Numerical Analysis of CFST Column with PBL Stiffeners under Axial Compression

Abstract

PBL stiffeners, made of thin-walled steel plates with circular openings and welded to the steel tube of a square concrete-filled steel tubular (CFST) column, can improve the combined effect effectively by co-carrying axial compressive forces and confining the concrete core. A numerical simulation study based on the previous test was conducted to study the ultimate strength of the CFST stub column with PBL stiffeners. Finite element models of CFST with different stiffeners were made and verified by the test results of typical failure modes and load–strain curves of specimens. The parameter study was conducted, including PBL stiffener detailing (i.e., material strength, stiffener thickness, opening diameter, and opening spacing). Finally, based on the study and analysis results, an ultimate bearing capacity prediction formula was proposed, which can reasonably predict the bearing capacity of a square CFST column with longitudinal or diagonal stiffeners, while the methods in ACI, BS5400, EC4, AIJ, and DBJ were more conservative.

1. Introduction

The concrete-filled steel tube (CFST) columns are widely used in modern structures, such as large-span bridges and high-rise buildings [1,2,3], because of the confining effect of steel tubes on the concrete core. However, compared with circular steel tubes, the confining effect is ineffective in square or rectangular tubes. Meanwhile, the square or rectangular CFST is also more prone to local buckling than the circular tubes (Figure 1a). To prevent local buckling of the steel tube wall and to increase the ultimate strength of the CFST, several kinds of stiffening schemes were proposed to improve the combined effect between the infilled concrete and square steel tube, enhancing the confining strength of the steel tube on the concrete core (Figure 1b–d). Meanwhile, Fu et al. [4] improved the ductility of rectangular CFST columns by replacing conventional sections with unequal-walled cross-sections; Nassiraei et al. [5,6] used Fiber Reinforced Polymer (FRP) and collar plate reinforcement to improve tubular X-joints’ ultimate load-carrying capacity and ductility. The stiffeners with some openings, called Perfobond Leiste (PBL) stiffeners (Figure 2), can enhance the connection between steel tube and concrete core and transfer the shear force [7,8].

Liu et al. [7,9] studied the influence of PBL stiffeners on the bond performance of steel tubes and concrete cores in CFST, and the results showed that PBL stiffeners could significantly improve the stiffness and shear bearing capacity of CFST, and the average bond strength was more than two times that of ordinary CFST.

Zhu et al. [10] studied the shear performance of five PBL connections set in rectangular CFST components. The results showed that the specimen has double-sided shear failure along both sides of the PBL connector; the shear bearing capacity of the PBL connector passively constrained by the steel tube was mainly determined by the passive lateral binding force provided by the steel tube, and adding the openings of PBL connector can improve the lateral binding force of steel tube.

The axial pressure tests on 17 CFST and two octagonal CFST specimens, of which the thickness ratio is between 50 and 150, were conducted in Ref. [11]; the influence of the thickness, opening size, and opening shape of the CFST with PBL diagonal stiffeners was studied.

The axial pressure tests on 21 CFST stub columns to study the thickness of the PBL diagonal stiffeners, discontinuous or positioning welding, and the welding position of the PBL diagonal stiffeners on steel tubes were conducted in Ref. [12]. The results showed that the PBL diagonal stiffeners combined with the steel tube and the concrete could effectively delay the local buckling of the steel tube and improve the bearing capacity and combined effect.

A parameter study was conducted for CFST with PBL diagonal stiffeners by the finite element method in Ref. [13]. The parameters were proposed, including the details of the PBL diagonal stiffeners (i.e., opening diameter, opening spacing, and welding position), material strength, and some reasonable range of parameter selection.

The bending and eccentric compression properties of the CFST beams and columns with PBL were studied in Ref. [14]. Three CFST beams and columns were fabricated, and the bending moment and eccentric compression were tested. The results showed that stiffeners with PBL can improve the mechanical properties of CFST specimens under bending and eccentric compression loads.

The authors designed and tested CFST columns with different stiffeners based on these studies. The test results were reported in the companion paper [15], which mainly focused on the performance of square CFST columns with diagonal ribs under axial forces. The ductility and ultimate strength were studied. A formula was proposed to predict the ultimate strength of CFST with diagonal ribs in the companion paper.

However, the experimental studies were limited, and the kinds of stiffening the steel tubular with PBL stiffeners are different (Figure 1), and many factors affect the ultimate strength of square CFST, including material strength, welding position of stiffeners, stiffener thickness, opening diameter, opening spacing, etc. Although some previous studies [8,11,12,13,15] have compared or studied the parameters of different stiffening ribs, they are not comprehensive or systematic. To facilitate the use of square CFST columns with different stiffeners and propose rational methods to predict the ultimate strength, finite element analysis (FEA) models of CFST with different stiffeners were made in this paper, followed by extensive parametric analysis. The main parameters including stiffener types (i.e., single longitudinal stiffeners, double longitudinal stiffeners, diagonal stiffeners), stiffener thickness, material strengths, opening diameter, and opening space were studied.

At last, based on the study and analysis results, an ultimate bearing capacity prediction formula was proposed in this paper, which can reasonably predict the bearing capacity of a square CFST column with longitudinal or diagonal stiffeners.

2. Description of the Previous Test

A total of 12 CFST columns were designed (Figure 3), tested under axial compression (Figure 4), and divided into five groups. The specimen details are listed in Ref. [15].

This paper aims to improve the ultimate strength and ductility performance of square CFST short columns by increasing stiffener thickness, increasing stiffener number, and using PBL stiffeners and diagonal binding ribs. Through the experiment, the effects of different stiffening methods on ultimate strength and ductility were compared; the better stiffening method with better comprehensive performance was studied and avoided the low ductility risk of Type B.

Square CFST columns with straight ribs are in Type A; square CFST columns with single PBL stiffeners are in Type B; square CFST columns with double PBL stiffeners are in Type C; square CFST columns with PBL diagonal binding ribs are in Type D; square CFST columns with no stiffeners are in Group E.

The length (L) of each stub column was designed to be 2.5 times the width (B), which was regarded as an appropriate length to include a representative pattern of residual stresses and geometric imperfections yet prohibit overall buckling. The width (B) and length (L) are 120 and 300 mm, respectively.

3. Finite Element Models

To study the compressive properties of CFST with different stiffeners, the finite element program ANSYS was used to numerically simulate the CFST specimens of Ref. [15]. The measured elasticity modulus of concrete $E_c$ and steel $E_s$ were adopted for the material properties. The steel yield strengths were 397.82 MPa and 349.72 MPa, corresponding to 2.75 mm thick and 4.75 mm thick steel plates, respectively. Moreover, Poisson’s steel ratios were taken as 0.28 and 0.31, respectively. The concrete cube compressive strength $f_{\mathit{cu}}$ was 59.8 MPa.

