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A Numerical Study addressing the Stress
Distribution in Circular Steel Tube Confined
Concrete Columns considering Various
Concrete Strengths
Hoang An Le
Ho Chi Minh City University of Transport, Vietnam
hoangan.le@ut.edu.vn
(corresponding author)
Received: 19 December 2022 | Revised: 30 January 2023 | Accepted: 1 February 2023
ABSTRACT
The axially compressive behavior of Steel Tube Confined Concrete (STCC) columns has been
experimentally investigated by many researchers throughout the world. However, it is extremely
complicated to measure the stresses of steel tubes and concrete core in real tests. Therefore, to investigate
the fundamental behavior of STCC columns under axial compression, this paper presents a numerical
study that explores the stress distribution in steel tubes and concrete core. The circular STCC columns
with the use of Normal Strength Concrete (NSC), High Strength Concrete (HSC), and Ultra-High Strength
Concrete (UHSC) were simulated in a Finite Element Model (FEM) in ABAQUS. The material model for
confined concrete incorporating a wide range of concrete strength values was developed in the simulation.
The obtained from FEM curves of load versus strain of circular STCC columns were compared with those
measured in real tests to verify the accurateness of the FEM. Deriving from the results of FEM, the stress
states and their distribution in outer steel tubes and concrete core along the column height were described.
Also, the longitudinal stresses on the cross-section of the concrete core were calculated corresponding with
the load stage to quantify the strength enhancement of the concrete core due to the confinement effect from
the steel tube. Furthermore, the confining pressure provided by the outer steel tube and impacting on the
concrete core was plotted. Based on the findings in this paper, the effect of various concrete strengths on
the stress distribution in circular STCC columns was investigated.
Keywords-steel tube confined concrete; NSC; HSC; UHSC; ABAQUS; FEM
I. INTRODUCTION
It is widely known that the confinement effect in Concrete
Filled Steel Tube (CFST) columns can result in remarkable
improvements in compressive strength and ductility [1-11].
Therefore, CFST columns have been extensively applied in
civil engineering projects. For this column type, when the steel
tube and the concrete core are loaded simultaneously, the
confining stress induced by the steel tube is less effective as
compared to the case where only the concrete core is loaded [5-
11]. If the ductility and strength of composite columns are
considered as critical design factors, it is suggested that the
load should be applied on the concrete core only in order to
form Steel Tube Confined Concrete (STCC) columns [9].
Many previous studies have conducted experimental tests on
the axial compressive behavior of STCC columns [1-4], but
they have mainly focused on the use of Normal Strength
Concrete (NSC) or High Strength Concrete (HSC). Recently,
several studies have reported the test results of circular STCC
columns infilled with Ultra-High Strength Concrete (UHSC)
under axial compression [5, 10, 11, 19]. According to them, the
confining stress level in STCC columns depends on concrete
strength. The dilation of concrete under compression is higher
with lower concrete strength, thus the interaction between the
concrete core and the steel tube becomes stronger. For STCC
columns, at the ultimate state, the concrete core is subjected to
triaxial stress, while the steel tube is under biaxial stress. In real
tests, it is complicated to measure all stresses in the concrete
core and in the steel tubes. Accordingly, the confining stress
induced by the steel tubes and imposed on the concrete core is
hardly calculated. Therefore, many studies have attempted to
use Finite Element Models (FEMs) to simulate the mechanism
in STCC columns. Based on the FEM results, all stresses on
STCC columns can be quantified. Authors in [12] presented an
FEM in ATENA-3D software to investigate the effect of
concrete strength on the compressive behavior of STCC
columns. Authors in [9] developed an FEM using ABAQUS
software to analyze the mechanisms of STCC short columns
under axial compression, however they only focused on NSC.
Authors in [13] introduced an analysis of circular STCC stub
columns with some modifications to the Drucker – Prager
Engineering, Technology & Applied Science Research Vol. 13, No. 2, 2023, 10347-10351 10348
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model in ABAQUS, and they quantified the lateral confining
pressure and the interface shear stress between the concrete
core and the steel tube. Authors in [15] investigated the effect
of friction on axially loaded STCC short columns by utilizing
an FEM. Equations considering the friction coefficient were
developed to predict the load bearing capacity of STCC short
columns.
