A Simple Method to Evaluate the Bearing Capacity of Concrete-Filled Steel Tubes with Rectangular and Circular Sections: Beams, Columns, and Beam–Columns

Abstract

This study investigates the behavior and capacity of concrete-filled steel tubes (CFTs) with rectangular and circular sections under various loading conditions: axial loads, pure-bending, and combined axial load-bending moments. A finite element-based numerical model is developed and validated against existing experimental data. Using an extensive databank generated from finite element analysis, this research proposes analytical empirical relations for determining the bearing capacity of rectangular and circular composite beams, columns, and beam–columns. These relations provide a direct, concise, and efficient method for calculating the ultimate strength of CFT members, making them valuable for engineering design. Comparisons between the proposed analytical results and previous experimental studies demonstrate the high accuracy of this method in analyzing rectangular and circular CFT member behavior. The findings contribute to a better understanding of CFT structural performance and offer practical tools for designers working with these composite elements.

1. Introduction

Concrete-filled steel tubes (CFT) exhibit a satisfactory load capacity, sufficient ductility, and adequate energy absorption capability. Additionally, the steel tube functions as a mold for pouring the concrete, leading to cost savings in construction. With the tube serving as both longitudinal and lateral reinforcement for the concrete core, no additional reinforcement is required. The enhanced structural characteristics of CFT members are primarily attributed to the combined effect of the steel hollow section and the core concrete. This composite action results in the core concrete being subjected to triaxial stress, while also preventing the steel hollow section wall from buckling inward [1]. Hatzigeorgiou [2] proposed analytical models for predicting the strength of circular CFT beam–columns, assuming that confinement enhances both the strength of the concrete core and its ductility. Furthermore, Zhang and Shahrooz [3], and Hajjar and Gourley [4] have proposed analytical models for predicting the strength of rectangular CFT beam–columns where these models assume that confinement only enhances the ductility of the concrete core, not the strength. Sakino et al. [5] proposed design formulas to predict the axial compressive load capacity and ultimate moment of CFT columns with circular or square cross-sections. Mukai and Nishiyama [6] suggested that the hoop stress on steel tubes for CFT columns is about 20% of the tensile yield stress of steel. Tang et al. [7] developed a model for circular tubes that considers the effect of geometry and material properties on strength enhancement and post-peak behavior. Susantha et al. [8] proposed appropriate stress–strain relationships for concrete confined by steel tubes of various shapes, adopting Tang et al.’s [7] model for circular columns. Watanabe et al. [9] conducted model tests to determine a stress–strain relationship for confined concrete and proposed a method to analyze the ultimate behavior of concrete-filled box columns, considering the local buckling of component plates and initial imperfections. However, there is limited research on the pure flexural behavior of concrete-filled hollow structural section beams. Furlong [10] proposed that the flexural strength derived solely from the capacity of steel tubes was approximately 50% lower than that of the tested specimens. Subsequently, Elchalakani et al. [11], Lu and Kennedy [12], and Prion and Boehme [13] conducted experimental investigations on numerous tests, revealing that the beam specimens exhibited adequate behavior and failed in a highly ductile manner.

This research offers a thorough and detailed examination aimed at forecasting the complete behavioral range of both rectangular and circular concrete-filled tube (CFT) members when subjected to a combination of axial forces and bending moments. The study is systematically organized into two distinct yet complementary sections, each significantly enhancing the overall comprehension and practical application of CFT structural components. The initial section focuses on the creation and application of various advanced finite element models. These models are intended to replicate a diverse array of loading conditions, including both concentric and eccentric axial loads on columns, as well as beams experiencing pure flexural loading. The main goal of this section is to establish a comprehensive and reliable response database that encapsulates the intricate behavior of CFT members under different loading scenarios. Several critical factors are meticulously considered and integrated into these advanced numerical models: the confinement effect, which is crucial for CFT behavior, is addressed by accounting for how it enhances the strength and ductility of the concrete core, while also considering the reduction in yield stress of the steel tube due to the biaxial stress state induced by confinement; the potential for slippage between the steel tube and the concrete core is also taken into account, which is essential for accurately predicting the composite action and load transfer mechanisms within the CFT member; finally, the models incorporate the possibility of local or global buckling of the steel tube, which is particularly significant for understanding the behavior of CFT members under high axial loads or slender configurations. By addressing these intricate interactions and behaviors, the finite element models deliver a comprehensive representation of CFT performance across a wide range of conditions. Therefore, this study advances the understanding of composite members beyond ultimate carrying capacity, offering a comprehensive behavioral prediction model for rectangular and circular CFTs under combined loads. This innovative approach integrates advanced finite element modeling with often-overlooked factors, such as confinement effects, material slippage, and steel tube buckling. By simulating diverse loading conditions, a comprehensive response database has been created that captures intricate CFT behavior. This holistic methodology, combining multiple critical factors in a single model, enhances the understanding of CFT behavior and provides valuable insights for practical design and analysis applications.

2. Materials and Methods

In this Section, the finite element analysis of CFTs is examined. Figure 1 depicts the composite members sections examined here, which are the most applied in engineering practice, i.e., rectangular and circular sections.

For the rectangular section, B, H, t_f, and t_w are the section width, height, flange thickness, and web thickness, respectively, as shown in Figure 1. Furthermore, for the circular section, D and t denote the diameter and thickness of the steel tube, respectively.

To develop an extended and reliable database for the behavior of CFT members with rectangular and circular sections, a 3D nonlinear finite element model was created, utilizing the ATENA [14] finite element analysis. To ensure precision in the simulation, factors such as the nonlinear behavior of confined concrete, the local buckling of steel tubes, and the interface action between concrete and steel tubes were considered. Furthermore, selecting the appropriate element type and mesh size that offer precise results within a reasonable computational time frame is crucial in the simulation procedure. This model takes into consideration both geometrical and material nonlinearities. Various element types were tested to determine the most suitable ones for simulating the behavior of CFT rectangular and circular columns. It was found that shell elements are ideal for modeling the steel tube, while solid elements are best suited for representing the concrete core [15,16]. These element types were subsequently utilized in the finite element analyses. The steel tube is represented using 8-node shell elements, while the concrete core is modeled using 20-node solid elements. To account for symmetry, only half of the column is modeled, as depicted in Figure 2. In all cases, this Figure is showing only half of each section, cut along the symmetry plane mentioned in the text. The meshes demonstrate the “refined discretization” described, especially at the base region. A refined discretization is applied to the base region of the column, with the height equal to the width of the cross-section for rectangular sections or to the diameter for circular sections. The nodes on the symmetry plane are appropriately constrained, while the remaining nodes are free to displace in any direction. To optimize computational efficiency, only one half of the column is modeled, resulting in a vertical plane of symmetry that bisects the column. The nodes situated along this symmetry plane are assigned specific movement limitations to accurately simulate the behavior of the entire column. These constraints generally encompass a) the prohibition of movement perpendicular to the symmetry plane, which ensures that the model functions as if the other half of the column were included; b) allowance for free movement parallel to the symmetry plane, which permits the model to deform naturally along the symmetry plane; and c) restriction on rotation around axes parallel to the symmetry plane, which prevents unrealistic twisting that would not occur in a complete column. To solve the nonlinear equations of motion, the Newmark step-by-step time integration method in conjunction with Newton–Raphson iterations is employed in ATENA [14]. A suitable constraint model has been used for the interface where the relative movement between the concrete core and the steel jacket is minimal. Similar discretization and modeling of CFTs with rectangular and circular sections have been applied by Skalomenos et al. [15] and Serras et al. [16].

