An Assessment of the Bearing Capacity of High-Strength-Concrete-Filled Steel Tubular Columns Through Finite Element Analysis

Abstract

This work aimed to evaluate the accuracy of analytical models for predicting the behavior of concrete-filled steel tubular (CFST) columns via finite element analysis coupled with physical nonlinearity. The methodology involved an extensive review of experimental tests from the literature, numerical modeling of columns with different configurations, and a comparison of the results obtained with available experimental data. Several characteristics were evaluated, such as the load capacity, confinement factor, and relative slenderness. The numerical model agreed well with the experimental results, with a less than 10% relative error. The results indicated that analytical models of the Chinese (GB 50936) and European (EC4) codes overestimated some load capacity values (up to 14.9% and 8.7%, respectively). In comparison, the American (AISC 360) and Brazilian (NBR 8800) standards underestimated the ultimate loads (23.3% and 31.6%, respectively). An approach coefficient β is proposed, contributing to safer and more efficient design practices in structural engineering.

1. Introduction

One recent advancement in the field relates to composite structures, which allow for the integration of different types of materials into a single structure, such as in concrete–steel systems. This approach has provided a series of benefits, including improved strength, increased durability, and reduced overall weight of the structure [1]. According to Dabaon et al. [2], a composite column is any structural element in which both materials resist a compressive load. There is a wide variety of composite columns, with the most employed ones being those formed by tubular sections and those composed of I-section profiles. In this regard, tubular profile columns, known as concrete-filled steel tubular (CFST) columns, exhibit high mechanical strength, high ductility, high fire resistance, and favorable practical construction considerations [1]. According to Elyoussef et al. [3], composite columns are widely employed in various applications, especially in large buildings and heavy engineering. Han et al. [4] mentioned their use in seismic-resistant structures, bridge structures subject to traffic impacts, piles, infrastructure, and oil and gas applications. Liew et al. [5] focused on high-rise buildings. The CFST column also performs very well under high axial compressive loads (Figure 1), making it a suitable solution for columns subjected to these forces. The steel tube in this system provides confinement to the concrete core, while the concrete filler reduces the local buckling of these tubes. In the initial elastic regime, the deformation of concrete is relatively small, resulting in a low confinement pressure [6].

In addition, the stiffness of CFST columns increases because the steel tube is located further away from the centroid of the cross-section, thereby increasing its inertia. The concrete inside contributes to forming an ideal structure that can withstand compressive loads, potentially delaying or preventing possible local instabilities of the steel tube [1,7]. Additionally, the lateral confinement provided by the tube significantly enhances the column’s strength [1].

Analytical models used to predict the ultimate compressive strength have been proposed by researchers and standards [8,9,10,11,12,13]; however, confined high-strength concrete may exhibit different behaviors than expected for conventional concrete [9,14,15,16]. High-strength concrete (HSC) has been broadly applied to various civil structures for its advantages including high compressive strength and excellent durability and creep resistance. However, the brittleness of HSC raises concern about its use in practice [17]. Steel tubes can provide greater ductility to the column, making them an ideal combination. Thus, measuring the accuracy of analytical models for high-strength-concrete-filled steel tubular columns is necessary.

The confinement effect is also expected to occur, as the literature suggests [18,19,20,21]. In the inelastic range, Poisson’s ratio is not constant but rather a function of axial deformation. In the initial loading stage, concrete’s Poisson’s ratio is lower than steel’s, allowing it to expand more rapidly in the radial direction than the concrete without constraining it. However, with increased loading, the tube walls begin to restrain the concrete core, resulting in stress on the steel walls. When subjected to axial loads, circular-section CFST columns distribute stress evenly [21]; however, rectangular-section columns provide more accessible beam–column connections and accommodate some specific engineering requirements. Thus, evaluating the behavior of both sections is essential.

A criterion of a length-to-diameter ratio less than four is adopted to define stub columns [22]; very slender columns are not included in this study because factors such as geometric nonlinearity are predominant, and the purpose of this research is to analyze only physical nonlinearity. According to Santini and Ramires [23], a general problem validating and verifying various analytical and numerical studies on tubular composite columns is that the range of experiments available in the literature is not standardized because each study amplifies information regarding the specific parameter of interest in that study. This study aimed to delimit the models subjected exclusively to axial compression; 677 tests were consulted to compose the review database (

Table 1. Literature database.

