Numerical Analysis and Resistance Design of UHPC- and UHTCC-Encased Rectangular Steel Tubular Columns Subject to Axial Compression

Abstract

Ultra-high performance concrete (UHPC) and ultra-high toughness cementitious composite (UHTCC) offer superior mechanical properties compared to normal concrete, with UHPC excelling in compressive strength and UHTCC in tensile ductility and crack resistance. This study focuses on UHPC/UHTCC-encased steel tubular (UEST) columns, establishing finite element (FE) models to simulate the axial behavior of UEST columns, conducting parametric studies on stud number, encasement thickness, steel yield strength, and width-to-thickness ratio, and developing a theoretical model considering thin-walled steel buckling to calculate the axial resistance of UEST columns. The proposed theoretical model predicts axial resistance with an average error of 3.4%, providing a reliable design method for engineering applications.

1. Introduction

Concrete-filled steel tubular (CFST) columns have gained widespread popularity in modern civil engineering over the past few decades, owing to their superior axial resistance, excellent seismic resistance, and convenient construction procedures [1,2,3,4,5]. The synergistic interaction between the steel tube and core concrete enables CFST columns to efficiently utilize the high tensile strength of steel and the high compressive strength of concrete, making them ideal for applications in high-rise buildings, bridges, and offshore structures [5,6,7,8,9]. However, a critical drawback of CFST columns lies in the direct exposure of the steel tube to the external environment, which significantly compromises their corrosion and fire resistance. In harsh environments such as marine settings, industrial zones, or areas with frequent acid rain, the steel tube is susceptible to chloride ion penetration and moisture-induced corrosion, leading to degradation of mechanical performance and increased maintenance costs over the service life [9,10,11,12,13,14]. This limitation restricts the safety and durability of CFST columns in extreme conditions.

To address the durability issues of CFST columns, concrete-encased steel tubular (CEST) columns have been developed as an alternative structural form. By encasing the steel tube with an outer layer of concrete, CEST columns not only retain the composite action advantages of CFST columns but also provide physical protection to the steel tube, thereby enhancing corrosion and fire resistance [3,15]. The outer concrete layer acts as a barrier against environmental aggressors, reducing direct contact between the steel tube and corrosive agents. However, CEST columns still face challenges due to the inherent limitations of normal concrete (NC). NC exhibits low tensile strength (1–3 MPa) and poor crack resistance, with peak tensile strain as low as 0.01% (

Table 1. Comparison of material properties between concrete, UHPC, and UHTCC [15].

Material	Ultimate Tensile Strength (MPa)	Ultimate Compressive Strength (MPa)	Modulus of Elasticity (GPa)	Peak Tensile Strain	Peak Compressive Strain
NC	1–3	20–50	28–34.5	0.01%	0.2%
UHPC	6–8	120–160	40–50	0.15%	0.2–0.6%
UHTCC	4.5–6	30–60	15–20	3–6%	0.5–0.6%

) [16,17,18]. Under service loads, the outer concrete layer is prone to cracking, creating pathways for chloride ions, water, and oxygen to penetrate and corrode the internal steel tube [19,20]. Over time, this degradation mechanism undermines the protective function of the concrete encasement, reducing the long-term durability of CEST columns, particularly in aggressive environments.

The application of fiber-reinforced cementitious composites has emerged as a promising solution to overcome the limitations of NC in CEST columns. By incorporating fibers (e.g., steel, polyvinyl alcohol (PVA), polyethylene (PE), or basalt fibers), these advanced materials exhibit enhanced mechanical properties, including improved tensile strength, ductility, and crack resistance [21]. Among them, ultra-high performance concrete (UHPC) and ultra-high toughness cementitious composite (UHTCC) have attracted significant attention in civil engineering due to their exceptional performance characteristics. UHPC is characterized by ultra-high compressive strength (typically 100–200 MPa), high elastic modulus (40–50 GPa), and excellent durability, attributed to its dense matrix and steel fiber reinforcement. These properties make UHPC well-suited for structural components requiring high axial resistance, such as bridge decks and composite columns [7,22,23,24]. UHTCC, on the other hand, is distinguished by its remarkable tensile ductility (peak tensile strain of 3–6%) and strain-hardening behavior, which enable it to control crack width at the microscale (≤100 μm) even under significant deformation. This makes UHTCC ideal for applications involving repeated loading, structural joints (e.g., wet joints in bridges), and retrofitting projects where crack resistance is critical [15]. A comparison of material properties (Table 1) highlights the advantages of UHPC and UHTCC over NC. UHPC offers 2–3 times higher ultimate tensile strength and 15 times higher peak tensile strain than NC, while UHTCC exhibits 1.5–2 times higher ultimate tensile strength and 300–600 times higher peak tensile strain. Both materials also demonstrate significantly lower permeability than NC. These properties make UHPC and UHTCC excellent candidates for replacing NC as the outer encasement in CEST columns, forming a new type of UHPC/UHTCC-encased steel tubular (UEST) column.

