## 1. Introduction

In the development of high-rise buildings and super high-rise buildings, structures with better performance are needed to improve the bearing capacity of buildings [

1]. The concrete-filled steel tube composite structure has emerged as a relatively new structure in the construction industry and has been widely used worldwide.

Many studies have been conducted on the circular and rectangular concrete-filled steel tube column. Han [

2] and Ding [

3] carried out an experimental study on the axial compression performance of circular concrete-filled steel tube short columns. Portoles [

4] and Liu [

5] conducted an experimental study on the bending performance of circular concrete-filled steel tube short columns. Serras [

6] and Zhou [

7] undertook experimental research on the seismic performance of circular concrete-filled steel tube short column. Ding [

8] and Ozbakkaloglu [

9] completed an experimental study on the axial compression performance of rectangular concrete-filled steel tube. Qu [

10] and Wang [

11] carried out an experimental study on the bending performance of a rectangular concrete-filled steel tube. Chen [

12] and Xu [

13] performed experimental research on the seismic performance of rectangular concrete-filled steel tube short columns. Zhang [

14] investigated the hysteretic behavior of concrete-filled thin-walled steel tubular (CFTST) columns under seismic load. The experimental results indicate that the CFTST columns under an axial load level below 0.5 exhibited plum hysteretic loops with a slight pinching effect, better ductility, and energy dissipation capacity. For the shear wall, it has two important sources of dissipation that affect greatly its dynamic behavior, one is the interaction between the steel and concrete, while the other is the natural dissipative phenomena occurring inside the concrete itself due to the micro-cracks in it [

15,

16].

With the development of concrete-filled steel tube columns, a special-shaped concrete-filled steel tube column has been increasingly favored by many researchers and designers due to the advantages of avoiding column protrusions from walls and increasing room space [

17,

18,

19,

20]. However, the steel tube in a special-shaped concrete-filled concrete column has a weak constraint on the core concrete, and there is a negative angle in the cross-section. When the load is applied, the negative-angle portion may undergo large deformation and damage, which in turn leads to a decrease in the bearing capacity of the special-shaped steel tube concrete column. In particular, for a T-shaped column, the separation of the steel tube and concrete at the inner corner may prevent the two materials from fully functioning and the large width-to-thickness ratio of the steel plate may cause local premature buckling.

In recent years, great efforts have been made on a special-shaped concrete-filled concrete column. Huang [

21] conducted an experimental study on the axial compression performance of five different ribbed and two unribbed T-shaped concrete-filled steel tube short column specimens and investigated the effect of the ribbed form on the mechanical properties of the T-shaped concrete filled-steel tube short column. The test results show that the installation of stiffeners improves the axial compression stiffness and axial compression bearing capacity of the T-shaped concrete-filled steel tube. Based on the elastic buckling theory of steel plate, Fu [

22] proposed a technical measure and design method of setting a T-shaped stiffener in a rectangular concrete-filled steel tube column with large plate width and thickness, and a T-shaped stiffener was arranged along the steel tube wall lengthwise. Since the flange of the T-shaped stiffener is partly enclosed in the concrete, the joint working performance of the stiffener and the concrete can be fully utilized, and the local buckling performance of the steel tube wall can be greatly improved. In order to delay the local buckling of steel tubes and improve the restraint performance of steel tubes on concrete, Liu [

23] welded tensile reinforcing bars on the inner surface of steel tubes. A modified fiber-based numerical modeling method was developed to study the specimens’ cyclic behavior, incorporating the effect of stiffeners on postponing the steel tube’s local buckling and the confinement for concrete. Yang [

24] conducted an experimental study on the surface of the tube welded with bars to improve the axial compression performance of the T-shaped CFST (concrete-filled-steel tube) column. However, it is costly to stiffen the steel bar to the steel plate and fold the steel plate into a hollow tube with a special-shaped cross section.

To overcome the shortcomings of the traditional special-shaped concrete-filled concrete column, this paper proposes a new type of combined square steel tube concrete composite shaped shear wall and T-shaped concrete-filled steel tube shear wall (CFSTSW) to improve the bearing capacity and meet the requirements of fast construction speed.

Based on the renovation project of Hongxinyuan urban shantytown in Hongguang Town, the authors used the methods of experimental research and numerical simulation to study the axial compression performance of T-shaped multi-cavity CFSTSW and propose the axial compression bearing capacity equation. The results calculated by the proposed equation are in good agreement with those calculated by the finite element method and can be used in the actual engineering design.