Reference	Tests	$\mathit{t} \left(\mathbf{m}\mathbf{m}\right)$	${\mathit{f}}_{\mathit{y}}$ (MPa)	${\mathit{f}}_{\mathit{c}}$ (MPa)	Geometry
Furlong [24]	8	-	294–420	21.4–35.6	Circ.
Furlong [24]	5	-	336–492	21.4–43.1	Rect.
Gardner & Jacobson [25]	7	1.70–4.10	363–605	21–34	Circ.
Knowles & Park [26]	11	-	369–614	21.2–34.9	Rect.
Knowles & Park [26]	6	-	324	39.9	Rect.
Bridge et al. [27]	22	0.86–2.82	185.7–363.3	47.5	Rect.
Schneider & Alostaz [28]	3	-	285–537	23.8–28.5	Circ.
Schneider & Alostaz [28]	11	-	312–430	23.8–30.5	Rect.
O’Shea & Bridge [29]	15	0.86–2.82	185.7–363.3	38.2–108	Rect.
Uy [30]	21	5	750	0–32	Rect.
Han [31]	24	2.8–7.6	198–228	59.3 1	Rect.
Sakino et al. [32]	114	2.96–9.45	262–835	25–91.1	Circ./Rect.
Zeghiche & Chaoui [33]	15	4.96–5.20	270–283	40–102	Circ.
Nardin & El Debs [34]	6	3.20–4.85	329.1–355	47.7–59.3	Circ./Rect.
Gupta et al. [35]	72	1.87–2.89	360	25.1–38.3	Circ.
Dabaon et al. [2]	15	2.0	285	34.8–61.9	Rect.
Oliveira et al. [21]	32	3.35–6.0	287.33	32.7–105.5	Circ.
Uenaka et al. [36]	12	0.9–2.14	221–308	18.7	Circ. 2
Han et al. [37]	80	3.62–3.72	319.6–380.6	60	Various 2
Zhao et al. [38]	9	1.7–6.0	394–454	63.4	Circ.
Liew & Xiong [39]	12	3.54–9.69	377–428	165–176	Circ.
Portolés et al. [40]	6	6.0	394–494	37.7–120.5	Circ.
Tao et al. [41]	13	3.6–8.0	321–372	42–81.8	Circ.
Tao et al. [41]	11	3.6–10.0	355–521	40.4–81.8	Rect.
Liew et al. [5]	27	3.6–16.0	374–779	51.6–193.3	Circ.
Xiong et al. [42]	18	3.6–10.0	300–428	51.6–193.3	Circ.
Xiong et al. [43]	2	10.0–16.0	374–412	180–186	Circ.
Chen et al. [44]	12	2.09–8.03	251.8–371.6	59.0–130.8	Circ./Rect.
Wang et al. [45]	2	2.77–5.50	375–419	30.1	Circ.
Rodrigues et al. [46]	23	2.86–2.94	276–300	40.5–115.6	Circ. 2
Ji et al. [47]	16	1.7–3.8	269–286	49.5–65.6	Rect. 3
Ren et al. [19]	18	1.25–11.87	242–496	40–70.9	Circ.
Liu et al. [48]	11	4.0–6.0	254–290	24.8	Rounded
Gao et al. [49]	8	3.68–20.19	261–279	59.02 1	Rect.
Range of values	677	0.86–20.19	185.7–780	18.7–193.3

¹ cubic strength; ² CFDST; ³ double section.

).

The ranges of typical values for thickness ($t$), steel yield strength ($f_y$) and concrete strength ($f_c$) are shown in Figure 2. A frequency density graph is displayed for the variables mentioned. These preliminary analyses allowed us to identify a trend toward using steel tube thicknesses between 2 and 4 mm, in addition to a steel yield strength and concrete compressive strength of approximately 400 and 40 MPa, respectively. The mean (µ) and variance $\left({\sigma }^2\right)$ obtained for these values were 4.464 and 6.746 for thickness, 416.10 and 34,397 for yield strength, and 61.40 and 1695.22 for concrete strength, respectively. This review presents valuable information about how tests on CFST columns are designed, showing the variability in tests by several authors. However, specific steel and high-strength-concrete standards restrict the materials’ mechanical resistance [8,50].

Despite the extensive development of analytical models aimed at predicting the ultimate compressive strength of concrete, existing research and standards primarily focus on conventional concrete behavior [8]. This oversight is particularly significant for confined high-strength concrete, which often demonstrates unique properties and performance characteristics that deviate from traditional expectations [51,52]. In this sense, this study aims to perform a comprehensive numerical analysis to investigate the mechanical behavior of composite elements incorporating confined high-strength concrete. By generating new data, this study will enhance our understanding of how these materials perform and provide critical insights that can refine existing predictive models. The findings will be instrumental for engineers and designers, contributing to safer and more efficient applications of high-strength concrete in real-world structures.

2. Numerical Model

Numerical modeling using finite elements involves employing constitutive relationships appropriate for the materials, boundary conditions according to the physical model studied, and sensitivity analysis of the parameters that govern the behavior of materials. Then, the results are compared to the experimentally obtained responses. The flowchart in Figure 3 summarizes the numerical modeling procedure developed in ABAQUS [49].