As illustrated in Figure 1, UEST columns feature a hollow steel tube encased in UHPC or UHTCC, with studs as shear connectors to ensure composite action between the steel tube and the outer layer. This structural form offers several key advantages: (1) the hollow design reduces self-weight compared to solid CFST columns, while maintaining bending and torsional stiffness; (2) the UHPC/UHTCC encasement provides superior protection against corrosion and cracking, enhancing long-term durability; (3) stud connectors facilitate efficient shear transfer and confinement, preventing premature buckling of the steel tube and improving overall axial resistance [25,26,27]. Despite these promising features, the mechanical behavior and design theory of UEST columns, particularly those with thin-walled steel tubes (large width-to-thickness ratio, λ), remain underdeveloped. For thin-walled steel tubes, local buckling under axial compression significantly affects the axial resistance, as the steel tube may deform inward before reaching its yield strength. Existing design codes for CFST or CEST columns (e.g., Eurocode 4 [28], AISC 360-16 [29]) do not adequately account for the buckling behavior of thin-walled steel tubes in UEST columns, leading to inaccurate predictions of axial resistance.

To address this knowledge gap, this study focuses on the numerical analysis and resistance design of UHPC- and UHTCC-encased steel tubular (UHPC-EST and UHTCC-EST) columns under axial load. An accurate finite element (FE) model is established to simulate the axial compressive behavior of UEST columns. Then, parametric studies are conducted to investigate the effects of some key design parameters (e.g., stud number, encasement thickness, steel yield strength, and width-to-thickness ratio) on axial resistance and stiffness. Finally, a theoretical model for predicting the axial resistance of UEST columns is proposed, incorporating the local buckling behavior of thin-walled steel tubes.

2. Materials and Constitutive Models

The concrete damaged plasticity (CDP) model was employed to simulate the mechanical behavior of UHPC and UHTCC materials. Based on experimental data from UHPC material tests and relevant research findings [22,30], the compressive stress–strain relationship of UHPC can be expressed by Equations (1)–(3), as shown in Figure 2a.

$${\sigma }_{\mathrm{c}}=\left\{\begin{array}{ll}{f}_{\mathrm{Uc}}{\displaystyle \frac{n\xi -{\xi }^2}{1+(n-2)\xi }}& {\epsilon }_{\mathrm{c}}\le {\epsilon }_0\\ f_{\mathrm{Uc}}{\displaystyle \frac{{\xi }^2}{2{(\xi -1)}^2+\xi }}& {\epsilon }_0<{\epsilon }_{\mathrm{c}}\le {\epsilon }_{\mathrm{cu}}\end{array}\right.$$

(1)

$$\xi ={\epsilon }_{\mathrm{c}}/{\epsilon }_0$$

(2)

$$n=E_{\mathrm{Uc}}/E_{\mathrm{Us}}$$

(3)

The tensile behavior of UHPC involves a bilinear stage (governed by stress–strain relationships) and a crack propagation stage (governed by stress–crack width relationships) [30]. Material tests indicate that the UHPC used in this study exhibits strain-hardening under tension, so the bilinear stage is further divided into an elastic phase and a strain-hardening phase (Figure 2b). The tensile stress–strain relationship is simplified as a three-segment polyline, as shown in Equation (4) [30].

$${\sigma }_{\mathrm{t}}=\left\{\begin{array}{ll}{E}_1{\epsilon }_{\mathrm{t}}& 0\le {\epsilon }_{\mathrm{t}}\le {\epsilon }_{\mathrm{Ute}}\\ f_{\mathrm{Ute}}+E_2({\epsilon }_{\mathrm{t}}-{\epsilon }_{\mathrm{Ute}})& {\epsilon }_{\mathrm{Ute}}\le {\epsilon }_{\mathrm{t}}\le {\epsilon }_{\mathrm{Utu}}\\ f_{\mathrm{Utu}}-E_3({\epsilon }_{\mathrm{t}}-{\epsilon }_{\mathrm{Utu}})& {\epsilon }_{\mathrm{Utu}}\le {\epsilon }_{\mathrm{t}}\le {\epsilon }_{\mathrm{Uts}}\end{array}\right.$$

(4)

where E₁, E₂, E₃ are the slopes of the tensile stress–strain curve in the elastic, strain-hardening, and strain-softening stages, respectively, calculated using Equations (5)–(7); f_Ute and ε_Ute denote the tensile strength at the proportional limit and the corresponding strain; f_Utu and ε_Utu are the ultimate tensile strength and peak strain; f_Uts and ε_Uts represent the post-cracking strength at the softening control point and the associated strain. Based on uniaxial tensile tests of existing tests of UHPC material [7,22], the typical parameters are assigned as f_Ute = 9.9 MPa, ε_Ute = 320 με, f_Utu = 10.6 MPa, ε_Utu = 2300 με, f_Uts = 2.1 MPa, and ε_Uts = 21,000 με.