From the literature review, it is apparent that numerical
studies on the mechanical performance of STCC columns
remain very limited with only a handful of studies considering
the STCC columns with the use of NSC or HSC. Therefore
additional research should be further carried out to better
understand the compressive behavior of STCC columns when
UHSC is infilled. Furthermore, the majority of the existing
studies were concerned with the comparison of load versus
displacement or load versus strain curves of STCC columns
between the FEMs and the test results and very few studies
dealt with the stress distribution in the steel tubes and the
concrete core of STCC columns. The interaction between the
concrete core and the steel tubes through the confining stress is
complex and can't be measured experimentally. To address the
aforementioned research gap, this paper aims to develop an
FEM in ABAQUS software to simulate the mechanism of
circular STCC stub columns with a wide range of concrete
strengths, i.e. NSC, HSC, and UHSC. The constitutives of
confined concrete were proposed based on some previous
studies and some modifications to Concrete Damaged Plasticity
Model (CDPM). The established FEM was verified against the
collected test results. Subsequently, the confining stress and the
distribution of axial load in the concrete core and the steel tube
were clarified. Finally, the biaxial stress state of the steel tube
was quantified by plotting the longitudinal and hoop stresses
along the height of the columns.
II. FINITE ELEMENT MODEL DESCRIPTION
A. General Description
The circular STCC short columns were simulated using
FEM in ABAQUS (version 6.11) [28]. Due to the symmetrical
nature of the circular column, only one-eighth of the column
was modelled. CAX4R elements with reduced integration and
4-node axisymmetric elements were used to model the steel
tubes and the concrete core. Based on the mesh convergence
studies, the element size was chosen as t/5, where t is the
thickness of the steel tube. For short columns with the ratio of
L/D (length to outer diameter) smaller than 4, the initial
imperfections can be ignored in FEM. Surface-to-surface
contact was employed to model the interaction between the
steel tubes and concrete core. A contact surface pair including
the inner surface of the steel tubes and the outer surface of the
concrete core was defined. The concrete surfaces were chosen
as master surfaces, while the steel surfaces were treated as
master. As suggested in [18], the friction coefficient between
the steel tube and concrete was taken as 0.6. Two Reference
Points (RPs) were created at the center of each upper and lower
cross-section. Boundary Conditions (BCs) were assigned to
both reference points, while the load was applied downward at
the upper reference point. The "rigid body" constraints were
employed to tie the reference points to both the end surfaces of
the concrete core. To ensure the load was applied only on the
concrete core, only the end surfaces of the concrete core were
fixed against all degrees of freedom except for the
displacement at the upper loaded end. Figure 1 shows the FEM
including mesh type, boundary conditions, and loading
application.
Fig. 1. Typical FEM of a circular STCC stub column.
B. Material Models
The stress – strain relationship ( - ) proposed in [22] was
adopted for the material model of the steel tube. This model
covers a wide range of steel yield strength (fy) from 200 to
800MPa. CDPM provided in ABAQUS was employed to
simulate the compressive behavior of confined concrete. The
key parameters in CDPM include the uniaxial compressive
stress – strain relationship, compressive damage variable dc
ratio of the second stress invariant on the tensile meridian to
that on the compressive meridian Kc, ratio of initial equibiaxial
compressive yield stress to initial uniaxial compressive yield
stress σbo/σco, dilation angle ψ, fracture energy Gf , viscosity,
and eccentricity α. The values of Kc, ψ, and σbo/σco were
calculated by using the equations suggested by [20-22], while
the remaining parameters such as α, dc, Gf, and viscosity were
defined as the default values in Abaqus. The stress – strain
relationship for NSC, HSC, and UHSC after taking into
account the confinement effect was developed for circular
STCC short columns, as can be seen in [22]. The - curve
includes three stages:
The ascending branch (OA) is expressed by the equations
proposed by [23]:
�
��′
� ���� ��
����
������ �����
0 � � � ��� (1)
where:
� � �� ������′ ; � �
��
��
�.�� � 1; �
�
���
(2)
The strain εc0 at the peak stress of the unconfined concrete
is calculated by [24]:
��� � ��0.067 � �#�′�� $ 29.9 � #�′ $ 1053� � 10) (3)
The confined strain at the point B (cc) is determined using
the equation proposed in [25]:
���
���
� 1 $ 17.4 � +�,��′ -
�.�)
(4)
Engineering, Technology & Applied Science Research Vol. 13, No. 2, 2023, 10347-10351 10349
www.etasr.com Le: A Numerical Study addressing the Stress Distribution in Circular Steel Tube Confined Concrete …
# �
�.�.��/
0�.�
�
1
2
(5)
where fB is the confining stress at the point B and determined
by (5) suggested by [26].
