concrete
compressive strengths and steel yield strengths on the cyclic load-deflection responses
of CFST slender beam-columns.
This chapter includes the following paper:
[1] Patel, V. I., Liang, Q. Q. and Hadi, M. N. S., “Numerical analysis of high-strength
concrete-filled steel tubular slender beam-columns under cyclic loading”, Journal
of Constructional Steel Research, 2013 (submitted).
228
229
Chapter 6: Rectangular CFST slender Beam-Columns under Cyclic Loading
V. I. Patel: Nonlinear Inelastic Analysis of Concrete-Filled Steel Tubular Slender Beam-Columns 230
Manuscript prepared for
Journal of Constructional Steel Research
January 2013
Numerical Analysis of High-Strength Concrete-Filled Steel
Tubular Slender Beam-Columns under Cyclic Loading
Vipulkumar Ishvarbhai Patela, Qing Quan Liang
a,*, Muhammad N. S. Hadi
b
aCollege of Engineering and Science, Victoria University, PO Box 14428, Melbourne, VIC 8001, Australia
bSchool of Civil, Mining and Environmental Engineering, University of Wollongong,
Wollongong, NSW 2522, Australia
Corresponding author:
Associate Professor Qing Quan Liang College of Engineering and Science Victoria University PO Box 14428 Melbourne VIC 8001 Australia Phone: +61 3 9919 4134 E-mail: [email protected]