3.1. Material Models

3.1.1. Steel

The stress and strain model of steel proposed by Tao et al. [16] was adopted (Equation (1)). Stress–strain relationship can be divided into three different stages: (1) elastic stage; (2) yield stage; (3) strain hardening stage, which was shown in Figure 5.

$$\sigma =\{\begin{array}{cc}{E}_s\epsilon & 0\le \epsilon <{\epsilon }_y\\ f_y& {\epsilon }_y\le \epsilon <{\epsilon }_p\\ f_u-\left(f_u-f_y\right){\left(\frac{{\epsilon }_u-\epsilon }{{\epsilon }_u-{\epsilon }_p}\right)}^p& {\epsilon }_p\le \epsilon <{\epsilon }_u\\ f_u& \epsilon \ge {\epsilon }_u\end{array}$$

(1)

$E_s$, $f_y$, $f_u$ represent the elastic modulus, yield strength, and ultimate tensile strength of steel tube; ${\epsilon }_y({\epsilon }_y=f_y/E_s)$ is yield strain; ${\epsilon }_p$ is the strain corresponding to the beginning of strain hardening which can be obtained by Equation (3); ${\epsilon }_u$ is the ultimate strain corresponding to the ultimate tensile strength which can be obtained by Equation (4). $p$ is the hardening strain index which can be obtained by Equation (2).

Figure 5. Stress–strain model of steel [16].

Figure 5. Stress–strain model of steel [16].

$$p=E_p\left(\frac{{\epsilon }_u-{\epsilon }_p}{f_u-f_y}\right)$$

(2)

$E_p$ is the initial elastic modulus corresponding to the beginning of strain hardening, which can be taken as $0.02 E_s$.

$${\epsilon }_p=\left\{\begin{array}{c}15{\epsilon }_y f_y\le 300 \mathrm{MPa} \\ \left[15-0.018\left(f_y-300\right)\right]{\epsilon }_y 300 \mathrm{MPa}<f_y\le 800 \mathrm{MPa}\hfill \end{array}\right.$$

(3)

$${\epsilon }_u=\left\{\begin{array}{c}100{\epsilon }_y f_y\le 300 \mathrm{MPa} \\ \left[100-0.15\left(f_y-300\right)\right]{\epsilon }_y 300 \mathrm{MPa}<f_y\le 800 \mathrm{MPa}\hfill \end{array}\right.$$

(4)

3.1.2. Concrete

The core concrete under compression was confined by the steel tube with stiffeners, so the stress–strain model proposed by Zhang et al. [8] was adopted, which studied the stress–strain model of square CFST columns with PBL stiffeners. The stress–strain model of core concrete is shown in Figure 6 and Equation (5).

$$y=\left\{\begin{array}{c}2x-x^2 x\le 1 \\ \frac{x}{\beta {\left(x-1\right)}^{\eta }+x} x>1 \hfill \end{array}\right.$$

(5)

$\beta$ can be obtained by Equation (11), and $\eta$ can be obtained by Equation (12)

$$x=\frac{\epsilon }{{\epsilon }_0}$$

(6)

${\epsilon }_0$ is the ultimate strain corresponding to the ultimate strength of the core concrete, which can be obtained by Equations (9) and (10).

$$y=\frac{\sigma }{{\sigma }_0}$$

(7)

${\sigma }_0$ is the ultimate stress corresponding to the ultimate strength of the core concrete, which can be obtained by Equation (8).

Figure 6. Stress–strain model of core concrete.

Figure 6. Stress–strain model of core concrete.

$${\sigma }_0=\left[0.89+\left(-0.012{\xi }^2+0.09\xi \right){\left(\frac{27}{f_{cu}}\right)}^{0.45}\right]f_{cu}$$

(8)

$\xi$ is the constraint effect coefficient of the CFST, which can be obtained by Equation (13).

$${\epsilon }_0=\left[{\epsilon }_{cc}+\left(570+28f_{cu}\right){\xi }^{0.2}\right]{10}^{-6}$$

(9)

$${\epsilon }_{cc}=\left(1300+11.1f_{cu}\right){10}^{-6}$$

(10)

$$\beta =\left\{\begin{array}{c}\frac{f_{cu}^{0.1}}{1.37\sqrt{1+\xi }} \xi \le 3\\ \frac{f_{cu}^{0.1}}{1.37\sqrt{1+\xi {\left(\xi -2\right)}^2}} \xi >3\hfill \end{array}\right.$$

(11)

$$\eta =1.6+\frac{1.5}{x}$$

(12)

$$\xi =\frac{A_t{f}_{yt}+A_s{f}_{ys}}{A_c{f}_{cu}}$$

(13)

where $A_t$, $A_s$, and $A_c$ are the total cross-sectional areas of the steel tube, stiffeners, and concrete, respectively. $f_{yt}$ is the yield strength of the steel tube; $f_{ys}$ is the yield strength of stiffeners; and $f_{cu}$ is the compressive strength of the concrete cube.

3.2. Finite Element Model

3.2.1. Finite Element Type

The solid element SOLID186 simulated the concrete. The element consists of 20 nodes, each with three degrees of freedom. The element has the characteristics of plastic, stress stiffness, large deformation, and large strain, which is suitable for simulating concrete.

Steel tubes and stiffeners were simulated by the solid element SOLID187 which is 3D with ten nodes. In order to better simulate the local buckling of the steel tube in the compression process, the “Solid shell” function during meshing in ANSYSWORKBENCH was used to make it more practical. Solid shell consists of eight nodes, and it has the following features:

(1)

It involves only displacement nodal DOFs and features an eight-node brick connectively. Thus, the transition problem between solid and shell elements can be eliminated.

(2)

Performs well in simulating steel tubes with a wide range of thicknesses.

(3)

Performs well for both flat-plate and curved shells.

3.2.2. Interface

The interaction between the inner surface of the steel tube and the outer surface of concrete can usually be simulated by the “frictional contact” model. Due to the large elastic modulus of the steel tube, the steel tube was defined as the “Contact Bodies”, and the concrete was defined as the “Target Bodies”. Due to the simultaneous load, there is little or no slip between the steel tube and concrete during the loading process. Therefore, the selection of friction coefficient between steel and concrete is not sensitive to the column’s property [17]. In this paper, the friction coefficient of 0.6 was taken to simulate the tangent contact between the steel tube and the core concrete. Meanwhile, the normal direction is set to “Pure Penalty” with a normal penalty stiffness (FKN) of 10, and the normal direction can be specified for the interface, which allows the separation of the interface in tension and no penetration of that in compression.

3.2.3. Boundary Conditions and Load Application

The upper end of the specimen was constrained freedoms of all directions. The lower end of the specimen was free in the longitudinal direction, and displacement-controlled loading was applied.

3.2.4. Mesh

According to the conclusion of Ref. [16], when the length–width ratio of the finite element was less than 3, the effect of the element length-width ratio on the load–strain curve can be ignored. So, width B (width of the steel tube) was divided into 15 parts laterally, and length L (length of column) was divided into 38 parts longitudinally. Finally, the whole model was meshed by “sweep”. The result of meshing is shown in Figure 7.