In this study, the following parameters have been investigated:

• Concrete grades: C20, C30, C40, and C50, with concrete compressive strength, f_c, equal to 20 MPa, 30 MPa, 40 MPa, and 50 MPa, respectively;
• Steel grades: S200, S300, S400, and S500, with steel yield stress, f_y, equal to 200 MPa, 300 MPa, 400 MPa, and 500 MPa, respectively;
• The diameter of the circular sections, D: 200 mm, 300 mm, 400 mm, 500 mm and 600 mm;
• The width of rectangular sections, B: 200 mm, 330 mm, 470 mm, and 600 mm;
• The height-to-width ratio of rectangular sections, H/B: 1.00 and 1.20;
• The diameter to thickness ratio of circular sections, D/t: 40, 55, 70, 85, and 100;
• The width to thickness ratio of rectangular sections, B/t_f: 40, 60, 80, and 100;
• The width thickness to height thickness ratio of rectangular sections, t_f/t_w: 1.00 and 1.20;
• The member length to section diameter ratio, L/D: 3.0, 6.0, 9.0 and 12.0.

Taking into account the aforementioned combinations of parameters for composite members with circular sections, suitable ones have (4 concrete grades) × (4 steel grades) × (5 diameters) × (5 diameter-to-thickness ratios) × (4 length-to-diameter ratios) = 1600 different models. Additionally, considering the aforementioned combinations of parameters for composite members with rectangular sections, suitable ones have (4 concrete grades) × (4 steel grades) × (4 widths) × (2 height-to-width ratios) × (4 width-to-thickness ratios) × (2 width thickness to height thickness ratios) × (4 length-to-diameter ratios) = 4096 different models. Therefore, the produced databank has 1600 + 4096 = 5696 finite element models, which have been examined under concentric or eccentric axial loads and pure flexural load.

3. Validation of Finite Element Results

In this Section, the validity of the aforementioned finite element analyses of circular and rectangular CFTs is examined. More specifically, some finite element models have been appropriately modified to reproduce the behavior of specific composite beams, columns, and beam–columns, taken from the pertinent literature. Figure 3 depicts the response of concrete-filled steel tubes under pure bending employing experimental results from Ref. [17] and finite element analysis.

Figure 4 shows the response of CFT beam–columns through experimental results from Ref. [18] and finite element analysis.

4. Axial Load-Bearing Capacity of Concrete-Filled Steel Tubes

4.1. Rectangular Composite Sections

For the rectangular composite section, the area for the concrete core can be computed from

$$A_c=\left(B-2t_w\right)\left(H-2t_f\right)$$

(1)

and the area for the steel tube is given by

$$A_s=BH-\left(B-2t_w\right)\left(H-2t_f\right)$$

(2)

The strict contribution of the concrete core to the maximum axial load can be found by

$$P_c={\mathrm{A}}_c{\mathrm{f}}_c=\left[\left(\mathrm{B}-2t_w\right)\left(\mathrm{H}-2t_f\right)\right]{\mathrm{f}}_c$$

(3)

while the corresponding contribution of steel tube is given by

$$P_s={\mathrm{A}}_s{\mathrm{f}}_y=\left[BH-\left(\mathrm{B}-2t_w\right)\left(\mathrm{H}-2t_f\right)\right]{\mathrm{f}}_y$$

(4)

To quantify the composite action for the aforementioned contribution of the discrete materials, the following empirical expressions can be expressed for the concrete core

$${P^{\prime }}_c=a_c{P}_c=\left[a_1+a_2\left({\displaystyle \frac{B}{t_f}}+{\displaystyle \frac{H}{t_w}}\right)+a_3\left({\displaystyle \frac{f_y}{f_c}}\right)\right]P_c$$

(5)

where a_c and a₁–a₃ are parameters that are presented in

Table 1. Parameters for the prediction of axial load–rectangular columns.

a1	a2	a3	b1	b2	b3	c1	c2
1.01837	0.002135	0.032575	1.882731	−0.00397	−0.01295	0.41722	0.038095

. Similarly, the effective axial force for the steel tube is given by

$${P^{\prime }}_s=b_s{P}_s=\left({\mathrm{b}}_1+{\mathrm{b}}_2\left({\displaystyle \frac{B}{t_f}}+{\displaystyle \frac{H}{t_w}}\right)+{\mathrm{b}}_3\left({\displaystyle \frac{f_y}{f_c}}\right)\right)P_s$$

(6)

where b_s and b₁–b₃ are parameters that are presented in Table 1. It should be mentioned that B and H are the rectangular section’s width and height, respectively, and t_f and t_w are the steel tube section thicknesses for the flange and web, respectively. Thus, according to this study, the predicted strength for axial load results from

$$P_{pred}=c\left[{P^{\prime }}_c+{P^{\prime }}_s\right]$$

(7)

where parameter c takes into account the effect of composite column length and can be

$$\mathrm{c}=\left(1-{\mathrm{c}}_1{\left({\displaystyle \frac{H}{L}}\right)}^{c_2}\right)$$

(8)

where c₁ and c₂ are the parameters that are presented in Table 1. Therefore, from Equations (5)–(8), one has

$$P_{pred}=\left[\left(a_1+a_2\left({\displaystyle \frac{B}{t_f}}+{\displaystyle \frac{H}{t_w}}\right)+a_3\left({\displaystyle \frac{f_y}{f_c}}\right)\right)P_c+\left(b_1+b_2\left({\displaystyle \frac{B}{t_f}}+{\displaystyle \frac{H}{t_w}}\right)+b_3\left({\displaystyle \frac{f_y}{f_c}}\right)\right)P_s\right]\left(1-c_1{\left({\displaystyle \frac{H}{L}}\right)}^{c_2}\right)$$

(9)

Table 2. Rectangular concrete-filled steel tube tests–axial load.

