Analysis of the Bearing Capacity of Concrete-Filled Thin-Walled Square Steel Tubes with Helical Stiffening Based on Local Buckling

Abstract

To address the issue of local buckling in thin-walled steel tube concrete columns, a form of helical stiffening ribs was proposed. Axial compression tests were conducted on five sections of square steel tube concrete column specimens. The research revealed that, compared to ordinary steel tube concrete columns, the axial compression bearing capacity and deformation capacity of steel tube concrete columns with helical rib constraints increased by 18.5% and 7.7%, respectively. The helical ribs effectively enhanced the buckling resistance of the thin-walled steel tube concrete components. The failure pattern of this new type of component was characterized by diagonal cracks in the encased concrete aligning with the direction of the helical ribs, and the buckling of the steel tube walls was concentrated between the helical stiffening ribs. Based on the experiments, an analysis of the buckling performance of thin-walled steel tubes with helical rib constraints was conducted, and this was incorporated into the bearing capacity calculation. The test, simulation, and theoretical calculations showed that the bearing capacity error of the composite columns for each specimen was within 10%. Ultimately, a formula for the critical buckling bearing capacity of the helical rib steel tubes was proposed. The research findings provide a foundation for the engineering application of this new type of component.

1. Introduction

With the development of industrialization and urbanization, the use of thin-walled steel tubes in concrete-filled steel tube columns has become important in the design of low-rise buildings [1]. These columns are lightweight, require less welding, and are cost-effective. However, their bearing capacity is relatively low, and local buckling often occurs in the thin-walled steel tubes. Currently, appropriate confinement methods to delay local buckling in thin-walled steel tubes are a key technology proposed for enhancing structural bearing capacity.

Regarding the local buckling issue of square concrete-filled steel tubes, numerous researchers have proposed various structural measures to enhance the local buckling resistance of thin-walled steel tubes while also improving the confining effect on the concrete. Lee [2] and Wang [3] investigated the impact of vertical stiffeners on the local stability performance of thin-walled square concrete-filled steel tubular columns. They found that the local buckling of specimens with vertical stiffeners was delayed. As the stiffness of the stiffeners increased, the development of local buckling slowed down even further. Xu [4] conducted a comparative analysis between ordinary ribs and longitudinal stiffeners (perforated bridge link ribs) on the confining effects of square concrete-filled steel tubular columns. The study revealed that the ordinary ribs significantly affected the ultimate strength of the components, while PBL ribs notably enhanced the deformation capacity. Zubaidi [5] and Sulaiman [6] conducted experiments by filling steel tubes with different types of lightweight aggregate concrete. Under axial compression, they found that the load-bearing capacity decreased with an increase in column height, while it increased with a larger cross-section and greater steel thickness. The concrete filling delayed the local buckling of the steel tubes, and the addition of steel plates enhanced the axial load-bearing capacity of the plain concrete columns, changing their failure mode from brittle to ductile. Additionally, numerous researchers have proposed the use of tie rod constraints [7,8,9]. Experimental studies on the axial and eccentric compression performance of concrete-filled steel tubular columns indicated that the tie rods’ restraining effect effectively reduces the lateral deformation of the components, thereby delaying their local buckling.

The inclusion of stirrups within reinforced concrete columns can also mitigate local buckling of steel tubes, enhancing both the plastic deformation capacity and the ultimate bearing capacity of the components [10]. Tan [11] conducted eccentric compression tests on square concrete-filled steel tubular columns with spiral reinforcement and ordinary square concrete-filled steel tubular columns. The results indicated that after the components entered the elastoplastic stage, the spiral reinforcement improved both the load-bearing capacity and deformation ability by constraining lateral deformation. Consequently, both the load-bearing capacity and ductility of the square concrete-filled steel tubular columns were enhanced. Moreover, incorporating high-strength spiral stirrups into concrete-filled steel tubular columns provides a stronger confining effect on the core concrete, addressing the issue of uneven constraint distribution in square concrete-filled steel tubular columns [12]. Through finite element simulation, Ding [13] analyzed the mechanical performance of short concrete-filled steel tubular columns with internal stirrups. The study found that when the overall reinforcement ratio remained constant, the effect of increasing steel pipe wall thickness on concrete restraint was lower than that of setting stirrup.

In various geometric configurations, rectangular concrete-filled steel tubes are more prone to local buckling compared to circular ones, thereby affecting the load-bearing capacity and ductility of the components. When the stress on the steel tube exceeds its elastic limit, it enters the elastoplastic range and subsequently undergoes elastoplastic buckling [14]. To address the buckling issues of thin-walled steel tubes, Liu [15] used the Fourier series method and the superposition principle to obtain exact solutions for the buckling of thin plates under complex boundary conditions. Through an improved Fourier series method, Batista [16] derived solutions for the bending problem of thin plates with four corner supports under uniformly distributed loads. Xu [17] applied the separation of variables method to the vibration problem of rectangular thin plates on an elastic foundation and obtained exact solutions for the natural frequencies under various boundary conditions. Li [18,19] proposed the symplectic superposition method to derive analytical solutions for the bending, buckling, and vibration problems of thin plates under various complex boundary conditions. Ullah [20] used the finite integral transform method and obtained exact solutions for the buckling problem of rectangular moderately thick plates with all edges clamped. Based on the generalized finite difference method, Tang [21] derived solutions for the bending deformation of parallelogram thin plates under four-edge fixed boundary conditions. Mijušković [22] and Nazarimofrad [23] utilized the Ritz method to determine the critical buckling loads of plates under different boundary conditions and edge loads.

