Behavior of multi-cell concrete-filled steel tube columns under axial load: Experimental study and calculation method analysis

Journal of Building Engineering 28 (2020) 101099

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Behavior of multi-cell concrete-filled steel tube columns under axial load: Experimental study and calculation method analysis Fei Yin , Su-Duo Xue , Wan-Lin Cao *, Hong-Ying Dong , Hai-Peng Wu College of Architecture and Civil Engineering Beijing University of Technology, Beijing, 100124, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Multi-cell concrete-filled steel tube Cross-section structures Axial compression Calculation model Constitutive relationship of concrete

Multi-cell concrete-filled steel tube (MCFST) columns are widely used in super high-rise buildings. However, related studies on their axial compression behavior are limited. In order to propose calculation model and design advice, three concrete-filled steel tube column samples were designed and constructed for testing. The samples varied in their cross-sections (i.e., basic type with reinforcement cage, simplified type without reinforcement cage, and reinforced type with circular steel tube). Damage evolution, load-bearing capacity, stiffness degra­ dation, ductility, and restoration capacity were analyzed to investigate the effect of different sections. Experi­ mental results showed that the reinforcement cage effectively improved the bearing capacity. Due to the excellent mechanical behavior of the concrete-filled circular steel tube, the comprehensive behaviors including the bearing capacity, ductility, and restoration capacity of the samples improved. The confinement mechanism of MCFST was analyzed based on the test results. A “separation model” was proposed, followed by calculating the load–deformation curves of 12 samples with various shapes and structures, which considered that the MCFST was composed of several single cells, and calculating the constitutive relationship of concrete in every cell individually. Bearing capacity errors were approximately � 5%, and errors of ultimate deformation were approximately � 15%, showing good agreement with the test results. This data indicates that the constitutive relationship of concrete based on the “separation model” is reasonable. Furthermore, the ultimate state of every individual MCFST cell was calculated based on the “separation model,” indicating that the model can be used to assist in MCFST design.

1. Introduction Super high-rise buildings have quickly developed across the world in recent years. Concrete-filled steel tube (CFST) columns are widely used in super high-rise buildings as longitudinal members to bear the axial load owing to their excellent bearing capacity, deformation capacity, and fire resistance. In 1960s, Klopple and Goder investigated CFST columns [1]. Until recently, the behaviors of square, circular, rectan­ gular, and some irregular shape CFST columns have been comprehen­ sively investigated [2,3]. In the CFST column, the steel tube provides confinement effect on concrete. Therefore, the constitutive relationship of concrete in CFST members, such as ultimate strength and ultimate strain, is different from that of unconfined concrete. Susantha et al. [4] established the uniaxial stress–strain relationship of confined concrete, considering the width-thickness ratio in circular, square, and octagonal tubes and pro­ posed a stress–strain equation for confined concrete based on the RC

column suggested by Mander [5]. Hu et al. [6] established a stress–strain relationship of confined concrete in circular and square tubes based on the uniaxial stress–strain relationship provided by Saenz [7]. Cai et al. [8–10] designed several rectangular (square) and T-shaped CFST col­ umns with binding bars and established a constitutive relationship of confined concrete considering the confinement effect of binding bars. Cross-section structures such as steel sections or reinforcement cages will affect the behavior of CFST columns under axial compression. Wang et al. [11] conducted axial compression experiments on 15 concrete-filled circular steel tube columns with steel sections and one column without steel sections, and proposed a calculation method of ultimate strength. The experimental results showed that the strength, ductility, and energy absorption of the CFST columns with steel sections were significantly enhanced. Dong et al. [12] investigated uniaxial compression behavior of concrete-filled rectangular steel tube columns with longitudinal stiffeners. The results showed that the longitudinal stiffeners can improve the stiffness, bearing capacity, ductility, and

* Corresponding author. E-mail address: [email protected] (W.-L. Cao). https://doi.org/10.1016/j.jobe.2019.101099 Received 17 September 2019; Received in revised form 28 November 2019; Accepted 29 November 2019 Available online 2 December 2019 2352-7102/© 2019 Elsevier Ltd. All rights reserved.

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energy dissipation. Wang et al. [13] designed four different arrange­ ments of reinforcement stiffener and analyzed the confinement mechanism. The design of CFST column should not only meet the requirements of working behavior, but also aesthetic criteria of buildings. Therefore, some special-shaped cross-sections have been suggested and investi­ gated. Liu et al. [14] conducted axial compression experiment of six L-shaped CFST stub columns and twelve T-shaped CFST stub columns. Based on the experimental results and finite element analysis, design formulae for sectional bearing capacity and stability bearing capacity were proposed. Some other shapes such as elliptical [15–17], hexagonal [18], and octagonal [19] were also suggested and analyzed. When used in super high-rise buildings, CFST columns usually have a large cross-sectional area. The hydration heat of massive concrete is a severe problem in case of single cell CFST columns. Therefore, multi-cell concrete-filled steel tube (MCFST) columns are used in many buildings such as The China Zun and 117 Tower in China (Fig. 1). In MCFST col­ umns, concrete is separated into several small cells, thus effectively reducing the hydration heat. Tu et al. [20] have studied the hysteretic behavior of multi-cell T-shaped CFST columns. The results showed that the multi-cell T-CFST showed better bearing capacity, ductility and energy dissipation capacity compared to conventional T-shaped CFST column. FEM models were established to analyze the effect of different parameters. Sui et al. [21] conducted experimental studies on the behavior of multi-cell T-shaped CFST columns subjected to biaxial eccentric compression. FEM models and fiber element models were established to analyze the factors influencing ultimate load. Tu et al. [22] experimentally studied the axial compression behavior of multi-cell T-shaped CFST columns and calculated the ultimate bearing capacity using different codes. The results showed that the calculated values were much smaller than the measured values. Liu et al. [23,24] researched the mechanical behavior of three-cell integrated and five-cell L-shaped multi-cell CFST columns under axial compression and established FEM model to analyze different parameters. Zhang et al. [25] conducted the experimental studies of hexagonal multi-cell CFST columns under axial compression and proposed a simplified method to estimate the bearing capacity. However, the sizes of samples in the previous studies were usually small and there was no specific stress-strain relationship for concrete in the MCFST. Hence, three large-size 13-cells MCFST columns were designed to investigate the effect of different structures on the axial compression behavior of MCFST columns and establish the calculation model of the deformation–load curves to provide useful insight for MCFST design.