The descending branch (BC) is expressed by an exponential
function proposed by [27]:
��� � +�0.067 � 3#�′4
� $ 29.9 � #�′ $ 1053- � 10) (6)
where α and β are factors determining the shape of the
descending branch.
III. FINITE ELEMENT MODEL VERIFICATION AND
MECHANISM ANALYSIS
The established FEM was used to simulate 3 circular STCC
short columns, taken from previous studies, under axial
compression. Specimens C2-30-3D-C and C1-100-3D-C
reported in [16] and NB4.0-UHFB reported in [17] were
considered. It is noted that, NSC (C30, fc = 32.7MPa) and HSC
(C90, fc = 105.5MPa) were used for the specimens C2-30-3D-C
and C1-100-3D-C, respectively, while UHSC (C150, fc =
174.2MPa) was used for NB4.0-UHFB. Dimensions and
material properties of the selected specimens are shown in
Figure 2.
(a)
(b)
(c)
Fig. 2. FEM verification and stress distribution on the cross section at different points. (a) Test C2-30-3D-C, (b) Test C1-100-3D-C, (c) Test NB4.0-UHFB.
Point 1
Point 2
Point 3
0
4
8
12
16
20
0
200
400
600
800
1000
1200
1400
0 0.005 0.01 0.015 0.02 0.025 0.03
C
o
n
fi
n
in
g
s
tr
e
ss
(M
P
a
)
L
o
a
d
N
(
k
N
)
Strain ε (mm/mm)
Test C2-30-3D-C FEM
Concrete core Steel tube
Confining stress
D = 114.3mm, t = 6mm, L = 342.9mm
fc = 32.7 MPa, fy = 342.95 MPa
Point 1
Point 2
Point 3
0
4
8
12
16
20
0
300
600
900
1200
1500
1800
2100
0 0.005 0.01 0.015 0.02 0.025 0.03
C
o
n
fi
n
in
g
s
tr
es
s
(M
P
a)
L
o
a
d
N
(
k
N
)
Strain ε (mm/mm)
Test C1-100-3D-C FEM
Concrete core Steel tube
Confining stress
D = 114.3mm, t = 3.35mm, L = 342.9mm,
fc = 105.5 MPa, fy = 287.33 MPa
Point 1
Point 2
Point 3
0
4
8
12
16
20
0
1000
2000
3000
4000
5000
6000
0 0.005 0.01 0.015 0.02 0.025 0.03
C
o
n
fi
n
in
g
s
tr
e
ss
(
M
P
a
)
L
o
a
d
N
(
k
N
)
Strain ε (mm/mm)
Test NB4.0-UHFB FEM
Concrete core Steel tube
Confining stress
D = 168.6mm, t = 3.9mm, L = 648mm
fc = 174.2 MPa, fy = 363.0 MPa
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In order to verify the accuracy of FEM, the load versus
strain (N-ε) curves of the 3 specimens obtained from FEM were
compared with those measured by the test results. It can be
seen in Figure 2 that, in the ascending phase, the FEM results
agree well with the measured results, but in the descending
phase, there is a slight difference between the predicted and the
measured curves. For the descending phase of specimens C1-
100-3D-C and NB4.0-UHFB, the (N-ε) curves in FEM exhibit
similar behaviors with the real tests. There is a loss of load
after the peak load, followed by a horizontal branch. Overall, it
can be observed that FEM predicts well the complete (N-ε)
curves of all tested specimens.