To delay the local buckling of thin-walled square concrete-filled steel tubular columns, this paper proposes a novel constraint form—spiral stiffening ribs. Five different cross-sectional specimens of square concrete-filled steel tubular columns were designed for comparison and analysis of the load-bearing capacity and failure modes, thereby elucidating the constraint mechanism of the spiral ribs. Combining analytical and numerical methods, a critical buckling stress formula for the spiral rib steel tube was proposed. This formula was used to verify the load-bearing capacity of different composite column components, demonstrating the potential application value of spiral stiffening ribs in thin-walled concrete-filled steel tubes.

2. Axial Compression Test

2.1. Specimen Design and Loading Device

To investigate the confinement effect of helical stiffening ribs on concrete columns under axial compression, five different cross-sectional concrete-filled square steel tube specimens were designed. These specimens were constructed using square steel tubes (Q235B), helical ribs (Q235B), vertical steel plates (Q235B), vertical rebars (HRB400E), and concrete (C30). Specific parameters are detailed in Figure 1 and Figure 2, and

Table 1. Parameters of the specimen (unit: mm) [26,27].

Specimen Number	Column (H × Bt)	Steel Tube (ds × ts)	Spiral Rib (br × tr)	Steel Plate (bp × tp)	Steel Bar (ns)	Constraint Measures
Z1	1000 × 340	340 × 1	-	-	-	None
Z2	1000 × 340	220 × 1	30 × 3	-	-	Spiral rib
Z3	1000 × 340	220 × 1	30 × 3	-	14	Spiral-ribbed steel bar
Z4	1000 × 340	220 × 1	30 × 3	30 × 5	-	Spiral-ribbed steel plate
Z5	1000 × 340	220 × 1	30 × 3	30 × 5	-	Spiral-ribbed steel plate (weided)

Note: H is specimen height; B_t is specimen width; d_s is steel tube width; t_s is steel thickness; b_r is spiral rib width; t_r is thickness of spiral rib; b_p is vertical steel plate width; t_p is vertical steel plate thickness; n_s is diameter of vertical steel bar.

. The mechanical properties of the various materials, determined through material tests, are listed in

Table 2. Steel mechanical property index [26,27].

Category	Yield Strength/MPa	Ultimate Strength/MPa	Yield Ratio	Elongation/%
Steel tube	278.70	388.93	0.72	27.3%
Spiral rib	256.94	363.79	0.71	25.7%
Steel plate	252.65	359.26	0.70	22.7%
Steel bar	462.95	625.30	0.74	24.6%

and

Table 3. Performance of concrete materials [26,27].

Strength Grade	Specimen Label	fcu,k/MPa	fck/MPa	Fc/MPa	Ec/GPa
C30	1	34.6	23.1	27.3	24.7
	2	33.8	22.6	26.7	24.4
	3	33.5	22.4	26.5	24.3
	Average	34.0	22.7	26.8	24.5

Note: f_cu,k is standard value of compressive strength of concrete cubes; f_ck is standard value of axial compressive strength of concrete; f_c is design value of axial compressive strength of concrete; E_c is modulus of elasticity for concrete.

[24,25].

A hydraulic servo-controlled long-column compression testing machine was selected to conduct axial compression tests on the specimens, with a total of five displacement gauges positioned to measure specimen deformations under axial loading. The loading device is shown in Figure 3. To ensure the proper operation of the loading device, a pre-load of 20 kN was applied to the specimens. The specimens were initially loaded up to the yield load under load control at a rate of 50 kN/min. After yielding, the loading was switched to displacement control at a rate of 0.1 mm/min. The test was terminated when the load decreased to 85% of the peak load.

2.2. Damage Phenomenon

From the failure processes of each specimen (Figure 4), the following observations were made: for specimen Z1, initial buckling of the upper steel tube occurred, forming inclined buckling ripples as the load increased. At peak load, the internal concrete was crushed and connected buckling ripples formed at the bottom of the structure. Specimen Z2 exhibited vertical cracks first at the outer concrete layer 150 mm below the top plate, gradually extending towards the centre. Upon reaching the ultimate load, the main concrete crack spiralled along the helical ribs, accompanied by surface concrete spalling. At an axial compression of 1130 kN for specimen Z3, vertical cracks appeared in both the top and bottom outer concrete layers. With increasing stress, the cracks extended towards the middle of the specimen and concrete spalling occurred at the corners of the top plate. Specimen Z4 developed a 200 mm crack at the top when loaded to 975 kN. After reaching peak load, the crack began to peel and exhibited slight spalling, with the width increasing as the load capacity decreased. Ultimately, the intersection of cracks led to concrete delamination and failure. Specimen Z5 showed minor cracks in the bottom outer concrete after loading to 930 kN. Continued loading caused these cracks to propagate upwards, forming multiple branching cracks. After the peak load, the load-bearing capacity continued to decrease, widening surface cracks and leading to concrete spalling and crushing.