2.1. Constitutive Model of Materials

The concrete damage plasticity (CDP) model is adopted based on the Drucker–Prager criterion. However, damage parameters are not used since the model is subjected exclusively to monotonic loads. The literature provides appropriate parameters for modeling structural concrete using ABAQUS. The parameters of the CDP model used in this study include the material dilation angle ($\psi$), eccentricity ($ϵ$), the ratio of biaxial compressive yield stresses to uniaxial yield stress ($\sigma b/\sigma c$), the ratio of the second invariant stress in the tension meridian to that in the compression meridian ($Kc$), and the viscosity parameter ($\mu$). It is also necessary to assume uniaxial compression and tension behaviors. Significant discrepancies exist among the values found for these quantities in the literature [53]; in this way, a sensitivity study is proposed for the dilation angle, and other values are adopted by default, as shown in

Table 2. Default values of CDP parameters [53].

$\mathit{\psi }$	$\mathit{ϵ}$	${\mathit{\sigma }}_{\mathit{b}}/{\mathit{\sigma }}_{\mathit{c}}$	${\mathit{K}}_{\mathit{c}}$	$\mathit{\mu }$
10°–56°	0.1	1.16	0.667	0.0001

.

The stress–strain ratio of the concrete-filled column is given by Tao et al. [54] through Equations (1)–(8).

$${\displaystyle \frac{\sigma }{f_c}}={\displaystyle \frac{A·X+B·X^2}{1+\left(A-2\right)·X+(B+1)·X^2}} ; \left(0<\epsilon \le {\epsilon }_{c0}\right)$$

(1)

where ${\epsilon }_{c0}$ = the strain at peak stress, given by Equation (2).

$${\epsilon }_{c0}=0.00076+\sqrt{\left(0.626·f_c-4.33\right)·{10}^{-7}}$$

(2)

The parameters X, A, and B are calculated as follows:

$$X={\displaystyle \frac{\epsilon }{{\epsilon }_{c0}}};A={\displaystyle \frac{{E_c·\epsilon }_{c0}}{f_c}};B={\displaystyle \frac{{\left(A-1\right)}^2}{0.55}}$$

(3)

$${\displaystyle \frac{{\epsilon }_{cc}}{{\epsilon }_{c0}}}={\epsilon }^k;k=(2.9224-0.00367·f_c)·{\left({\displaystyle \frac{f_B}{f_c}}\right)}^{0.3124+0.002·f_c}$$

(4)

where $f_B$ = the confining stress, calculated by Equation (5).

$$\left\{\begin{array}{c}\mathrm{R}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r} \mathrm{s}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \to f_B={\displaystyle \frac{0.25·\left(1+0.027·f_y\right)·e^{-\frac{0.02·\sqrt{B^2+D^2}}{t}}}{1+1.6·e^{-10}·f_c^{4.8}}} \\ \mathrm{C}\mathrm{i}\mathrm{r}\mathrm{c}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r} \mathrm{s}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \to f_B={\displaystyle \frac{(1+0.027·f_y)·e^{-\frac{0.02·D}{t}}}{1+1.6·e^{-10}·f_c^{4.8}}} \end{array}\right.$$

(5)

$$\sigma =f_r+(f_c-f_r)·exp\left[-{\left({\displaystyle \frac{\epsilon -{\epsilon }_{cc}}{\alpha }}\right)}^{\beta }\right]; \left(\epsilon \ge {\epsilon }_{cc}\right)$$

(6)

$$\left\{\begin{array}{c}\mathrm{R}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r} \mathrm{s}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \to f_r=0.1·f_c\hfill \\ \mathrm{C}\mathrm{i}\mathrm{r}\mathrm{c}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r} \mathrm{s}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \to f_r=0.1·\left(1-e^{-1.38·\xi }\right)·f_c\le 0.25·f_c \hfill \end{array}\right.$$

(7)

$$\left\{\begin{array}{c}\mathrm{R}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r} \mathrm{s}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \to \alpha =0.005+0.075·\xi \\ \mathrm{C}\mathrm{i}\mathrm{r}\mathrm{c}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r} \mathrm{s}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \to \alpha =0.04-{\displaystyle \frac{0.036}{1+e^{6.08·\xi -3.49}}};\end{array}\right.$$

(8)

The parameter $\beta$ is equal to 1.2 for circular sections and 0.92 for rectangular sections [54].

$\xi$ = the confinement factor (Equation (9)).

$$\xi ={\displaystyle \frac{A_s·f_y}{A_c·f_{ck} }}$$

(9)

The curve described through the presented equations is represented in Figure 4. The model captures the confinement effect through residual stress.