4. Model Verification

The test results were compared with the finite element analysis results to verify the rationality of the numerical analysis results and provide a basis for the subsequent parametric and theoretical analysis.

4.1. Typical Failure Modes

The FEA results, including the local buckling failure mode and the location of the maximum equivalent stress of CFST with different stiffeners, are shown in Figure 8, Figure 9, Figure 10 and Figure 11. The values of ultimate strength $N_{ep}$ are listed in

Table 1. Ultimate strength of finite element analysis and test results.

Specimens	${\mathit{N}}_{\mathit{u}\mathit{e}}$ (kN)	${\mathit{N}}_{\mathit{e}\mathit{p}}$ (kN)	$\frac{{\mathit{N}}_{\mathit{e}\mathit{p}}}{{\mathit{N}}_{\mathit{u}\mathit{e}}}$
SC-3-A	1348	1414	1.049
SC-5-A	1326	1418	1.069
SC-3-B	1332	1376	1.033
SC-5-B	1462	1416	0.969
SC-3-C	1189	1283	1.079
SC-5-C	1262	1351	1.071
SC-3-D1	1262	1245	0.987
SC-3-D2	1318	1261	0.957
SC-5-D1	1541	1478	0.959
SC-5-D2	1638	1568	0.957
Average			1.013
Standard Deviation			0.049

.

The stiffeners in Type A had no openings; the overall lateral expansion of the column was more uniform under the axial pressure; the equivalent stress of the upper and lower ends was small; and high equivalent stress was in most of the middle area of the column (Figure 8).

The local buckling areas of specimens in Type B and Type C were mainly in the middle part of the columns, accompanied by high stress (Figure 9a and Figure 10a), which was consistent with the actual situation of the test. The openings of the PBL stiffeners were also deformed, resulting in the smaller diameter of openings, and the concrete in the openings was squeezed and greatly deformed too (Figure 9c,d and Figure 10c,d).

The local buckling areas of specimens in Type D were mainly in the position close to the end of the column and accompanied high stress (Figure 11a), consistent with the actual situation of the test. The openings and concrete in the openings were similar to the Type B and Type C (Figure 11c,d).

4.2. The Comparison of Test and FEA

The test results in the companion paper [15] were compared with the results of the FEA model in this paper to verify the feasibility of the finite element analysis (FEA) model. It can be seen that the finite element simulation was accurate in terms of the failure modes. The comparison of typical failure models of CFST columns with different stiffeners between test and FEA models is shown in Figure 12.

4.3. Axial Load–Strain Curves

The comparisons of axial load–strain curves of test results in Ref. [15] and FEA models are shown in Figure 13. It can be seen that the FEA models could simulate the axial performance well before the ultimate load. The curve slopes of the elastic stage predicted by FEA models are very close to the ones of the test, and the load–strain curves of FEA decreased faster than those of the test after the ultimate load.

The ultimate bearing capacity of ANSYS simulation ($N_{ep}$) and test result ($N_{ue}$) are listed in Table 1, and the comparison between $N_{ep}$ and $N_{ue}$ is shown in Figure 14. The mean ratio of $\frac{N_{ep}}{N_{ue}}$ was 1.013, and the standard deviation was 0.049. Therefore, the FEA model could predict the strength and strain development of the tested specimens with satisfactory accuracy. Hence, this model is acceptable.

5. Parametric Studies

The parameters of CFST specimens were studied by using the finite element program ANSYS, including the characteristic cube strength ($f_{cu}$) of concrete, the wall thickness ($t_t$) of steel tube, the diameter ($d$) of openings on PBL, and the opening spacing ($s$), width ($b_s$), and thickness ($t_s$) of steel stiffeners. The width (B) and length (L) of all the specimens were 120 mm and 300 mm, respectively. $N_{\mathit{ep}}$ was the ultimate strength predicted by FE models. Other detailed parameters are shown in

Table 2. Parametric analysis of CFST column under axial compression.