After completing the test loading and removing the external concrete casing, observations of the buckling patterns of the internal steel tubes revealed the following: the thin-walled steel tube of specimen Z2 buckled between two stiffening ribs, with no apparent deformation observed in the spiral stiffening ribs. Compared to specimen Z1, there was an improvement in localized buckling. The number of localized buckles in the steel tube wall of specimen Z3 had significantly reduced, with no noticeable deformation observed in the spiral ribs or vertical steel bars. In specimen Z4, localized buckling occurred in the middle and upper parts of the steel tube, with buckling appearing after the instability of the vertical steel plates. The surface of the steel tube in specimen Z5 exhibited slight buckling, which was minor in severity, and there was no noticeable buckling of the vertical plates.

2.3. Load–Displacement Curve

The load–displacement curves for each specimen are depicted in Figure 5a, showing three distinct phases: elastic, elastoplastic, and failure. During the loading phase, the curves exhibit similar slopes, indicating consistent initial stiffness of the components. Upon reaching peak load, the curves enter the descending phase. Compared to specimen Z1, the curves for specimens Z3, Z4, and Z5 exhibit a more gradual descent, indicating the significantly enhanced ductile behaviour of the components.

Comparing the performance variations of each specimen (Figure 5b), it was found that the ordinary concrete-filled steel tube specimen (Z1) had a buckling load of 2642 kN and a peak load of 3248 kN, with a ductility coefficient (µ) of 1.83. Compared to specimen Z1, the ordinary spiral stiffener-reinforced specimen (Z2) showed a 3.3% increase in ductility coefficient, while the yield load and peak load decreased by 18% and 11.5%, respectively. This reduction was attributed to the severe damage to the encased concrete, which weakened the specimen’s load-bearing performance. In contrast, the spiral rib-vertical reinforcement specimen (Z3) exhibited a significant improvement over Z2, with the yield and peak loads increasing by 37.9% and 33.9%, respectively, and an increase in ductility coefficient of 4.2%. This indicates a notable enhancement in the overall load-bearing performance of the specimen. Specimens Z4 and Z5, which incorporate spiral ribs and vertical plates, showed improved deformation capacity compared to Z3. However, both the yield load and peak load decreased by approximately 20%.

2.4. Mechanism of Spiral Rib Confinement

Comparison of the experimental phenomena, failure modes, buckling states of the steel tubes, and load–displacement relationships of various specimens under an axial compressive load revealed that the spiral rib-reinforced concrete-filled steel tube composite column (Z3) exhibits the best overall mechanical performance. The analysis of its working mechanism is as follows: the composite column is equipped with spirally distributed stiffening ribs on the exterior of the steel tube, dividing the concrete and steel tube into several helical belt-like structures. Under vertical loads, the spiral ribs, arranged at an angle, transform the vertical force into shear forces along the direction of the spiral ribs and vertical compressive forces. The concrete, subjected to this combined compressive-shear action, experiences diagonal cracking. The thin-walled steel tube under the constraint of the spiral ribs behaves similarly, undergoing buckling under the combined compressive-shear action. Additionally, the steel tube is subjected to circumferential compression and tensile forces from the internal concrete. The horizontal tensile forces are also decomposed along the angle of the spiral ribs, transforming them into tensile and shear stresses. The stress distribution and load-bearing mechanism are illustrated in Figure 6.

3. Spiral Rib Steel Tube Buckling Analysis

3.1. Theoretical Analysis Method

3.1.1. Mechanical Model

Currently, common methods for analyzing structural stability involve using analytical approximate techniques [28]. Based on the axial compression failure mode and the confinement mechanism of the spiral rib-reinforced components, local buckling of the steel tube in specimen Z3 primarily occured within the parallelogram-shaped regions formed by the spiral ribs, as illustrated in Figure 7a. To analyze the local buckling performance of the thin-walled steel tube, the most deformed rectangular thin steel plate region was selected for study, as shown in Figure 7b.

Consider a rectangular thin steel plate with length (a), width (b), and thickness (t) as the analysis unit. The boundary conditions and loading characteristics of this rectangular unit are determined based on the loading conditions of the specimen. The dimensions of this rectangular calculation unit are related to the geometric dimensions of the specimen as follows:

$$\{\begin{array}{l}a=\frac{d_{\mathrm{s}}\sin \theta }{\tan \theta }\\ b=\frac{d_{\mathrm{s}}}{\tan \theta \cos \theta }-\frac{d_{\mathrm{s}}\cos \theta }{\tan \theta }\end{array}$$

(1)

where d_s is the width of the steel tube, θ is the angle between the spiral stiffening rib and the longitudinal axis of the steel tube’s outer wall.

In the buckling analysis of the thin plate model, consider the combined effects of the compressive load (σ_x), tensile load (σ_y), and shear load (τ). The axial load ratio is given by β = σ_y/σ_x = −0.2. The mechanical calculation unit and the coordinate system used are shown in Figure 8. For different aspect ratios (γ), 0.8, 1.2, 1.5, 2.0, 3.0, 4.0, and stress ratios (S = σ_x/τ_xy), 1.28, 1.52, 1.79, 2.13, a total of 24 thin plates were analyzed. The Ritz method was employed to calculate the critical buckling load coefficients for each plate.