For the tensile behavior of concrete, the concept of fracture energy ( $G_F$) is used, with the formulation presented by Equation (10) [54].

$$G_F=\left(0.0469·d_{máx}^2-0.5·d_{máx}+26\right)·{\left({\displaystyle \frac{f_c}{10}}\right)}^{0.7}$$

(10)

Various researchers have used different stress–strain curves for steel tubes, including the perfect elastic–plastic and elastic–plastic models with linear hardening or multilinear hardening. Steel does not exhibit significant work hardening at stresses of general structural interest (normally less than 5%). Approximate curves of axial load (N) versus axial strain (ε) are obtained using different stress–strain models for steel [54]. Therefore, the bilinear constitutive model (Figure 5) is adopted, and the model is described by Equations (11) and (12). The elastic modulus ${(E}_s)$ and Poisson’s ratios were 205 GPa [55] and 0.3, respectively.

$$\sigma =\left\{\begin{array}{c}{E}_s·\epsilon \left(\sigma \le f_y\right)\\ f_y+0.01·E_s \left(\sigma >f_y\right)\end{array}\right.$$

(11)

$$f_u=\left\{\begin{array}{c}\left[1.6-2·{10}^{-3}\left(f_y-200\right)\right]·f_y \left(200 \mathrm{M}\mathrm{P}\mathrm{a}\le f_y\le 400 \mathrm{M}\mathrm{P}\mathrm{a}\right)\\ \left[1.2-3.75·{10}^{-3}\left(f_y-400\right)\right]·f_y \left(400 \mathrm{M}\mathrm{P}\mathrm{a}\le f_y\le 800 \mathrm{M}\mathrm{P}\mathrm{a}\right)\end{array}\right.$$

(12)

2.2. Finite Element Model Establishment

Mesh convergence analysis was performed to determine the optimal element mesh to provide a relatively accurate solution in a low computational time. Based on the mesh convergence studies, the element size in the cross-section was chosen to be D/15 for a circular column or B/15 for a rectangular column, as observed by Tao et al. [54], where D and B are the total diameter of the circular tube and the total width of the rectangular tube, respectively. In this way, the mesh selected for the samples contains a total of more than 10,000 elements with an average size between 7 and 10 mm (Figure 6). For finite element analysis, ABAQUS v.1 software was used, and the Riks method coupled with the arc length method was used. The study was conducted on a computer with an Intel Core I7 2.5 GHz processor and 16 GB of RAM. Less than 1000 increments were used to obtain the model response for a total processing time of approximately 50 min. Solid elements C3D8R (eight nodes and three degrees of freedom per node, with reduced integration to improve computational efficiency) were used for steel and concrete. C3D8R elements support analysis of plastic with large deformations, and the reduced integration offers better computational cost [57]. Other studies use the same approach [54,58,59,60,61].

2.3. Boundary Conditions and Contact Properties

Initial imperfections and residual stresses influence the behavior of hollow steel tubes. However, the effects of local imperfections and residual stresses are minimized by filling the concrete [54]. Therefore, they were ignored in the simulation. Surface-to-surface contact was generally used to simulate the interaction between the steel tube and the concrete. A contact surface pair comprises the steel tube’s inner surface and the concrete core’s outer surface. According to the Coulomb friction model, this interface used two contact properties: a normal rigid contact and a tangential contact. The behavior is not sensitive to the friction coefficient between the steel and the concrete since they are loaded simultaneously, resulting in little or no sliding between the steel tube and the concrete. However, the typical recommended values in the literature range between 0.2 and 0.6 for the friction coefficient [54], and a value of 0.6 was adopted. The boundary conditions were modeled as depicted in Figure 7.

3. Standard Procedures

The standards of the American Institute of California, 360 (2022), Eurocode 4 (2004), GB 50936 (2014), and NBR 8800 (2024) were considered to evaluate the analytical models. Each standard considers the materials’ column slenderness, form, reduction, confinement factors, and mechanical properties. The European and Chinese codes consider the effect of confinement through coefficients or factors. The American, Brazilian, and European standards also address reduction factors due to column instabilities. The main equations are presented in

Table 3. Main equations for predicting axial strength via standard methods.