## 5. Axial Load–Strain Relationship Curve and Ductility Coefficient

The load-longitudinal strain curve of a multi-cavity concrete-filled steel tube can intuitively reflect the mechanical behavior of a concrete-filled steel tube.

Figure 6 shows the load–strain curve of a T-shaped multi-cavity CFSTSW.

The bearing capacities of the T-shaped multi-cavity CFSTSW for the test specimens TA4-600, TA5-600, and TA6-600 were 625.67 kN, 768.98 kN, and 913.60 kN, respectively.

Based on the test load–longitudinal strain curve, the peak load ${N}_{\mathrm{u}}$ (the highest point of the curve) of the multi-cavity CFSTSW can be obtained, and the ductility coefficient u can be obtained by the drawing method.

Taking TA4-600 as an example, the method of calculating the ductility coefficient is described in detail, as shown in

Figure 7.

First, we used the Origin software to make the load–longitudinal strain relationship curve of TA4-600, then used the plugin Tangent in Apps to make the load–longitudinal strain relationship curve through the origin tangent, and the line intersection tangent line parallel to the abscissa axis through point B with point A, extending AB to cross the ordinate axis at point J. The ordinate of point J is

${N}_{\mathrm{u}}$, and 85% of the peak load was taken. The parallel curve of AB is the cross section of the LG load–longitudinal strain curve at point G. The vertical line through point G crosses the axis of the abscissa at point H, the abscissa of point H is

${\epsilon}_{0.85}$, the vertical axis through point A intersects the axis of abscissa at point C, the cross-load–longitudinal strain relationship curve at point D connects OD, extends OD on line AB, and crosses point E. The vertical axis through point E intersects the abscissa axis at point F, and the abscissa of point F is

${\epsilon}_{\mathrm{y}}$.

${\epsilon}_{\mathrm{u}}$ is the peak strain. The expression of the ductility coefficient is expressed as:

Finally, the ductility coefficient of each test specimen is given in

Table 4.

**Table 4.**
Ductility coefficient.

**Table 4.**
Ductility coefficient.

No. | Specimen Number | N_{y} (kN) | N_{u} (kN) | N_{y}/N_{u} | Ductility Coefficient |
---|

1 | TA4-600 | 594.39 | 625.67 | 0.95 | 2.27 |

2 | TA5-600 | 738.22 | 768.98 | 0.96 | 2.15 |

3 | TA6-600 | 886.19 | 913.60 | 0.97 | 1.79 |

It can be seen from

Table 4 that the ductility coefficient of the multi-cavity CFSTSW specimen decreases with the increase in the number of cavities.

## 6. Load–Stress Relationship Curve

The partial strain data of the steel tube measured by the test were converted into corresponding transverse stress ${\sigma}_{\mathrm{h}}$, longitudinal stress ${\sigma}_{\mathrm{v}}$, and converted stress ${\sigma}_{\mathrm{z}}$ to obtain the axial load–stress relationship curve of the specimen, and the development law of the steel tube stress during the entire loading process was analyzed to clarify the interaction mechanism between the steel tubes and concrete. The conversion equations are shown in Equations (2)–(4).

In the elastic stage, the stress-strain relationship of steel conforms to Hooke’s law, namely:

In the equation, $\Delta {\sigma}_{\mathrm{h}}$ and $\Delta {\epsilon}_{\mathrm{h}}$ are the transverse stress and strain of the steel, respectively.

In the elastoplastic phase, the elastoplastic incremental theory is used.

${E}_{\mathrm{s}}^{\mathrm{t}}$ is the tangent modulus of the steel in the elastoplastic stage.

${\mu}_{\mathrm{sp}}$ is the Poisson’s ratio at the elastoplastic stage of the steel.

In the plastic strengthening stage, the steel tube complies with the von Mises yield criterion.