Ref.	L (mm)	B (mm)	tf (mm)	H (mm)	tw (mm)	fy (MPa)	fc (MPa)	Pexp (kN)	Ppred (kN)	Ppred/Pexp
[19]	1318.3	329.9	4.47	329.9	4.47	370.3	31.6	4363.3	4864.3	1.11
	1328.4	331.0	4.47	331.0	4.47	370.3	27.4	4411.7	4510.8	1.02
	1320.8	331.0	4.50	331.0	4.50	370.3	27.4	4656.8	4520.3	0.97
	1318.3	331.0	4.50	331.0	4.50	370.3	31.6	4411.7	4901.2	1.11
	1318.3	333.0	6.38	333.0	6.38	444.7	31.6	5862.8	6207.0	1.06
	1318.3	331.0	6.30	331.0	6.30	444.7	31.6	5843.2	6113.8	1.05
	1318.3	331.0	6.32	331.0	6.32	444.7	27.4	5833.4	5742.2	0.98
	1320.8	331.0	6.32	331.0	6.32	444.7	27.4	5637.2	5742.5	1.02
[20]	800.0	150.0	1.40	150.0	1.40	247.3	22.5	711.5	655.5	0.92
	800.0	150.0	2.10	150.0	2.10	248.5	22.5	792.8	710.2	0.90
	800.0	200.0	1.40	150.0	1.40	247.0	22.5	881.0	868.3	0.99
	800.0	200.0	2.10	150.0	2.10	248.3	22.5	843.8	914.3	1.08
	800.0	150.0	1.40	150.0	1.40	247.3	35.3	973.1	911.1	0.94
	480.0	200.0	1.40	150.0	1.40	247.0	33.7	1191.7	1167.4	0.98
	800.0	200.0	2.10	150.0	2.10	248.3	35.3	1288.4	1231.2	0.96
[21]	485.1	323.1	4.37	323.1	4.37	261.3	25.4	3365.5	3465.6	1.03
	485.1	323.1	4.37	323.1	4.37	261.3	41.0	4949.1	4720.8	0.95
	485.1	323.1	4.37	323.1	4.37	261.3	41.0	4828.5	4720.8	0.98
	485.1	323.6	4.37	323.6	4.37	261.3	80.1	7478.4	7816.8	1.05
	477.5	319.0	6.35	319.0	6.35	616.4	25.4	6318.7	6213.6	0.98
	477.5	318.5	6.35	318.5	6.35	616.4	41.0	7777.3	7604.1	0.98
	477.5	318.3	6.35	318.3	6.35	616.4	41.0	7470.3	7597.2	1.02
	477.5	318.5	6.35	318.5	6.35	616.4	84.9	10,353.7	10,875.0	1.05
	396.2	264.9	6.48	264.9	6.48	832.9	25.4	6544.2	5873.3	0.90
	396.2	263.9	6.48	263.9	6.48	832.9	41.0	7114.5	7064.8	0.99
	396.2	264.4	6.48	264.4	6.48	832.9	41.0	7169.6	7082.4	0.99
	396.2	264.9	6.48	264.9	6.48	832.9	84.9	8987.2	9505.3	1.06
[22]	660.4	222.8	3.00	222.8	3.00	313.7	30.1	2412.7	1995.7	0.83
[23]	558.0	186.0	3.00	186.0	3.00	300.0	32.0	1555.0	1489.5	0.96
	738.0	246.0	3.00	246.0	3.00	300.0	38.0	3095.0	2727.2	0.88
	918.0	306.0	3.00	306.0	3.00	300.0	38.0	4003.0	4145.4	1.04
	918.0	306.0	3.00	306.0	3.00	281.0	47.0	4253.0	4793.5	1.13
	918.0	306.0	3.00	306.0	3.00	281.0	47.0	4495.0	4793.5	1.07
	918.0	306.0	3.00	306.0	3.00	281.0	47.0	4658.0	4793.5	1.03
[24]	600.0	152.3	3.00	76.6	3.00	430.0	30.5	50.8	50.8	1.00
[25]	630.0	210.0	5.00	210.0	5.00	750.0	30.0	3710.0	3709.0	1.00
	630.0	210.0	5.00	210.0	5.00	750.0	30.0	3483.0	3709.0	1.06
[26]	599.4	199.9	3.20	199.9	3.20	317.9	24.8	1577.8	1534.5	0.97
	749.3	249.9	3.20	249.9	3.20	317.9	24.8	2123.1	2221.1	1.05
	899.2	300.0	3.20	300.0	3.20	317.9	24.8	2749.9	3059.1	1.11
	599.4	199.9	3.20	199.9	3.20	317.9	30.3	2463.0	1707.7	0.69
	899.2	300.0	3.20	300.0	3.20	317.9	30.3	4590.6	3484.4	0.76
[27]	730.0	220.0	5.00	220.0	5.00	761.0	20.0	3609.0	3345.4	0.93
	880.0	270.0	5.00	270.0	5.00	761.0	20.0	3950.0	4353.1	1.10

presents data from the experimental works of various researchers. Furthermore, the evaluation of maximum axial load using Equation (9) is also examined. It is found that the proposed empirical Equation (9) can predict the axial load capacity in a simple, yet effective and accurate manner. This is also obvious in Figure 5, where the predicted axial forces are close to the experimental ones.

4.2. Circular Composite Sections

For the circular composite section, the area for the concrete core can be computed from

$${\mathrm{A}}_c={\displaystyle \frac{\pi {\left(\mathrm{D}-2t\right)}^2}{4}}$$

(10)

and the area for the steel tube is given by

$${\mathrm{A}}_s={\displaystyle \frac{\pi {\mathrm{D}}^2}{4}}-{\displaystyle \frac{\pi {\left(\mathrm{D}-2t\right)}^2}{4}}$$

(11)

The strict contribution of the concrete core to the maximum axial load can be found by

$$P_c={\mathrm{A}}_c{\mathrm{f}}_c=\left[{\displaystyle \frac{\pi {\left(\mathrm{D}-2t\right)}^2}{4}}\right]{\mathrm{f}}_c$$

(12)

while the corresponding contribution of steel tube is given by

$$P_s={\mathrm{A}}_s{\mathrm{f}}_y=\left[{\displaystyle \frac{\pi {\mathrm{D}}^2}{4}}-{\displaystyle \frac{\pi {\left(\mathrm{D}-2t\right)}^2}{4}}\right]{\mathrm{f}}_y$$

(13)

To quantify the composite action for the aforementioned contribution of the discrete materials, the following empirical expressions can be expressed for the concrete core

$${P^{\prime }}_c=a_c{P}_c=\left[a_1+a_2\left({\displaystyle \frac{D}{t}}\right)+a_3\left({\displaystyle \frac{f_y}{f_c}}\right)\right]P_c$$

(14)

where $a_c$ and $a_1$–$a_3$ are parameters that are presented in

Table 3. Parameters for the prediction of axial load–rectangular columns.

a1	a2	a3	b1	b2	b3	c1	c2
0.935594	0.000474	25.19892	20.0523	−6.29828	−0.00406	1.07973	2.378821