jin 117 tower (Fig. 1) as well. Therefore, investigating the generic rule according to this type of MCFST will be a good choice. Three MCFST columns were designed based on The China Zun. The outside dimensions of the three samples were the same. The long axis was 1060 mm, the short axis was 476 mm, and the cross-sectional area was 3.78 � 105 mm2. The three samples mainly differed in their cross-section structures. Sample CFST-R was the basic type with reinforcement cages; sample CFST-S was the simplified type without reinforcement cages; sample CFST-C was the reinforced type with a circular steel tube set in the corner compared to sample CFST-R. The multi-cell steel tubes were welded with 4 mm steel plates. The longitudinal stiffener was 23 mm wide and 3 mm thick. The longitudinal reinforcement and stirrup of the reinforcement cages was Φ2mm. Several studies on concrete-filled double-tube columns have been reported [26,27]. In these studies, the ratio of the diameter of the circular steel tube and the width of the outer steel tube was in the range 0.5–0.8 and the diameter–thickness ratio of the circular steel tube was in the range 29.2–92.2. The size of the cir­ cular steel tube in this study was Φ90 mm � 4 mm, which is close to the lower limit in order to highlight the effect of the circular steel tube. Horizontal diaphragms were set in every cell, and the vertical space was 300 mm. The horizontal diaphragms were 4-mm steel plates, and 10-mm circular holes were set in the horizontal diaphragms for the longitudinal reinforcements to pass through. The net width of the horizontal di­ aphragms was 30 mm. Headed studs were set in every cell to enhance the bonding behavior between concrete and steel tube. The details of the cross-sections are shown in Fig. 2, and the steel ratios of the cross-sections are listed in Table 1. In order to prevent local damage and ensure that the axial load was uniformly transferred, two load-ends were set and the column embedded in the load-ends. The load-end was a steel box with di­ mensions of 1120 � 640 � 250 mm3 filled with concrete, and the centroid point of the box was coincident with the cross-section. Two spherical hinges were set at the load-ends. Fig. 3 shows the details of external design. 2.2. Measured mechanical properties of materials The actual size of the steel was different from the nominal size. Actual thickness of the 4 mm steel plate was 3.75 mm and actual thickness of the 3 mm steel plate was 2.85 mm. Actual diameter of the Φ2mm rebar was 2.16 mm and actual size of the circular steel tube was Φ89 mm � 3.85 mm. The average cubic compressive strength of the concrete was measured using a 150 mm � 150 mm � 150 mm cube. The measured value was fcu,m ¼ 51.2 MPa. The relationship between the cylinder strength and fcu,m was f’c ¼ 0.815fcu,m. Measured mechanical properties of all materials are listed in Table 2. Here, fy is the measured yield strength of steel. fu is the measured ultimate strength of steel. Es and Ec are the measured elastic modulus values of steel and concrete. δ is the elongation of steel.

2. Experimental design 2.1. Sample design MCFST consists of several single cells with different shapes. Hence, the shape of MCFST is arbitrary. A particular type of MCFST column has been used in The China Zun, and a similar cross-section was used in Tian

Fig. 1. MCFST columns in super high-rise buildings. 2

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Fig. 2. Details of cross-sections of the CFST samples. Table 1 Steel ratio of cross-sections. Sample

CFSTR CFST-S CFST-C

Features

Area/ � 104mm2(Steel ratio) Total Area (At)

Structural steel (As)

Longitudinal stiffener (Al)

Longitudinal reinforcement (Ar)

Circular steel tube (Acs)

Concrete (Ac)

Reinforcement cage

37.8

2.3 (6.59%)

0.41 (1.17%)

0.15 (0.41%)

0

34.9

Without reinforcement cage Circular steel tube

37.8

2.3 (6.59%)

0.41 (1.17%)

0

0

35.1

37.8

2.3 (6.59%)

0.41 (1.17%)

0.15 (0.41%)

0.21 (0.60%)

34.8

2.3. Loading program and measuring equipment

two types of strain gauges; the vertical strain gauge was used to measure the vertical compressive strain, and the horizontal strain gauge was used to measure the lateral tensile strain. Strain gauge arrangement is shown in Fig. 3, where h is the horizontal strain gauge. A hydraulic jack with the maximum load of 40,000 kN was used for the test. The loading program adopted load–displacement control [28, 29], as shown in Fig. 4. The load control was adopted before the samples exhibited an obvious yield (obvious bending in load–deformation curves). The load was increased by 2000 kN per level (~10% of the

Four displacement meters (DMs) labeled 1, 2, 3, and 4 were set be­ tween the two loading-ends to measure the integral deformation to monitor the collapse of samples (Fig. 3). However, due to the cyclo-hoop effect of the loading-ends, DM5 and DM6 were set in the middle span (1200 mm height) of the samples. Each sample was separated into eight vertical layers by the hori­ zontal diaphragms. Strain gauges were set in every layer. There were 3

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Fig. 3. Details of external design of the samples and measured equipment arrangement.

ultimate load). Then, the loading program was changed to displacement control. The increment between the two levels was Δy (Δy is the displacement when obvious yield appeared). The load was unloaded to 2000 kN before testing the next level to prevent collapse of samples. The test was completed when the load decreased to 85% of the ultimate load.

Table 2 Measured mechanical properties of materials. Materials

Application

Actual thickness (diameter)/ mm

fy/ MPa

fu(fcu, m)/ MPa

Es(Ec)/ MPa

δ/%

4 mm steel plate

Vertical steel plate/ Horizontal diaphragm Reinforcement cage Circular steel tube

3.75

393.1

527.8

1.9 � 105

28.6

2.2 � 105 2.0 � 105

16.4

Φ2mm bar Φ90 mm �4 mm circular steel tube 3 mm steel plate concrete

3. Experimental results and analysis 3.1. Damage evolution

2.16

438.1

457.2

3.85

361.3

531.7

Longitudinal stiffened

2.95

338.3

471.0

2.0 � 105

27.0

Core concrete

–

–

51.2

3.45 � 104

–

The cross-section of the three samples was an octagonal shape comprising 13 cells. The numbers of every side and cell are shown in Fig. 5. Sample CFST-R and sample CFST-S displayed similar damage evolution, while the damage evolution of sample CFST-C had several differences compared to sample CFST-R.

