Experimental and numerical investigations of square concrete-filled double steel tubular stub columns

Abstract

In this paper, the structural behavior of concrete-filled double steel tubular (CFDST) stub columns composed of square hollow sections is investigated experimentally and numerically. The experimental program comprises compression tests on short columns loaded concentrically. The test parameters mainly focused on the influences of the width-to-thickness ratios of steel tubes and concrete strength on the axial behavior of CFDST stub columns. Finite element (FE) models are also developed to investigate the influences of a wide range of structural parameters on their axial performance. It is observed that square CFDST columns have improved strength and ductility compared to their CFST and DCFST counterparts. Finally, a calculation formula is proposed to predict their ultimate compressive strengths under the axial compression load.

Keywords

axial load concrete-filled steel tube double steel FE model stub columns

Introduction

Because of their higher bearing capacity, ductility, and ease of connection to the beams, square concrete-filled steel tubular (CFST) columns are widely used in the construction of composite structures (Wang et al., 2018; Yuan et al., 2019; Zhu et al. 2019). The square CFST columns are often made of thin-walled steel tubes that are particularly susceptible to the localized buckling which affects their structural performance significantly. Furthermore, the application of the high-strength concrete in CFST columns even further compensate their ductility compared to their circular counterpart. To overcome these issues researchers proposed different forms of CFST columns that offer the benefits of a square CFST column but with increase load-bearing capacity (Ahmed and Liang, 2020a, 2020b; Ding et al., 2015, 2016; Hassanein and Patel, 2018, He et al., 2019, Liew and Xiong, 2012, Wang et al., 2020; Xu et al., 2016, Yu et al., 2013).

Square concrete-filled double steel tubular (CFDST) columns are made of two thin-walled square steel tubes sitting concentrically and filled with concrete as illustrated in Figure 1(a). The infilled concrete prevents the inner tube from buckling both outwards and inwards, which improves the strength and ductility. Furthermore, the bending stiffness, fire resistance, and seismic resistance of such columns are higher than conventional CFST and double-skin concrete-filled steel tubular (DCFST) column (Ekmekyapar and Al-Eliwi, 2017, Hassanein et al., 2013, 2017; Romero et al., 2015; Wan and Zha, 2016). The hollow outer and inner steel tubes of CFDST columns can have different geometric and material properties and be filled with different strengths of concrete to obtain specific project criteria. Moreover, the square CFDST column can be also used to strengthen existing CFST columns. The earliest applications of CFDST columns found in the literature were the city hall in Wuppertal, Germany reported by Roik and Bergmann (1985). In recent years, the CFDST column was utilized to strengthen the damaged columns of a large glass furnace in southern China (Lin, 2013). However, compared with the wide application of the traditional CFST column, the engineering application of the CFDST column is still limited due to the lack of guidance of design codes. This is mainly due to the relatively few experimental and numerical studies performed on the behavior analysis of CFDST columns.

Figure 1.

Schematic view of circular CFDST column:(a) cross-section view and (b) elevation view.

Wan and Zha (2016), Ekmekyapar and Al-Eliwi (2017), Romero et al. (2015), Ahmed et al. (2019a), and Chang et al. (2013) studied the performance of circular CFDST columns whereas Peng et al. (2011), Wang et al. (2017), Qian et al. (2011, 2014), Ahmed et al. (2019b) investigated the behavior of square CFDST columns with an inner circular hollow section. Numerical models were also developed by researchers such as Wan and Zha (2016), Ahmed et al. (2018, 2019a, 2019b, 2019c, 2020), Wang et al. (2017), Qian et al. (2011, 2014), Ci et al. (2020a, 2020b), and Hassanein et al. (2013, 2017) for such columns. Furthermore, Ci et al. (2021) carried out a series of tests on circular CFDST short columns with an inner square tube. However, research studies on square CFDST columns with an inner square hollow section (SHS) are very limited. Experimental programs on short square CFDST columns were only carried out by Ahmed et al. (2020) and Xiong et al. (2017). Ahmed et al. (2020) tested twenty short square CFDST columns that were either loaded axially or eccentrically. However, the width-to-thickness ratio of the outer tube ( $B_{o} / t_{o}$ ), the width-to-thickness ratio of the inner tube ( $B_{i} / t_{i}$ ) and the concrete strength of the specimens were very low. Furthermore, development of numerical modeling of such columns has been very rare. The only numerical model reported on short square CFDST columns was by Ahmed et al. (2020) based on the fiber model to analyze their eccentric behavior.

Thus, this paper fills the knowledge gaps by investigating the performance of square CFDST stub columns with SHS as inner tube under axial compression load through experimental and numerical analysis. Axial compression tests were carried out on short columns made of high-strength concrete. The sensitivities of important column parameters are examined. The FE model is also developed to study the extensive parameter study and proposed a strength calculation formula that can provide an accurate estimation of their ultimate compressive strengths.