Figure 2 also illustrates the distribution of the axial load
between the concrete core and the steel tube and the confining
stress provided by the steel tube. For NSC filled in steel tube
(C2-30-3D-C), at the ultimate state, the concrete core
contributed approximately 70% to the total load. However, it is
observed that the concrete core carries almost the entire load
for the case of HSC and UHSC. At the ultimate state, 90% of
the total load was carried by the HSC core in specimen C1-
100-3D-C and UHSC core in specimen NB4.0-UHFB.
Furthermore, the steel tube in the case of HSC and UHSC core
tends to carry less axial load before the peak stress as compared
to the case of NSC. Accordingly, in the ascending branch of the
(N-ε) curves, the confining stress in the case of NSC core is
more intense and develops faster than in the case of HSC and
UHSC. In the plastic stage, the confining stress steadily
increases after reaching a stable value. Figure 2 demonstrates
the distribution of longitudinal stress in the concrete section (at
the top surface of the column) corresponding to three points (1,
2, and 3) in the (N-ε) curves. For the NSC core, the confining
stress reaches the maximum value of 8.2MPa, which is twice
the maximum value of the confining stress in the case of HSC
and UHSC core. It can be seen in Figure 2 that the longitudinal
stress across the concrete section is uniform along the axis of
cylinder and increases gradually when the distance apart from
the center of section increases. The longitudinal stress in the
concrete section for the NSC core was significantly larger than
that for HSC and UHSC cores. This implies that the
confinement is more effective with lower concrete strength.
These findings are in good agreement with the research results
in [1, 8, 12-15, 22].
Figure 3 shows the distribution of longitudinal stress and
hoop stress in the steel tube at the ultimate state. The ratios of
stress to yield strength of steel tube (σ/fy) were plotted along the
height of the columns (h/H). It can be observed that the
distribution of stresses in the steel tube is non-uniform. The
longitudinal stress of the steel tube is zero and the hoop stress
is relatively high at both ends of the columns. The longitudinal
stress increases and the hoop stress decreases towards the
column mid-height. Also, at the mid-height of the columns, the
maximum value of longitudinal stress is achieved, while the
hoop stress is significantly reduced. The hoop stress was
maximum at H/10, which is quite near to the end surface of the
column. The maximal values of longitudinal stress and hoop
stress were about 0.55fy and 0.9fy, 0.50fy, and 0.9fy, and 0.40fy
and 0.9fy for NSC, HSC, and UHSC cores, respectively. The
distribution of longitudinal stress in the steel tube along the
column height is in line with the results reported in [9, 13, 14],
however there is a difference in the distribution of hoop stress
in the steel tube between this study and the results in [9, 13,
14]. The maximum value of hoop stress in this study is
generally smaller than that in [9, 13, 14]. For example, authors
in [9] stated that the hoop stress is highest at the column ends.
It should be noted that the confining stress depends mainly on
the hoop stress. Based on the FEM results, it can be identified
that confining stress is maximum at H/10 for circular STCC
columns.
Fig. 3. Distribution of stresses in the steel tube at the ultimate state.
IV. CONCLUSIONS
The main conclusions deriving from the FEM results of the
current study are:
The established FEM with some modifications to CDPM in
ABAQUS can predict well the compressive behavior of
circular STCC columns with the use of NSC, HSC, and
UHSC. The distribution of stress in the concrete core and
steel tube, and the confining stress can be quantified by
using this FEM.
The confining stress or confinement effect is more effective
with lower concrete strength.
The axial load carried by the concrete core becomes smaller
with increasing concrete strength.
The longitudinal stress of the steel tube is zero at both ends
of the columns and increases towards the mid-height of the
column, while the hoop stress is maximal at the position of
H/10 and decreases towards the mid-height of the column.
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REFERENCES
[1] M. Johansson and K. Gylltoft, "Mechanical Behavior of Circular Steel–
Concrete Composite Stub Columns," Journal of Structural Engineering,
vol. 128, no. 8, pp. 1073–1081, Aug. 2002, https://doi.org/10.1061/
(ASCE)0733-9445(2002)128:8(1073).
[2] L.-H. Han, G.-H. Yao, Z.-B. Chen, and Q. Yu, "Experimental
behaviours of steel tube confined concrete (STCC) columns," Steel and
Composite Structures, An International Journal, vol. 5, no. 6, pp. 459–
484, 2005, https://doi.org/10.12989/scs.2005.5.6.459.