17.1

3.1.1. Damage evolution of CFST-R and CFST-S When the equivalent strain reached 2.4 � 103 με, 45� slip lines first appeared on S6, then extended along the long axis. The 45� slip lines appeared not only in the middle of the layer, but also at the horizontal diaphragm. Fig. 6 shows the failure model of CFST-R and CFST-S. After the 45� slip lines appeared, obvious buckling of the steel tube first appeared between the third and sixth layers in cell 6, and then devel­ oped vertically. This indicates that the cyclo-hoop effect of the load-end significantly affected the damage evolution. Therefore, it is reasonable to set the displacement meter in the middle span (layer 3 to layer 6) of

Fig. 4. Loading program. Fig. 5. Numbers of every cell and side. 4

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Fig. 6. Failure model of CFST-R and CFST-S.

samples. The next obvious buckling appeared in cell 5, still between layers 3 and 6. Obvious buckling of S1 was earlier than that of S6 because S1 was longer than S6, and the width–thickness ratio of S1 was also greater than S6. Therefore, S1 would be easier to buckle. As the moment of inertia of the cross-section along the short axis was smaller than that along the long axis, initial damage appeared along the short axis. Stress was redistributed when cells 5 and 6 were gradually damaged, which then developed along the long axis. When the sample reached ultimate point, the weld in cell 1 cracked, crushing the filled concrete, which indicated that the steel tube and concrete cooperated well.

ultimate point. In sample CFST-C, cell 1 was at the protruded position of the cross-section with stress concentration, while the circular steel tube decreased the effect of stress concentration. Fig. 7 shows the failure model of CFST-C. The final damage plane of every sample was an inclined plane along the short axis. It was similar to the failure model of stub columns, indicating that the concrete conjoined well with steel tube, and the strengths of the concrete and the steel were fully utilized. The obvious buckling was concentrated in the middle span of the sample, while some buckling appeared at the interface of the load-end and column, indi­ cating that this type of load-end can uniformly distribute the load in the steel tube and concrete. 45� slip lines appeared both in the middle of the layers and the position of horizontal diaphragm. 45� slip lines are related to the tensile yield of steel. Under the axial compression load, the steel tube suffered vertical compression and lateral tension, while the horizontal diaphragm suffered lateral tension, indicating that vertical steel plate and horizontal diaphragm have high tension stress levels and provide an effective confinement effect on the concrete.

3.1.2. Damage evolution of CFST-C The final damage plane of the CFST-C was also an inclined plane. The main difference between CFST-C and CFST-R was the damage evolution of cell 1. 45� slip lines first appeared at S6, while the first slip line of CFST-C appeared concurrently at S6 and S3. It is likely that the circular steel tube increases the integrity of the concrete, and the steel tube in cell 1 provides confinement effect on the concrete earlier. The weld of cell 1 in CFST-R cracked when the sample reached ultimate point. However, the weld of cell 1 in CFST-C showed buckling only at the

Fig. 7. Failure model of CFST-C. 5

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3.2. Load-deformation curves

3.3. Bearing capacity analysis

The F–Δ curves of the three samples are listed in Fig. 8. The bottom abscissa axis (Δ) is the average value of DMs 5 and 6. The top abscissa axis (Δ/1200 � 106) is equivalent strain. The fitted lines show that loading stiffness and unloading stiffness did not decrease significantly during the multiple loading and unloading process, indicating the stable axial behavior of the three samples. The skeleton curves in Fig. 8(d) show that the initial stiffness and ultimate load of CFST-C, CFST-R, and CFST-S decreased in turn, indi­ cating that the reinforcement cage and circular steel tube had a positive effect. The circular steel tube exhibited a better effect. Secant stiffness K ¼ Fi/Δi is defined to describe stiffness degenera­ tion, where Fi is the peak load of every load level and Δi is the peak displacement. K–Δ curves in Fig. 8(d) show clear decay laws. Before the samples yielded, stiffness degeneration rate was slow. At this time, the samples were still in the elastic stage. After the yield point, the rate of stiffness degeneration started to accelerate. At this time, the samples reached the elastic–plastic stage. When the load reached Fd, the samples started to enter the plastic stage. At this stage, stiffness slowly decreased and gradually approached to zero. The F–Δ curve of CFST-R declined faster than those of CFST-S and CFST-R. When the samples reached the elastic–plastic stage, the stiffness of CFST-R decreased quickly and approached CFST-S. This was likely because the buckling of reinforcement cages accelerates concrete dam­ age. The stiffness of CFST-C was larger than that of CFST-R. When the samples entered the plastic stage, the stiffness of CFST-C slowly approached that of CFST-S. The circular steel tube had a positive in­ fluence on stiffness.

The experimental results of the feature points of the three samples are listed in Table 3, where Fy is the yield load and Δy is the yield displacement, Fy and Δy are calculated by the energy method [30]. Fu is the ultimate load and Δu is the ultimate displacement. Fd ¼ 85% Fu is the damage load and Δd is the damage displacement. η ¼ Fy/Fu is the yield ratio. A lower η equates to a greater strength reserve from the yield state to the ultimate state. μ ¼ Δd/Δy is the ductility coefficient. A larger μ equates to better ductility. The experimental results in Table 3 show that 1) Compared to samples CFST-S, the Fy of CFST-R and Fu of CFST-R increased by 9.25% and 8.32% (2371 kN), respectively. The direct contribution of the longitudinal reinforcement to Fu was Fr ¼ Ar � fyr ¼ 759 kN < 2371 kN, where fyr is the yield stress of the longitudinal reinforcement. Compared to CFST-R, the Fy of CFST-C and the Fu of CFST-C increased by 1.63% and 5.68% (1754 kN), respectively. The direct contribution of the circular steel tube to Fu was Fcs ¼ As � fycs ¼ 744 kN < 1754 kN, where fycs is the yield stress of the circular steel tube, indicating that the contribution of the circular steel tube and the reinforcement cage to Fu was not only the direct contribution, but also provided the confinement effect to the concrete and thus indi­ rectly contributed to Fu. 2) The CFST-R load reached its ultimate load quickly after the yield point. Therefore, the η of CFST-R was significantly greater than the corresponding values of CFST-S and CFST-C. The yield load of CFSTC was similar to that of CFST-R, but there was more rising space after the yield point, and the η of CFST-C was lower. This is related to the