Experimental program

Test specimens

A total of ten specimens were designed for the axial compression test, including seven square CFDST columns, two square CFST columns, and one square double-skin concrete-filled steel tubular (DCFST) column without filling concrete in the core section. The parameters used for the test are as follows: (1) $B_{o} / t_{o}$ ratio of the outer tube ( $B_{o} / t_{o}$ = 36.36, 44.44, and 57.14); (2) $B_{i} / t_{i}$ ratio of the inner tube ( $B_{i} / t_{i}$ = 28.57, 40, and 58.82); (3) width ratio ( $B_{i} / B_{o}$ = 0.3 and 0.5); and (4) concrete strength ( $f_{cu}$ = 60 MPa and 68 MPa). In naming rules of test specimens in Table 1 (i) the letters “CD” at the beginning indicates that the tube type is CFDST column (“CF”-CFST and “CS”-DCFST), (ii) the letters “OT” in the middle represents the change of the outer tube thickness (“IT”-the inner tube thickness, “ID”-internal steel width and “CS”-concrete strength), and (iii) the last number is the sequence number of the parameter change. The nominal dimension ( $B_{o} \times$ L) of all test specimens is 200 mm × 650 mm respectively, where $B_{o}$ is the width of the columns and L is the length of specimens. The height-to-width ratio was designed to be 3.25 to eliminate the influences of end restraint and overall buckling. The geometric dimensions of the test specimens are summarized in Table 1.

Table 1.

Test data of CFDST stub columns.

No.	Specimenlabel	Outer tube			Inner tube			L [mm]	$f_{cu}$ [MPa]	$P_{u, \exp}$ [kN]	$P_{u, FE}$ [kN]	$\frac{P_{u, FE}}{P_{u, \exp}}$
No.	Specimenlabel	$B_{o}$ [mm]	$t_{o}$ [mm]	$B_{o} / t_{o}$	$B_{i}$ [mm]	$t_{i}$ [mm]	$B_{i} / t_{i}$	L [mm]	$f_{cu}$ [MPa]	$P_{u, \exp}$ [kN]	$P_{u, FE}$ [kN]	$\frac{P_{u, FE}}{P_{u, \exp}}$
1	CF-1	200	4.5	44.4	–	–	–	650	60	3307.54	2912	0.88
2	CF-2^#	200	4.5	44.4	–	–	–	650	60	3249.31	2912	0.90
3	CS	200	4.5	44.4	100	2.5	40.0	650	60	2814.55	2885	1.03
4	CD-E	200	4.5	44.4	100	2.5	40.0	650	60	3475.52	3443.72	0.99
5	CD-OT-1	200	3.5	57.1	100	2.5	40.0	650	60	2839.13	3130.59	1.10
6	CD-OT-2	200	5.5	36.4	100	2.5	40.0	650	60	3422.07	3457.27	1.01
7	CD-IT-1	200	4.5	44.4	100	1.7	58.8	650	60	3202.01	3152.82	0.98
8	CD-IT-2	200	4.5	44.4	100	3.5	28.6	650	60	3604.55	3521.69	0.98
9	CD-IW-1	200	4.5	44.4	60	2.5	24	650	60	2974.58	3282.45	1.10
10	CD-CS-1	200	4.5	44.4	100	2.5	40	650	68	3569.33	3645.18	1.02
Mean												1.00
Standarddeviation (SD)												0.07
Coefficient ofvariation (CoV)												0.07

^#Indicates that it was a repeated test specimen.

To construct a CFDST column, two cold-formed square hollow steel tubes were placed concentrically and welded with three steel bars to ensure their concentricity. The bottom of the steel tube was fixed with wood formwork and sealed with soft glue to prevent the exposure of concrete liquid. The formwork of a CFDST column is shown in Figure 2.

Figure 2.

The formworks of square CFDST columns.

Material properties

In order to determine the properties of the tubes, three standard tensile coupons were prepared from each tube and performed the tensile test following the Chinese standard (2010). Table 2 presents the measured yield strength $f_{sy}$ , modulus of elasticity $E_{s}$ , tensile strength $f_{u}$ , and tensile strain $ε_{u}$ , expressed as the average value ± standard deviation of three test coupons. Figure 3 presents the measured stress-strain curves for one coupon sample for each tube.

Table 2.

Material properties of steel tubes obtained from tensile coupon tests.

Tube type	No.	Geometry dimensionof tube B×B×t [mm]	$f_{sy}$ (MPa)	$E_{s}$ (GPa)	$f_{u}$ (MPa)	$ε_{u}$
Outer tube	1	200 × 200 × 3.5	345.4 ± 4.8	204.5 ± 5.2	457.0 ± 1.5	0.19 ± 0.00
	2	200 × 200 × 4.5	356.6 ± 7.2	202.6 ± 6.1	481.5 ± 0.6	0.15 ± 0.01
	3	200 × 200 × 5.5	309.3 ± 4.5	203.1 ± 3.2	437.4 ± 3.5	0.20 ± 0.01
Inner tube	4	60 × 60 × 2.5	338.8 ± 12.3	196.1 ± 7.2	420.9 ± 2.9	0.16 ± 0.00
	5	100 × 100 × 1.7	304.0 ± 1.2	200.8 ± 3.6	376.8 ± 3.9	0.21 ± 0.03
	6	100 × 100 × 2.5	328.8 ± 10.0	194.6 ± 6.1	459.1 ± 8.1	0.15 ± 0.02
	7	100 × 100 × 3.5	334.9 ± 1.2	202.9 ± 4.2	440.0 ± 1.9	0.18 ± 0.01

Figure 3.