3.1.2. Assumptions of Thin Plate Stability Theory

To analyze a thin plate, we established a coordinate system with the xoy plane as the plane within the plate and the z-axis represents the thickness direction of the plate, as illustrated in Figure 9.

Due to the small deflection bending issues in thin plates, the following computational assumptions are introduced:

(1)

The linear strain of the mid-surface normal direction is not taken into consideration, which results in ε_z = 0:

$$w=w\left(x,y\right)$$

(2)

(2)

The minor stress components τ_zx, τ_zy, and σ_z are significantly smaller compared to the other stress components. Therefore, the shear strains (γ_zx, γ_zy) and normal strain (ε_z) induced by these components are neglected. This simplification transforms the problem of small deflection bending of the thin plate into a plane stress problem, governed by the following physical equations:

$$\{\begin{array}{l}{\epsilon }_{\mathrm{x}}=\frac{1}{E}\left({\sigma }_{\mathrm{x}}-\mu {\sigma }_{\mathrm{y}}\right)\\ {\epsilon }_{\mathrm{y}}=\frac{1}{E}\left({\sigma }_{\mathrm{y}}-\mu {\sigma }_{\mathrm{x}}\right)\\ {\gamma }_{\mathrm{xy}}=\frac{2\left(1+\mu \right)}{E}{\tau }_{\mathrm{xy}}\end{array}$$

(3)

(3)

In the context of small deflection bending of a thin plate, longitudinal displacements along the mid-surface of the plate are neglected. When the mid-surface of the plate undergoes bending deformation, its projected shape, including its line segments and area, remains unchanged on the xoy plane:

$${\left({\epsilon }_{\mathrm{x}},{\epsilon }_{\mathrm{y}},{\gamma }_{\mathrm{xy}}\right)}_{\mathrm{z}=0}=0$$

(4)

3.1.3. Thin Plate Bending Equation

To calculate the buckling load of a plate using the energy method, we constructed the total potential energy function for the plate under the assumptions of small deflection theory. We neglected the internal forces caused by deviations from the mid-plane of the plate:

$$\prod =U-V$$

(5)

$$U=\frac{D}{2}{\displaystyle \underset{0}{\overset{b}{\int }}{\displaystyle \underset{0}{\overset{a}{\int }}\left[{\left(\frac{{\partial }^{2}w}{\partial x^2}+\frac{{\partial }^{2}w}{\partial y^2}\right)}^2-2\left(1-\mu \right)\left(\frac{{\partial }^{2}w}{\partial x^2}\frac{{\partial }^{2}w}{\partial y^2}\right)-{\left(\frac{{\partial }^{2}w}{\partial x\partial y}\right)}^2\right]dxdy}}$$

(6)

$$D=\frac{E{t}^3}{12\left(1-{\mu }^2\right)}$$

(7)

where ∏ is the total potential energy, U is the strain energy, V is the external potential energy (Formula (8)), D is the bending stiffness of the plate, E is the modulus of elasticity of the material, μ is Poisson’s ratio; m and n are the half-wavelengths in the X and Y directions, respectively, after the buckling of the thin plate; A_mn is coefficients, and ω is the deflection surface function that satisfies the boundary conditions of the plate (Equation (9)).

$$V=\frac{t}{2}{\displaystyle \underset{0}{\overset{b}{\int }}{\displaystyle \underset{0}{\overset{a}{\int }}\left[{\sigma }_{\mathrm{x}}{\left(\frac{\partial w}{\partial x}\right)}^2+{\sigma }_{\mathrm{y}}{\left(\frac{\partial w}{\partial y}\right)}^2-2{\tau }_{\mathrm{xy}}\left(\frac{\partial w}{\partial x}\frac{\partial w}{\partial y}\right)\right]dxdy}}$$

(8)

$$w\left(x,y\right)={\displaystyle \sum _{m=1}{\displaystyle \sum _{n=1}{A}_{\mathrm{mn}}\sin \frac{m\pi x}{b}\sin \frac{n\pi y}{a}}}$$

(9)

Substituting Equations (6), (8) and (9) into Equation (5) and applying the principle of stationary potential energy establishes a system of linear algebraic equations. When this system has several sets of solutions, the smallest value among them represents the buckling load.

$$\{\begin{array}{l}\frac{\partial \prod }{\partial A_{11}}=0\\ \frac{\partial \prod }{\partial A_{12}}=0\\ \dots \dots \\ \frac{\partial \prod }{\partial A_{\mathrm{mn}}}=0\end{array}$$

(10)

3.1.4. Critical Buckling Stress

For a simply supported thin plate subjected to compressive, tensile, and shear loads, the total strain energy can be expressed as follows when calculating the critical elastic buckling stress using theoretical methods:

$$U=\frac{{\pi }^{4}abD}{8}{\displaystyle \sum _{m=1}^{\infty }{\displaystyle \sum _{n=1}^{\infty }{A}_{\mathrm{mn}}^2{\left(\frac{m^2}{a^2}+\frac{n^2}{b^2}\right)}^2}}$$

(11)

The total work done by the external loads acting on the thin plate includes three components; axial forces in the x and y directions, and the work done by shear stresses in the xy direction.