Fig. 8. F–Δ curves and K–Δ curves of each sample Notes Line 1, Fitted initial loading line; Line 2, Fitted loading line under Fu; Line 3, Fitted unloading line under Fu; Line 4, Fitted loading line under Fd; Line 5, Fitted unloading line under Fd. 6

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Table 3 Experimental results of the feature points. Sample

Fy/ � 10 kN

Δy/mm

μεy

Fu/ � 10 kN

Δu/mm

μεu

Fd/ � 10 kN

Δd/mm

μεd

η

μ

CFST-R CFST-S CFST-C

2815.9 2518.0 2861.7

3.35 3.60 3.43

2792 3000 2858

3087.1 2850.0 3262.5

5.21 5.20 5.44

4342 4333 4533

2624.0 2422.5 2773.1

9.02 10.35 10.81

7517 8625 9008

0.912 0.884 0.885

2.69 2.87 3.15

confinement mechanism of the circular steel tube. The concrete in the circular steel tube will have a larger peak strain than that of the concrete in the non-circular steel tube. Therefore, when the strain of the samples exceeded the strain of the concrete in the non-circular steel tube, the concrete in the circular steel tube still did not reach its peak strain.

restore once the bonding interface separated. Hence, it is suggested to use reinforcement, which has better bonding behavior with concrete in order to minimize the decrease in the restoration capacity. Since the concrete-filled circular steel tube had an excellent deformation capacity, the restoring capacity of sample CFST-C significantly increased. 3.6. Strain development

3.4. Ductility analysis

The measured strains in Fig. 10 show that when the samples reached the yield point, the strain in every location uniformly developed and agreed with the assumption of the plane section. Therefore, the calcu­ lation models established in the later section are based on the assump­ tion of a plane section. After the yield point, the strain development in different locations was different. The strain in the middle of the steel plate developed faster, because of buckling in the center. The circular steel tube affected the strain development of cell 1. The strain measured by d4, d5, d6, d7, and d8 of CFST-C uniformly developed when the sample reached its ultimate point, while the strain of CFST-R was not as uniform. The strain measured by d6, d7, and d8 of sample CFST-R developed faster than the strain at d4 and d5, indicating that the circular steel tube caused the strains to develop more uniformly. The confinement effect of the steel tube on concrete can be reflected by the development of lateral strain. Two types of lateral strain gauges were set at layer 6, as shown in Fig. 11. One was set in the middle of the layer, including d1h, d2h, d5h, d7h, d9h, and d11h. These strain gauges were used to measure the confinement effect of the steel tube. The second type including c1, c2, c3, c4, c5 and c6 was set in the horizontal diaphragm. These strain gauges estimated whether the horizontal di­ aphragms could provide effective confinement effect. The measured strains in Fig. 11 show that before the samples reached the yield point, lateral strain slowly developed. When samples reached their yield points, lateral strain developed quickly. The strain in the middle of the layer and horizontal diaphragms reached or were close to the yield strain. After the ultimate point, lateral strain did not have significant degeneration, indicating that the steel plate and horizontal diaphragm provided an effective confinement effect to the concrete, and the confinement effect was persistent and stable. The steel tube was in the plane stress state under axial compression load, suffering vertical compression and lateral tension. According to the von Mises yield criterion, the vertical stress fv and lateral stress fl obey Equation (1):

The ductility coefficients of CFST-R, CFST-S and CFST-C were in turn increased, indicating that CFST-C had the greatest ductility and CFST-R had the least ductility of the three samples. The F–Δ curve of CFST-R decreased faster than that of CFST-S in the later stage, most likely because the space of the stirrup in the rein­ forcement cage was large, and the restriction effect of buckling of lon­ gitudinal reinforcement was less [31]. Therefore, the buckling of the longitudinal reinforcement accelerated the damage of concrete in the later stage. The ductility coefficient of CFST-C was larger than that of CFST-R and similar to that of CFST-S, indicating that the circular steel tube effectively enhanced the ductility of CFST-R. This was possibly because the circular steel tube has better confinement effect than the noncircular steel tube, which can enhance the peak strain and delay the damage more effectively. 3.5. Restoration capacity analysis When the external load disappeared, the samples returned to their original state partially after deformation. The deformation restoring coefficient (γ) is defined in Fig. 9 to present the restoration capacity of the samples, where Δi is the maximal displacement of every load level, and Δj is the residual displacement when unloaded to 2000 kN. The sample which has larger γ will have smaller residual deformation and better restoration capacity than the sample which has smaller γ. The γ–Δ curves in Fig. 9 show that the deformation capacity of CFSTS was better than that of CFST-R, most likely because the diameter of the longitudinal reinforcement was small in this study, and the reinforce­ ment adopted plain reinforcement. Thus, the bonding behavior between the reinforcement and concrete was weak and separated easily. The bond interface between the reinforcement and concrete would not

f 2l þ f 2v

fl ⋅fv ¼ f 2y

(1)

The lateral and vertical strains in the middle of the steel plate at layer 6 were used to analyze whether the steel plate reached its yield state. The fl – fv curves in Fig. 12 show that all the steel plates reached the yield state before the samples reached their ultimate point. 4. Calculation methods of axial compression bearing capacity in different codes and related studies The calculation methods of axial compression bearing capacity of concrete-filled steel tube columns are provided in the codes of many countries and related studies. EC4-2004 [32], AISC-LRFD [33], GB 50936-2014 [34], AIJ [35] and related studies [36,37] were used to calculate the ultimate bearing capacity of CFST-R, CFST-S and CFST-C in this study. For the calculation methods which did not mention the

Fig. 9. γ–Δ curves. 7

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Fig. 10. Longitudinal strain at layer 6.