Measured stress-strain curves of steel tube materials: (a) outer tube and (b) inner tube.

Ready-mixed concrete was used to construct the CFDST columns. The average compressive concrete strength ( $f_{cu}$ ) was measured from three 150 mm × 150 mm × 150 mm cubes cast at the same time of CFDST columns according to the Chinese standards GB/T 50081-2019 (2019). The mean values of the measured compressive concrete cube strengths ( $f_{cu}$ ) are 60 MPa and 68 MPa.

Test set-up and measurement

The compression experiments of all stub column specimens were carried out by 400-t compression testing machine in the structural laboratory of Beijing University of Technology, China. Figure 4 illustrates the test set-up of specimens and instrumentations. Before commencing loading, the test specimens were hoisted to the loading frame and the center of the test specimens were aligned with the center of the testing machine. Four 600 mm range precise draw-wire displacement sensors were used to measure the end shortenings of the specimens. These displacement sensors were installed between the top and bottom loading plates, respectively at the four corners of the test specimens. To measure the strain distributions of the test specimens, a total of four pairs of strain rosettes were pasted in different positions of mid-height cross-section and different vertical and lateral positions in the inner and outer tubes. The locations of the strain gauges are shown in Figure 5.

Figure 4.

Test setup and instrumentation for test specimens.

Figure 5.

Arrangement and numbering of strain gauges in the test specimens.

The ends of the square CFDST columns were confined by typical steel fixtures to prevent the elephant foot buckling failure which may affect the true performance of such columns. The surface of the top end of the test specimens was not completely flat due to the shrinkage of the concrete and the cutting error of the steel tubes. Therefore, after the test specimens were hoisted to the loading device, the top surface of the column was layered with super hard gypsum and levelled.

An initial load of approximately 300 kN was applied to the specimens to eliminate any possible gap and initial bending effect and then unloaded to 10 kN before finally started collecting the data. At the same time, it is also to check whether the measuring instruments are in good contact. The displacement control method was adopted for all specimens, and the loading rate was 1 mm/min. The applied load, strain, and displacement values were acquired by the DH3821 quasi-static strain test acquisition system.

Experimental results and discussion

Performance index

In order to better quantify the influences of different parameters on the performance of CFDST columns, strength index (SI), ductility index (DI), and residual strength ratio ( $P_{res} / P_{u, \exp}$ ) have been used to evaluate the performance of composite columns (Han et al., 2005, Ekmekyapar and Hasan, 2019). The strength index (SI) is defined by the ratio of the ultimate axial strength ( $P_{u}$ ) of CFDST columns to the sum of the strengths of the individual components (i.e. the core concrete, sandwiched concrete, inner steel tube, and outer steel tube), as expressed in equation (1)

S I = P_{u} / [f_{s y, i} A_{s i} + f_{s y, o} A_{s o} + 0.85 * (f_{c, o}^{'} A_{c, o} + f_{c, i}^{'} A_{c, i})]

(1)

where $A_{so}$ , $A_{si}$ , $A_{c, o}$ , and $A_{c, i}$ are the cross-sectional areas of outer steel tube, inner steel tube, sandwiched concrete, and core concrete, respectively. $f_{sy, o}$ and $f_{sy, i}$ represent the yield strength of the outer steel tube and inner steel tube, respectively. $f_{c, o}^{'}$ and $f_{c, i}^{'}$ represent the cylinder compressive strength of sandwiched concrete and core concrete, respectively.

The ductility index (DI) is defined as the ratio of the $ε_{u, 0.9}$ to $ε_{y}$ ; where $ε_{y}$ is the axial yield strain. $ε_{u, 0.9}$ is the axial strain at 90% of the $P_{u}$ after the specimens experience its ultimate axial strength, as expressed in equation (2). The higher value of the DI represents the better ductility of the specimens.

DI = ε_{u, 0.9} / ε_{y}

(2)

The residual strength ratio ( $P_{res} / P_{u, \exp}$ ) is the ratio of the lowest post-peak strength to the ultimate strength of the column. The higher the value of $P_{res} / P_{u, \exp}$ , the lower the attenuation of the strength of the column.

CFDST columns

The failure modes of all tested CFDST columns are shown in Figure 6(a-g). The local buckling of all CFDST columns occurred after the columns reached the ultimate load. All CFDST columns failed by local outward buckling of the outer square steel tube and the crushing of the concrete at the corresponding position. Figure 6 demonstrates that steel fixtures successfully prevented the columns from the elephant foot buckling failure.

Figure 6.

Failure modes of CFDST specimens: (a) CD-E, (b) CD-OT-1, (c) CD-OT-2, (d) CD-IT-1, (e) CD-IT-2, (f) CD-IW-1, and (g) CD-CS-1.