$$\begin{array}{l}V=\frac{t{\sigma }_{\mathrm{x}}}{2}{\displaystyle \underset{0}{\overset{a}{\int }}{\displaystyle \underset{0}{\overset{b}{\int }}{\left(\frac{\partial w}{\partial x}\right)}^2}}dxdy+\frac{t{\sigma }_{\mathrm{y}}}{2}{\displaystyle \underset{0}{\overset{a}{\int }}{\displaystyle \underset{0}{\overset{b}{\int }}{\left(\frac{\partial w}{\partial y}\right)}^2}}dxdy-t{\tau }_{\mathrm{xy}}{\displaystyle \underset{0}{\overset{a}{\int }}{\displaystyle \underset{0}{\overset{b}{\int }}\left(\frac{\partial w}{\partial x}\frac{\partial w}{\partial y}\right)}}dxdy\\ =\frac{abt{\pi }^2}{8}{\displaystyle \sum _{m=1}^{\infty }{\displaystyle \sum _{n=1}^{\infty }{A}_{\mathrm{mn}}^2\left({\sigma }_{\mathrm{x}}\frac{m^2}{a^2}+{\sigma }_{\mathrm{y}}\frac{n^2}{b^2}\right)}}-4t{\tau }_{\mathrm{xy}}{\displaystyle \sum _{m=1}^{\infty }{\displaystyle \sum _{n=1}^{\infty }{\displaystyle \sum _{p=1}^{\infty }{\displaystyle \sum _{q=1}^{\infty }{A}_{\mathrm{mn}}{A}_{\mathrm{pq}}\frac{mnpq}{\left(m^2-p^2\right)\left(n^2-q^2\right)}}}}}\end{array}$$

(12)

In the Formula, m, n, p, and q are all integers that must satisfy the condition that they are odd. Assuming the aspect ratio of the plate is γ = a/b, and letting k₁ = σ_xb²t/Dπ², k₂ = σ_yb²t/Dπ², k₃ = τ_xyb²t/Dπ², simplifying Equation (12) and substituting it into Equation (5) gives

$$\begin{array}{l}\prod ={\displaystyle \sum _{m=1}^M{\displaystyle \sum _{n=1}^N{A}_{\mathrm{mn}}^2\left[{\left(m^2+{\gamma }^2{n}^2\right)}^2-k_1{\gamma }^2{m}^2-k_2{\gamma }^4{n}^2\right]}}\\ -\frac{32k_3{\gamma }^3}{{\pi }^2}{\displaystyle \sum _{m=1}^{\infty }{\displaystyle \sum _{n=1}^{\infty }{\displaystyle \sum _{p=1}^{\infty }{\displaystyle \sum _{q=1}^{\infty }{A}_{\mathrm{mn}}{A}_{\mathrm{pq}}\frac{mnpq}{\left(m^2-p^2\right)\left(n^2-q^2\right)}}}}}\end{array}$$

(13)

M and N represent the minimum values required for the convergence of the results. When the derivative of A_mn equals zero, we obtain a homogeneous linear system of equations. The solution to this system gives the critical buckling stress of the thin plate.

$$A_{\mathrm{mn}}\left[{\left(m^2+{\gamma }^2{n}^2\right)}^2-k_1{\gamma }^2{m}^2-k_2{\gamma }^4{n}^2\right]-\frac{32k_3{\gamma }^3}{{\pi }^2}{\displaystyle \sum _{p=1}^{\infty }{\displaystyle \sum _{q=1}^{\infty }{A}_{\mathrm{pq}}\frac{mnpq}{\left(m^2-p^2\right)\left(n^2-q^2\right)}}}=0$$

(14)

$${\left[C\right]}_{L\times L}{\left\{A\right\}}_{L\times 1}=0$$

(15)

The equation [C]_L×L is an L-order matrix where L = M × N; {A}_L×1 represents a column vector of M × N unknown coefficients A_mn; by solving |[C]_L×L| = 0, we obtain the critical buckling coefficients. However, solving the determinant of the M × N order matrix is complex. Therefore, for the thin plates subjected to biaxial loads and shear loads, the corresponding critical stress relationships are proposed.

$$\{\begin{array}{l}\frac{k_1}{k_{1cr}}+\frac{k_2}{k_{2cr}}+{\left(\frac{k_3}{k_{3cr}}\right)}^2=1\hfill \\ \frac{k_1}{k_{1cr}}+\frac{k_2}{k_{2cr}}+{\left(\frac{k_3}{k_{3cr}}\right)}^2=e^{\frac{\beta \gamma }{20}\cdot {\left(\frac{{\tau }_{xy}}{{\tau }_{cr}}\right)}^2}\hfill \end{array}\begin{array}{l}\left(\frac{{\tau }_{\mathrm{xy}}}{{\tau }_{cr}}\right)<0.25 \mathrm{or} \gamma \le 1.6\\ \left(\frac{{\tau }_{\mathrm{xy}}}{{\tau }_{cr}}\right)\ge 0.25 \mathrm{and} \gamma >1.6\end{array}$$

(16)

In the equation, β represents the ratio of axial loads, and the half-wavelengths in the x and y directions are determined according to Figure 10. The critical buckling solutions for the plate under 24 different loading combinations are computed and summarized in

Table 4. Formula calculation results.