Code	Main Equations
AISC 360 [62]	$N_{AISC}=\left\{\begin{array}{c}N·({0.658}^{{\displaystyle \frac{N}{N_{cr}}}}); {\displaystyle \frac{N}{N_{cr}}}\le 2.25\\ 0.877·N_{cr} ; {\displaystyle \frac{N}{N_{cr}}}>2.25\end{array}\right.$ $N=\left\{\begin{array}{c}{N}_P ; \lambda <{\lambda }_P N_P-{\left({\displaystyle \frac{\lambda -{\lambda }_P}{{\lambda }_r-{\lambda }_P}}\right)}^2·\left(N_P-N_y\right) ; {\lambda }_P\le \lambda <{\lambda }_r\hfill \\ N_y ; \lambda \ge {\lambda }_r\hfill \end{array}\right.$ $N_P=A_s{·f}_y+\alpha ·A_c{·f}_{ck}$ ${ N}_y=A_s{·f}_{cr}+0.7·A_c{·f}_c$ $f_{cr}=\left\{\begin{array}{c}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}\to {\displaystyle \frac{9·E_s}{{\left(\lambda \right)}^2}}\hfill \\ \mathrm{c}\mathrm{i}\mathrm{r}\mathrm{c}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}\to {\displaystyle \frac{0.72·f_y}{{\left[\frac{D}{t}·\frac{f_y}{E_s}\right]}^{0.2}}}\hfill \end{array}\right.$ $\left\{\begin{array}{c}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}\to \lambda ={\displaystyle \frac{b}{t}} ; {\lambda }_P\le 2.26·\sqrt{{\displaystyle \frac{E_s}{f_y}}} ; {\lambda }_r\ge 3.0·\sqrt{{\displaystyle \frac{E_s}{f_y}}}\\ \mathrm{c}\mathrm{i}\mathrm{r}\mathrm{c}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}\to \lambda ={\displaystyle \frac{D}{t}} ; {\lambda }_P\le 0.15·{\displaystyle \frac{E_s}{f_y}} ; {\lambda }_r\ge 0.19·{\displaystyle \frac{E_s}{f_y}}\end{array}\right.$ ${\left(EI\right)}_e=E_s·I_s+{C_3·E}_c·I_c$ $C_3=0.45+3·\left({\displaystyle \frac{A_s}{A_g}}\right)\le 0.9$
EUROCODE 4 [63]	$N_{EC4}={\eta }_s·A_s{·f}_y+A_c{·f}_{ck}·\left(1+{\eta }_c{\displaystyle \frac{t}{D}}·{\displaystyle \frac{f_y}{f_{ck}}}\right)$ ${\eta }_s=0.25·(3+2·{\lambda }_0) \le 1$ ${\eta }_c=4.9-18.5·{\lambda }_0+{17·{\lambda }_0}^2\ge 0$ ${\lambda }_0=\sqrt{{\displaystyle \frac{N_p}{N_{cr}}}}$ $N_p=A_s{·f}_y+A_c{·f}_{ck}$ $N_{cr}=\frac{{\pi }^2·{\left(EI\right)}_e}{{\left(L_e\right)}^2}$ ${\left(EI\right)}_e=E_s·I_s+{0.6·E}_{cm}·I_c$
GB 50936 [64]	$N_{GB}=\left(1.212+B·\xi +C·{\xi }^2\right)·{(A}_s+A_c){·f}_{ck}$ $\left\{\begin{array}{c}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}\to B={\displaystyle \frac{0.1381·f_y}{235}}+0.7646\\ C={\displaystyle \frac{{-0.0727·f}_{ck}}{20 }}+0.0216 \\ \mathrm{c}\mathrm{i}\mathrm{r}\mathrm{c}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}\to B={\displaystyle \frac{0.176·f_y}{213}}+0.974\\ C={\displaystyle \frac{{-0.104·f}_{ck}}{14.4 }}+0.031 \end{array}\right.$
NBR 8800 [65]	${ N}_{NBR}=\chi {·N}_p$ ${\lambda }_0\le 1.5\to \chi ={0.658}^{{\lambda }_0^2}$ ${\lambda }_0>1.5\to \chi ={\displaystyle \frac{0.877}{{\lambda }_0^2}}$ ${\lambda }_0=\sqrt{{\displaystyle \frac{N_p}{N_{cr}}}}\le 2.0$

.

3.1. Modulus of Elasticity

Concrete with higher compressive strength usually deforms less than low-resistance concrete and consequently has a higher elastic modulus [14]. The modulus of elasticity expression is applied only to the straight part of the stress–strain curve; alternatively, when there is no linear stretch, the expression is applied to the tangent of the curve at the origin. The time factor is not predominant in rapid tests, as in axial compression tests, loading is immediate. Then, the initial tangent modulus is employed in the analyses. Each standard is formulated according to its criteria, and several analytical models for predicting concrete stiffness are presented in

Table 4. Modulus of elasticity of concrete.

Code	${\mathit{E}}_{\mathit{c}}$ (MPa)
AISC 360 [62]	${\mathrm{E}}_c=0.043·{}^1\rho ^{1.5}·\sqrt{f_{ck}}\to 21\le f_{ck}\le 69$
Eurocode 2 [66]	${\mathrm{E}}_c=22000{\left({\displaystyle \frac{f_{ck}+8}{10}}\right)}^{0.3}\to f_{ck}\le 90$
GB50010 [67]	According to Table 5.
NBR 6118 [68]	${\mathrm{E}}_c=21500·{\alpha }_E·{\left({\displaystyle \frac{f_{ck}}{10}}+1.25\right)}^{{\displaystyle \frac{1}{3}}}\to 55\le f_{ck}\le 90$ ${\mathrm{E}}_c=5600·{\alpha }_E·\sqrt{f_{ck}}\to 20\le f_{ck}\le 55$

¹ concrete density, usually equals 2400 kg/m³; α_E the parameter related to the aggregate used in concrete usually equals 1.0.