The measured load-axial strain ( $N - ε_{a}$ ) curves of CFDST columns are presented in Figure 7(a-d) and ultimate loads are tabulated in Table 1. The axial strain ( $ε_{a}$ ) shown in Figure 7(a-d) is the ratio of the average value of the longitudinal displacement (Δ) of the specimens measured by the four displacement sensors to the total height (L) of the specimen. It appears from Figure 7(a-d) that CFDST columns have a large deformation capability. When analyzing the influences of different parameters, it is found that the change of the outer tube thickness from 3.5 mm to 5.5 mm increases the ultimate loads by 20%. However, it should be noted that the ultimate strength of specimen CD-OT-2 was found to be lower than specimen CD-E regardless of having a thicker outer tube. This may be due to the fact that specimen CD-OT-2 was tested a bit earlier than specimen CD-E during the testing program, hence, the actual strength of concrete of specimen CD-E may have achieved higher strength than specimen CD-OT-2 due to additional days of curing. Similarly, when the inner tube thickness increases from 1.7 mm to 3.5 mm, the increase in the maximum axial load is determined as 12.6%.

Figure 7.

Load-axial strain curves of square CFDST specimens: (a) effects of outer tube thickness, (b) effects of inner tube thickness, (c) effects of the width of the inner tube, and (d) effects of the concrete strength.

Comparison with CFST and DCFST stub columns

The specimens CF-1, CF-2, and CS were used for comparison purposes. The modes of failure of the CFST and DCFST columns were the same as that of the CFDST column as shown in Figure 8(a-d). However, in the test process, after the CFST column reached the ultimate load, the outer steel tube had a rapid local buckling deformation associated with shear failure. The failure of the DCFST columns is mainly due to the buckling of the internal tube inwardly as shown in Figure 8(c-d) which reduced their ultimate strength remarkably.

Figure 8.

Failure modes of CFST and DCFST specimens: (a) CF-1, (b) CF-2, (c) CS, and (d) CS.

The measured $N - ε_{a}$ curves for the three types of tested columns including two CFST columns, one CFDST column, and one DCFST column are plotted in Figure 9. The initial stiffness of the CFDST column is higher than that of the CFST and DCFST columns. The maximum axial load of the CFDST column is 6% higher than that of the CFST column and 23.5% larger than that of the DCFST column. Furthermore, the residual strength of square CFDST columns is remarkably higher compared to its CFST and DCFST counterparts.

Figure 9.

Comparisons of the load-axial strain graphs of square CFST, CFDST, and DCFST columns.

In addition, it can be seen from the performance indexes given in Table 3 that almost all CFDST columns have higher performance indexes than CFST and DCFST columns, which also reflects the excellent performance of CFDST columns.

Table 3.

Performance index of the tested columns.

LabelIndex	CF-1	CF-2^#	CS	CD-E	CD-OT-1	CD-OT-2	CD-IT-1	CD-IT-2	CD-IW-1	CD-CS-1
DI	1.45	1.43	1.32	2.86	2.74	2.93	2.86	2.90	3.20	2.34
SI	1.16	1.14	1.03	1.25	1.13	1.22	1.21	1.23	1.02	1.21
$P_{res}$	2265.06	2160.64	1590.36	2801.34	2346.16	2709.21	2353.68	2833.33	2490.38	2760.38
$P_{res} / P_{u, \exp}$	0.68	0.66	0.57	0.81	0.83	0.79	0.74	0.79	0.84	0.77

Finite element (FE) modeling

General

Due to the cost and time associated with the experimental research, it is hard to investigate the sensitivities of a wide range of structural parameters that influence the performance of square CFDST stub columns loaded concentrically. Therefore, finite element models have been developed using the FE program ABAQUS/Standard to investigate the behavior of square CFDST columns. The material laws of concrete and steel adopted in the finite element modeling are described in the following sections.

Constitutive models of concrete

Figure 10 shows the material laws of concrete proposed by Hu et al. (2003) adopted to simulate the material behavior of confined concrete. As shown in the figure, the stress-strain curve is expressed in two parts. The first part is the parabola rising section (i.e. segment OA) which can be modeled using the expressions proposed by Saenz (1964). It is shown in equations (3) to (5)

Figure 10.

Stress-strain curves for the confined concrete model in CFDST columns.

σ_{c} = \frac{E_{c} ε_{c}}{1 + (R + R_{E} - 2) (\frac{ε_{c}}{ε_{cc}^{'}}) - (2 R - 1) {(\frac{ε_{c}}{ε_{cc}^{'}})}^{2} + R {(\frac{ε_{c}}{ε_{cc}^{'}})}^{3}}

(3)

R = \frac{R_{E} (R_{σ} - 1)}{{(R_{ε} - 1)}^{2}} - \frac{1}{R_{ε}}

(4)

R_{E} = \frac{E_{c} ε_{cc}^{'}}{f_{cc}^{'}}

(5)

where $R_{σ}$ and $R_{ε}$ is suggested as four by Hu et al. (2003). $f_{cc}^{'}$ and $ε_{cc}^{'}$ are the concrete compressive strength and the corresponding strain. $E_{c}$ is the modulus of concrete, suggested by Lim and Ozbakkaloglu (2014) as:

E_{c} = 4400 \sqrt{f_{c, e}^{'}} (MPa)

(6)

where $f_{c, e}^{'}$ represents the strength of unconfined concrete, which is affected by the size of the column, the quality of concrete, and the loading rate. $f_{c, e}^{'}$ is expressed as $f_{c, e}^{'} = r_{c} f_{c}^{'}$ , where $f_{c}^{'}$ is the concrete cylinder strength and $r_{c}$ is the strength reduction factor (Liang and Fragomeni 2009), taken as:

r_{c} = 1.85 D_{c}^{- 0.135} (0.85 \leq r_{c} \leq 1)

(7)

where $D_{c}$ is the width of concrete taken as $(B_{i} - 2 t_{i})$ and ( $B_{o} - 2 t_{o} - B_{i}$ )/2, respectively for the core and sandwiched section.

The expressions given by Lim and Ozbakkaloglu (2014) are used to estimate the compressive strength of confined sandwiched concrete $f_{cc, o}^{'}$ and corresponding strain $ε_{cc, o}^{'}$ defined as :

f_{cc, o}^{'} = r_{c} f_{c}^{'} + 5.2 {(r_{c} f_{c}^{'})}^{0.91} {(\frac{f_{l, o}}{r_{c} f_{c}^{'}})}^{a} where a = (r_{c} f_{c}^{'})^{- 0.06}

(8)

ε_{cc, o}^{'} = ε_{c}^{'} + 0.045 {(\frac{f_{l, o}}{r_{c} f_{c}^{'}})}^{1.15}

(9)

ε_{c}^{'} = \frac{{(r_{c} f_{c}^{'})}^{0.225}}{1000}

(10)

where $ε_{c}^{'}$ is the strain at $f_{c}^{'}$ , $f_{l, o}$ is the lateral confining pressure on the sandwiched concrete and quantified using the formulas proposed by Hu et al. (2003) written as

f_{l, o} = {\begin{matrix} [0.055048 - 0.001885 (\frac{B_{o}}{t_{o}})] f_{y, o} for 17 \leq \frac{B_{o}}{t_{o}} \leq 29.2 \\ 0 for 29.2 \leq \frac{B_{o}}{t_{o}} \leq 150 \end{matrix}

(11)

The compressive strength of confined core concrete $f_{cc, i}^{'}$ and corresponding strain $ε_{cc, i}^{'}$ are proposed as

\begin{matrix} f_{cc, i}^{'} = r_{c} f_{c}^{'} + 5.2 {(r_{c} f_{c}^{'})}^{0.91} {(\frac{f_{l, o} + f_{l, i}}{r_{c} f_{c}^{'}})}^{a} \\ where a = (r_{c} f_{c}^{'})^{- 0.06} \end{matrix}

(12)

ε_{cc, i}^{'} = ε_{c}^{'} + 0.045 {(\frac{f_{l, o} + f_{l, i}}{r_{c} f_{c}^{'}})}^{1.15}

(13)

$f_{l, i}$ is the lateral confining pressure to the core concrete and its basic form is the same as equation (11), but it needs to replace $B_{o} / t_{o}$ and $f_{y, o}$ with $B_{i} / t_{i}$ and $f_{y, i}$ respectively.

The part AB of the stress-strain curve depicted in Figure 10 is determined by:

σ_{c} = r k_{3} f_{cc}^{'} + (\frac{11 ε_{cc}^{'} - ε_{c}}{11 ε_{cc}^{'} - ε_{cc}^{'}}) (f_{cc}^{'} - r k_{3} f_{cc}^{'})

(14)

where $k_{3}$ denotes the concrete material degradation parameters and proposed by Hu et al. (2003) as:

k_{3} = {\begin{matrix} 1 f o r 21.7 \leq \frac{B_{o}}{t_{o}} \leq 40 \\ 0.000039 {(\frac{B_{o}}{t_{o}})}^{2} - 0.010085 (\frac{B_{o}}{t_{o}}) + 1.3491 f o r 40 \leq \frac{B_{o}}{t_{o}} \leq 150 \end{matrix}

(15)

The value of parameter r is 1.0 and 0.5 for concrete cube strengths of 30 MPa and 100 MPa, respectively. The value for cube strength between 30 MPa and 100 MPa can be calculated using the linear interpolation.

In FE analysis, the concrete material is considered elastic until it reaches 0.5 $f_{cc}^{'}$ as shown in Figure 10. This part is represented by the tangent modulus from calculation and Poisson’s ratio of 0.2 in the FE. The Drucker-Prager plasticity model is used for simulating the confined concrete material behavior after the linear response. The angle of friction and the flow stress ratio are taken as 20° and 0.8, respectively (Ci et al., 2020a, 2020b).

Constitutive models of steel

The material laws of structural steel for CFDST columns were simulated using the model suggested by Ramberg and Osgood (1943) expressed as:

ε = \frac{σ}{E} + 0.002 {(\frac{σ}{F_{ty}})}^{n}

(16)

The elastic modulus ( $E_{0}$ ) and Poisson’s ratio ( $μ_{s}$ ) of steel tube are taken as 203 GPa and 0.3 GPa, respectively.