. Similarly, the effective axial force for the steel tube is given by

$${P^{\prime }}_s=b_s{P}_s=\left({\mathrm{b}}_1+{\mathrm{b}}_2\left({\displaystyle \frac{D}{t}}\right)+{\mathrm{b}}_3\left({\displaystyle \frac{f_y}{f_c}}\right)\right)P_s$$

(15)

where b_s and b₁–b₃ are parameters that are presented in Table 3. It should be mentioned that D is the circular section’s diameter and t is the steel tube section thickness. Thus, according to this study, the predicted strength for axial load results from

$$P_{pred}=\mathrm{c}\left[{P^{\prime }}_c+{P^{\prime }}_s\right]$$

(16)

where parameter c takes into account the effect of composite column length and can be computed by

$$\mathrm{c}=\left(1-{\mathrm{c}}_1{\left({\displaystyle \frac{D}{L}}\right)}^{c_2}\right)$$

(17)

where $c_1$ and $c_2$ are parameters that are presented in Table 3. Therefore, from Equations (14)–(17), one has

$$P_{pred}=\left[\left({\mathrm{a}}_1+{\mathrm{a}}_2\left({\displaystyle \frac{D}{t}}\right)+{\mathrm{a}}_3\left({\displaystyle \frac{f_y}{f_c}}\right)\right)P_c+\left({\mathrm{b}}_1+{\mathrm{b}}_2\left({\displaystyle \frac{D}{t}}\right)+{\mathrm{b}}_3\left({\displaystyle \frac{f_y}{f_c}}\right)\right)P_s\right]\left(1-{\mathrm{c}}_1{\left({\displaystyle \frac{D}{L}}\right)}^{c_2}\right)$$

(18)

Table 4. Circular concrete-filled steel tube tests–axial load.

Ref.	L (mm)	D (mm)	T (mm)	fy (MPa)	fc (MPa)	Pexp (kN)	Ppred (kN)	Pexp/Ppred
[28]	914.4	152.4	1.55	331	21	682.4	666.0	0.98
	914.4	152.4	1.55	331	25.9	721.5	751.9	1.04
	914.4	152.4	1.55	331	25.9	733.1	751.9	1.03
[20]	480	150	0.7	248.2	22.5	538.0	487.3	0.91
	800	150	0.7	248.2	22.5	513.5	512.2	1.00
	480	150	0.7	248.2	33.7	743.8	676.2	0.91
[29]	750	250	2	260	59.5	3400.0	3101.0	0.91
[30]	1348.7	450.1	2.97	283.4	25.4	4413.5	5140.8	1.16
	1348.7	449.8	2.97	283.4	41	6867.6	7393.0	1.08
	1348.7	450.1	2.97	283.4	41	6983.3	7401.4	1.06
	716.3	238.5	4.55	578.5	25.4	3034.1	3172.3	1.05
	713.7	238.3	4.55	578.5	40.4	3582.2	3775.3	1.05
	713.7	238	4.55	578.5	40.4	3646.2	3769.6	1.03
	713.7	237.7	4.55	578.5	76.8	5576.3	5134.2	0.92
	1082	360.7	4.55	578.5	25.4	5631.4	5616.4	1.00
	1082	360.7	4.55	578.5	41	7257.7	7069.3	0.97
	1079.5	360.2	4.55	578.5	41	7043.3	7052.5	1.00
	1082	360.4	4.55	578.5	84.9	11,501.8	10,955.3	0.95
	1010.9	336.8	6.48	834.3	25.4	8472.5	8146.4	0.96
	1008.4	336.6	6.48	834.3	41	9665.5	9534.0	0.99
	1010.9	336.8	6.48	834.3	41	9832.3	9545.2	0.97
[31]	661.5	190	1.55	315.3	113.5	3260.0	3297.2	1.01
	660	190	1.15	184.8	113.5	3058.0	3176.1	1.04
	662	190	0.95	211.2	113.5	3070.0	3234.9	1.05
[32]	662	190	1.11	203.1	110	3030.0	3106.4	1.03
	656	190	1.11	203.1	110	2940.0	3102.4	1.06
	662	190	1.11	203.1	110	3140.0	3106.4	0.99
	662	190	1.11	203.1	94.7	2462.0	2699.4	1.10
	664	190	1.11	203.1	110	3055.0	3107.7	1.02
	660	190	1.11	203.1	110	3000.0	3105.1	1.04
	664	190	0.86	210.7	91.7	2553.0	2661.7	1.04
	661.5	190	1.13	185.7	113.6	3220.0	3183.4	0.99
	662.5	190	1.13	185.7	91.7	2630.0	2602.5	0.99
	658	190	1.52	306.1	91.7	2830.0	2720.2	0.96
[33]	664.5	190	1.52	306.1	48.3	1695.0	1600.6	0.94
	664.5	190	1.13	185.7	41	1377.0	1255.7	0.91
	659	190	0.86	210.7	41	1350.0	1271.5	0.94
	663.5	190	1.52	306.1	80.2	2602.0	2426.2	0.93
	662.5	190	1.13	185.7	80.2	2295.0	2297.0	1.00
	663.5	190	0.86	210.7	74.7	2451.0	2196.0	0.90
	661.5	190	1.52	306.1	108	3260.0	3143.3	0.96
	660	190	1.13	185.7	108	3058.0	3033.8	0.99
	662	190	0.86	210.7	108	3070.0	3106.7	1.01

presents data from the experimental works of various researchers. Furthermore, the evaluation of maximum axial load using Equation (18) is also examined. It is found that the proposed empirical Equation (18) can predict the axial load capacity in a simple, yet effective and accurate manner. This is also obvious in Figure 6, where the predicted axial forces are close to the experimental ones.

5. Flexural Bearing Capacity of Concrete-Filled Steel Tubes

In this Section, the flexural capacity of CFTs is examined. More specifically, the bearing capacity of concrete-filled steel tubes with rectangular and circular sections is investigated under the action of pure bending.

5.1. Rectangular Composite Sections

For the rectangular composite section, the plastic section modulus for the concrete core results from

$$W_c={\displaystyle \frac{\left(B-2t_w\right){\left(H-2t_f\right)}^2}{4}}$$

(19)

and the plastic section modulus for the steel tube is given by

$$W_s={\displaystyle \frac{B{H}^2-\left(\mathrm{B}-2t_w\right){\left(\mathrm{H}-2t_f\right)}^2}{4}}$$

(20)

The contribution of the concrete core to the maximum pure bending moment can be found by

$$M_c={\mathrm{W}}_c{\mathrm{f}}_c=\left[{\displaystyle \frac{\left(\mathrm{B}-2t_w\right){\left(\mathrm{H}-2t_f\right)}^2}{4}}\right]{\mathrm{f}}_c$$

(21)

while the corresponding contribution of steel tube is given by

$$M_s={\mathrm{W}}_s{\mathrm{f}}_y=\left[{\displaystyle \frac{B{H}^2-\left(\mathrm{B}-2t_w\right){\left(\mathrm{H}-2t_f\right)}^2}{4}}\right]{\mathrm{f}}_y$$