.

3.2. Limitations

According to Thai et al. [8], except for the Australian standard ASNZS 2327 (2017) [69], which is permissive for the use of steel and high-strength concrete, the other standards are prepared based on the conventional strengths of the materials. Therefore, assessing the analytical models proposed by these standards for high-strength materials is essential.

Table 6. Limits.

Code	${\mathit{f}}_{\mathit{y}}$ (MPa)	${\mathit{f}}_{\mathit{c}}$ (MPa)	${\mathbf{\lambda }}_{\mathit{m}\mathit{á}\mathit{x}}$ (Circ.)	${\mathbf{\lambda }}_{\mathit{m}\mathit{á}\mathit{x}}$ (Rect.)	Confinement
AISC 360 [62]	$f_y$  $\le$ 525	$f_{ck}$  $\le$ 100	$0.31·{\displaystyle \frac{E_s}{f_y}}$	$5.00·\sqrt{{\displaystyle \frac{E_s}{f_y}}}$	-
Eurocode 4 [63]	$f_y$  $\le$ 460	$f_{ck}$  $\le$ 50	$90·{\displaystyle \frac{235}{f_y}}$	$52·\sqrt{{\displaystyle \frac{235}{f_y}}}$	-
GB 50936 [64]	$f_y$  $\le$ 420	$f_{cu}$  $\le$ 80	$135·{\displaystyle \frac{235}{f_y}}$	$60·\sqrt{{\displaystyle \frac{235}{f_y}}}$	$0.5\le \xi \le 2.0$
NBR 8800 [65]	$f_y$  $\le$ 460	$f_{ck}\le 90$	$0.31·{\displaystyle \frac{E_s}{f_y}}$	$5.00·\sqrt{{\displaystyle \frac{E_s}{f_y}}}$	-

summarizes the usual limits of resistance, slenderness, and other requirements for each instruction. The current standard models are invalid for high-strength classes and require further analyses.

4. Validation of Finite Element Model

Load–displacement curves and resistance capacities were considered to validate the numerical model. Based on the previous analysis, two experiments were selected from the literature: one with a circular section and the second with a rectangular section. Thus, four tests from Oliveira [21] and four from Han [31] were selected, and the analyzed parameters are presented in

Table 7. Parameters from tests.

ID	Geom.	${\mathit{f}}_{\mathit{c}}$ (MPa)	${\mathit{f}}_{\mathit{y}}$ (MPa)	D/B (mm)	H (mm)	t (mm)	$\mathit{\xi }$	D/t	${\mathit{N}}_{\mathit{e}\mathit{x}\mathit{p}}$ (kN)
C1	Circ.	58.7	287.3	114.3	342.9	3.35	0.63	34.1	952.0
C2	Circ.	58.7	287.3	114.3	342.9	6.0	1.22	19.1	1329.1
C3	Circ.	88.8	287.3	114.3	342.9	6.0	0.8	19.1	1496.0
C4	Circ.	105.5	287.3	114.3	342.9	6.0	0.68	19.1	1683.4
R1	Rect.	50.7	228	100 × 100	300	2.86	0.7	35.0	780.0
R2	Rect.	50.7	228	90 × 70	270	2.86	0.93	24.5	565.0
R3	Rect.	50.7	228	150 × 135	450	2.86	0.5	47.2	1380.0
R4	Rect.	50.7	228	140 × 80	420	2.86	0.7	28.0	810.0

. Eight tests were validated with different concrete strengths ( $f_c)$, confinement factors ( $\xi )$, and relative slenderness (D/t or B/t). In this way, it was possible to compare them with standard analytical models.

5. Results and Discussion

5.1. Sensitivity Analysis

Model calibration showed that the use of analytical equations to measure the concrete modulus of elasticity provided very similar results, with the ultimate load being slightly more significant for the model calculated through NBR 6118 (Figure 8a). This result may be due to the consideration of the coefficient ${\alpha }_E$, which takes into account the type of aggregate. Figure 8b displays the sensitivity analysis for model C3 (C80 concrete). The model presented a disturbance in the numerical curve, probably due to the buckling effect. The results show that the formulations for predicting the concrete modulus of elasticity agree. This consistency supports the reliability of these formulations.

The calibration for support analyses is presented in Figure 8c. Elastic support was also considered, with a spring k coefficient of 0.5∙ ${10}^5$ providing the best model fit, closely matching the physical data. However, for each study, the coefficient k changes due to the conditions of the experiment. The coefficient k is only a calibrator of the initial stiffness, so it was not considered in the simulation to obtain the ultimate load of the numerical model.