The engineering stress-strain curves were converted into the true stress-strain curves and introduced in the FE modeling. The conversions were made using the following equations (17) and (18)

σ_{true} = σ_{nom} (1 + ε_{nom})

(17)

ε_{pl} = \ln (1 + ε_{nom}) - \frac{σ_{true}}{E_{0}}

(18)

Model development

Surface-to-surface interactions were used in the ABAQUS models. Steel tube surfaces and the concrete surfaces were chosen as the master and slave surface, respectively. The contact pressure-overclosure model was used in the normal direction, and the “normal behavior” was selected as “hard contact”. Coulomb friction model with a coefficient of friction of 0.25 (Hassanein and Patel, 2018) in tangential direction was employed. Figure 11 shows a total of three boundary conditions were used for modeling each CFDST column l. The first boundary condition RP2 is coupled with the bottom surface of the column to fix all degrees of freedom. The second boundary condition was used to enable movement The second boundary condition coupled with the top surface of the columns is used to impose a displacement of −35 mm in the Z direction of the specimen, which fixes the displacement in the X and Y directions. The third boundary condition was used to simulate the constraint effect of steel fixture on both ends of the column, and to fix the X and Y displacements of the two ends of the column.

Figure 11.

Load and boundary condition of square CFDST columns.

Considering that the shell element applied to steel tube is easy to cause convergence problems, the eight-node reduced integration linear solid elements were adopted for steel tubes while three translation degrees of freedom, C3D8R were used to mesh concrete elements. From the sensitivity analysis, it is found that the exact results and reasonable calculation time can be obtained when the size for the mesh for steel tube and concrete is about 16 mm and 18 mm respectively.

Validation of the FE model

The accuracy of the developed FE model is validated against the test results reported in the paper. It should be noted that the concrete cylinder strength $f_{c}^{'}$ is used in the FE model, while the concrete cube strength ( $f_{cu}^{'}$ ) is given in the experimental study. Therefore, the conversion between concrete cube strength and cylinder strength is carried out through equation (19).

f_{c}^{'} = [0.76 + 0.2 lo g_{10} \frac{f_{cu}^{'}}{19.6}] f_{cu}^{'}

(19)

The maximum compressive loads of the stub columns obtained from the FE analysis ( $P_{u, FE}$ ) are compared with test results ( $P_{u, Exp}$ ) in Table 1 where it is observed that the FE model can yield the test ultimate loads with reasonable accuracy. However, the CFST specimens obtained much higher ultimate load capacities than the predicted values based on the FE analysis. This may be due to the fact that these CFST specimens were tested at the very last during the testing program, hence, the actual strength of concrete for these specimens may have been much higher than 60 MPa due to the additional days of curing. The average $P_{u, FE} / P_{u, Exp}$ is estimated at 1.00 with a corresponding CoV value of 0.07. The $N - ε_{a}$ curves of the test columns are compared against the FE results in Figure 12. Good agreement can be observed with the test results.

Figure 12.

Comparisons between experimental and FE load-axial strain curves: (a) CD-E, (b) CD-OT-1, (c) CD-OT-2, (d) CD-IT-1, (e) CD-IT-2, (f) CD-IW-1, (g) CD-CS-1, (h) CF, and (i) CS.

Parametric study

The validated FE model is used to analyze the performance of the CFDST stub columns for a wide range of column parameters. To avoid the influence of overall instability and end conditions of columns, the length of all models is three times the width ( $B_{o}$ ) of columns cross-section. Table 4 presents the details of the columns used for parametric studies. The concrete strength used for the parameter study is considered to be cylindrical strength which covers both the range of normal concrete and high-strength concrete according to EN 1993-1-1 (2004). The yield strength of the steel used is taken as 235 MPa, 355 MPa, and 460 MPa respectively, which correspond to the yield strength of the steel commonly used in the industry, such as S235, S355, and S460 in the European code EN 10027-1 (2005) and Q235, Q355, and Q460 in the Chinese code GB/T 1591-2018 (2018).

Table 4.

The material properties and geometric dimensions of reference columns used for parametric study.