γ	S	R	k3	k1	σx/MPa
0.8	1.28	−0.256	3.45	4.44	49.30
1.2	1.28	−0.256	3.84	4.97	77.96
1.5	1.28	−0.256	3.13	4.08	87.15
2.0	1.28	−0.256	3.00	3.95	125.85
3.0	1.28	−0.256	2.91	3.91	517.47
4.0	1.28	−0.256	2.87	3.93	812.63
0.8	1.52	−0.304	2.97	4.61	51.21
1.2	1.52	−0.304	3.12	4.88	77.91
1.5	1.52	−0.304	2.77	4.36	93.05
2.0	1.52	−0.304	2.67	4.24	135.11
3.0	1.52	−0.304	2.62	4.24	561.83
4.0	1.52	−0.304	2.59	4.27	884.10
0.8	1.79	−0.358	2.57	5.02	55.80
1.2	1.79	−0.358	2.74	5.39	84.48
1.5	1.79	−0.358	2.44	4.83	103.12
2.0	1.79	−0.358	2.37	4.73	150.80
3.0	1.79	−0.358	2.34	4.76	630.12
4.0	1.79	−0.358	2.31	4.78	989.05
0.8	2.13	−0.426	2.19	6.16	68.43
1.2	2.13	−0.426	2.38	6.73	105.49
1.5	2.13	−0.426	2.12	6.02	128.68
2.0	2.13	−0.426	2.07	5.92	188.90
3.0	2.13	−0.426	2.05	5.96	789.15
4.0	2.13	−0.426	2.04	6.02	1245.01

.

3.2. Numerical Analysis

We performed numerical analysis on the buckling of thin plates under complex combined loading conditions, then compared and validated the results with formulaic calculations.

3.2.1. Linear Buckling Model

Using the eigenvalue method, a model was established with ABAQUS 6.14 software to analyze the buckling behaviour of a thin plate. The plate was analyzed using 3D shell elements. Compressive (tensile) loads were applied through edge loads on the shell—normal edge traction, while shear forces were applied through edge loads on the shell—shear edge traction. Figure 11 illustrates the finite element model of the thin plate (using a plate with an aspect ratio of γ = 1.2 as an example for buckling analysis). Each geometric edge was subjected to 1 N/mm edge loads, with the boundary conditions set as simply supported on all four sides. The dimensions of the rectangular thin plate were 495.5 × 436 × 4 mm, with an elastic modulus of E = 205 GPa and a Poisson’s ratio of μ = 0.3.

3.2.2. Buckling Mode Calculation

By performing eigenvalue buckling analysis, the first four buckling modes and the corresponding critical buckling load factors of the thin plate were obtained, as shown in Figure 12. As the number of modes increases, the eigenvalues (critical loads) also increase accordingly. The critical buckling loads of the thin plate model were determined from the buckling eigenvalues shown in the figure and are summarized in

Table 5. The first four order critical loads.

Mode	Eigenvalue	Critical Buckling Load (MPa)
First mode	296.98	74.25
Second mode	329.88	82.47
Third mode	542.30	135.58
Fourth mode	812.71	203.18

.

Simulations were conducted separately for 24 combinations of loading (γ = 0.8, 1.2, 1.5, 2, 3, 4; S = 1.28, 1.52, 1.79, 2.13) to analyze the buckling of the thin plate.

(1)

Under loading conditions with S = 1.28 and β = −0.2, the buckling modes are shown in Figure 13. The critical buckling stress values were computed based on the eigenvalues extracted from the output file, as summarized in

Table 6. S = 1.28, β = −0.2 thin plate buckling stress.

γ	Eigenvalue	Critical Buckling Load (MPa)
0.8	198.68	49.67
1.2	283.68	70.92
1.5	362.05	90.51
2.0	509.43	127.36
3.0	2130.20	532.55
4.0	3405.20	851.30

.

(2)

Under loading conditions with S = 1.52 and β = −0.2, the buckling modes of the thin plate are shown in Figure 14. The corresponding critical buckling stresses are listed in

Table 7. S = 1.52, β = −0.2 thin plate buckling stress.

γ	Eigenvalue	Critical Buckling Load (MPa)
0.8	202.83	50.71
1.2	296.98	74.25
1.5	376.26	94.07
2.0	535.24	133.81
3.0	2249.50	562.38
4.0	3586.30	896.58

.

(3)

Under loading conditions with S = 1.79 and β = −0.2, the buckling modes of the thin plate are shown in Figure 15. The corresponding critical buckling stresses are listed in

Table 8. S = 1.79, β = −0.2 thin plate buckling stress.

γ	Eigenvalue	Critical Buckling Load (MPa)
0.8	205.98	51.50
1.2	338.56	84.64
1.5	387.62	96.91
2.0	616.82	154.21
3.0	2776.40	694.10
4.0	4094.60	1023.65

.

(4)

Under loading conditions with S = 2.13 and β = −0.2, the buckling modes of the thin plate are shown in Figure 16. The corresponding critical buckling stresses are listed in

Table 9. S = 2.13, β = −0.2 thin plate buckling stress.

γ	Eigenvalue	Critical Buckling Load (MPa)
0.8	208.50	52.13
1.2	345.73	86.43
1.5	529.73	132.43
2.0	840.45	210.11
3.0	2847.2	1423.6
4.0	4792.6	1198.15

.