The concrete dilation angle (ψ) was also tested across 10° to 40° values, as shown in Figure 8d. A value of 25° most accurately reflected the physical model. Circular sections are more sensitive to this parameter; for rectangular sections, there is a consensus in the literature that this parameter is 40° [54]. These findings emphasize that precise calibration of the elasticity modulus, support (stiffness), and dilation angle is critical to enhancing model accuracy and reliability.

5.2. Results of Validation Study

The values obtained for the ultimate resistance capacity through numerical simulation and the corresponding errors observed for the standard procedures are shown in

Table 8. Errors observed between experimental/standards and predicted via FE.

ID	Numerical (kN)	Experimental	EC4	AISC 360	GB 50936	NBR 8800
C1	957.6	1.006	1.087	0.880	1.144	0.880
C2	1344.4	1.012	1.005	0.777	0.943	0.777
C3	1547.5	1.034	1.023	0.827	1.055	0.827
C4	1686.4	1.002	1.016	0.836	1.089	0.836
R1	713.2	0.914	0.987	0.892	1.061	0.795
R2	512.0	0.906	0.929	0.849	0.962	0.711
R3	1250.1	0.906	1.048	0.934	1.149	0.803
R4	742.9	0.917	1.057	0.955	1.088	0.684
Mean error		0.962	1.019	0.869	1.061	0.789
Mean error (%)		3.79%	1.90%	13.1%	6.10%	21.1%
Standard deviation		5.57%	4.82%	5.85%	7.55%	6.48%
CV (%)		5.80%	4.73%	6.74%	7.11%	8.22%

. Regarding the standard predictions, the results were as expected: EC4 exhibited greater accuracy than the other models, as observed by Uslu and Taşkın [70], with a slight overestimation of the ultimate load. GB 50936 also overestimated some results, considering the high gains for confined concrete. AISC 360 and NBR 8800 presented higher errors, probably due to not considering the gains from the confinement effect, underestimating the ultimate load by up to 23.3% and 31.6%, respectively. The following topics better present the discussion.

Figure 9 shows the mean error, standard deviation, and adjusted R square observed, and the coefficient of the line was 1.016. The values predicted by the numerical model are close to the experimental values, indicating a good fit.

Figure 10 presents the load–displacement curves of the proposed studied cases. There is a close match between them, particularly in the initial portion, where the behavior is linear and elastic. Despite capturing the confinement effect, the post-peak behavior of the nonlinear branch exhibited some divergence but was still very similar to that of the experimental tests.

A validation study of the FEM indicated very high model accuracy for the ultimate loads. The higher loads reached for the experimental curves R1–R4 indicate that higher concrete strengths were obtained by the Han [31] tests because the strength considered for the numerical model was the average of the prototypes tested by the author, a possible effect of experimental variability. In addition, the model proposed by Tao et al. [54] is based on regression analysis of uniaxial compression test results from 17 references, in which the concrete strength ranged from 10 MPa to 100 MPa; thus, the C4 model curve is slightly outside the expected behavior. Moreover, the experimental curve shows some perturbation.

In addition to the presented results, the deformed configurations obtained for the numerical models also agreed with the experimentally tested physical models, as shown in Figure 11. The observed failure mode was the local instability of the steel tube in conjunction with concrete crushing. The difference is that the numerical model captured the instability close to the section’s middle height, as Rodrigues et al. noted [46]. Han [31] presented a lower instability in his tests. This effect could be due to a concentration of stresses, orthotropy of the concrete, or imperfections; however, when observing the load–displacement curve, the behaviors are similar. The stress distribution and plastification observed in the numerical model also agreed with expectations (Figure 12).

As expected, the results for the C1–C4 models also showed that the maximum stress at the end of the analysis occurred in the steel tube. Stress distribution showed that the section was entirely plasticized for nearly half the height of the specimen (Figure 13 and Figure 14). It is important to note that a higher concrete strength causes less stress in the outer steel tube when the load is applied across the entire section. In general, the numerical models represented the physical models well.

5.3. Confinement Factor

The discrepancy between the numerical model and the standard prediction was slightly more significant for higher confinement factors, as shown in the graph in Figure 15. Regarding circular sections, which present more effective confinement, the American (AISC 360) and Brazilian (NBR 8800) standards were conservative, estimating loads smaller than those observed in experimental tests. In contrast, the Chinese (GB 50936) and European (EC4) standards [63] overestimated some results, especially for low-confinement factor values. Despite the rectangular sections presenting less effective confinement, the tendency was the same as that observed for the circular sections. These results were expected since standards considering the confinement effect achieved greater precision, suggesting that coefficients could be adopted.

5.4. Relative Slenderness

By increasing the relative slenderness, the loads predicted by the standard models also increased, indicating an effect of section size, as shown in the graph in Figure 16, as observed by Liu et al. [48] and Gao et al. [49]. It is worth mentioning that the models with lower relative slenderness presented higher confinement factors, corroborating the results of the previous research. The results for both sections also agreed. Both the European code (EC4) and the Chinese code (GB 50936) overestimated the load-bearing capacity for larger specimens, which also agrees with the findings of Gao et al. [49].