(22)

To quantify the composite action for the aforementioned contribution of the discrete materials, the following empirical expressions can be expressed for the concrete core

$${M^{\prime }}_c=a_c{M}_c=\left[a_1+a_2\left({\displaystyle \frac{B}{t_f}}+{\displaystyle \frac{H}{t_w}}\right)+a_3\left({\displaystyle \frac{f_y}{f_c}}\right)\right]M_c$$

(23)

where a_c and a₁–a₃ are parameters that are presented in

Table 5. Parameters for the prediction of the maximum bending moment–rectangular columns.

a1	a2	a3	b1	b2	b3	c1	c2
0.610416	0.005127	1.768951	12.923533	−0.181790	−0.079599	0.878259	0.006304

. Similarly, the effective axial force for the steel tube is given by

$${M^{\prime }}_s=b_s{M}_s=\left({\mathrm{b}}_1+{\mathrm{b}}_2\left({\displaystyle \frac{B}{t_f}}+{\displaystyle \frac{H}{t_w}}\right)+{\mathrm{b}}_3\left({\displaystyle \frac{f_y}{f_c}}\right)\right)M_s$$

(24)

where b_s and b₁–b₃ are parameters that are presented in Table 5. It should be mentioned that B and H are the rectangular section’s width and height, respectively, and t_f and t_w are the steel tube section thicknesses for the flange and web, respectively, as shown in Figure 1. Thus, according to this study, the maximum pure bending can be estimated by

$$M_{pred}=\mathrm{c}\left[{M^{\prime }}_c+{M^{\prime }}_s\right]$$

(25)

where parameter c takes into account the effect of composite column length and can be computed by

$$\mathrm{c}=\left(1-{\mathrm{c}}_1{\left({\displaystyle \frac{H}{L}}\right)}^{c_2}\right)$$

(26)

where c₁ and c₂ are parameters that are presented in Table 5. Therefore, from Equations (23)–(26), one has

$$M_{pred}=\left[\left({\mathrm{a}}_1+{\mathrm{a}}_2\left({\displaystyle \frac{B}{t_f}}+{\displaystyle \frac{H}{t_w}}\right)+{\mathrm{a}}_3\left({\displaystyle \frac{f_y}{f_c}}\right)\right)M_c+\left({\mathrm{b}}_1+{\mathrm{b}}_2\left({\displaystyle \frac{B}{t_f}}+{\displaystyle \frac{H}{t_w}}\right)+{\mathrm{b}}_3\left({\displaystyle \frac{f_y}{f_c}}\right)\right)M_s\right]\left(1-{\mathrm{c}}_1{\left({\displaystyle \frac{H}{L}}\right)}^{c_2}\right)$$

(27)

Table 6. Rectangular concrete-filled steel tube tests–bending moment.

Ref.	L (mm)	B (mm)	tf (mm)	H (mm)	tw (mm)	fy (MPa)	fc (MPa)	Mexp (kNm)	Ppred (kNm)	Mexp/Mpred
[34]	1400.0	200.0	1.90	200	1.90	282	81.3	42.3	51.4	1.22
	1400	200.0	1.90	200	1.90	282	81.3	54.9	51.4	0.94
	1400	200.0	1.90	200	1.90	282	81.3	56.7	51.4	0.91
	1200	100.0	1.90	100	1.90	282	69.1	10.8	10.6	0.98
	1200	100.0	1.90	100	1.90	282	69.1	10.0	10.6	1.06
	1200	100.0	1.90	100	1.90	282	69.1	10.3	10.6	1.03
	1200	200.0	1.90	200	1.90	282	69.1	42.3	47.1	1.11
	1200	200.0	1.90	200	1.90	282	69.1	54.9	47.1	0.86
	1200	200.0	1.90	200	1.90	282	69.1	56.7	47.1	0.83
[35]	2000	150.0	2.00	150	2.00	397	56.0	31.1	30.2	0.97
[25]	2250	110.0	5.00	110	5.00	750	30.0	66.0	69.5	1.05
	2250	160.0	5.00	160	5.00	750	30.0	141.0	146.2	1.04
	2250	210.0	5.00	210	5.00	750	32.0	228.0	247.8	1.09
[36]	1460	150.2	4.92	150.2	4.92	438	87.5	98.4	90.1	0.92
	1460	150.2	4.84	150	4.84	438	87.5	101.4	88.5	0.87
	1460	150.3	5.93	200.2	5.93	495	87.5	181.7	185.4	1.02
	1460	150.2	5.82	200.2	5.82	495	87.5	175.2	182.1	1.04
	1460	149.4	5.91	250.1	5.91	409	70.6	215.5	210.2	0.98
	1460	148.7	5.89	250.1	5.89	409	70.6	212.4	208.9	0.98
	1460	149.3	5.88	250.2	5.88	409	90.9	219.8	217.1	0.99
	1460	148.8	5.85	250.1	5.85	409	90.9	217.7	215.5	0.99
[37]	1200	150.0	4.00	150	4.00	435	87.2	70.1	73.8	1.05
	1200	150.0	4.00	150	4.00	430	87.2	65.9	73.1	1.11
	1200	150.0	5.00	150	5.00	420	87.2	90.3	88.6	0.98
	1200	150.0	5.00	150	5.00	416	87.2	82.4	87.8	1.07
	1200	150.0	6.00	150	6.00	430	87.2	99.7	107.3	1.08
	1200	150.0	6.00	150	6.00	436	87.2	106.6	108.6	1.02
[38]	1500	150.0	5.00	150	5.00	537	81.2	106.1	108.1	1.02
	1500	150.0	5.00	150	5.00	748	81.2	137.7	145.4	1.06
	1500	150.0	5.00	150	5.00	820	81.2	152.0	157.9	1.04
	1500	150.0	5.00	150	5.00	886	81.2	156.8	169.3	1.08
[39]	1700	80.0	5.00	140	5.00	709	96.0	94.0	87.7	0.93
	1700	80.0	5.00	160	5.00	687	89.0	111.6	102.7	0.92
	1700	80.0	5.04	180	5.04	696	92.0	131.9	124.0	0.94
	1700	80.0	3.50	140	3.50	682	96.0	65.3	60.0	0.92
	1700	80.0	4.00	140	4.00	722	96.0	80.4	71.9	0.89
	1700	80.0	5.00	140	5.00	709	35.0	89.8	80.5	0.90
	1700	80.0	5.00	160	5.00	687	34.0	106.3	94.2	0.89
	1700	80.0	5.04	180	5.04	696	35.0	122.5	113.2	0.92
	1700	80.0	4.91	140	4.91	272	90.0	43.8	36.2	0.83
	1700	80.0	4.91	160	4.91	272	89.0	54.6	44.2	0.81
	1700	80.0	4.91	180	4.91	272	94.0	67.3	53.1	0.79
	1700	80.0	4.91	140	4.91	272	35.0	44.0	32.9	0.75
	1700	80.0	4.91	160	4.91	272	34.0	53.9	39.8	0.74
	1700	80.0	4.91	180	4.91	272	35.0	63.6	47.1	0.74
[40]	1700	80.0	5.00	140	5.00	709	100.0	95.1	88.0	0.93
	1700	80.0	5.00	140	5.00	709	96.0	94.0	87.7	0.93
	1700	80.0	5.00	140	5.00	709	35.0	89.8	80.5	0.90
	1700	80.0	4.91	140	4.91	272	90.0	43.8	36.2	0.83
	1700	80.0	4.91	140	4.91	272	35.0	44.0	32.9	0.75

presents data from the experimental works of various researchers. Furthermore, the evaluation of the maximum bending moment using Equation (27) is also examined. It is found that the proposed empirical Equation (27) can predict the flexural capacity in a simple, yet effective and accurate manner. This is also obvious in Figure 7, where the predicted values of the maximum bending moment are close to the experimental ones.