3.3. Critical Buckling Capacity

Based on the theoretical formulation and finite element eigenvalue analysis, the critical buckling stresses of a simply supported rectangular thin plate under combined compressive–tensile–shear loads were compared. The corresponding curves of elastic buckling stress are shown in Figure 17.

From Figure 17, it can be observed that the error between theoretical values and simulated values is less than 15%. Formula (16) can compute the elastic buckling load of a simply supported rectangular plate under biaxial and shear loads. Building upon this geometric relationship transformation, we can derive the critical buckling stress (σ_cr) formula for the steel pipes constrained by spiral ribs, as well as the formula for calculating the elastic buckling capacity (N_s,y) of spiral rib steel pipes.

$${\sigma }_{\mathrm{cr}}=k_1\frac{{\pi }^{2}E}{12\left(1-{\mu }^2\right)}\left(\frac{t_{\mathrm{s}}^2{\tan }^2\theta {\cos }^2\theta }{{d_{\mathrm{s}}}^2{\sin }^5\theta }\right)$$

(17)

$$N_{\mathrm{s},\mathrm{y}}=f_{\mathrm{y}1}\times A_{\mathrm{sr}}+{\sigma }_{\mathrm{cr}}\times A_{\mathrm{st}}$$

(18)

where k₁ is the critical buckling coefficient of a four-sided simply supported rectangular thin plate under compression–tension–shear load; t_s and d_s are steel pipe width and steel pipe thickness, θ is the vertical angle between the spiral stiffener and the outer wall of the steel pipe, f_y1 is the yield strength of spiral rib, A_sr is the cross-sectional area of the spiral rib, and A_st is the cross-sectional area of the steel pipe.

4. Bearing Capacity

4.1. Impact Parameters

The axial compressive capacity of spiral rib-reinforced steel tube concrete columns is influenced by the combined effects of the concrete, steel tube, spiral ribs, and steel reinforcement. Considering the impacts of spiral ribs, steel tubes, and steel reinforcement, the influence of parameters such as the rib width-to-thickness ratio (LX), steel tube width-to-thickness ratio (GH), pitch (LJ), and the vertical steel bar diameter (GJ) on the axial performance of composite columns is analyzed. Numerical analyses provided load–displacement curves for spiral rib-reinforced steel tube concrete columns under different parameter conditions [26,27], as illustrated in Figure 18.

4.2. Bearing Capacity of Outsourced Concrete

We can fit the correlation between influencing parameters and the stress in external concrete (Figure 19) and introduce a reduction factor for concrete strength to reflect the loss of load-bearing capacity due to the cracking and spalling of the concrete (Formulas (19) and (20)). We can then calculate the external concrete bearing capacity of spiral rib-reinforced steel tube concrete columns using Formula (21).

$${\alpha }_1=0.0003b_{\mathrm{r}}/t_{\mathrm{r}}+0.0011d_{\mathrm{s}}/t_{\mathrm{s}}+0.0033p+0.0002n_{\mathrm{s}}$$

(19)

$$\alpha ={\alpha }_0+{\alpha }_1=0.61+\left(0.0003b_{\mathrm{r}}/t_{\mathrm{r}}+0.0011d_{\mathrm{s}}/t_{\mathrm{s}}+0.0033p+0.0002n_{\mathrm{s}}\right)$$

(20)

$$N_{\mathrm{oc}}=\alpha \times A_{\mathrm{oc}}\times f_{\mathrm{c}}$$

(21)

where α is the reduction coefficient of external concrete strength, α₁ is the offset value produced by increasing the contact surface, and α₀ is the strength loss caused by the spalling of concrete. b_r/t_r is the ratio of helical rib width to thickness, d_s/t_s is the width and thickness ratio of steel pipe, p is the ratio of pitch to column height, n_s is the vertical bar diameter, N_oc is the clad concrete bearing capacity, A_oc is the external concrete cross-section area, and f_c is the design value of the compressive strength of the concrete axis.

4.3. Core Concrete Bearing Capacity

We used the spiral rib steel tube confinement factor to measure the steel confinement effect on the core concrete of steel tube concrete sections (Formulas (22) and (23)), and fitted the curves as shown in Figure 20. We introduced the enhancement factor of core concrete under hoop confinement (Formula (24)) to assess the reinforcing effect of spiral rib steel tube confinement on core concrete materials, as depicted in Figure 21. We calculated the core concrete bearing capacity using Formula (25):

$$\varsigma =\frac{\left(A_{\mathrm{s}1}{f}_{\mathrm{y}1}+A_{\mathrm{s}2}{f}_{\mathrm{y}2}\right)\times \left(1+\chi \right)}{A_{\mathrm{c}2}{f}_{\mathrm{c}}}$$

(22)

$$\chi =0.004b_{\mathrm{r}}/t_{\mathrm{r}}+0.255p-0.10d_{\mathrm{s}}/t_{\mathrm{s}}+0.005n_{\mathrm{s}}$$

(23)

$$\eta =1+0.784{\varsigma }^{0.383}$$

(24)

$$N_{\mathrm{cc}}=\eta \times A_{\mathrm{cc}}\times f_{\mathrm{c}}$$

(25)

where ς is the spiral rib steel pipe restraint coefficient, χ is the comprehensive influence coefficient of steel pipe strength, η is the core concrete strengthening coefficient, N_cc is the core concrete bearing capacity, A_cc is the core concrete cross-section area, and f_c is the design value of compressive strength of concrete axis.