${\sigma}_{\mathrm{h}}^{\prime}$ is the transverse stress deviation of the steel. ${\sigma}_{\mathrm{h}}^{\prime}={\sigma}_{\mathrm{h}}-{\sigma}_{\mathrm{cp}}$

${\sigma}_{\mathrm{v}}^{\prime}$ is the lateral stress deviation of the steel. ${\sigma}_{\mathrm{v}}^{\prime}={\sigma}_{\mathrm{v}}-{\sigma}_{\mathrm{cp}}$

${\sigma}_{\mathrm{cp}}$ is the mean stress of the steel. ${\sigma}_{\mathrm{cp}}=\frac{1}{3}({\sigma}_{\mathrm{h}}+{\sigma}_{\mathrm{v}})$

p is a calculation parameter. $p=\frac{2{H}^{\prime}}{9{E}_{\mathrm{s}}}{\sigma}_{\mathrm{z}}^{2}$

H is a calculation parameter. ${H}^{\prime}=\frac{d\sigma}{d{\epsilon}_{\mathrm{p}}}=3\times {10}^{-3}{E}_{s}$

${\sigma}_{\mathrm{z}}$ is the mean stress of the steel. ${\sigma}_{\mathrm{z}}=\sqrt{{\sigma}_{\mathrm{h}}^{2}+{\sigma}_{\mathrm{v}}^{2}+{\sigma}_{\mathrm{h}}{\sigma}_{\mathrm{v}}}$

Q is calculation parameters. $Q={\sigma}_{\mathrm{h}}^{{}^{\prime}2}+{\sigma}_{\mathrm{v}}^{{}^{\prime}2}+2{\mu}_{\mathrm{s}}{\sigma}_{\mathrm{h}}^{\prime}{\sigma}_{\mathrm{v}}^{\prime}+\frac{2{H}^{\prime}(1-{\mu}_{\mathrm{s}}){\sigma}_{\mathrm{z}}^{2}}{9G}$

The test specimen TA4-600 was used as an example to illustrate the load–stress curve, as shown in

Figure 8.

As shown from

Figure 8a, in the initial loading, the lateral stress of the steel tube was almost zero, and the longitudinal stress was large and increased linearly with the increase in load. The Poisson’s ratio of concrete was smaller than that of steel, and the lateral expansion of concrete in the elastic stage was smaller than that of steel tubes, so the interaction between the steel tubes and concrete was small. After entering the elastoplastic stage, the lateral deflection of the test specimen increased, the concrete expanded, the steel tube began to be tensioned, the lateral stress gradually increased in the positive direction, and the restraining effect gradually manifested. After the test bearing capacity, the lateral deflection of the test specimen increased rapidly. As the load continued to increase, the transverse stress of the steel tube also increased rapidly, and the steel tube had a strong restraining effect on the concrete. As the load decreased, the longitudinal stress decreased somewhat. The stress development law of the remaining several measuring points was basically the same as that of the measuring point of the “1–1” surface. However, for the measuring points on the “2–3” surface, before the test bearing capacity, due to the buckling of steel tube near the measuring point, the steel tube at the location of the measuring point is compressed. The concrete expanded rapidly, and the steel tube began to be pulled. When the converted stress reached the yield strength, the steel tube yielded.

## 8. Bearing Capacity Equation Deduction

Using the finite element modeling method above, the bearing capacities of the multi-cavity CFSTSW with different concrete strength, steel yield strength, and steel content were obtained under the action of axial compression load. The calculated results are plotted in

Figure 13.

The tick label in the x-axis of

Figure 13 has its naming rule, taking “4-Q235-2-C40” as an example, 4 represents a four-cavity test specimen with the steel yield strength of 235 MPa, a thickness of 2 mm, and standard compressive strength of concrete of 40 MPa.

According to the literature [

30], the average value of the ratio of the longitudinal compression stress of the steel tube to the yield strength of the steel tube is expressed as:

Among them, ${\sigma}_{\mathrm{L},\mathrm{S}}$ is the longitudinal compression stress of the steel tube, and ${f}_{\mathrm{s}}$ is the yield strength of the steel tube.

According to the von Mises yield criterion,

${\sigma}_{\mathsf{\theta},\mathrm{S}}$ can be expressed as:

Among them, ${\sigma}_{\mathrm{r},\mathrm{s}}$ is the average value of the transverse stress of the steel tube under tension.

The multi-cavity steel tube mainly restrains the four corners and the center of the concrete. The reinforced area and the non-reinforced area can be derived according to the multi-cavity CFSTSW cloud map of the ABAQUS finite element analysis results. The calculation diagram is shown in

Figure 14 where the semicircle of b is the length of a single cavity of a multi-cavity steel tube.