5.2. Circular Composite Sections

For the circular composite section, the plastic section modulus for the concrete core results from

$${\mathrm{W}}_c={\displaystyle \frac{{\left(\mathrm{D}-2t\right)}^3}{6}}$$

(28)

and the plastic section modulus for the steel tube is given by

$$W_s={\displaystyle \frac{D^3-{\left(\mathrm{D}-2t\right)}^3}{6}}$$

(29)

The contribution of the concrete core to the maximum pure bending moment can be found by

$$M_c={\mathrm{W}}_c{\mathrm{f}}_c=\left[{\displaystyle \frac{{\left(\mathrm{D}-2t\right)}^3}{6}}\right]{\mathrm{f}}_c$$

(30)

while the corresponding contribution of steel tube is given by

$$M_s={\mathrm{W}}_s{\mathrm{f}}_y=\left[{\displaystyle \frac{D^3-{\left(\mathrm{D}-2t\right)}^3}{6}}\right]{\mathrm{f}}_y$$

(31)

To quantify the composite action for the aforementioned contribution of the discrete materials, the following empirical expressions can be expressed for the concrete core

$${M^{\prime }}_c=a_c{M}_c=\left[a_1+a_2\left({\displaystyle \frac{D}{t}}\right)+a_3\left({\displaystyle \frac{f_y}{f_c}}\right)\right]M_c$$

(32)

where a_c and a₁–a₃ are parameters that are presented in

Table 7. Parameters for the prediction of maximum bending moment–circular columns.

a1	a2	a3	b1	b2	b3	c1	c2
7.868303	−0.05441	271.8801	207.2656	−45.2648	−0.30532	0.96368	0.002373

. Similarly, the effective axial force for the steel tube is given by

$${M^{\prime }}_s=b_s{M}_s=\left({\mathrm{b}}_1+{\mathrm{b}}_2\left({\displaystyle \frac{D}{t}}\right)+{\mathrm{b}}_3\left({\displaystyle \frac{f_y}{f_c}}\right)\right)M_s$$

(33)

where b_s and b₁–b₃ are parameters that are presented in Table 7. It should be mentioned that D and t are the steel tube’s diameter and thickness, respectively, as shown in Figure 1. Thus, according to this study, the maximum pure bending can be estimated by

$$M_{pred}=\mathrm{c}\left[{M^{\prime }}_c+{M^{\prime }}_s\right]$$

(34)

where parameter c takes into account the effect of composite column length and can be computed by

$$\mathrm{c}=\left(1-{\mathrm{c}}_1{\left({\displaystyle \frac{H}{L}}\right)}^{c_2}\right)$$

(35)

where c₁ and c₂ are parameters that are presented in Table 7. Therefore, from Equations (23)–(26), one has

$$M_{pred}=\left[\left({\mathrm{a}}_1+{\mathrm{a}}_2\left({\displaystyle \frac{B}{t_f}}+{\displaystyle \frac{H}{t_w}}\right)+{\mathrm{a}}_3\left({\displaystyle \frac{f_y}{f_c}}\right)\right)M_c+\left({\mathrm{b}}_1+{\mathrm{b}}_2\left({\displaystyle \frac{B}{t_f}}+{\displaystyle \frac{H}{t_w}}\right)+{\mathrm{b}}_3\left({\displaystyle \frac{f_y}{f_c}}\right)\right)M_s\right]\left(1-{\mathrm{c}}_1{\left({\displaystyle \frac{H}{L}}\right)}^{c_2}\right)$$

(36)

Table 8. Circular concrete-filled steel tube tests–bending moment.

Ref.	L (mm)	D (mm)	T (mm)	fy (MPa)	fc (MPa)	Mexp (kN)	Mpred (kN)	Mexp/Mpred
[28]	914.4	152.4	1.55	331	21	682.4	666.0	0.98
	914.4	152.4	1.55	331	25.9	721.5	751.9	1.04
	914.4	152.4	1.55	331	25.9	733.1	751.9	1.03
[20]	480	150	0.7	248.2	22.5	538.0	487.3	0.91
	800	150	0.7	248.2	22.5	513.5	512.2	1.00
	480	150	0.7	248.2	33.7	743.8	676.2	0.91
[31]	750	250	2	260	59.5	3400.0	3101.0	0.91
[30]	1348.7	450.1	2.97	283.4	25.4	4413.5	5140.8	1.16
	1348.7	449.8	2.97	283.4	41	6867.6	7393.0	1.08
	1348.7	450.1	2.97	283.4	41	6983.3	7401.4	1.06
	716.3	238.5	4.55	578.5	25.4	3034.1	3172.3	1.05
	713.7	238.3	4.55	578.5	40.4	3582.2	3775.3	1.05
	713.7	238	4.55	578.5	40.4	3646.2	3769.6	1.03
	713.7	237.7	4.55	578.5	76.8	5576.3	5134.2	0.92
	1082	360.7	4.55	578.5	25.4	5631.4	5616.4	1.00
	1082	360.7	4.55	578.5	41	7257.7	7069.3	0.97
	1079.5	360.2	4.55	578.5	41	7043.3	7052.5	1.00
	1082	360.4	4.55	578.5	84.9	11,501.8	10,955.3	0.95
	1010.9	336.8	6.48	834.3	25.4	8472.5	8146.4	0.96
	1008.4	336.6	6.48	834.3	41	9665.5	9534.0	0.99
	1010.9	336.8	6.48	834.3	41	9832.3	9545.2	0.97
[31]	661.5	190	1.55	315.3	113.5	3260.0	3297.2	1.01
	660	190	1.15	184.8	113.5	3058.0	3176.1	1.04
	662	190	0.95	211.2	113.5	3070.0	3234.9	1.05
[32]	662	190	1.11	203.1	110	3030.0	3106.4	1.03
	656	190	1.11	203.1	110	2940.0	3102.4	1.06
	662	190	1.11	203.1	110	3140.0	3106.4	0.99
	662	190	1.11	203.1	94.7	2462.0	2699.4	1.10
	664	190	1.11	203.1	110	3055.0	3107.7	1.02
	660	190	1.11	203.1	110	3000.0	3105.1	1.04
	664	190	0.86	210.7	91.7	2553.0	2661.7	1.04
	661.5	190	1.13	185.7	113.6	3220.0	3183.4	0.99
	662.5	190	1.13	185.7	91.7	2630.0	2602.5	0.99
	658	190	1.52	306.1	91.7	2830.0	2720.2	0.96
[33]	664.5	190	1.52	306.1	48.3	1695.0	1600.6	0.94
	664.5	190	1.13	185.7	41	1377.0	1255.7	0.91
	659	190	0.86	210.7	41	1350.0	1271.5	0.94
	663.5	190	1.52	306.1	80.2	2602.0	2426.2	0.93
	662.5	190	1.13	185.7	80.2	2295.0	2297.0	1.00
	663.5	190	0.86	210.7	74.7	2451.0	2196.0	0.90
	661.5	190	1.52	306.1	108	3260.0	3143.3	0.96
	660	190	1.13	185.7	108	3058.0	3033.8	0.99
	662	190	0.86	210.7	108	3070.0	3106.7	1.01