Fig. 11. Lateral strain at layer 6.

reinforced bars, only the direct contribution of the reinforced bars was considered by using superposition method. EC4-2004 (Equation (2) – Equation (3)):

Nu ¼ χNpl,Rd

8

(2)

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Fig. 12. Development path of stress.

Npl,Rd ¼ Aa fyd þ Ac fcd þ As fsd

(3)

where, Aa is area of the steel plate, fyd is the yield strength of steel plate, Ac is the area of concrete, fcd is the cylinder strength of concrete, As is the area of longitudinal reinforcement, and fsd is the yield strength of the longitudinal reinforcement. χ is the reduction factor related to slenderness. AISC-LRFD (Equation (4) – Equation (6)): Nu ¼ AsFcr � � ( 2 Fcr ¼ 0:658λc Fmy λc � 1:5 � � Fcr ¼ 0:877 λ2c Fmy λc > 1:5 Fmy ¼ Fy þ Fyr ðAr = As Þ þ 0:85f ’c ðAc = As Þ

F ¼ min(fy, 0.7fu)

where, As is the area of the steel plate, fy is the yield strength of the steel plate, fu is the ultimate strength of the steel plate, f’c is the cylinder strength of the concrete, and Ac is the area of the concrete. Reference [36,37] (Equation (14) – Equation (15)):

(4)

Nu ¼ Asfyþ(1 þ k)fcAc . qffiffiffiffi�ffiffiffiffiffiffiffiffiffi k ¼ 0:5668 0:0039h t fy 235

(5)

where, As is the area of steel plate, Fy is the yield strength of steel plate, Ar is the area of longitudinal reinforcement, Fyr is the yield point of the longitudinal reinforcement, Ac is the area of concrete, and f’c is the cyl­ inder strength of the concrete. λc is the column slenderness parameter. GB 50936-2014 (Equation (7) – Equation (10)): (7)

fsc¼(1.212.BθþCθ2)fcu,m

(8)

αsc ¼ As/Ac

(9)

θ ¼ αscf/fc

(10)

where, Asc is the area of cross-section, B and C are the parameters related to the yield strength of the steel plate and the shape of cross-section, As is the area of steel plate, Ac is the area of the concrete, f is the design strength of the steel plate, and fc is the design strength of concrete. AIJ (Equation (11) – Equation (13)): Nu ¼ AsFþ0.85f’cAc (square steel tube)

(11)

0.27)AsFþ0.85f’cAc

(12)

Nu¼(1 þ

(circular steel tube)

(14) (15)

where, As is the area of the steel plate, fy is the yield strength of the steel plate, fc is the compressive strength obtained from 150 � 150 � 300 mm prismatic concrete, Ac is the area of the concrete, and h/t is the width-tothickness ratio of the steel tube. The previous studies have proposed constitutive relationships of concrete in CFST columns which can be used to estimate the deformation-load curves. Han et al. [38] proposed a constitutive rela­ tionship of concrete in the CFST columns as shown in Equation (16) and the main parameter is the confinement effect coefficient as shown in Equation (17). Hu et al. [6] proposed a constitutive relationship of concrete in the CFST columns as shown in Equation (18). The main parameter is the width-thickness ratio of the wall, which is the average value of every wall in this study. In Equations (16)–(18), x ¼ ε/ε0. The details of the other parameters are shown in literature [6,38]. 8 2x x2 x � 1 σ < ¼ x (16) σ0 : x>1 βðx 1Þη þ x

(6)

Nu ¼ Ascfsc

(13)

ξ ¼ As fy

9

��

ðAc fck Þ

(17)

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F. Yin et al.

8 > <

σ¼

> :

1 þ ðR þ RE

Ec ε 2Þx ð2R ½ðk3

1Þðx

1Þx2 þ Rx3

x�1

(18)

1Þ=10 þ 1�σ0 x > 1

In generally, the calculation values listed in Table 4 were signifi­ cantly lower than the experimental values. The calculation curves of several specimens with different shapes and structures were also significantly lower than the experimental curves as shown in Fig. 20. It indicates that the constitutive relationships of concrete and the codes used in this paper are conservative to predict the F-Δ curves and bearing capacity of MCFST columns in this study. Tu [22] and Xu [39] conducted experimental studies on compression behaviors of T-shaped MCFST columns and polygonal MCFST columns, respectively. Several codes were used to calculate the bearing capacity and the results also showed that the calculation results were smaller than the experimental results. However, the studies on the calculation method to predict the F-Δ curves and bearing capacity of MCFST are still limited. Hence, it is necessary to establish the constitutive relationship suitable for MCFST.

Fig. 13. Effective confined region of the single cell.

5. Calculation model of axial compression bearing capacity of MCFST Mander et al. [5] provided a concrete constitutive relationship in the RC column, considering the confinement effect of stirrups. Assuming that the steel tube is the stirrup with zero space, the concrete includes the effective strong confined region and the weak confined region. The separatrix is a parabola with an initial slope of 1. Fig. 13 shows the confined region of the single cell concrete steel tube. The inner steel plate in MCFST is restricted by concrete on both sides and can be regarded as having no weak confined region in the inner steel plate. Moreover, the inner plate separated one weak confined region into several weak confined regions. Fig. 14 shows the difference between the confined region of MCFST and single cell CFST. The area of the weak confined region in Figs. 13 and 14 is expressed by Equations (19) and (20), respectively. � n �2 P ai A’ ¼ i¼1 (19) 6

Fig. 14. Effective confined region of the multi-cell.

the confinement effect is shared by concrete on the two cells and the proportion is difficult to ensure, necessitating the need for a simplified confined region model of concrete. The individual cells in MCFST have different shapes and areas. Therefore, the ultimate state of individual cell is different. Therefore, determining the ultimate state of every in­ dividual cell is necessary. Based on the analysis above, MCFST was separated into several individual cells and every individual cell enjoyed the inner steel plate alone. This separation method equated to increasing the thickness of the inner steel plate. Therefore, the added confinement effect caused by adding steel plate thickness can be decreased by the models in Fig. 15. The experimental analysis in Section 2.3 shows that the reinforce­ ment cage and the circular steel tube can provide an extra confinement effect on concrete. The steel ratio of the circular steel tube was 0.6%, but concentrated in the two cells, and thus it will have significant influence locally. Therefore, the concrete constitutive relationship in the circular steel tube is needed. The steel ratio of longitudinal reinforcement was 0.41%, and the space of stirrups was 60 mm distributed across 13 cells. According to the Mander model, there is a slight confinement effect of reinforcement cage. However, the experimental results show that the confinement ef­ fect of the reinforcement does objectively exist. Hence, the stirrups were simplified into horizontal diaphragms that have equal steel ratio in this study. It has been analyzed that the buckling of longitudinal reinforce­ ment may accelerate the damage of concrete. However, this effect will be obvious in the later stage. Hence, the buckling of longitudinal rein­ forcement is neglected in the calculation of constitutive relationship of concrete. The analysis in Section 2.5 showed that the damage of the bonding interface between concrete and reinforcement cage will affect

n P

a2i A ¼ i¼1 6

(20)