(a)
Specimens (Type A)	${\mathit{f}}_{\mathit{c}\mathit{u}}$	${\mathit{t}}_{\mathit{t}}$	$\mathit{d}$	${\mathit{b}}_{\mathit{s}}$	${\mathit{t}}_{\mathit{s}}$	$\mathit{s}$	${\mathit{N}}_{\mathit{e}\mathit{p}}$
SC-3-A-1	40	2.75	0	42	2.75	0	1179
SC-3-A-2	50	2.75	0	42	2.75	0	1297
SC-3-A-3	60	2.75	0	42	2.75	0	1414
SC-3-A-4	70	2.75	0	42	2.75	0	1531
SC-3-A-5	40	1.2	0	42	2.75	0	899
SC-3-A-6	50	1.2	0	42	2.75	0	1017
SC-3-A-7	60	1.2	0	42	2.75	0	1134
SC-3-A-8	70	1.2	0	42	2.75	0	1249
SC-5-A-1	40	2.75	0	42	4.75	0	1188
SC-5-A-2	50	2.75	0	42	4.75	0	1303
SC-5-A-3	60	2.75	0	42	4.75	0	1418
SC-5-A-4	70	2.75	0	42	4.75	0	1533
SC-5-A-5	40	1.2	0	42	4.75	0	909
SC-5-A-6	50	1.2	0	42	4.75	0	1019
SC-5-A-7	60	1.2	0	42	4.75	0	1135
SC-5-A-8	70	1.2	0	42	4.75	0	1247
(b)
Specimens (Type B)	${\mathit{f}}_{\mathit{c}\mathit{u}}$	${\mathit{t}}_{\mathit{t}}$	$\mathit{d}$	${\mathit{b}}_{\mathit{s}}$	${\mathit{t}}_{\mathit{s}}$	$\mathit{s}$	${\mathit{N}}_{\mathit{e}\mathit{p}}$
SC-3-B-1	40	2.75	16	42	2.75	50	1127
SC-3-B-2	50	2.75	16	42	2.75	50	1248
SC-3-B-3	60	2.75	16	42	2.75	50	1376
SC-3-B-4	70	2.75	16	42	2.75	50	1486
SC-3-B-5	40	1.2	16	42	2.75	50	843
SC-3-B-6	50	1.2	16	42	2.75	50	965
SC-3-B-7	60	1.2	16	42	2.75	50	1087
SC-3-B-8	70	1.2	16	42	2.75	50	1211
SC-5-B-1	40	2.75	16	42	4.75	50	1182
SC-5-B-2	50	2.75	16	42	4.75	50	1297
SC-5-B-3	60	2.75	16	42	4.75	50	1416
SC-5-B-4	70	2.75	16	42	4.75	50	1533
SC-5-B-5	40	1.2	16	42	4.75	50	905
SC-5-B-6	50	1.2	16	42	4.75	50	1024
SC-5-B-7	60	1.2	16	42	4.75	50	1146
SC-5-B-8	70	1.2	16	42	4.75	50	1267
SC-3-B-d1	60	2.75	0	42	2.75	50	1414
SC-3-B-d2	60	2.75	$0.2 b_s$	42	2.75	50	1397
SC-3-B-d3	60	2.75	$0.38 b_s$	42	2.75	50	1376
SC-3-B-d4	60	2.75	$0.6 b_s$	42	2.75	50	1334
SC-3-B-d5	60	2.75	$0.8 b_s$	42	2.75	50	1301
SC-5-B-d1	60	2.75	0	42	4.75	50	1418
SC-5-B-d2	60	2.75	$0.2 b_s$	42	4.75	50	1451
SC-5-B-d3	60	2.75	$0.38 b_s$	42	4.75	50	1416
SC-5-B-d4	60	2.75	$0.6 b_s$	42	4.75	50	1361
SC-5-B-d5	60	2.75	$0.8 b_s$	42	4.75	50	1316
SC-3-B-s1	60	2.75	$0.38 b_s$	42	2.75	23	1358
SC-3-B-s2	60	2.75	$0.38 b_s$	42	2.75	27.3	1359
SC-3-B-s3	60	2.75	$0.38 b_s$	42	2.75	33.3	1360
SC-3-B-s4	60	2.75	$0.38 b_s$	42	2.75	42.86	1355
SC-3-B-s5	60	2.75	$0.38 b_s$	42	2.75	50	1376
SC-3-B-s6	60	2.75	$0.38 b_s$	42	2.75	60	1364
SC-3-B-s7	60	2.75	$0.38 b_s$	42	2.75	75	1365
SC-3-B-s8	60	2.75	$0.38 b_s$	42	2.75	100	1364
(c)
Specimens (Type C)	${\mathit{f}}_{\mathit{c}\mathit{u}}$	${\mathit{t}}_{\mathit{t}}$	$\mathit{d}$	${\mathit{b}}_{\mathit{s}}$	${\mathit{t}}_{\mathit{s}}$	$\mathit{s}$	${\mathit{N}}_{\mathit{e}\mathit{p}}$
SC-3-C-1	40	2.75	8	21	2.75	50	1023
SC-3-C-2	50	2.75	8	21	2.75	50	1145
SC-3-C-3	60	2.75	8	21	2.75	50	1283
SC-3-C-4	70	2.75	8	21	2.75	50	1385
SC-3-C-5	40	1.2	8	21	2.75	50	748
SC-3-C-6	50	1.2	8	21	2.75	50	873
SC-3-C-7	60	1.2	8	21	2.75	50	998
SC-3-C-8	70	1.2	8	21	2.75	50	1124
SC-5-C-1	40	2.75	8	21	4.75	50	1106
SC-5-C-2	50	2.75	8	21	4.75	50	1230
SC-5-C-3	60	2.75	8	21	4.75	50	1351
SC-5-C-4	70	2.75	8	21	4.75	50	1469
SC-5-C-5	40	1.2	8	21	4.75	50	813
SC-5-C-6	50	1.2	8	21	4.75	50	936
SC-5-C-7	60	1.2	8	21	4.75	50	1056
SC-5-C-8	70	1.2	8	21	4.75	50	1148
SC-3-C-d1	60	2.75	$0.2 b_s$	21	2.75	50	1303
SC-3-C-d2	60	2.75	$0.38 b_s$	21	2.75	50	1283
SC-3-C-d3	60	2.75	$0.6 b_s$	21	2.75	50	1248
SC-3-C-d4	60	2.75	$0.8 b_s$	21	2.75	50	1216
SC-5-C-d1	60	2.75	$0.2 b_s$	21	4.75	50	1368
SC-5-C-d2	60	2.75	$0.38 b_s$	21	4.75	50	1351
SC-5-C-d3	60	2.75	$0.6 b_s$	21	4.75	50	1271
SC-5-C-d4	60	2.75	$0.8 b_s$	21	4.75	50	1222
(d)
Specimens (Type D1)	${\mathit{f}}_{\mathit{c}\mathit{u}}$	${\mathit{t}}_{\mathit{t}}$	$\mathit{d}$	${\mathit{b}}_{\mathit{s}}$	${\mathit{t}}_{\mathit{s}}$	$\mathit{s}$	${\mathit{N}}_{\mathit{e}\mathit{p}}$
SC-3-D1-1	40	2.75	16	42	2.75	50	902
SC-3-D1-2	50	2.75	16	42	2.75	50	1023
SC-3-D1-3	60	2.75	16	42	2.75	50	1245
SC-3-D1-4	70	2.75	16	42	2.75	50	1343
SC-3-D1-5	40	1.2	16	42	2.75	50	731
SC-3-D1-6	50	1.2	16	42	2.75	50	849
SC-3-D1-7	60	1.2	16	42	2.75	50	975
SC-3-D1-8	70	1.2	16	42	2.75	50	1094
SC-5-D1-1	40	2.75	16	42	4.75	50	1249
SC-5-D1-2	50	2.75	16	42	4.75	50	1368
SC-5-D1-3	60	2.75	16	42	4.75	50	1478
SC-5-D1-4	70	2.75	16	42	4.75	50	1596
SC-5-D1-5	40	1.2	16	42	4.75	50	937
SC-5-D1-6	50	1.2	16	42	4.75	50	1060
SC-5-D1-7	60	1.2	16	42	4.75	50	1180
SC-5-D1-8	70	1.2	16	42	4.75	50	1300
(e)
Specimens (Type D2)	${\mathit{f}}_{\mathit{c}\mathit{u}}$	${\mathit{t}}_{\mathit{t}}$	$\mathit{d}$	${\mathit{b}}_{\mathit{s}}$	${\mathit{t}}_{\mathit{s}}$	$\mathit{s}$	${\mathit{N}}_{\mathit{e}\mathit{p}}$
SC-3-D2-1	40	2.75	21	55	2.75	50	1026
SC-3-D2-2	50	2.75	21	55	2.75	50	1145
SC-3-D2-3	60	2.75	21	55	2.75	50	1261
SC-3-D2-4	70	2.75	21	55	2.75	50	1374
SC-3-D2-5	40	1.2	21	55	2.75	50	778
SC-3-D2-6	50	1.2	21	55	2.75	50	903
SC-3-D2-7	60	1.2	21	55	2.75	50	1024
SC-3-D2-8	70	1.2	21	55	2.75	50	1144
SC-5-D2-1	40	2.75	21	55	4.75	50	1341
SC-5-D2-2	50	2.75	21	55	4.75	50	1457
SC-5-D2-3	60	2.75	21	55	4.75	50	1568
SC-5-D2-4	70	2.75	21	55	4.75	50	1682
SC-5-D2-5	40	1.2	21	55	4.75	50	1060
SC-5-D2-6	50	1.2	21	55	4.75	50	1186
SC-5-D2-7	60	1.2	21	55	4.75	50	1306
SC-5-D2-8	70	1.2	21	55	4.75	50	1422
SC-3-D2-d1	60	2.75	$0.2 b_s$	55	2.75	50	1305
SC-3-D2-d2	60	2.75	$0.38 b_s$	55	2.75	50	1261
SC-3-D2-d3	60	2.75	$0.6 b_s$	55	2.75	50	1216
SC-3-D2-d4	60	2.75	$0.8 b_s$	55	2.75	50	1181
SC-5-D2-d1	60	2.75	$0.2 b_s$	55	4.75	50	1638
SC-5-D2-d2	60	2.75	$0.38 b_s$	55	4.75	50	1568
SC-5-D2-d3	60	2.75	$0.6 b_s$	55	4.75	50	1510
SC-5-D2-d4	60	2.75	$0.8 b_s$	55	4.75	50	1452

.