$$E_1={\displaystyle \frac{f_{\mathrm{Ute}}}{{\epsilon }_{\mathrm{Ute}}}}$$

(5)

$$E_2={\displaystyle \frac{f_{\mathrm{Utu}}-f_{\mathrm{Ute}}}{{\epsilon }_{\mathrm{Utu}}-{\epsilon }_{\mathrm{Ute}}}}$$

(6)

$$E_3={\displaystyle \frac{f_{\mathrm{Utu}}-f_{\mathrm{Uts}}}{{\epsilon }_{\mathrm{Uts}}-{\epsilon }_{\mathrm{Utu}}}}$$

(7)

To characterize damage evolution in UHPC under tension and compression, damage variables d_c (compressive damage) and d_t (tensile damage) are introduced. Using the energy equivalence principle [30], these variables are derived by equating the strain energy of the damaged material to that of the undamaged material, as shown in Equations (8) and (9).

$$d_{\mathrm{c}}=1-\sqrt{{\displaystyle \frac{{\sigma }_{\mathrm{c}}}{E_0{\epsilon }_{\mathrm{c}}}}}$$

(8)

$$d_{\mathrm{t}}=1-\sqrt{{\displaystyle \frac{{\sigma }_{\mathrm{t}}}{E_0{\epsilon }_{\mathrm{t}}}}}$$

(9)

where d_c0 and d_t0 are the initial damage variables under compression and tension, respectively. According to the existing research [30], d_c0 and d_t0 are typically both set to 0.01.

The mechanical behavior of UHTCC is also simulated using the CDP model, with its stress–strain relationship illustrated in Figure 3. For tensile characterization, a bilinear strain-hardening formulation proposed by Maalej et al. [31] is adopted as expressed in Equation (10), which describes post-cracking stress transfer through linear elastic and strain-hardening responses, differing from conventional UHPC models.

$$\sigma =\left\{\begin{array}{ll}{E}_{\mathrm{c}}\epsilon & 0<\epsilon \le {\epsilon }_{\mathrm{tc}}\\ f_{\mathrm{tc}}+{\displaystyle \frac{f_{\mathrm{tu}}-f_{\mathrm{tc}}}{{\epsilon }_{\mathrm{tu}}-{\epsilon }_{\mathrm{tc}}}}(\epsilon -{\epsilon }_{\mathrm{tc}})& {\epsilon }_{\mathrm{tc}}<\epsilon <{\epsilon }_{\mathrm{tu}}\end{array}\right.$$

(10)

where E_c is the elastic modulus of UHTCC; ε_tc and f_tc are the first-cracking strain and strength; ε_tu and f_tu are the ultimate tensile strain and strength.

The uniaxial compressive response of UHTCC follows the triphasic constitutive law established by Zhou et al. [32]: the ascending branch uses a parabolic function to describe pre-peak hardening, with peak stress at strain ε₀; the post-peak phase is modeled by a bilinear softening curve, where the initial slope m governs early damage evolution, and the secondary slope n controls stress degradation after crack penetration, as expressed in Equation (11).

$$\sigma =\left\{\begin{array}{ll}{E}_{\mathrm{c}}\epsilon & 0<\epsilon \le {\epsilon }_{0.4}\\ (1-\alpha )E_{\mathrm{c}}\epsilon & {\epsilon }_{0.4}<\epsilon \le {\epsilon }_0\\ m(\epsilon -{\epsilon }_0)+f_{\mathrm{cp}}& {\epsilon }_0<\epsilon \le {\epsilon }_{\mathrm{i}}\\ n(\epsilon -{\epsilon }_{\mathrm{i}})+{\sigma }_{\mathrm{i}}& {\epsilon }_{\mathrm{i}}<\epsilon \le {\epsilon }_{\mathrm{cu}}\end{array}\right.$$

(11)

where f_cp is the cylindrical compressive strength; ε_0.4 is the strain at 40% peak stress (transition from elastic to nonlinear phase); ε₀ is the peak compressive strain; ε_i and σ_i are the knee-point parameters for bilinear descent, and they are decided as ε_i = 1.5ε₀, σi = 0.5f_cp based on statistical analysis of the test results conducted by Zhou et al. [32]; coefficients α, m and n are derived from empirical equations as expressed in Equations (12)–(13), with constants a = 0.308, b = 0.124 [30]. The end point of compressive constitute curve of UHTCC is determined as ε_i = 4.5ε₀, σi = 0.3f_cp, which is determined from the data analysis of Zhou et al. [32], using their experimental tests.

$$\alpha =a\cdot {\displaystyle \frac{\epsilon E_{\mathrm{c}}}{f_{\mathrm{cp}}}}-b$$

(12)

$$m=-{\displaystyle \frac{f_{\mathrm{cp}}}{{\epsilon }_0}}=15n$$

(13)

Similarly to UHPC, UHTCC damage variables are derived using the energy equivalence approach expressed in Equations (8) and (9).