’’

Equation (21) is the difference between A’ and A”: a1 A’

A’’ ¼

n P i¼2

ai þ a2

n P

ai þ ⋯ þ an

n P 1

ai

i¼n

i¼3

3

(21)

>0

Equation (21) indicates that the weak confined region in MCFST is smaller than that of the single cell CFST, indicating that the confinement effect of MCFST is more efficient than that of the single cell CFST. Therefore, establishing a model suitable for concrete constitutive rela­ tionship in MCFST is necessary. Although there is no weak confined region of the inner steel plate, Table 4 Comparison between experimental results and calculated results. Samples

Experimental value/ � 10 kN

CFST-R

3087.1

CFST-S

2850.0

CFST-C

3262.5

EC4-2004

AISC-LRFD

GB50936-2014

AIJ

Literature [36,37]

Error

Error

Error

Error

Error

2422.4 21.5% 2368.7 16.9% 2492.7 23.6%

2246.2 27.2% 2027.6 28.9% 2286.9 29.9%

2634.4 14.7% 2547.8 10.1% 2683.1 17.8%

2340.4 24.2% 2107.1 26.1% 2389.8 26.7%

2490.7 19.3% 2425.0 14.9% 2629.1 19.4%

Notes: error ¼ (calculated value - experimental value)/(experimental value) � 100%. 10

F. Yin et al.

8 > > > > > < Ao ¼

Z

Journal of Building Engineering 28 (2020) 101099

x0

y1 0

� y2 dx; 0 � y2 �

ðtan α sin β þ cos βÞ2 L2 L2 L2 cos β ; ðx0 ; y0 Þ is on y2 ; þ þ 4 tan α tan β sin β 2 sin β 2 tan α sin2 β

1

Z > > > > > :

x0

y1 0

C y2 C Adx þ

Z

2 L2 ðtan α sin βþcos βÞ 4 tan α sin β

x0

yþ 2

�

y2 dx; y2 >

ðtan α sin β þ cos βÞ2 L2 L2 L2 cos β ; ðx0 ; y0 Þ is on yþ þ þ 2: 4 tan α tan β sin β 2 sin β 2 tan α sin2 β

the restoration capacity. However, this will only affect the unloading process. In the loading process, the bonding interface between concrete and steel tube is closed. Hence, the damage of bonding interface can also be neglected in the calculation model. The equation of stress–strain in MCFST adopted Equations (22)–(26) suggested by Mander et al. [5]: fc ¼ x¼

r¼

r

fcc ⋅x⋅r 1 þ xr

Ec

5.2. Lateral confining pressure f’l In order to simplify the calculation, the lateral confining pressure of the steel tube on the concrete was assumed to be even (Fig. 16). Ac­ P cording to Equation (31), f ’l ¼ L=t2fh 2. Lt ¼ 1n n1 Ltii is the average value of the

width–thickness ratio of every side or diameter–thickness ratio of the circular steel tube. 8 ’ > < f l ðL 2tÞ ¼ 2fh ⋅tnon circular ​ cross section Z π dθ πðL 2tÞ > : sin θ ¼ f ’l ðL 2tÞ ¼ 2fh ⋅t ​ ​ circular ​ cross section f ’l 2 π 0 (31)

(23) Ec fcc =εcc

(24)

�

εcc ¼ εc0 1 þ ξ fcc f ’c fcc ¼ f ’c

The confinement effectiveness coefficient is ke ¼ Ae/Ac.

(22)

εc εcc

�� 1

(25)

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7:94fl 1:254 þ 2:254 1 þ ’ fc

Under the axial compressive load, longitudinal compression stress and lateral tension stress of the steel tube meet the von Mises yield criterion, as shown in Fig. 12. The width–thickness ratio (W) of the steel tube, as shown in Equation (32), is the main parameter affecting the failure model [40]. According to Equation (33), when W > 0.85, samples will experience local buckling failure; when W � 0.85, local buckling is negligible. For the circular steel tube, local buckling can be also ignored. rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisffiffiffiffi L 12ð1 μ2 Þ fy W¼ (32) t 4π 2 Es

! 2

fl f ’c

(26)

5.1. “Separation model” of MCFST Equation (27) is the equation of curve y1: tan α 2 x þ tan α⋅x L1

y1 ¼

(30)

(27)

Equation (28) is the equation of curve y2:

8 > > > > > > > < y2 ¼

> > > > > > > :

x L2 þ þ tan β 2 sin β x L2 þ þ tan β 2 sin β

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 tan α sin β ðtan α sin β þ cos βÞ2 x � ðtan α sin β þ cos βÞ2 L2 L2 L2 cos β L2 ; y2 > ; yþ þ þ 2 4 tan α tan β sin β 2 sin β 2 tan α sin2 β 2 2 tan α sin β rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 tan α sin β L2 ðtan α sin β þ cos βÞ2 x � ðtan α sin β þ cos βÞ2 L2 L2 L2 cos β L2 ; 0 � y2 � ; y þ þ 4 tan α tan β sin β 2 sin β 2 tan α sin2 β 2 2 tan α sin2 β

L2 cos β þ L2

L2 cos β

8 > < fv ¼ 1:2 0:3 � 1W > 0:85 W W2 fy > : fh ¼ 0:21fy ; fv ¼ 0:89fy W � 0:85 ​ or ​ circular ​ steel ​ tube

The area of the effective confined region of typical cell is calculated by Equation (29): Ae ¼ A c

A1

A2

A3

L2 A4 þ Ao ; where Ai ¼ i 6

(29)

(28)

(33)

ξ is the modified parameter of strain at peak concrete stress. The reference equation of single cell CFST columns was proposed [41]. However, it is not very accurate for the MCFST columns in this study and several related studies [42–44]. Considering that MCFST has a better confinement effect than single cell CFST, modified Equation (34) is

When α � β < 2α, make y1 ¼ y2, the point of intersection was (0, 0) and (x0, y0). Ao is calculated by Equation (30):

11

F. Yin et al.

Journal of Building Engineering 28 (2020) 101099

Fig. 15. Separation model of MCFST columns.