[3] A. A. Fam, F. S. Qie, and S. Rizkalla, "Concrete-Filled Steel Tubes
Subjected to Axial Compression and Lateral Cyclic Loads," Journal of
Structural Engineering, vol. 130, no. 4, pp. 631–640, Apr. 2004,
https://doi.org/10.1061/(ASCE)0733-9445(2004)130:4(631).
[4] M. D. O’Shea and R. Q. Bridge, "Design of Circular Thin-Walled
Concrete Filled Steel Tubes," Journal of Structural Engineering, vol.
126, no. 11, pp. 1295–1303, Nov. 2000, https://doi.org/10.1061/(ASCE)
0733-9445(2000)126:11(1295).
[5] M.-X. Xiong, D.-X. Xiong, and J. Y. R. Liew, "Axial performance of
short concrete filled steel tubes with high- and ultra-high- strength
materials," Engineering Structures, vol. 136, pp. 494–510, Apr. 2017,
https://doi.org/10.1016/j.engstruct.2017.01.037.
[6] P. C. Nguyen, D. D. Pham, T. T. Tran, and T. Nghia-Nguyen, "Modified
Numerical Modeling of Axially Loaded Concrete-Filled Steel Circular-
Tube Columns," Engineering, Technology & Applied Science Research,
vol. 11, no. 3, pp. 7094–7099, Jun. 2021, https://doi.org/
10.48084/etasr.4157.
[7] A. N. Hassooni and S. R. A. Zaidee, "Behavior and Strength of
Composite Columns under the Impact of Uniaxial Compression
Loading," Engineering, Technology & Applied Science Research, vol.
12, no. 4, pp. 8843–8849, Aug. 2022, https://doi.org/10.48084/
etasr.4753.
[8] Z. H. Abdulghafoor and H. A. Al-Baghdadi, "Static and Dynamic
Behavior of Circularized Reinforced Concrete Columns Strengthened
with Hybrid CFRP," Engineering, Technology & Applied Science
Research, vol. 12, no. 5, pp. 9336–9341, Oct. 2022, https://doi.org/
10.48084/etasr.5162.
[9] Q. Yu, Z. Tao, W. Liu, and Z.-B. Chen, "Analysis and calculations of
steel tube confined concrete (STCC) stub columns," Journal of
Constructional Steel Research, vol. 66, no. 1, pp. 53–64, Jan. 2010,
https://doi.org/10.1016/j.jcsr.2009.08.003.
[10] A. L. Hoang, E. Fehling, B. Lai, D.-K. Thai, and N. V. Chau,
"Experimental study on structural performance of UHPC and UHPFRC
columns confined with steel tube," Engineering Structures, vol. 187, pp.
457–477, May 2019, https://doi.org/10.1016/j.engstruct.2019.02.063.
[11] A. Le Hoang, E. Fehling, D.-K. Thai, and C. Van Nguyen, "Evaluation
of axial strength in circular STCC columns using UHPC and UHPFRC,"
Journal of Constructional Steel Research, vol. 153, pp. 533–549, Feb.
2019, https://doi.org/10.1016/j.jcsr.2018.11.001.
[12] A. Le Hoang and E. Fehling, "Numerical study of circular steel tube
confined concrete (STCC) stub columns," Journal of Constructional
Steel Research, vol. 136, pp. 238–255, Sep. 2017, https://doi.org/
10.1016/j.jcsr.2017.05.020.
[13] A. Haghinejada and M. Nematzadeh, "Three-Dimensional Finite
Element Analysis of Compressive Behavior of Circular Steel Tube-
Confined Concrete Stub Columns by New Confinement Relationships,"
Latin American Journal of Solids and Structures, vol. 13, pp. 916–944,
May 2016, https://doi.org/10.1590/1679-78252631.
[14] P. K. Gupta and H. Singh, "Numerical study of confinement in short
concrete filled steel tube columns," Latin American Journal of Solids
and Structures, vol. 11, pp. 1445–1462, Dec. 2014,
https://doi.org/10.1590/S1679-78252014000800010.