3. Numerical Analysis

3.1. Mesh Condition and Interaction Settings

A finite element (FE) model of the rectangular steel tubular column was developed using ABAQUS 2021. The FE model adopts incremental displacement control to simulate axial compression, where a prescribed displacement is applied at the top reference node, and the reaction force (axial load) is measured. Consistent SI units are used throughout the model: length (m), force (N), stress (Pa), strain (dimensionless), modulus of elasticity (Pa). Static general step is used with automatic time stepping. The initial increment size is set to 0.01 mm, the minimum increment size to 1 × 10⁻⁵ mm, and the maximum increment size to 0.02 mm. Force balance convergence criterion with a tolerance of 1 × 10⁻³ (relative error). If force balance is not achieved within 10 iterations for a given increment, the increment size is automatically reduced until convergence is reached.

The mesh conditions, element types, and interaction settings are detailed in Figure 4. The steel tube is modeled with S4R [5,33,34,35,36], longitudinal rebars with B31, and UHTCC/UHPC and studs with C3D8R elements. S4R is a four-node quadrilateral shell element with bilinear interpolation for in-plane displacement and linear interpolation for out-of-plane displacement, containing six degrees of freedom (DOFs) per node (three translations: UX, UY, UZ; three rotations: URX, URY, URZ), and it uses reduced integration (1 × 1 Gauss points) for in-plane stresses and 2 × 2 Gauss points for bending stresses. B31 is a two-node linear beam element with linear interpolation for axial and bending displacement, containing six DOFs per node (same as S4R element), and it uses two Gauss points along the element length. C3D8R is an eight-node hexahedral solid element with trilinear interpolation for displacement, containing three DOFs per node (only translations: UX, UY, UZ; rotations are neglected for solid elements), and it uses reduced integration (1 × 1 × 1 Gauss points).

For S4R (shell) and C3D8R (solid) elements, reduced integration is adopted to avoid shear locking—a common issue in full integration for elements subjected to bending or compression. Shear locking occurs when full integration overestimates shear stiffness, leading to unconservative (stiffer) simulation results. Reduced integration mitigates this by using fewer Gauss points, ensuring accurate capture of plastic deformation and buckling behavior of the steel tube and UHPC/UHTCC layer. For the S4R element, the combination of 1 × 1 in-plane and 2 × 2 bending integration balances computational efficiency and accuracy for thin-walled steel tubes. For C3D8R elements, 1 × 1 × 1 integration is sufficient for the UHPC/UHTCC matrix (which exhibits nonlinear damage) and studs (which undergo localized shear), as it reduces computational cost without compromising the prediction of interface shear transfer and material crushing.

The top end plate is kinematically coupled to a reference node with constrained translations in the X and Y directions (UX = UY = 0) but free movement in the Z direction (UZ) and rotations (URX, URY, URZ). The bottom end plate is fully fixed, with all six degrees of freedom constrained (UX = UY = UZ = URX = URY = URZ = 0). Axial loading is applied via displacement control in a “static” step, with initial, minimum, and maximum increment sizes set to 0.01, 1 × 10⁻³⁵, and 0.02, respectively. Interaction settings include embedded constraints for rebars and studs within the UHTCC/UHPC matrix; tie constraints between the edges of studs and the steel tube; and surface-to-surface contact between the steel tube and UHTCC/UHPC, with “hard” contact in the normal direction and a penalty friction model (friction coefficient = 0.2) in the tangential direction.

3.2. Sensitivity Analysis of Mesh Size

A mesh sensitivity analysis was performed to determine the optimal discretization for the FE model. As shown in Figure 5, axial resistance varies significantly with element sizes between 25 mm and 50 mm, peaking at 50 mm. Coarser meshes improve computational efficiency but compromise accuracy, while finer meshes enhance precision but increase computational cost. Based on the analysis, a mesh size of 25 mm was selected to ensure both reliability and efficiency.

3.3. FE Model Validation

To validate the FE modeling methodology, comparative analyses were conducted between experimental data and simulation results for three structural configurations [37]: UHTCC-encased rectangular laminated columns (Figure 6a), UHTCC-encased rectangular composite columns (Figure 6b), and UHTCC-encased circular composite columns (Figure 6c). The axial load–strain curves from simulations show good agreement with experimental measurements, capturing both elastic and post-yield behavior. The buckling patterns observed in simulations (Figure 6e) also match the experimental observations (Figure 6d), with inward deformation of the steel tube and localized crushing of the UHTCC layer in composite columns.

Table 2. Comparison results between experimental and simulated results.

Specimen	Type of Result	Ultimate State		RMSE 1
		Load (kN)	Error
Laminated column	Test	3595	−3.6%	270.3
	FE	3464
Rectangular UEST columns	Test	3461	0.1%	287.3
	FE	3463
Circular UEST columns	Test	3331	1.3%	81.0
	FE	3373

¹ RMSE is the root mean square error between test and FE results.

summarizes the ultimate load comparisons, revealing minimal errors (maximum 3.6%), confirming the effectiveness of the FE model in predicting the axial compressive behavior of UEST columns.

3.4. Parametric Study

A parametric study was conducted to investigate the effects of key design parameters on the axial resistance of UHPC- and UHTCC-encased rectangular steel tubular columns.