Fig. 16. Lateral confining pressure model.

suggested for MCFST columns. ξ ¼ 52:765ð1 þ ke Þ

qffiffiffiffi�ffiffiffiffiffiffi! fv f ’c

assumed that there is no weak confined region in the vertical direction (Fig. 17). The horizontal diaphragms and the steel plate combine into Tshaped cross-section and have large flexural rigidity, thus horizontal buckling does not occur easily. Hence, assuming that the lateral pressure of the horizontal diaphragms is effective, a small weak confined region is

2:0531

W 0:36

(34)

5.3. Effective lateral confining pressure fl The analysis of lateral strain in Section 2.6 shows that horizontal diaphragms have a high stress level. Therefore, the contribution of horizontal diaphragms to the effective lateral confining pressure should be considered. Effective lateral confining pressure comprises the steel tube and horizontal diaphragm, as shown in Equation (35). fl ¼ fls þ flh, where fls ¼ ke f’l

(35)

The lateral confining pressure of the horizontal diaphragm has two paths to transfer into the concrete. The first path is similar to the stirrup in RC, and confining pressure is transferred through concrete. Another path is transferred through the vertical steel plate. Therefore, it can be considered that the lateral pressure of the horizontal diaphragm can be transferred to concrete effectively and uniformly. Hence, it can be Fig. 18. Weak confined region in the lateral direction.

Fig. 17. Weak confined region in the vertical direction.

Fig. 19. Ideal elastic-plastic model of steel. 12

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Journal of Building Engineering 28 (2020) 101099

Fig. 20. Calculation of F–Δ curves of samples from varius studies.

13

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Journal of Building Engineering 28 (2020) 101099

Table 5 Calculated results of different MCFST columns. Samples

CFST-R

CFST-S

CFST-C

Wang-1

Wang-2

Wang-3

Fu/10 kN (Experiment) Fu/10 kN (Calculation) Error Δu/mm (Experiment) Δu/mm (Calculation) Error

3087.1 3017.4 2.3% 5.21 5.19 0.4%

2850.0 2874.5 0.9% 5.20 5.19 0.2%

3262.5 3171.3 2.8% 5.44 5.59 2.8%

2623.3 2727.6 4.0% 5.31 5.30 0.2%

3211.9 3296.7 2.6% 6.48 7.39 14.0%

3424.8 3368.7 1.6% 9.04 9.02 0.2%

Samples

Yang-1

Yang-2

Yang-3

Cao-1

Cao-2

Cao-3

Fu/10 kN (Experiment) Fu/10 kN (Calculation) Error Δu/mm (Experiment) Δu/mm (Calculation) Error

3748.2 3570.0 4.8% 2.86 5.00 74.8%

3795.2 3822.9 0.73% 4.21 5.32 26.4%

3955.2 3899.8 1.4% 3.60 4.90 36.1%

1480.4 1521.3 2.8% 6.15 5.19 15.6%

1740.6 1698.8 2.4% 3.67 4.12 12.3%

1668.7 1814.3 8.7% 5.46 5.46 0

observed in the lateral direction (Fig. 18). The lateral confining pressure of horizontal diaphragms can be calculated by Equation (36). flh ¼

Ash fh LH

Wang-3 was a 4-cell CFST with the reinforcement cage. Yang et al. [43] conducted the experiment of three bifurcated MCFST columns with horizontal diaphragms. Sample Yang-1 had no reinforcement cage; Sample Yang-2 had the reinforcement cage; and Sample Yang-3 had the circular steel tube. Cao et al. [44] conducted the axial compression experiment on three hexagonal MCFST columns. Sample Cao-1 had the reinforcement cage, and the strength of the concrete was C30; Sample Cao-2 had no reinforcement cage with the strength of the concrete as C40; and Sample Cao-3 had the reinforcement cage with the strength of the concrete as C40. These samples include different shapes, cell numbers, and cross-section structures. The calculated F–Δ curves are shown in Fig. 20. The error comparison is listed in Table 5 and Fig. 21. In generally, the calculation curves of the 12 samples showed good agreement with the measured curves. The ultimate bearing capacity errors were between 5% and 5%, and the ultimate displacement errors were mostly between 15% and 15%. The only exceptions were Yang’s three samples because the displacements of the samples were obtained by considering both the measured values and calculation values, and some errors may be introduced from the calculation values. The above results indicate that the “separation model” can accurately predict the load–deformation curves of MCFST columns under axial compression.

(36)

where L is the average side length of cell, and H is the space of the horizontal diaphragms. 5.4. Stress-strain relationship of steel The stress–strain relationship of all types of steel follows an ideal elastic–perfectly plastic model [6,36]. The yield stress of the steel tube is fv, the yield stress of the longitudinal reinforcement is fr and the yield stress of the longitudinal stiffener is fl, as shown in Fig. 19. 5.5. Calculation results of “separation model” The superposition method was used to calculate F–Δ curves, as shown in Equation (37): F ¼ Asfv þ ΣAcifci þ Arfr þ Alfl

(37)

6. Discussion

In order to prove the accuracy of the “separation model” proposed in this study, in addition to the three samples herein, another nine MCFST samples from the previous studies of our group were used as examples [42–44]. Wang et al. [42] conducted the axial compression experiment of three pentagonal MCFST columns with horizontal diaphragms. Sample Wang-1 was a single cell CFST with reinforcement cage; Sample Wang-2 was a 4-cell CFST without the reinforcement cage; and Sample

The above analysis shows that setting the circular steel tube in MCFST is an effective method to improve the axial compression behavior of the MCFST column. A reasonably positioned circular steel tube set in the MCFST column will more effectively enhance the ad­ vantages of the circular steel tube. The failure model in Section 2.1

Fig. 21. Error distribution of the ultimate bearing capacity and ultimate displacement for 12 samples. 14

F. Yin et al.

Journal of Building Engineering 28 (2020) 101099

showed that protruded cross-section (small angle), such as cell 1, will have a greater degree of damage. Therefore, setting a circular steel tube with an abrupt change in cross-section is ideal. The “separation model” in this study can be used to estimate the weak cell in the cross-section in order to ideally place the circular steel tube. The effect of different setting positions for the circular steel tube in MCFST will be a direction of further research. The “separation model” in this study can present the ultimate state of every cell and can be used to design structures more effectively and realize damage control through reasonable design. The design method of MCFST as the foundation of “separation model” is expected to be another promising direction of further research.