Group	Specimen	Outer tube			Inner tube			L (mm)	B_i/B_o	$f_{c, i}^{'}$ $(MPa)$	$f_{c, o}^{'}$ (MPa)	$f_{sy, i}$ $(MPa)$	$f_{sy, o}$ $(MPa)$
Group	Specimen	$B_{o}$ (mm)	$t_{o}$ (mm)	$B_{o} / t_{o}$	$B_{i}$ (mm)	$t_{i}$ (mm)	$B_{i} / t_{i}$	L (mm)	B_i/B_o	$f_{c, i}^{'}$ $(MPa)$	$f_{c, o}^{'}$ (MPa)	$f_{sy, i}$ $(MPa)$	$f_{sy, o}$ $(MPa)$
1 (CS)	CS1	500	8	62.5	270	6	45	1500	0.54	30	30	355	355
	CS2	500	8	62.5	270	6	45	1500	0.54	50	50	355	355
	CS3	500	8	62.5	270	6	45	1500	0.54	70	70	355	355
	CS4	500	8	62.5	270	6	45	1500	0.54	90	90	355	355
	CS-I1	500	8	62.5	270	6	45	1500	0.54	30	50	355	355
	CS-I2	500	8	62.5	270	6	45	1500	0.54	70	50	355	355
	CS-I3	500	8	62.5	270	6	45	1500	0.54	90	50	355	355
	CS-O1	500	8	62.5	270	6	45	1500	0.54	50	30	355	355
	CS-O2	500	8	62.5	270	6	45	1500	0.54	50	70	355	355
	CS-O3	500	8	62.5	270	6	45	1500	0.54	50	90	355	355
2 (OT)	OT-W1	600	8	75	270	6	45	1800	0.45	50	50	355	355
	OT-W2	550	8	68.75	270	6	45	1650	0.49	50	50	355	355
	OT-W3	450	8	56.25	270	6	45	1350	0.60	50	50	355	355
	OT-T1	500	4	125	270	6	45	1500	0.54	50	50	355	355
	OT-T2	500	6	83.33	270	6	45	1500	0.54	50	50	355	355
	OT-T3	500	10	50	270	6	45	1500	0.54	50	50	355	355
3 (IS)	IT-T1	500	8	62.5	321.8	5	135	1500	0.64	50	50	355	355
	IT-T2	500	8	62.5	233.3	7	67.5	1500	0.47	50	50	355	355
	IT-T3	500	8	62.5	206.0	8	33.75	1500	0.41	50	50	355	355
4 (SS)	SS1	500	8	62.5	270	6	45	1500	0.45	50	50	235	235
4 (SS)	SS2	500	8	62.5	270	6	45	1500	0.45	50	50	460	460

Effects of the strength of concrete ( $f_{c}^{'}$ )

The specimens in Group 1 had the concrete increased from 30 MPa to 90 MPa. The $N - ε$ graphs of CFDST columns with varying strength of concrete are shown in Figure 13(a-c). Considering that different strengths of concrete can be poured within the hollow sections of CFDST columns, three cases are considered. For the first case (CS1–CS4), the strength of concrete is the same for both core and sandwiched sections, and the concrete strength increased from 30 MPa to 50 MPa, 70 MPa, and 90 MPa. The $N - ε$ curves displayed in Figure 13(a) illustrate that the maximum axial load increases with the increase of the strength of concrete. The ultimate strength of columns increased by 31.0%, 61.9%, and 92.7% respectively when $f_{c}^{'}$ increased from 30 MPa to 50 MPa, 70 MPa, and 90 MPa. However, as $f_{c}^{'}$ increases from 30 MPa to 90 MPa, the ductility index of columns decreases from 4.45 to 2.63. In the second case (CS-I1, CS2, CS-I2, and CS-I3), the strength of sandwiched concrete remains constant at 50 MPa, and the strength of the concrete at the core section ( $f_{c, i}^{'}$ ) changes. When $f_{c, i}^{'}$ increases from 30 MPa to 90 MPa, the maximum axial load increases only by 20.3%. But the ductility of the column hardly changes. In the last case, while $f_{c, i}^{'}$ was remained constant at 50 MPa, the strength of concrete for sandwiched section ( $f_{c, o}^{'}$ ) for columns CS-O1, CS2, CS-O2, and CS-O3 increased from 30 MPa to 90 MPa. From Figure 13(c), it is observed that the maximum axial load increased significantly by increasing $f_{c, o}^{'}$ . The ductility of the column CS-O3 is only 2.62 while it is 4.06 for column CS-O1.

Figure 13.

Sensitivities of the strength of concrete on the $N - ε$ curves of CFDST columns: (a) changing concrete strength for sandwiched and core concrete together, (b) changing concrete strength for core concrete only, and (c) changing concrete strength for sandwiched concrete only.

Effects of width-to-thickness ratio ( $B_{o} / t_{o}$ ) of the outer tube

The changes in the $B_{o} / t_{o}$ ratio of the outer tube is investigated by changing the width and thickness of the column. Figure 14(a) presents the $N - ε$ curves of the columns utilized for the first case (columns OT-W1, OT-W2, CS2, and OT-W3) where the width of the tube was changed. The maximum axial load of the column increases by 52.5% by changing the width of the outer tube from 450 mm to 600 mm. However, for the second case (columns OT-T1, OT-T2, CS2, and OT-T3), when the outer tube thickness increased from 4 mm to 10 mm, the maximum axial load of the column only increased by 21.9%, as shown in Figure 14(b). Increasing the outer tube thickness also increases the ductility index of columns from 3.05 to 4.07.

Figure 14.

Sensitivities of $B_{o} / t_{o}$ ratio of outer tube on the $N - ε$ curves of CFDST columns: (a) by changing the width of the outer tube, and (b) by changing the thickness of the outer tube.