Table 3. Parameter selection and numeric range.

Group №	Key Design Parameters
	Outer Layer Material	nst	tc (mm)	fy	λ
1	UHPC	0–3	50	355	30–75
2	UHPC	2	50–80	355	30–75
3	UHPC	2	50	355–460	30–75
4	UHTCC	2	50	355	30–75
5	UHTCC	2	50–80	355	30–75
6	UHTCC	2	50	355–460	30–75

lists the parameters and their ranges: outer layer material (UHPC or UHTCC), number of studs (n_st), encasement thickness (t_c), steel tube yield strength (f_y), and width-to-thickness ratio of steel tube within the UEST columns (λ). In this section, the width of all steel tubes is determined as 300 mm, and the thickness of steel tubes is changed among 10, 7.5, 5, and 4 mm to realize the width-to-thickness ratio of 30–75.

Figure 7 illustrates the influence of key parameters on the axial resistance (N_u) of UHPC-EST column, with Figure 7a–c focusing on the number of studs (n_st), encasement thickness (t_c), and steel yield strength (f_y), respectively. In Figure 7a, N_u exhibits a nonlinear relationship with n_st: increasing from 0 to 1 stud significantly enhances resistance due to improved interface bonding and confinement, but further increasing n_st to 2 or 3 results in marginal gains, as the UHPC–steel interaction reaches saturation. This trend is consistent across all width-to-thickness ratios (λ = 30 to 75), with thinner steel tubes (λ = 75) showing slightly greater sensitivity to stud count. It can be seen that the axial resistance N_u achieves its maximum value when stud number n_st is 1, and maintains constant regardless of the further increase in n_st. This phenomenon is primarily attributed to the saturation of the interface interaction between the steel tube and the outer layer. When there are no studs (n_st = 0), the shear transfer between the steel tube and outer layer is relatively weak. This insufficient shear transfer leads to interface slip under axial compression, reducing the lower axial resistance. When stud is added (n_st = 1), it acts as an effective shear connector, significantly enhancing the interface bonding. However, when the number of studs increases further to 2 or 3, the interface interaction between the steel tube and outer layer has already reached a saturated state, and the axial resistance will be insensitive to the increase in the stud number. As shown in Figure 7a, there is a slight decrease in axial resistance (N_u) when the stud number (n_st) increases from 1 to 2 or 3 for slender steel tubes. This may be attributed to local stress concentration in the outer layer caused by excessive studs, reducing the outer layer’s bearing ability.

Figure 7b demonstrates a clear positive correlation between t_c (50–80 mm) and N_u, as thicker UHPC layers provide stronger lateral restraint against steel buckling and contribute additional compressive capacity. The rate of increase is more pronounced for columns with larger λ, highlighting the critical role of UHPC in mitigating thin-walled steel instability. In Figure 7c, N_u increases linearly with f_y (355–460 MPa) due to the direct contribution of higher-strength steel to axial load-bearing, with the effect being most significant for columns with smaller λ, where steel yielding dominates over buckling.

Figure 8 examines the axial resistance of UHTCC-encased rectangular steel tubular columns under varying design parameters, with Figure 8a–c analyzing the effects of stud number (n_st), encasement thickness (t_c), and steel yield strength (f_y). Similarly to UHPC-EST columns, Figure 8a shows that N_u peaks at n_st = 1 across all λ values, but the magnitude of improvement is smaller than in UHPC systems, as UHTCC’s lower elastic modulus reduces its confinement efficiency. For columns with λ = 75, the benefit of adding stud is more pronounced, as UHTCC layer helps delay buckling in thin-walled steel tubes. Figure 8b reveals that increasing t_c (50–80 mm) enhances N_u, but the rate of increase is slower than in UHPC columns due to UHTCC’s lower compressive strength; notably, the effect of t_c diminishes for λ = 30 (thicker steel tube), where steel resistance dominates. Figure 8c confirms a linear relationship between f_y (355–460 MPa) and N_u. These findings highlight that UHTCC-encased columns rely more on steel strength and stud-induced bonding than on encasement thickness, reflecting UHTCC’s role as a ductile restraint rather than a high-strength load-bearing component. Figure 8a–c confirm a basically linear relationship between f_y and N_u.

Figure 9 compares the impact of stud number (n_st) on the axial stiffness of UHPC-EST and UHTCC-EST columns across varying steel width-to-thickness ratios (λ = 30 to 75). Axial stiffness, defined as the initial slope of the load–strain curve, is shown in Figure 9, highlighting percentage changes relative to n_st = 0. In Figure 9a, UHPC-EST columns exhibit higher baseline stiffness than UHTCC-EST columns due to UHPC’s higher elastic modulus. Increasing n_st from 0 to 3 enhances stiffness by 2.0–2.1%, with the greatest gain observed at λ = 30 (thicker steel tube), where studs effectively transfer load between the rigid UHPC and steel, reducing interface slip. Conversely, increasing λ from 30 to 75 decreases stiffness by 2.1–6.5%, as thinner steel tubes undergo greater early buckling, lowering overall system rigidity. In Figure 9b, UHTCC-EST columns show similar trends but with smaller stiffness gains (1.6–1.9%) from adding studs. The reduction in stiffness with λ is similarly severe (2.1–6.5%) to UHPC systems. These results indicate that studs moderately enhance axial stiffness, while minimizing λ (using thicker steel tube) is more critical for maximizing rigidity in both UHPC-EST and UHTCC-EST columns.