All of the FE specimens were divided into five types, which were based on the specimens in Ref. [15]. The names of specimens starting with SC mean Stiffened CFST columns with longitudinal ribs or diagonal ribs. Following the SC, the number represents the thickness of ribs $t_s$; the following letter represents the Type Classification. For example, SC-3-A-5 means Type A, which is shown in Figure 15a; the last number is the serial number. The specimens of Type D consist of two types of specimens, which have different welding positions of diagonal ribs (Figure 15d,e).

5.1. Concrete Strength

Specimens with four different concrete strengths ($f_{cu}=$ 40, 50, 60, 70 MPa) were designed. The ultimate strength of CFST columns increased with increasing concrete strength.

In Figure 16a, as the thickness of the steel tube and stiffener $t_t=t_s$ = 2.75 mm, the ultimate strength of specimens of Type A, Type B, Type C, and Type D2 increased linearly as $f_{cu}$ increased. For SC-3-D1, the ultimate strength also increased as the $f_{cu}$ increased. However, as $f_{cu}$ increased from 50 MPa to 60 MPa, the ultimate strength of SC-3-D1 increased faster than in other stages. Five groups ranked from largest to smallest in order of strength value were: Group A > Group B > Group C > GroupD1 > GroupD2. It can be seen that the effect of $f_{cu}$ on ultimate strength was different for different stiffeners. It was most significant for Type D1.

In Figure 16b, the wall thickness of steel tube $t_s$ = 4.75 mm; the thickness of stiffener $t_t=2.75 \mathrm{mm}$; the ultimate strength of all groups also increased linearly with the $f_{cu}$ increasing, and it can be seen that the effect of $f_{cu}$ on ultimate strength was almost the same for all groups. Compared with Figure 16a, there is an obvious difference in Figure 16b. The ultimate strengths of Type D1 and Type D2 were more significant than those of Type A, Type B, and Type C. Type D1 and Type D2 were diagonal stiffeners, while Type A, Type B, and Type C were longitudinal stiffeners, which indicated the thickness of stiffener had more effect on the ultimate strength of CFST columns with diagonal stiffener. Compared with Figure 16a, another difference was that the ultimate strengths of SC-5-A (ordinary stiffener with no openings) and SC-5-B (PBL stiffener) were almost the same as $f_{cu}$ changed, which indicated that as the thickness of stiffeners $t_s$ increased, the openings on stiffeners could not affect the ultimate strength of CFST columns. Five groups rank from largest to smallest in order of strength value: Type D2 > Type D1 > Type A > Type B > Type C.

The characteristic of the curves in Figure 16c,d were similar to the ones in Figure 16a,b, respectively. The ultimate strength of specimens in Type A, Type B, Type C, Type D1, and Type D2 increased linearly as $f_{cu}$ increased. The members of types D1 and D2 are more sensitive to the thickness of the stiffeners. When the thickness of the stiffeners is the same as the outer wall, the effect is not as good as other stiffeners; when the thickness of the stiffeners is greater than the outer wall, there is a significant increase in ultimate load capacity. Five groups rank from largest to smallest in order of strength value ($N_p$), which were: Type A > Type B > Type D2 > Type C > Type D1 in Figure 16c. Compared with Figure 16c, as the $t_s$ increased from 2.75 mm to 4.75 mm in Figure 16d, the ultimate strengths of Types D1 and D2 were bigger than those of Type A, Type B, and Type C, and the five groups ranked from largest to smallest in order of strength value were: Type D2 > Type D1 > Type A > Type B > Type C.

5.2. Thickness $t_t$ and $t_s$

The effect of steel tube thickness ($t_t$) and stiffener ($t_s$) on the ultimate strength of CFST columns with different stiffeners was studied.

As the thickness ($t_t$) of the steel tube increases and $t_s$ remains unchanged, the ultimate strength of CFST columns increases. The columns with longitudinal ribs (Type A, Type B, Type C) and columns with diagonal ribs (Type D1, Type D2) increased obviously. For example, $f_{\mathit{cu}}$ = 60 MPa, $t_s$ = 2.75 mm, the $t_t$ increased from 1.2 mm to 2.75 mm; the $N_{\mathit{ep}}$ increased by 24.7%, 26.6%, 28.6%, 27.7%, and 23.1%, respectively, from Type A to Type D2; as the $f_{\mathit{cu}}$ = 60 MPa, $t_s$ = 4.75 mm, and $t_t$ increased from 1.2 mm to 2.75 mm, the $N_{\mathit{ep}}$ increased by 25%, 23.6%, 27.9%, 25.3%, and 20.1%, respectively, from Type A to Type D2, which is shown in Figure 17. It was shown that the effects of the thickness of steel tube $t_t$ on the ultimate strength of CFST with different stiffeners were all significant. The $N_{\mathit{ep}}$ of Type C increased the most, and Type D2 increased least, so the thickness of steel tube $t_t$ had the most effect on the ultimate strength of Type C (with two longitudinal ribs on each side) and had the least effect on the ultimate strength of Type D2 (with diagonal rib).

As the thickness ($t_s$) of stiffeners increases and $t_t$ remains unchanged, the ultimate strength of CFST columns increases. However, columns with longitudinal ribs (Type A, Type B, Type C) increased little, and columns with diagonal ribs (Type D1, Type D2) increased obviously. For example, $f_{\mathit{cu}}$ = 60 MPa, $t_t$ = 1.2 mm, the $t_s$ increased from 2.75 mm to 4.75 mm, and the $N_{\mathit{ep}}$ increased by 0%, 5.4%, 5.8%, 21%, and 27%, respectively, from Type A to Type D2; as the $f_{\mathit{cu}}$ = 60 MPa, $t_t$ = 2.75 mm, the $N_{\mathit{ep}}$ increased by 0.3%, 2.9%, 5.3%, 18.7%, and 24.3%, respectively, from Type A to Type D2, which is shown in Figure 18. So, the ultimate strength of CFST columns with diagonal ribs was greatly affected by the change in $t_s$.

5.3. Opening Diameter

The effect of opening diameter on the ultimate strength of CFST columns with different stiffeners was studied. The opening diameter d was minimal, so the PBL stiffener can be used as an ordinary stiffener [8,18,19]. If the openings were large enough, the strengthening effect of PBL on the steel tube wall decreased due to the stiffener section decreasing, but PBL can enhance the connection between steel tube and concrete and transfer the shear force.