The concrete area of the non-enhanced area is expressed as:

The concrete area of the enhanced area is expressed as:

Among them, ${A}_{\mathrm{c}1}$ is the concrete area of the non-enhanced area; ${A}_{\mathrm{c}2}$ is the concrete area of the enhanced area, and ${A}_{\mathrm{c}}$ is the concrete area.

Assuming that the lateral stress of the concrete in the non-enhanced area is not considered, it can be known from the limit equilibrium theory that the lateral restraint force received by the reinforced zone concrete and the lateral force received by the multi-cavity steel tube are in equilibrium. The relationship of the transverse stress of the concrete in the reinforced area can be expressed as:

where

${\sigma}_{\mathrm{r},\mathrm{c}}$ is the lateral restraint force of concrete in the non-enhanced area.

In the non-enhanced area, the axial compressive strength of concrete is approximately equal to the axial compressive strength

f_{c} of plain concrete. The axial compression strength of core concrete in constrained area is:

where

${\sigma}_{\mathrm{L},\mathrm{c}}$ is the axial compressive strength of the filled concrete in the enhanced area;

k is the lateral compression coefficient, and the coefficient is 3.4 according to reference [

31].

The T-shaped multi-cavity CFSTSW is divided into three parts: end column

${N}_{\mathrm{d}}$, middle column

${N}_{\mathrm{z}}$ and web interspace

${N}_{\mathrm{f}}$.

${N}_{\mathrm{u}}$ can be expressed as:

where

${N}_{\mathrm{d}}$ is the axial compression bearing capacity of the end column.

${N}_{\mathrm{z}}$ is the axial compression bearing capacity of the middle column.

${N}_{\mathrm{f}}$ is the axial compression bearing capacity of the web interspace.

The axial bearing capacity of the end column

${N}_{\mathrm{d}}$ is composed of three parts and can be expressed as:

The first part ${A}_{\mathrm{c}1,\mathrm{d}}{f}_{\mathrm{c},\mathrm{d}}$ is the axial bearing capacity of the concrete in the non-enhanced area. The second part ${\sigma}_{\mathrm{L},\mathrm{c},\mathrm{d}}{A}_{\mathrm{c}2,\mathrm{d}}$ is the axial bearing capacity of the enhanced concrete. The third part ${\sigma}_{\mathrm{L},\mathrm{s},\mathrm{d}}{A}_{\mathrm{s},\mathrm{d}}$ is the axial bearing capacity of the multi-cavity steel tube.

Here, ${A}_{\mathrm{c}1,\mathrm{d}}$ is the area of the non-enhanced area of the end column; ${f}_{\mathrm{c},\mathrm{d}}$ is the axial compressive strength of the concrete of the end column; ${\sigma}_{\mathrm{L},\mathrm{c},\mathrm{d}}$ is the axial compressive strength of the concrete in the enhanced area of the end column; ${A}_{\mathrm{c}1,\mathrm{d}}$ is the area of the enhanced area of the end column; ${\sigma}_{\mathrm{L},\mathrm{s},\mathrm{d}}$ is the longitudinal compression stress of the steel tube of the end column; and ${A}_{\mathrm{s},\mathrm{d}}$ is the section area of the steel tube of the end column.

Substituting Equations (7)–(12) into Equation (14),

${N}_{\mathrm{d}}$ can be simplified as:

The axial bearing capacity of the web interspace

${N}_{\mathrm{f}}$ is composed of three parts and can be expressed as:

The first part ${A}_{\mathrm{c}1,\mathrm{f}}{f}_{\mathrm{c},\mathrm{f}}$ is the axial compression concrete bearing capacity of the non-enhanced area. The second part ${\sigma}_{\mathrm{L},\mathrm{c},\mathrm{f}}{A}_{\mathrm{c}2,\mathrm{f}}$ is the axial bearing capacity of the enhanced concrete. The third part ${\sigma}_{\mathrm{L},\mathrm{s},\mathrm{f}}{A}_{\mathrm{s},\mathrm{f}}$ is the axial compression bearing capacity of the multi-cavity steel tube. Here, ${A}_{\mathrm{c}1,\mathrm{f}}$ is the area of the non-enhanced area of the web interspace. ${f}_{\mathrm{c},\mathrm{f}}$ is the axial compressive strength of the concrete of the web interspace. ${\sigma}_{\mathrm{L},\mathrm{c},\mathrm{f}}$ is the axial compressive strength of the concrete in the enhanced area of the web interspace. ${A}_{\mathrm{c}1,\mathrm{f}}$ is the area of the enhanced area of the web interspace. ${\sigma}_{\mathrm{L},\mathrm{s},\mathrm{f}}$ is the longitudinal compression stress of the steel tube of the web interspace. ${A}_{\mathrm{s},\mathrm{f}}$ is the section area of the steel tube of the web interspace.