The FE model uses ideal tie constraints and embedded constraints to simulate perfect interface bonding. The ideal tie constraint assumes perfect force transfer between studs and the steel tube, ensuring studs fully restrict steel tube buckling. In practice, this constraint is violated by two key imperfections: incomplete stud welding and stud shear failure/pull-out if the studs with non-ideal ties cannot provide continuous lateral restraint to the steel tube. For thin-walled steel tubes, this causes earlier onset of local buckling, reducing the effective load-bearing area of the steel tube. The theoretical model assumes ideal embedded constraints between the studs and UHPC/UHTCC layer. In practice, non-ideal embedding allows studs/rebars to slip relative to the UHPC/UHTCC layer. Under axial compression, the UHPC/UHTCC layer carries less load (due to slip-induced loss of composite action), forcing the steel tube to bear additional stress—accelerating yielding and buckling. To improve model accuracy for detailed design, future studies are needed to be incorporated: investigating cohesive zone model for simulating interface slip between steel tube and UHPC/UHTCC; investigating imperfect stud welding to model stud detachment or partial bonding; investigating time-dependent slip to account for creep-induced slip in UHPC/UHTCC over service life.

In addition, the current study focuses on the short-term axial behavior of UEST columns under ideal environmental conditions. In real-world applications, UEST columns are exposed to aggressive environments (e.g., marine, industrial, cold regions), where chloride ions penetrate the UHPC/UHTCC layer and corrode the steel tube/studs. These factors degrade the long-term axial resistance and durability of UEST columns, which are not captured in the current parametric study. To address this, future research should conduct accelerated corrosion tests on UEST column specimens to measure steel corrosion rate and UHPC/UHTCC chloride permeability. Moreover, it is needed to develop a corrosion-dependent FE model to establish a durability-based analysis method.

4. Axial Resistance Design of UEST Columns

4.1. Elastic Buckling Analysis of Thin-Walled Steel Plates

The preceding sections have adequately described the axial compression behavior and axial resistance of UEST columns. It is recognized that for UEST columns with thin-walled steel tubes featuring large width-to-thickness ratios, their axial load resistance depends heavily on the buckling performance of the steel tube. Thus, a theoretical analysis is required to calculate the axial resistance of UEST columns considering the buckling of thin-walled steel tubes. To simplify the theoretical analysis of the elastic buckling behavior of thin-walled steel plates in UEST columns, the following assumptions are adopted in this section:

(1)

Only in-plane stresses (σ_x, σ_y, τ_xy) are considered, with out-of-plane stresses neglected due to the thin-walled nature;

(2)

The plate undergoes elastic deformation, adhering to Hooke’s law;

(3)

Membrane stresses induced by minor stretching are ignored;

(4)

The bond stress between UHPC/UHTCC and the steel tube is negligible compared to stud-induced shear transfer.

The studs are considered as a fixed boundary condition in this theoretical model, which is verified based on both experimental data and mechanical analysis. In the FE model validation, the axial load–strain curves and buckling patterns of the simulated UEST columns with studs show good agreement with the experimental results, indicating that the studs effectively restrict the relative displacement between the steel tube and the outer UHPC/UHTCC layer, preventing significant interface slip. If the studs cannot provide sufficient boundary conditions, the constraint effect of the outer UHPC/UHTCC layer on the steel tube will be weakened, leading to an earlier occurrence of local buckling of the steel tube, reducing the effective load-bearing area of the steel tube, thereby decreasing the axial resistance of the UEST column.

Based on assumptions (1)–(3), the mechanical equilibrium of the thin-walled steel plate is illustrated in Figure 10 [38], and the corresponding equilibrium equation is given by

$$D\left({\displaystyle \frac{{\partial }^{4}w}{\partial x^4}}+2{\displaystyle \frac{{\partial }^{4}w}{\partial x^2\partial y^2}}+{\displaystyle \frac{{\partial }^{4}w}{\partial y^4}}\right)=N_x{\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}+2N_{xy}{\displaystyle \frac{{\partial }^{2}w}{\partial x\partial y}}+N_y{\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}$$

(14)

where D is the bending stiffness of the plate per unit width, defined in Equation (15); w is the buckling wave function; N_x, N_y, and N_xy are the internal axial and shear forces per unit length of the plate; E_s is Young’s modulus of steel; t is the thickness of the steel plate; and ν is Poisson’s ratio.

$$D={\displaystyle \frac{E_{\mathrm{s}}{t}^3}{12(1-{\nu }^2)}}$$

(15)

According to the existing research, the critical buckling stress of thin-walled steel plates is characterized by the critical buckling coefficient k [39], expressed as

$${\sigma }_{\mathrm{cr}}={\displaystyle \frac{k{\mathrm{\pi }}^{2}D}{b^{2}t}}=k{\displaystyle \frac{{\mathrm{\pi }}^2{E}_{\mathrm{s}}}{12-(1-{\nu }^2)}}{\displaystyle \frac{t^2}{b^2}}$$

(16)

where b is the width of the steel plate; and σ_cr is the critical buckling stress.