The stiffeners in Type A have no openings, which can be considered the opening diameter 0 times $b_s$ in Type B. The welding position of diagonal ribs was different between Type D1 and Type D2. The welding position of diagonal ribs in D2 was proposed by Zhou et al. [13], which performed well on composite effect and ductility capacity. So, 26 specimens with opening diameter d varied from 0 to 0.8 times the stiffener width ($b_s$) based on specimens SC-3-B-3, SC-5-B-3, SC-3-C-3, SC-5-C-3, SC-3-D2-3, and SC-3-D2-5 which were designed and correspond to Type B, Type C, and Type D. The FEA results are shown in Figure 19.

The ultimate strength of the CFST column decreased as the opening diameter d increased in all groups. For Type B (Figure 19a), the strength differences in the specimens with d smaller than 0.4 b_s were both within the range of 5.0% for SC-3-B and SC-5-B. After 0.4 b_s, the $N_{\mathit{ep}}$ decreased faster. As d was small (0.2 b_s), the strength values of SC-3-B and SC-5-B were different; the value of SC-5-B was larger than the one of SC-3-B due to larger thickness, but as d increased, especially as d was 0.8 b_s, the strengths values of SC-3-B and SC-5-B was very close. Increasing the thickness was useless for ultimate strength as the opening diameter were large. So, it was proposed that the opening diameters was not larger than 0.4 b_s. Otherwise, the effect of stiffeners decreases significantly.

For Type C, the situation was similar to Type B; For Type D2, the ultimate strength of the CFST column decreased almost linearly as the opening diameter d increased. The ultimate strength of SC-3-D2 and SC-5-D2 decreased by 9.5% and 11.4%, respectively, as d increased from $0.2 b_s$ to $0.8 b_s$, but if d was not larger than $0.4 b_s$, the decreases in ultimate strength were both within 5%. It was consistent with the result in Ref. [13]. So, it was proposed that the opening diameter d was within the ranges of 0.2–0.4 times the width of the diagonal stiffeners (i.e., d = 0.2–0.4 b_s).

5.4. Opening Spacing $s$

Opening spacing $s$ was another variable parameter. In Ref. [13], the opening spacing was studied, and the effect on the CFST column with PBL diagonal ribs was insignificant. Zhang et al. [8] designed four specimens, with s varying from 1.5 d to 2.5 d, to study the effect of opening spacing on CFST column with PBL longitudinal rib and found the ultimate strength increased first and then decreased with the opening spacing increasing. There was no apparent regularity.

To further study the effect of opening spacing on the ultimate strength of CFST columns with PBL longitudinal rib, eight FE specimens with opening spacing $s$ varied from about 1.5 d to 6.25 d were designed; d (d = 16 mm) was the opening diameter. The eight FE specimens were based on SC-3-B, and one PBL longitudinal rib on each side of the steel tube.

As shown in Figure 20, the effect of opening spacing $s$ on ultimate strength is insignificant. The ultimate strength changes between 1350 MPa and 1375 MPa with the s increasing; the magnitude of the change is within 3%. The peak value corresponds to s equals 3 d. Openings of PBL stiffener can form concrete dowels, which have benefits for increasing the ultimate strength of the CFST columns. According to Ref. [20], the stress fields from the concrete dowels would overlap as the opening spacing reduces, resulting in highly stressed regions and a decrease in strength. So, the opening spacing $s$ should not be too small. As the opening spacing $s$ increases, the number of openings would be decreased, and the effect of concrete dowels decreases, but the steel ratio of PBL stiffener increases. So, the ultimate strength would not change significantly as the opening spacing changes.

It is proposed that the opening spacing of PBL diagonal rib should be at least 2.25 times the opening diameter (i.e., $s\ge 2.25 d$) by Ref. [13], and the opening spacing of PBL longitudinal rib should be at least three times the opening diameter in this paper.

6. Prediction of Bearing Capacity

6.1. Existing Design Codes

There are several widely used design codes or specifications for calculating the ultimate strength of CFST columns under axial loads, such as Architecture Institute of Japan (AIJ) (2008) [21], BS 5400 (2005) [22], ACI Committee 318 (2014) [23], and DBJ13-51-2010 [24], EC4 (2004) [25]. The bearing capacity of CFST columns generally consists of two parts: concrete and steel tube. For the CFST column with stiffener, the contribution of stiffeners on bearing capacity should be considered by adding a term $f_{ys}{A}_s$ ($f_{ys}$ is the yield strength of stiffener, $A_s$ is the cross-sectional area of stiffeners). However, for PBL stiffeners, the effect of openings on the stiffeners should be considered. So, some more accurate models or calculation methods were studied for CFST columns with PBL stiffeners, mainly divided into two types: PBL longitudinal stiffeners and PBL diagonal stiffeners. The stiffeners of Type A are longitudinal stiffeners; the stiffeners of Type B and Type C are PBL longitudinal stiffeners; the stiffeners of Type D1 and Type D2 are PBL diagonal stiffeners.

6.2. Zhang J G [8]

The square CFST columns with PBL longitudinal stiffeners were studied by Zhang J G [8]; the method predicted the bearing capacity of CFST columns with PBL longitudinal stiffeners could be expressed by Equation (14):

$$N_p=A_t{f}_{yt}+{\gamma }_s{f}_{ys}{A}_s+{\gamma }_c{f}_c{A}_c$$

(14)

where $f_{yt}$, and $f_{ys}$, are the yield strength of steel tube and PBL ribs, respectively; $f_c$ is the prism compressive strength of concrete; $A_c$, $A_t$, and $A_s$ are the cross-sectional areas of concrete, steel tube, and stiffeners, respectively; ${\gamma }_s$ is the reduction coefficient about PBL ribs; ${\gamma }_c$ is coefficient about core concrete. d is the opening diameter; $b_s$ is the width of PBL ribs. $\alpha$ is the steel ratio of CFST column.

$${\gamma }_s=1-d/b_s, {\gamma }_c=\left\{\begin{array}{c}1.0 {\gamma }_s=0\\ \\ 0.56\times ln\left(\alpha +1\right) {\gamma }_s=1.0\\ \\ 0.4+0.46 ln\alpha 0<{\gamma }_s<1.0\end{array}\right.$$

Zhang predicted the bearing capacity of the specimens of his experiment and some other researchers by the method above; a mean ratio $N_p/N_{ue}$ ($N_p$ was the result calculated by Equation (14), $N_{ue}$ was the test result) of 1.06 was obtained with a standard deviation of 0.075. Compared with other results predicted by ACI, BS 5400, EC4, and DBJ, the result predicted by Equation (14) was close to his test result [8].