[15] J. Liu, X. Zhou, and D. Gan, "Effect of friction on axially loaded stub
circular tubed columns," Advances in Structural Engineering, vol. 19,
no. 3, pp. 546–559, Mar. 2016, https://doi.org/10.1177/
1369433216630125.
[16] W. L. A. de Oliveira, S. De Nardin, A. L. H. de C. El Debs, and M. K.
El Debs, "Evaluation of passive confinement in CFT columns," Journal
of Constructional Steel Research, vol. 66, no. 4, pp. 487–495, Apr. 2010,
https://doi.org/10.1016/j.jcsr.2009.11.004.
[17] H. Schneider, "Zum tragverhalten kurzer, umschnürter, kreisförmiger,
druckglieder aus ungefasertem UHFB," Ph.D. dissertation, University of
Leipzig, Leipzig, Germany, 2006.
[18] L.-H. Han, G.-H. Yao, and Z. Tao, "Performance of concrete-filled thin-
walled steel tubes under pure torsion," Thin-Walled Structures, vol. 45,
no. 1, pp. 24–36, Jan. 2007, https://doi.org/10.1016/j.tws.2007.01.008.
[19] J. Wei, Z. Xie, W. Zhang, X. Luo, Y. Yang, and B. Chen, "Experimental
study on circular steel tube-confined reinforced UHPC columns under
axial loading," Engineering Structures, vol. 230, Mar. 2021, Art. no.
111599, https://doi.org/10.1016/j.engstruct.2020.111599.
[20] T. Yu, J. G. Teng, Y. L. Wong, and S. L. Dong, "Finite element
modeling of confined concrete-I: Drucker–Prager type plasticity model,"
Engineering Structures, vol. 32, no. 3, pp. 665–679, Mar. 2010,
https://doi.org/10.1016/j.engstruct.2009.11.014.
[21] V. K. Papanikolaou and A. J. Kappos, "Confinement-sensitive plasticity
constitutive model for concrete in triaxial compression," International
Journal of Solids and Structures, vol. 44, no. 21, pp. 7021–7048, Oct.
2007, https://doi.org/10.1016/j.ijsolstr.2007.03.022.
[22] Z. Tao, Z.-B. Wang, and Q. Yu, "Finite element modelling of concrete-
filled steel stub columns under axial compression," Journal of
Constructional Steel Research, vol. 89, pp. 121–131, Oct. 2013,
https://doi.org/10.1016/j.jcsr.2013.07.001.
[23] A. K. Samani and M. M. Attard, "A stress–strain model for uniaxial and
confined concrete under compression," Engineering Structures, vol. 41,
pp. 335–349, Aug. 2012, https://doi.org/10.1016/j.engstruct.2012.03.
027.
[24] M. A. Tasdemir, C. Tasdemir, S. Akyuz, A. D. Jefferson, F. D. Lydon,
and B. I. G. Barr, "Evaluation of strains at peak stresses in concrete: A
three-phase composite model approach," Cement and Concrete
Composites, vol. 20, no. 4, pp. 301–318, Jan. 1998,
https://doi.org/10.1016/S0958-9465(98)00012-2.
[25] Q. G. Xiao, J. G. Teng, and T. Yu, "Behavior and Modeling of Confined
High-Strength Concrete," Journal of Composites for Construction, vol.
14, no. 3, pp. 249–259, Jun. 2010, https://doi.org/10.1061/(ASCE)
CC.1943-5614.0000070.
[26] D.-D. Pham and P.-C. Nguyen, "Finite Element Modelling for Axially
Loaded Concrete-Filled Steel Circular Tubes," in 5th International
Conference on Geotechnics, Civil Engineering Works and Structures,
Singapore, Singapore, 2020, pp. 75–80, https://doi.org/10.1007/978-981-
15-0802-8_8.
[27] B. Binici, "An analytical model for stress–strain behavior of confined
concrete," Engineering Structures, vol. 27, no. 7, pp. 1040–1051, Jun.
2005, https://doi.org/10.1016/j.engstruct.2005.03.002.
[28] Abaqus/Cae Users Manual. USA: Hibbitt, Karlsson & Sorensen Inc,
2000.