4.2. Elastic Buckling Analysis of UEST Column

Building on the buckling analysis of thin-walled steel plates, this section proceeds to analyze the buckling of thin-walled steel tubes in UEST columns. As shown in Figure 11a, due to symmetry, the internal force in the cross-sectional direction (yz-plane) is zero, and the shear force in the xy-plane is also zero. Thus, the following relationship holds

$$\left\{\begin{array}{l}{N}_y=N_{xy}=0\\ N_x=-{\sigma }_{x}t\end{array}\right.$$

(17)

where N_y and N_xy are the internal axial force and shear force in the y-direction, respectively. Substituting Equation (17) into Equation (14), the bending strain energy (U_b) and external work potential energy (V) of the thin-walled steel tube in UEST columns are derived as

$$U_{\mathrm{b}}={\displaystyle \frac{D}{2}}{\displaystyle {\int }_0^a{\displaystyle {\int }_0^b\left[{\left(\frac{{\partial }^{2}w}{\partial x^2}\right)}^2+{\left(\frac{{\partial }^{2}w}{\partial y^2}\right)}^2+2v_{\mathrm{s}}\frac{{\partial }^{2}w}{\partial x^2}\frac{{\partial }^{2}w}{\partial y^2}+2\left(1-v_{\mathrm{s}}\right){\left(\frac{{\partial }^{2}w}{\partial x\partial y}\right)}^2\right]}} \mathrm{d}x\mathrm{d}y$$

(18)

$$V=-{\displaystyle \frac{1}{2}}{\displaystyle {\int }_0^a{\displaystyle {\int }_0^b{\sigma }_{\mathrm{x}}t{\left(\frac{\partial w}{\partial x}\right)}^2}}\mathrm{d}x\mathrm{d}y$$

(19)

where σ_x is the in-plane stress in the x-direction; a is the length of the buckling wave in the axial direction.

As shown in Figure 11a, in the cross-sectional direction (yz-plane), the thin-walled steel tube cannot rotate or translate at the studs and the four corners, so these points are treated as fixed supports. In the longitudinal direction (xz-plane), the steel tube is also constrained at the studs (Figure 11b), which act as fixed supports. Therefore, the buckling wave function of the thin-walled steel tube in UEST columns must satisfy the following boundary conditions:

$$\left\{\begin{array}{ll}w=0, {\displaystyle \frac{\partial w}{\partial x}}=0& (x=0, a)\\ w=0, {\displaystyle \frac{\partial w}{\partial y}}=0& (y=0, {\displaystyle \frac{b}{n_{\mathrm{st}}+1}})\end{array}\right.$$

(20)

where n_st is the number of studs on one side of the cross-section. Based on these boundary conditions, the buckling wave function is assumed as

$$w=f\left(1-\cos {\displaystyle \frac{2\mathrm{\pi }x}{a}}\right)\left(1-\cos {\displaystyle \frac{2\mathrm{\pi }\left(n_{\mathrm{st}}+1\right)y}{b}}\right)$$

(21)

where f is the amplitude of the buckling wave; b is the width of the steel plate between two adjacent corners.

The total potential energy equation of the thin-walled steel tube in UEST columns can be expressed as

$$\mathrm{\Pi }=U_{\mathrm{b}}-V$$

(22)

where Π is the total potential energy; U_b is the bending strain energy; V is the external work potential energy. Substituting Equations (18)–(21) into Equation (22) and applying the principle of stationary potential energy (i.e., ∂Π/∂f = 0 with f ≠ 0), the critical buckling stress is derived as

$${\sigma }_x=\left(4{\displaystyle \frac{a^2}{b^2}}{\left(n_{\mathrm{st}}+1\right)}^4+{\displaystyle \frac{4b^2}{a^2}}+{\displaystyle \frac{8}{3}}{\left(n_{\mathrm{st}}+1\right)}^2\right){\displaystyle \frac{{\mathrm{\pi }}^2\mathrm{D}}{b^{2}t}}=k{\displaystyle \frac{{\mathrm{\pi }}^2\mathrm{D}}{b^{2}t}}$$

(23)

where k is the buckling coefficient for the UEST column. Letting β = a/b (the ratio of buckling wave length to plate width), the buckling coefficient can be expressed as

$$k=4{\beta }^2{\left(n_{\mathrm{st}}+1\right)}^4+{\displaystyle \frac{4}{{\beta }^2}}+{\displaystyle \frac{8}{3}}{\left(n_{\mathrm{st}}+1\right)}^2$$