4.4. Axial Compressive Capacity of Composite Column

Through the principle of superposition, the axial compressive capacity of the spiral rib-reinforced steel tube concrete composite columns is determined by the combined contributions of spiral rib steel tubes, external concrete, core concrete, and vertical steel reinforcement:

$$N_{\mathrm{y}}=N_{\mathrm{s},\mathrm{y}}+N_{\mathrm{oc}}+N_{\mathrm{cc}}+N_{\mathrm{b}}$$

(26)

$$N_y=(f_{y1}\times A_{\mathrm{sr}}+{\sigma }_{cr}\times A_{st})+\alpha \mathrm{x}(A_{oc}\times f_c)+\eta \mathrm{x}(A_{cc}\times f_c)+A_{sb}\times f_{y2}$$

(27)

where N_y is the axial compressive capacity of the composite column, N_s,y is the buckling capacity of spiral ribbed steel pipe, N_oc is the external concrete bearing capacity, N_cc is the core concrete bearing capacity, and N_b is the bearing capacity of the vertical reinforcement.

The buckling capacity of specimen Z3 was calculated to be 2560 kN, with an error of 5.2% compared to the experimental value of 2700 kN. Furthermore, the axial buckling capacities under compressive loads for various composite columns obtained through finite element parametric analysis are summarized in

Table 10. Comparison of simulated values (test values) and formula calculation results.

Specimen Number	Buckling Capacity/kN		Error Value/%
	Test or Simulation	Formula Calculation
Z3	2700.0	2560.0	−5.2%
LX1	27,992.0	28,831.8	3.0%
LX2	26,302.0	27,406.7	4.2%
LX3	27,117.6	27,470.1	1.3%
LX4	24,889.7	23,645.2	−5.0%
LX5	26,739.1	25,589.3	−4.3%
LX6	27,736.0	28,679.1	3.4%
GH1	24,933.1	26,628.6	6.8%
GH2	25,798.9	25,102.3	−2.7%
GH3	27,992.0	30,007.5	7.2%
GH4	28,033.7	28,314.1	1.0%
GH5	28,391.4	27,454.5	−3.3%
LJ1	25,831.2	23,764.7	−8.0%
LJ2	28,033.7	28,314.0	1.0%
LJ3	28,936.8	29,891.7	3.3%
LJ4	29,324.3	28,503.2	−2.8%
GJ1	26,833.6	27,423.9	2.2%
GJ2	28,936.8	27,750.4	−4.1%
GJ3	29,172.5	28,880.7	−1.0%
GJ4	30,748.3	31,763.0	3.3%
GJ5	30,944.2	32,429.6	4.8%

. The errors between the formula calculated values and the simulated values are within 10%, indicating that the formula is suitable for estimating the elastic buckling capacity of composite structures constrained by spiral rib reinforcement under axial compression loads.

5. Conclusions

To investigate the confinement effect of spiral ribs on concrete columns, axial compression tests were conducted to observe the failure modes of square steel tube concrete columns with different cross-sections. The confinement mechanism provided by spiral ribs was analyzed. By integrating theoretical and numerical methods, a formula for calculating the critical buckling capacity of spiral rib-reinforced thin-walled structures was derived. The axial load-bearing capacities of each specimen were validated accordingly. The specific conclusions are as follows:

(1)

The failure mode of steel tube concrete columns under axial compression typically involves the crushing of the core concrete and the outward buckling of the steel tube. In the case of spiral rib-reinforced composite columns, the external concrete casing tends to crack diagonally along the direction of the spiral ribs. The buckling of the steel tube wall occurs predominantly between the spiral ribs. Spiral ribs significantly enhance the ability of thin-walled steel tube concrete components to resist local buckling;

(2)

In comparison to ordinary steel tube concrete columns, using a single spiral rib restraint enhances the ductility of the structure but can reduce its load-bearing capacity. However, when spiral ribs are combined with steel bars (welded), the load-bearing capacity and ductility of the structure increase by 18.5% and 7.7%, respectively. Spiral ribs effectively delay the buckling of thin-walled steel tubes, thereby enhancing both the load-bearing capacity and deformation capability of the structure to a certain extent;

(3)

The mechanism of spiral rib confinement works as follows: spiral ribs on the outer wall of the steel tube divide the concrete and the steel tube into several spiral band-like structures. Under an axial compressive load, the vertical force is converted into shear force along the direction of the spiral ribs and vertical pressure. The component undergoes a combined compression–shear stress pattern, thereby restricting the buckling of the steel tube and lateral deformation;

(4)

The axial compressive load capacity of spiral rib-reinforced steel tube concrete composite columns is influenced by the spiral ribbed steel tube, external concrete, core concrete, and vertical steel bars. Based on the eigenvalue buckling mode method, a formula for the critical buckling stress of spiral rib-reinforced steel tubes is proposed. By applying the superposition principle, the experimental and numerical simulations and the theoretical calculations show that the load-bearing capacities of the composite columns are generally consistent.