Effects of width-to-thickness ratio ( $B_{i} / t_{i}$ ) of inner tube

Considering the inner tube is restrained from inward and outward buckling, the influences of the $B_{i} / t_{i}$ ratio of the inner steel tube is investigated for the same inner steel area. For columns IT-T1, CS2, IT-T2, and IT-3 in Table 4, the thickness ranged from 5 mm to 8 mm. However, the area of the column remains unchanged, which leads to the variation of the diameter thickness ratio of the inner tube between 25.75 and 64.36. From the N-ε curve of the column shown in Figure 15, it can be seen that increasing the $B_{i} / t_{i}$ ratio of the inner tube has little effect on the overall performance of the column. When the $B_{i} / t_{i}$ ratio of the column increases from 25.75 to 64.36, the maximum axial load increases by only 1.84%.

Figure 15.

Sensitivities of $B_{i} / t_{i}$ ratio of inner tube on the $N - ε$ curves of CFDST columns.

Effects of steel yield stress ( $f_{y}$ )

Columns SS1, CS2, and SS2 in Table 4 had yield strength of steel tube varied from 235 MPa to 460 MPa to examine the influences of the steel yield stress. The $N - ε$ graphs of these columns are given in Figure 16. The $N - ε$ curves show that the increase in steel yield strength significantly increases the stiffness and maximum axial load of the columns. The ultimate strength of column CS2 and SS2 is 16.0% and 29.5% higher than column SS1.

Figure 16.

Sensitivities of steel yield stress on the $N - ε$ curves of CFDST columns.

Strength enhancement compared to CFST columns

To compare the strength enhancement of CFDST columns due to the utilization of the inner steel tube, comparisons are made for CFDST columns with various column parameters listed in Table 4 with the corresponding CFST columns for the same steel area. The comparisons between the ultimate strengths of CFDST and CFST columns are presented in Figure 17. It is seen that CFDST columns have a higher ultimate capacity than CFST columns in all cases.

Figure 17.

Strength enhancement of CFDST columns compared to CFST columns for various column parameters.

Design formula

Based on the numerical analysis and the existing studies of CFDST columns Ci et al. (2020a, 2020b), a new calculation formula for predicting the maximum axial load of square CFDST stub columns is suggested as

P_{u, cal} = f_{sy, i} A_{s, i} + f_{sy, o} A_{s, oe} + f_{cc, o}^{'} A_{c, o} + f_{cc, i}^{'} A_{c, i}

(20)

where $A_{s, i}$ , $A_{c, o}$ , and $A_{c, i}$ are the cross-sectional area of the inner steel tube, sandwiched concrete and core concrete, respectively, while $A_{s, oe}$ is the effective cross-sectional area of the outer steel tube.

The accuracy of the calculation formula proposed is verified by comparing the maximum axial load of the tested columns reported in this paper, test specimens SS1 and SS6 reported in reference Ahmed et al. (2020) as well as the reference columns used for the parameter study. The comparisons of the columns are presented in Figure 18. A good correlation with an average value of $P_{u, cal} / P_{u, \exp / FE}$ is 0.95 obtained from statistical analysis.

Figure 18.

Verification of the calculation formula.

Conclusions

This study examines the axial performance of square CFDST columns loaded concentrically. A series of tests carried out on such columns are reported and discussed. A FE model is also developed to investigate the influences of various structural parameters. The numerical model is validated against the experimental results and used to study the effects of different parameters on the column performance. Based on this investigation, the following conclusions can be drawn:

The experimental and FE results indicate that the square CFDST columns with inner SHS have improved ultimate strength and ductility when compared with the conventional CFST column and DCFST column.

Based on the parametric study, it is found that the strength of concrete especially the strength of sandwiched concrete has a more obvious effect on the column axial performance.

The $B_{o} / t_{o}$ ratio has a significant influence on the column axial performance. However, the influences of $B_{i} / t_{i}$ is less significant compared to the $B_{o} / t_{o}$ ratio.

The steel tube with higher steel yield stress can significantly improve the overall performance of the columns.

The proposed calculation formula can provide reasonable predictions of their ultimate axial load, therefore, can be used to design such a column.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Mizan Ahmed

Shicai Chen

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Experimental and numerical investigations of square concrete-filled double steel tubular stub columns

Abstract

Keywords

Introduction

Experimental program

Test specimens

Material properties

Test set-up and measurement

Experimental results and discussion

Performance index

CFDST columns

Comparison with CFST and DCFST stub columns

Finite element (FE) modeling

General

Constitutive models of concrete

Constitutive models of steel

Model development

Validation of the FE model

Parametric study

Effects of the strength of concrete ( f c ′ )

Effects of width-to-thickness ratio ( B o / t o ) of the outer tube

Effects of width-to-thickness ratio ( B i / t i ) of inner tube

Effects of steel yield stress ( f y )

Strength enhancement compared to CFST columns

Design formula

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

References

Effects of the strength of concrete ( $f_{c}^{'}$ )

Effects of width-to-thickness ratio ( $B_{o} / t_{o}$ ) of the outer tube

Effects of width-to-thickness ratio ( $B_{i} / t_{i}$ ) of inner tube

Effects of steel yield stress ( $f_{y}$ )