(24)

To find the minimum critical buckling stress, the derivative of k with respect to β is set to zero (∂k/∂β = 0), yielding the minimum buckling coefficient as

$$k_{\min }={\displaystyle \frac{32}{3}}{\left(n_{\mathrm{st}}+1\right)}^2$$

(25)

4.3. Axial Resistance of UEST Column

The preceding section established the method for calculating the buckling load of thin-walled steel tubes. Based on effective width method (Figure 12) [40,41], the contribution of the central (more buckling-prone portion) of the steel tube is neglected, and only the contribution of the boundary regions (less susceptible to buckling) is considered. This approach uses an average effective width to evaluate the axial resistance of the steel tube, defined as

$$\rho ={\displaystyle \frac{b_{\mathrm{e}}}{b}}=\sqrt{{\displaystyle \frac{{\sigma }_{\mathrm{cr}}}{f_{\mathrm{y}}}}}\left(1-0.22\sqrt{{\displaystyle \frac{{\sigma }_{\mathrm{cr}}}{f_{\mathrm{y}}}}}\right)\le 1$$

(26)

where ρ is the effective width ratio; b_e is the effective width, a virtual concept to help understand the effective bearing capacity of thin-walled steel tube, which can be back-calculated from the buckling stress of steel tube, using Equation (26).

Using the superposition principle, the total axial resistance of the UEST column, incorporating contributions from UHPC/UHTCC and longitudinal rebars, can be expressed as

$$N_{\mathrm{u}}=N_{\mathrm{s}}+N_{\mathrm{rebar}}+N_{\mathrm{c}}=\sum {\rho }_{\mathrm{i}}{A}_{\mathrm{si}}{f}_{\mathrm{y}}+A_{\mathrm{r}}{f}_{yr}+A_{\mathrm{c}}{f}_{\mathrm{ck}}$$

(27)

where N_u is the ultimate axial resistance of the UEST column; A_s is the cross-sectional area of the steel tube; A_st is the total cross-sectional area of the studs; A_c is the cross-sectional area of the UHPC/UHTCC layer; f_ck is the characteristic compressive strength of UHPC/UHTCC; A_r is the cross-sectional area of longitudinal rebars; and f_yr is the yield strength of longitudinal rebars.

4.4. Theory Validation

The proposed resistance calculation formula is validated using finite element model results and existing experimental data. Comprehensive evaluations are conducted on UEST columns with steel tubes of varying width-to-thickness ratios. The results show that the proposed formula accurately predicts the axial resistance of UEST columns across a range of width-to-thickness ratios, with an average error of only 3.4% (Figure 13). This confirms the reliability of the theoretical model in capturing the buckling behavior and composite action of UEST columns. The theoretical model uses the effective width method to account for inelastic buckling, which approximates the plastic deformation captured in the FE model. The good agreement confirms that the theoretical model’s elastic buckling framework, when combined with the effective width method, effectively captures the plastic behavior of the UEST column system. The FE model includes initial geometric imperfections to simulate real-world manufacturing defects, which are critical for accurate buckling prediction. However, the theoretical model does not explicitly include imperfections, but it implicitly accounts for them via the critical buckling coefficient, which is calibrated against FE results with imperfections.

The proposed model can be adapted to Eurocode or AISC frameworks by introducing the local buckling analysis and effective width method. Moreover, it is needed to add UHPC/UHTCC as acceptable encasement materials in EC4 [28] and AISC 360-16 [29], with specified properties. The core assumptions of these two codes, superposition principle can be reserved for use as long as the resistance contributions of steel tube and outer layer can be calculated via the proposed method of this study.

5. Conclusions

This study focuses on UHPC- and UHTCC-encased steel tubular (UHPC-EST and UHTCC-EST) columns, developing FE models and conducting parametric studies on some key parameters. Based on the FE simulation and existing public test results, this paper proposes an axial resistance design theory of UHPC- and UHTCC-encased rectangular steel tubular columns. Some key conclusions can be drawn as follows.

• Axial resistance: UHPC-EST columns exhibit higher resistance than UHTCC-EST columns. Increasing stud number from 0 to 1 stud significantly enhances resistance due to improved interface bonding and confinement, but further increasing stud number to 2 or 3 results in marginal gains, as the steel–UHPC/UHTCC interaction reaches saturation. An increase in steel yield strength from 355 to 460 MPa linearly boosts resistance, with the effect most significant for the columns with thinner steel tube.
• Axial stiffness: UHPC-EST columns have higher baseline stiffness than UHTCC-EST columns. Stud number increase from 0 to 3 enhances stiffness by 2.0–2.1% for UHPC and 1.6–1.9% for UHTCC. Stiffness decreases by 2.1–6.5% as width-to thickness ratio of steel tube increases from 30 to 75 for both encasement materials.
• Proposed theory: An axial resistance design theory integrating buckling analysis and effective width method is proposed, predicting axial resistance with an average error of 3.4%, providing a reliable design method for UEST columns, particularly those with thin-walled steel tubes.