6.3. Zhou Z [11] and Gan D [12]

The square CFST column with PBL diagonal stiffeners was studied by Zhou et al. [11,12]; the method predicted the bearing capacity of CFST columns with PBL diagonal stiffeners could be expressed by Equation (15):

$$N_p=A_c{\sigma }_{cc}+A_t{\sigma }_{tv}+A_s{\sigma }_{sv}$$

(15)

$A_c$, $A_t$, and $A_s$ were effective cross-sectional areas of concrete, steel tube, and diagonal ribs, respectively, and the openings in the diagonal ribs were taken into consideration when calculating the cross-sectional areas of diagonal ribs. As ${\sigma }_{cc}$ was the compressive strength of confined concrete calculated by Mander et al. [26], ${\sigma }_{tv}$ and ${\sigma }_{sv}$ were vertical stresses of steel tube and diagonal ribs, respectively.

6.4. Guo J H, Diao Y [15]

Based on Refs. [8,11,12], the authors made a further study and modified Equation (15). The new method proposed in Ref. [15], which can be mainly expressed by Equation (16), takes into account the characteristics of the diagonal binding ribs and the outer steel tube undertaking the load together and the influence of the local buckling of the outer steel tube on the ultimate strength, which was in good agreement with the test results; a mean ratio $N_p/N_{ue}$ of 0.976 was obtained with a standard deviation of 0.041; $N_p$ was the result calculated by Equation (16); $N_{ue}$ was the test result. Compared with the results predicted by other methods, Equation (16) gave a more accurate prediction about the bearing capacity of a square CFST column with PBL diagonal ribs.

$$N_p=A_c{f}_{cc}+A_t{\sigma }_{tv}+A_s{\sigma }_{sv}$$

(16)

6.5. This Paper

The square CFST column consists of three parts: concrete, steel tube, and stiffeners; the ultimate bearing capacity came from the contribution of concrete, steel tube, and stiffeners. Based on the parameter study above and considering the effect of PBL on the ultimate strength of CFST columns, Equation (17) was proposed as the predicted formula for the ultimate strength of square CFST with stiffeners.

$$N_p={\beta }_c{f}_c{A}_c+f_{yt}{A}_t+{\beta }_s{f}_{ys}{A}_s$$

(17)

where $f_{yt}$ and $f_{ys}$ are the yielding strength of steel tube and PBL ribs, respectively; $f_c$ is the prism compressive strength of concrete; $A_c$, $A_t$, and $A_s$ are the cross-sectional areas of concrete, steel tube, and stiffeners, respectively; ${\beta }_s$ is the reduction coefficient about PBL ribs, which can be expressed by Equation (18).

$${\beta }_s=1-{\left(\frac{d}{b_s}\right)}^3$$

(18)

where $d \mathrm{is} \mathrm{the} \mathrm{opening} \mathrm{diameter}; b_s \mathrm{is} \mathrm{the} \mathrm{width} \mathrm{of} \mathrm{PBL} \mathrm{ribs}.$

${\beta }_c$ is the coefficient of core concrete, and $\alpha$ ($\alpha$ = $A_t/A_c$) is the steel ratio of CFST columns. According to stiffener types, the data fitting of ${\beta }_c$ was divided into four types: Type A (with no openings), Type B (one rib on each side of the steel tube), Type C (two ribs on each side of the steel tube), and Type D (diagonal ribs, including Type D1 and Type D2). Using test results and FEA results through regression analyses, the expression of ${\beta }_c$ was obtained as follows (Equation (19)):

$${\beta }_c=\left\{\begin{array}{l}1.05+0.024\ln \alpha Type A\\ 0.76+0.24\ln \alpha Type B\\ 1.05+0.01\ln \alpha Type C\\ 1.08+0.04\ln \alpha Type D\end{array}\right.$$

(19)

Formulas are divided into two categories; Types D1 and D2 width-to-thickness ratios should conform to $B/t_s\le 1.7\sqrt{E_s/f_{ys}}$ [15], and other types of width-to-thickness ratios should conform to $60\sqrt{235/f_{yt}}$ [24].

6.6. Comparison and Recommendation

To verify the model in this paper, the bearing capacity of specimens in Refs. [8,11,12,15] were calculated by the methods in ACI, BS 5400, EC4, AIJ, and DBJ and method (Equation (17)) in this paper; the results were compared and listed in

Table 3. Predicted values of bearing capacity of test specimens.

Specimen	${\mathit{N}}_{\mathit{u}\mathit{e}}$	$\frac{{\mathit{N}}_{\mathit{A}\mathit{C}\mathit{I}}}{{\mathit{N}}_{\mathit{u}\mathit{e}}}$	$\frac{{\mathit{N}}_{\mathit{B}\mathit{S}}}{{\mathit{N}}_{\mathit{u}\mathit{e}}}$	$\frac{{\mathit{N}}_{\mathit{E}\mathit{C}}}{{\mathit{N}}_{\mathit{u}\mathit{e}}}$	$\frac{{\mathit{N}}_{\mathit{A}\mathit{I}\mathit{J}}}{{\mathit{N}}_{\mathit{u}\mathit{e}}}$	$\frac{{\mathit{N}}_{\mathit{D}\mathit{B}\mathit{J}}}{{\mathit{N}}_{\mathit{u}\mathit{e}}}$	$\frac{{\mathit{N}}_0}{{\mathit{N}}_{\mathit{u}\mathit{e}}}$	${\mathit{N}}_{\mathit{p}}$	$\frac{{\mathit{N}}_{\mathit{p}}}{{\mathit{N}}_{\mathit{u}\mathit{e}}}$	Source
SC-3-A	1348	0.894	0.614	0.960	0.964	0.982	0.961	1358	1.008	[15]
SC-5-A	1326	0.970	0.618	1.036	1.067	1.059	1.037	1437	1.083
SC-3-B	1332	0.904	0.622	0.972	0.976	0.994	0.972	1475	1.107
SC-5-B	1462	0.880	0.560	0.940	0.968	0.960	0.940	1548	1.059
SC-3-C	1189	1.013	0.697	1.089	1.093	1.113	1.089	1339	1.126
SC-5-C	1262	1.019	0.649	1.089	1.121	1.112	1.089	1417	1.123
SC-3-D1	1262	0.955	0.656	1.026	1.030	1.049	1.026	1388	1.100
SC-3-D2	1318	0.953	0.625	1.020	1.031	1.042	1.021	1434	1.088
SC-5-D1	1541	0.835	0.532	0.892	0.918	0.911	0.892	1460	0.948
SC-5-D2	1638	0.832	0.496	0.884	0.921	0.903	0.885	1529	0.933
SCB-20-1	4096	0.791	0.557	0.849	0.806	0.856	0.845	3620	0.884	[8]
SCC-20-1	4122	0.786	0.553	0.844	0.800	0.851	0.840	3842	0.932
SCC-20-2	3874	0.837	0.589	0.898	0.852	0.905	0.893	3842	0.992
SCC-20-3	3955	0.819	0.577	0.879	0.834	0.887	0.875	3842	0.971
SCB-30-1	6798	0.882	0.616	0.963	0.895	0.961	0.957	6841	1.006
SCC-30-1	7793	0.769	0.537	0.840	0.781	0.839	0.835	7123	0.914
SCC-30-2	7088	0.846	0.590	0.924	0.858	0.922	0.918	7064	0.997
SCC-30-3	6954	0.862	0.602	0.942	0.875	0.940	0.936	6833	0.983
SCB-30-2	5984	0.854	0.591	0.948	0.872	0.946	0.941	5944	0.993
SCC-30-4	5089	1.004	0.695	1.115	1.025	1.113	1.107	5997	1.178
SCC-30-5	5969	0.856	0.593	0.950	0.874	0.949	0.943	5957	0.998
SCC-30-6	6111	0.836	0.579	0.928	0.854	0.927	0.921	5803	0.950
SCB-30-3	10,997	0.735	0.519	0.782	0.773	0.802	0.779	8933	0.812
SCC-30-7	10,063	0.804	0.567	0.855	0.845	0.877	0.851	9722	0.966
SCC-30-8	9566	0.845	0.597	0.899	0.889	0.922	0.895	9615	1.005
SCC-30-9	8777	0.921	0.651	0.980	0.968	1.005	0.976	9193	1.047
SS-2-2-B	5501	0.767	0.512	0.882	0.797	0.895	0.888	5384	0.979	[11]
SS-2-2-B3	5188	0.814	0.543	0.936	0.845	0.949	0.942	5349	1.031
SS-2-3-B	5605	0.819	0.499	0.932	0.864	0.944	0.937	5719	1.020
SS-2-6-B	5871	0.849	0.469	0.955	0.917	0.966	0.960	6056	1.032
SS-3-2-B	6184	0.811	0.580	0.913	0.867	0.918	0.917	6235	1.008
SS-3-3-B	6677	0.807	0.535	0.900	0.871	0.905	0.904	6565	0.983
SS-3-6-B	6754	0.855	0.522	0.946	0.940	0.951	0.950	6896	1.021
SS-6-2-B	7374	0.792	0.598	0.873	0.880	0.897	0.877	7122	0.966
SS-6-3-B	7540	0.822	0.583	0.901	0.919	0.925	0.905	7438	0.986
SS-2-2-H	5427	0.770	0.520	0.888	0.799	0.900	0.894	5336	0.983	[12]
SS-2-3-H	5843	0.763	0.481	0.872	0.800	0.883	0.877	5552	0.950
SS-3-2-H	6015	0.805	0.576	0.909	0.883	0.917	0.914	6049	1.006
SS-3-3-H	6468	0.812	0.554	0.909	0.873	0.914	0.914	6397	0.989
Average		0.851	0.575	0.930	0.903	0.943	0.931	—	1.004
Standard Deviation		0.069	0.053	0.070	0.086	0.071	0.069	—	0.069