5.5. Approach Coefficient Proposal

Based on the results, an approximation coefficient is proposed through linear regression. $\beta$ is due to the confinement factor; in the sequence, $\beta$ is presented for each standard analyzed in Equations (13)–(16). ${\beta }_1$ and ${\beta }_2$ are related to the circular and rectangular sections, respectively.

$$N=\left(1+\beta \right)·N_{AISC}\to \left\{\begin{array}{c}{\beta }_1=-0.3089·{\xi }^2+0.7268·\xi -0.2043\\ {\beta }_2=0.6312·{\xi }^2-0.7049·\xi +0.2607\end{array}\right.$$

(13)

$$N=\left(1+\beta \right)·N_{EC4}\to \left\{\begin{array}{c}{\beta }_1=0.084·\ln \xi -0.0145\\ {\beta }_2=0.152·\ln \xi +1.0039\end{array}\right.$$

(14)

$$N=\left(1+\beta \right)·N_{GB}\to \left\{\begin{array}{c}{\beta }_1=-0.4178·{\xi }^2+1.0945·\xi -0.6569\\ {\beta }_2=0.2712·{\xi }^2-0.047·\xi -0.2403 \end{array}\right.$$

(15)

$$N=\left(1+\beta \right)·N_{NBR}\to \left\{\begin{array}{c}{\beta }_1=-0.3089·{\xi }^2+0.7268·\xi -0.2043\\ {\beta }_2=-0.4502·{\xi }^2+0.8577·\xi -0.1193\end{array}\right.$$

(16)

When multiplied by the axial bearing capacity, these coefficients provide more accurate results, increasing or penalizing the resistance due to the confinement parameter. Even when resistance is added, these coefficients are conservative and are within the range 0.5 ≤ ξ ≤ 2.0. The coefficient β is similar to the Chinese standard procedure (GB 50936). Only for EC4 was the equation obtained through regression of a natural logarithmic function due to the better accuracy of this standard.

Table 9. Impact of coefficient β on codes.

ID	EC4	AISC 360	GB 50936	NBR 8800
C1	1.029	0.995	0.992	0.995
C2	1.007	0.950	0.996	0.950
C3	0.989	0.975	1.004	0.975
C4	0.968	0.959	0.974	0.959
R1	0.937	0.960	0.912	1.002
R2	0.945	0.977	0.914	0.916
R3	0.949	0.996	0.924	0.961
R4	1.182	1.028	0.935	0.862
Mean error	1.001	0.980	0.956	0.953
Mean error (%)	0.1%	2.0%	4.4%	4.7%
Standard deviation	8.0%	2.6%	3.9%	4.5%
CV (%)	8.0%	2.6%	4.1%	4.8%

shows the impact of the coefficient on the codes, demonstrating a significant improvement in prediction for bearing capacity. The most impactful result is for NBR 8800, which, when applying the coefficient, reduced the prediction error from 21.1% to 4.7% on average. The coefficient also reduced variability, as shown by the deviation and the coefficient of variation. Any safety coefficient was applied in the analyses. Therefore, the coefficient $\beta$ by itself presents more conservative predictions.

For more reliable results, a more comprehensive database should be analyzed. With the numerical model already validated, the next step in the research is an extensive parametric study through automation. Additionally, there is a limitation to this approach: since global slenderness was not evaluated, global instabilities may occur in the column, which are not considered. Thus, the proposed approach coefficient is valid only for compact sections.

6. Conclusions

This study developed a finite element model to evaluate the behavior of high-strength-concrete-filled composite columns under axial loading. The key findings and potential paths for future research are summarized below:

•

The numerical results aligned with the experimental data with a relative error of less than 10% for load-bearing capacity.

•

The model did not address global slenderness (length-to-diameter ratio), an important variable for comprehensive structural assessment.

•

The analytical models were primarily conservative, missing the concrete core’s confinement strength gains—only the Chinese and European standards aligned closely with the experimental results.

•

The Chinese (GB 50936) and European (EC4) codes overestimated some load capacity values (up to 14.9% and 8.7%, respectively), while the American (AISC 360) and Brazilian (NBR 8800) standards underestimated the ultimate loads (to 23.3% and 31.6%, respectively).

•

An approach coefficient was proposed to improve standard predictions; however, it is valid only for compact sections, and the results were impactful for predicting bearing capacity. The most significant improvement in prediction was for NBR.

It is essential to advance this research to gather additional experimental data for reliability analysis for the proposed approach coefficient and enhance its applicability across different scenarios. Further investigation should also assess the influence of slenderness on the load-bearing capacity and structural response of high-strength-concrete-filled composite columns.