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Yang, et al., Study on mechanical behavior of integrated multicell concrete-filled steel tubular stub columns under concentric compression, Int. J. Civ. Eng. 17 (3) (2019) 361–376. [24] Jiepeng Liu, Hua Song, Yuanlong Yang, Research on mechanical behavior of Lshaped multi-cell concrete-filled steel tubular stub columns under axial compression, Adv. Struct. Eng. 22 (2) (2019) 427–443. [25] Yong-Bo Zhang, Lin-Hai Han, Kan Zhou, et al., Mechanical performance of hexagonal multi-cell concrete-filled steel tubular (CFST) stub columns under axial compression, Thin-Walled Struct. 134 (2019) 71–83. [26] Z.B. Wang, Z. Tao, Q. Yu, Axial compressive behaviour of concrete-filled doubletube stub columns with stiffeners, Thin-Walled Struct. 120 (2017) 91–104. [27] Yufen Zhang, Junhai Zhao, et al., Study on compressive behavior capacity of concrete-filled square steel tube column reinforced by circular steel tube inside, J. Civ. Eng. Manag. 19 (6) (2013) 787–795. 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7. Conclusion Three samples were used to investigate the axial compression behavior of MCFST columns and effect of different structures. The working mechanism of the three samples was studied by analyzing the damage evolution, bearing capacity, deformation ability, stiffness degeneration, and stress development. A “separation model” suitable for predicting F–Δ curves of the MCFST column under axial compression was established based on the experimental results. 1) The reinforcement cage provided confinement effect to concrete and effectively improved the initial stiffness and bearing capacity. 2) The circular steel tube significantly improved the comprehensive axial behavior of the MCFST column. Compared to CFST-R, the cir­ cular steel tube not only improved the initial stiffness and bearing capacity significantly, but also improved the ductility and deforma­ tion ability. Therefore, setting circular steel tube in MCFST column is a good choice to improve the working behavior. 3) Based on the analysis of experimental results, a “separation model” was established and used to calculate the concrete constitutive relationship of MCFST. The calculation results of 12 samples with different cross-sectional shapes and structures showed good agree­ ment with the experimental results. The errors of ultimate load were essentially between 5% and 5%, whereas those of ultimate displacement were essentially between 15% and 15%. 4) The “separation model” can reflect the difference in the axial be­ haviors among different cells, easily determining the weakest cell in the cross-section. Therefore, the “separation model” can provide useful guidance for MCSFT column design. Declaration of competing interest The authors declare that there have no conflicts of interest to this work, the work described is original research that has not been pub­ lished previously, and not under consideration for publication else­ where, in whole or in part. CRediT authorship contribution statement Fei Yin: Methodology, Investigation, Data curation, Writing - review & editing. Su-Duo Xue: Writing - review & editing. Wan-Lin Cao: Conceptualization, Methodology. Hong-Ying Dong: Validation, Writing - review & editing. Hai-Peng Wu: Writing - review & editing. Acknowledgments The research carried out in this paper was financially supported by the Beijing Natural Science Foundation (8182005) and National Natural Science Foundation of China (No. 51808014). Wan-Lin Cao conceived and designed the test. Fei Yin analyzed the data and wrote the article. Su-Duo Xue, Hong-Ying Dong, and Hai-Peng Wu edited and reviewed all the contents.

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[35] AIJ, Recommendations for Design and Construction of Concrete Filled Steel Tubular Structures, Architectural institute of Japan, Tokyo, 1997. [36] Y. Du, Z. Chen, M.X. Xiong, Experimental behavior and design method of rectangular concrete-filled tubular columns using Q460 high-strength steel, Constr. Build. Mater. 125 (2016) 856–872. [37] Y. Du, Z. Chen, J.Y. Richard Liew, et al., Rectangular concrete-filled steel tubular beam-columns using high-strength steel: experiments and design, J. Constr. Steel Res. 131 (2017) 1–18. [38] L.H. Han, Fire performance of concrete filled steel tubular beam-columns, J. Constr. Steel Res. 57 (6) (2001) 697–711. [39] L.H. Xu, P. Xu, Y.J. Hou, et al., Experimental study on axial compression behavior of short polygonal multi-cell and self-compacting high-strength CFST columns, China Civ. Eng. J. 50 (1) (2017) 37–45 (In Chinese).

[40] H.B. Ge, T. Usami, Strength analysis of concrete-filled thin-walled steel box columns, J. Constr. Steel Res. 30 (3) (1994) 259–281. [41] C. Jian, Constitutive relationship of concrete core confined by square steel tube, J. South China Univ. Technol. 36 (1) (2008) 105–109 (in Chinese). [42] Lichang WANG, Wanlin CAO, XU Meng meng, et al., Experimental research on compression behavior of pentagonal cross-section CFST mega-columns, J. Build. Struct. 35 (1) (2014) 77–84 (in Chinese). [43] Guang YANG, COMPRESSIVE BEHAVIOR OF SPECIAL–SHAPED MULTI-CELL CONCRETE-FILLED STEEL TUBE (CFST) TREE MEGA COLUMNS, Beijing University of Technology, Beijing, China, 2009 (in Chinese). [44] Wanlin Cao, Zhihui Wang, bin Peng, et al., Experimental study on axial compression performance of multi-cell CFST mega-columns with reinforcement cage inside, Struct. Eng. 28 (03) (2012) 135–140 (in Chinese).

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