. The comparison of test and prediction by Equation (17) of the specimens in Refs. [8,11,12,15] was shown in Figure 21.

For the bearing capacity of CFST columns with longitudinal ribs predicted by the method in ACI, BS 5400, EC4, AIJ, and DBJ, the mean ratios of 0.867, 0.598, 0.939, 0.905, and 0.949 were obtained with a standard deviation of 0.076, 0.044, 0.085, 0.096, and 0.086, respectively. For the method in this paper, which can be expressed by Equation (17), a mean ratio $N_p/N_{ue}$ of 1.009 was obtained with a standard deviation of 0.075.

For the bearing capacity of CFST columns with diagonal ribs predicted by the method in ACI, BS 5400, EC4, AIJ, and DBJ, the mean ratios of 0.828, 0.546, 0.919, 0.896, and 0.933 were obtained with a standard deviation of 0.051, 0.048, 0.044, 0.067, and 0.045, respectively. For the method in this paper, which can be expressed by Equation (17), a mean ratio $N_p/N_{ue}$ of 1.004 was obtained with a standard deviation of 0.069.

It can be seen that the method (Equation (17)) proposed in this paper can reasonably predict the bearing capacity of square CFST columns with different stiffeners. In contrast, the methods in ACI, BS 5400, EC4, AIJ, and DBJ were more conservative.

The method in Ref. [8] can only predict the CFST column with longitudinal ribs. The method in Refs. [11,12] can only predict the CFST column with diagonal ribs and requires many parameters, such as ${\sigma }_{cc}$, ${\sigma }_{tv}$, ${\sigma }_{sv}$, $f_r$,$k_e$, etc., needed to be calculated and solve equations in the calculation process. Some parameters much also be distinguished according to force analysis, which makes the calculation process too cumbersome. The method proposed in this paper does not need to calculate many parameters, can accurately predict the bearing capacity, and is suitable for different types of PBL stiffeners.

7. Conclusions

Through the numerical simulation of CFST specimens, finite element models of CFST with different stiffeners were made and verified by the test results. The parameter study was conducted, and the effect of different parameters on ultimate bearing capacity was analyzed. An ultimate bearing capacity prediction formula was proposed and compared with existing ultimate bearing capacity prediction formulas. Based on the study and analysis results, the following conclusions can be drawn:

• The FEA model could predict the tested specimens’ failure modes, ultimate strength, and strain development with satisfactory accuracy. The mean ratio of $N_{ep}/N_{ue}$ was 1.013, and the standard deviation was 0.049.
• The ultimate strength of specimens in Type A, B, C, D1, and D2 increased linearly as $f_{cu}$ increased. The values of ultimate strength of Type A, B, and C were generally more significant than that of Type D1 and Type D2 as stiffener thickness was small ($t_s$ = 2.75 mm), but the ultimate strengths of Type D1 and D2 were bigger than Type A, B, and C as the stiffener thickness increased ($t_s$ = 4.75 mm).
• The effects of the thickness of steel tube $t_t$ on the ultimate strength of CFST with different stiffeners were all significant. As $t_t$ increased and $t_s$ remained unchanged, the ultimate strength of CFST columns with longitudinal and diagonal stiffeners increased significantly. The effects of the thickness of stiffener $t_s$ on the ultimate strength of CFST with diagonal stiffeners were significant but insignificant for CFST with longitudinal stiffeners.
• The ultimate strength of CFST columns decreased as the opening diameter d increased. The ultimate strength differences in the specimens with d smaller than 0.4 b_s were 5.0%. So, it was proposed that the opening diameter d was within the ranges of 0.2–0.4 times the width of stiffeners (i.e., d = 0.2–0.4 b_s) for both longitudinal and diagonal stiffeners.
• The effect of opening spacing $s$ on ultimate strength is not significant. The stress fields from the concrete dowels would overlap as the opening spacing reduces, resulting in highly stressed regions and a decrease in strength. It is proposed that the opening spacing of PBL diagonal ribs should be at least 2.25 times the opening diameter (i.e., $s\ge 2.25 d$), and the opening spacing of PBL longitudinal ribs should be at least three times the opening diameter.
• The ultimate bearing capacity prediction method proposed in this paper can reasonably predict the bearing capacity of square CFST columns with PBL stiffeners, while the methods in ACI, BS 5400, EC4, AIJ, and DBJ are more conservative. The Equation is suitable for different types of PBL stiffeners, and the Equation of the formula is relatively simple compared to [12,15].