presents data from the experimental works from previous studies where the evaluation of the maximum bending moment using Equation (36) is also shown. It is found that the proposed empirical Equation (36) can effectively predict the flexural capacity. This is also obvious in Figure 8, comparing Equation (36) to experimental results.

6. Concrete-Filled Steel Tubes under Combined Axial Force–Bending Moment

In everyday engineering practice, structural members are under the action of combined axial force–bending moment loading conditions. Thus, the reliable description of the members appears to be vital. This Section examines the behavior of concrete-filled steel tubes under the action of combined axial force–bending moment, i.e., the behavior of CFT beam–columns.

Lai and Varma [41] and Lai et al. [42], have elaborated on the development of slenderness limits that are utilized to classify concrete-filled tubular (CFT) columns into the categories of compact, noncompact, or slender, contingent upon the governing slenderness ratio (width-to-thickness b/t or D/t ratio, λ) of the steel tube in detail in their publications. The purpose of this does not elaborate on the use of these ratios. Their studies have shown that one of the most important parameters that affect the behavior of CFT beam–columns is the relative strength ratio, ξ, which can be defined as

$$\xi ={\displaystyle \frac{{\mathrm{A}}_s{\mathrm{f}}_s}{{\mathrm{A}}_c{\mathrm{f}}_c}}$$

(37)

In the following, this parameter is used to form the axial force–bending moment curve of CFT beam–columns, i.e., the P-M curve.

6.1. Rectangular Composite Sections

For the rectangular composite section, the relative strength ratio is given by (see Equations (1), (2) and (37))

$$\mathsf{\xi }={\displaystyle \frac{\left[\mathrm{B}\mathrm{H}-\left(\mathrm{B}-2{\mathrm{t}}_{\mathrm{w}}\right)\left(\mathrm{H}-2{\mathrm{t}}_{\mathrm{f}}\right)\right]{\mathrm{f}}_{\mathrm{s}}}{\left[\left(\mathrm{B}-2{\mathrm{t}}_{\mathrm{w}}\right)\left(\mathrm{H}-2{\mathrm{t}}_{\mathrm{f}}\right)\right]{\mathrm{f}}_{\mathrm{c}}}}$$

(38)

For specific values of axial load, P, and relative strength ratio, ξ, the corresponding bending moment of the P-M curve can be defined by the following empirical relation

$${\displaystyle \frac{\mathrm{M}}{{\mathrm{M}}_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}}}={\displaystyle \frac{{\mathrm{m}}_1+{\mathrm{m}}_2\ln \left(\mathsf{\xi }\right)+{\mathrm{m}}_3\left(\frac{\mathrm{P}}{{\mathrm{P}}_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}}\right)}{1+{\mathrm{m}}_4\ln \left(\mathsf{\xi }\right)+{\mathrm{m}}_5\left(\frac{\mathrm{P}}{{\mathrm{P}}_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}}\right)+{\mathrm{m}}_6{\left(\frac{\mathrm{P}}{{\mathrm{P}}_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}}\right)}^2+{\mathrm{m}}_7{\left(\frac{\mathrm{P}}{{\mathrm{P}}_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}}\right)}^3}}$$

(39)

where m₁–m₇ are parameters that are presented in

Table 9. Parameters for the prediction of P-M curve–rectangular columns.

m1	m2	m3	m4	m5	m6	m7
0.892687	−0.000510	−0.897224	0.127727	−2.215097	3.462437	−1.789411

.

The applicability and accuracy of the proposed empirical Equation (39) are shown in Figure 9, comparing this relation to experimental results that have been taken from previous studies. The corresponding data for the studies examined here appear in

Table 10. Data for rectangular concrete-filled steel tube beam–column members.

Ref.	No.	L (mm)	B (mm)	tf (mm)	H (mm)	tw (mm)	fy (MPa)	fc (MPa)	Pexp (kN)	Mexp (kNm)
[43]	I	969	323	4.38	323	4.38	262	41.1	3306	201
	II	969	323	4.38	323	4.38	262	41.1	1479	297
	III	954	318	6.36	318	6.36	618	41.1	4100	408
	IV	957	318	6.36	318	6.36	618	41.1	1967	593
[44]	V	600	200	3.17	200	3.17	310	119	1049	136
	VI	600	200	3.17	200	3.17	310	119	2108	136
	VII	600	200	3.17	200	3.17	310	119	2108	139
	VIII	600	200	3.09	200	3.09	781	119	1294	195
	IX	600	200	3.09	200	3.09	781	119	2569	143
[25]	X	630	210	5	210	5	750	32	3106	77.7
	XI	630	210	5	210	5	750	32	2617	130.9
[27]	XII	2416	220	5	220	5	761	20	2939	58.8
	XIII	2817	270	5	270	5	761	20	3062	76.6

.

6.2. Circular Composite Sections

For the circular composite section, the relative strength ratio is given by (see Equations (1), (10) and (37))

$$\xi ={\displaystyle \frac{\left[{\mathrm{D}}^2-{\left(\mathrm{D}-2t\right)}^2\right]{\mathrm{f}}_s}{\left[{\left(\mathrm{D}-2t\right)}^2\right]{\mathrm{f}}_c}}$$

(40)

For specific values of axial load, P, and relative strength ratio, ξ, the corresponding bending moment of the P-M curve can be also defined from empirical Equation (41)

$${\displaystyle \frac{M}{M_{pred}}}={\displaystyle \frac{m_1+m_2\mathit{ln}\left(\xi \right)+m_3\left(\frac{P}{P_{pred}}\right)}{1+m_4\mathit{ln}\left(\xi \right)+m_5{\mathit{ln}}^2\left(\xi \right)+m_6\left(\frac{P}{P_{pred}}\right)+m_7{\left(\frac{P}{P_{pred}}\right)}^2}}$$

(41)

where, in this case, the m₁—m₇ parameters are presented in

Table 11. Parameters for the prediction of P-M curve–circular columns.

m1	m2	m3	m4	m5	m6	m7
0.954363	0.006437	−0.946844	0.050883	−0.024529	−1.591625	0.880543

.