Substituting Equations (7)–(12) into Equation (16),

${N}_{\mathrm{f}}$ can be simplified as:

The axial bearing capacity of the middle column

${N}_{\mathrm{f}}$ is composed of three parts and can be expressed as:

The first part ${A}_{\mathrm{c}1,\mathrm{z}}{f}_{\mathrm{c},\mathrm{z}}$ is the axial compression bearing capacity of the non-enhanced area concrete. The second part ${\sigma}_{\mathrm{L},\mathrm{c},\mathrm{z}}{A}_{\mathrm{c}2,\mathrm{z}}$ is the axial bearing capacity of the reinforced concrete. The third part ${\sigma}_{\mathrm{L},\mathrm{s},\mathrm{z}}{A}_{\mathrm{s},\mathrm{z}}$ is the axial compression bearing capacity of the multi-cavity steel tube.

Here, ${A}_{\mathrm{c}1,\mathrm{z}}$ is the area of the non-enhanced area of the middle column; ${f}_{\mathrm{c},\mathrm{z}}$ is the axial compressive strength of the concrete of the middle column; ${\sigma}_{\mathrm{L},\mathrm{c},\mathrm{z}}$ is the axial compressive strength of the concrete in the enhanced area of the middle column; ${A}_{\mathrm{c}2,\mathrm{z}}$ is the area of the enhanced area of the middle column; ${\sigma}_{\mathrm{L},\mathrm{s},\mathrm{z}}$ is the longitudinal compression stress of the steel tube of the middle column; and ${A}_{\mathrm{s},\mathrm{z}}$ is the section area of the steel tube of the middle column.

Substituting Equations (7)–(12) into Equation (18),

${N}_{\mathrm{z}}$ can be simplified as:

Substituting Equations (15), (17) and (19) into Equation (13),

${N}_{\mathrm{u}}$ can be expressed as:

The calculated value by Equation (20) was about 18% larger than the test value. The main reason is that, first of all, the initial defects of the multi-cavity steel tube were not considered in the calculation. In the process of processing and transportation, the tube will be irregularly deformed due to manual cutting errors, welding, and collisions, and there are certain initial defects. As a result, the bearing capacity of the shear wall specimen is reduced. Second, the weld effect of the steel tube was not considered. Third, due to the limitation of the size and the complex shape of the specimen, the concrete pouring compactness could be affected to some extent, so the correction coefficient of 0.82 was introduced.

Considering the correction coefficient, finally,

${N}_{\mathrm{u}}$ can be expressed as:

The calculated bearing capacity results by Equation (21) and the finite element simulation under different parameters are shown in

Appendix A. The calculated results by the equation were in good agreement with the finite element results, the error between them was less than 4% and can therefore meet the requirements of actual engineering. Thus, the equation can provide references for the design of such components in actual engineering.

For the calculation equation of the axial compression bearing capacity of the concrete-filled steel tube column, the widely used design codes mainly include the American code (ANSI/AISC 341-10) [

32] and the European standard EC4 (2004) [

33]. These design codes consider the bearing capacity of the CFST column as the simple sum of the bearing capacities of the concrete and the steel tube, without considering the interaction between the steel tube and the concrete as well as the uneven constraint of the steel tube on the concrete. The equation established in this article considers the interaction between the steel tube and concrete and the uneven constraint of the steel tube on the concrete as well as the number of cavities of the multi-cavity steel tube, so it is more complicated and accurate to calculate the axial compression bearing capacity of a T-shaped multi-cavity CFSTSW when compared with the American code and the European standard. However, only three T-shaped multi-cavity CFSTSW test specimens were designed and were manufactured according to 1/5 of the original size in this paper, so the deduced equation and reduction coefficient in the proposed equation may be improved later through full-scale experiments and more test specimens.