The applicability and accuracy of the proposed empirical Equation (39) for composite circular sections are shown in Figure 10. More specifically, this figure depicts the proposed axial force–bending moment interaction curves for various circular CFTs beam–columns in comparison to the experimental results that have been taken from previous studies. The corresponding data for these studies examined here, appear in

Table 12. Data for circular concrete-filled steel tube beam–column members.

Ref.	No.	L (mm)	D (mm)	t (mm)	fy (MPa)	fc (MPa)	Pexp (kN)	Mexp (kNm)
[45]	I	2000.0	300.0	4.25	438	66	3070	329.0
	II	2000.0	300.0	5.83	420	64	1932	348.0
	III	2000.0	300.0	5.83	420	62	3217	326.0
	IV	2000.0	300.0	5.65	403	62	2981	318.0
	V	2000.0	300.0	5.65	419	66	4560	256.0
	VI	2000.0	300.0	6.23	436	66	1961	418.7
	VII	2000.0	300.0	5.65	419	66	3256	402.0
	VIII	2000.0	300.0	6.16	406	66	3285	405.0
	IX	2000.0	300.0	5.70	431	66	3305	391.0
	X	2000.0	300.0	5.70	431	66	4629	327.0
	XI	2000.0	165.2	4.35	588	66	1285	99.0
	XII	2000.0	165.2	4.35	588	66	1285	99.0
[31]	XIII	2120.0	152.0	1.70	328	92	470	29.7
	XIV	2120.0	152.0	1.70	328	92	570	32.1
	XV	2120.0	152.0	1.70	328	92	270	30.1
	XVI	2120.0	152.0	1.70	328	92	270	30.8
	XVII	2120.0	152.0	1.70	328	92	670	34.8
	XVIII	1071.0	152.0	1.70	328	92	1273	21.4
	XIX	1071.0	152.0	1.70	328	92	1451	13.8
	XX	1071.0	152.0	1.70	328	92	1309	15.9
[33]	XXI	580.5	165.0	2.82	363	48	1525	10.7
	XXII	661.0	190.0	1.94	256	41	1533	13.2
	XXIII	580.0	165.0	2.82	363	48	1123	19.3
	XXIV	664.0	190.0	1.94	256	41	1284	20.8
	XXV	662.0	190.0	1.52	306	48	1260	19.5
	XXVI	579.5	165.0	2.82	363	80	1940	18.2
	XXVII	662.5	190.0	1.94	256	75	2203	22.0
	XXVIII	579.5	165.0	2.82	363	80	1653	29.6
	XXIX	663.0	190.0	1.94	256	75	1730	36.0
	XXX	578.5	165.0	2.82	363	113	2246	15.3

.

7. Discussion

This study examined the behavior of concrete-filled steel tubes, with rectangular and circular sections, under pure axial/compressive loads, pure bending moments, and combined axial load-bending moment loads. From the statistical analysis of finite element results of Section 2, the following empirical expressions have been proposed:

• Equation (9): the maximum compressive bearing capacity for composite rectangular sections;
• Equation (18): the maximum compressive bearing capacity for composite circular sections;
• Equation (27): the maximum flexural bearing capacity for composite rectangular sections;
• Equation (36): the maximum flexural bearing capacity for composite circular sections;
• Equation (39): the normalized axial force–bending moment interaction curve for composite rectangular sections;
• Equation (41): the normalized axial force–bending moment interaction curve for composite circular sections.

It is important to note that the previously mentioned equations were among the simplest equations that accurately represented the numerical data exhibiting concave curves in every direction. These equations were obtained through the utilization of Table Curve 3D software (Version 5.01, Systat, Richmond, CA, USA) after extensive testing of approximately 8000 mathematical equations. The selection of these equations was based on the criterion for achieving the minimum absolute residual error using the Pearson VII limit, specifically minimizing the sum of ln[√(1 − residual²)].

Equations (9), (18), (27) and (36) take into account the ratio of external section dimensions to the tube thickness, e.g., B/t_f, H/t_w (rectangular section) and D/t (circular section). The proposed empirical equations can be used for the following range: 30 ≤ (B/t_f or H/t_w or D/t) $\le$ 120. Furthermore, these equations also consider the material strength ratio, f_y/f_c, which can range: 3.5 ≤ f_y/f_c ≤ 18.

Moreover, Equations (39) and (41) describe the axial force–bending moment curves for rectangular and circular sections, respectively, and they are expressed in normalized form, i.e., P/Pmax − M/Mmax. They take into account the relative strength ratio (see Equation (37)), which can range: 0.20 ≤ ξ ≤ 2.0.

The suggested equations eradicate the necessity of the existing section classification to estimate the compressive strength of the steel tube section caused by local buckling. It is worth noting that these empirical equations were derived from an advanced finite element model, capable of effectively simulating local buckling phenomena (refer to [15,16] for further details).

As detailed in Section 1, the Finite Element Method (FEM) was utilized to simulate the system being studied. The findings from this analysis are depicted graphically throughout the manuscript, offering a visual summary of the principal results. In particular, the figures illustrate the outcomes of the FEM analysis. These visual representations provide an extensive overview of the system’s behavior under varying conditions, as anticipated by the FEM model. While the section on formula derivation emphasizes the theoretical framework, it is crucial to recognize that the FEM results serve to directly inform and validate these mathematical formulations. The interaction between the theoretical model and the FEM analysis represents a significant strength of this research.

8. Conclusions

A simple and effective model was developed to estimate the bearing capacity of the concrete-filled rectangular and circular tubes under axial load, pure bending, and combination axial load-bending moment load. Using appropriate constitutive models for steel and concrete, the finite element method is used to build an expanded database from which the ultimate behaviors of the CFT beams, columns, and beam–columns were determined by statistical analysis. This database took into account the local buckling of the steel tube, the confinement of the concrete core, and the likelihood of slippage between the steel tube and the concrete core. It is also shown that the suggested method’s ability to predict analytically the capacity of the rectangular and circular CFT beam, columns, and beam–columns is practical and reliable by comparing the proposed empirical expressions for the bearing capacity of composite beams, columns, and beam–columns with numerous experimental test results. Finally, the proposed empirical expressions eliminate the necessity of the section classification according to traditional structural codes by introducing a continuous function to estimate the compressive strength of the steel tube section caused by local buckling, taking into account that these equations were derived from an advanced finite element model, capable of effectively simulating local buckling phenomena.