Evolution of Confinement Stress in Axially Loaded Concrete-Filled Steel Tube Stub Columns: Study on Enhancing Urban Building Efficiency

Abstract

In the context of green building and sustainable urban development, understanding the mechanical behavior of structural components like concrete-filled steel tube (CFST) columns is crucial due to their improved load-bearing capacity, energy efficiency, and optimized material usage, which enhance structural resilience and sustainability. This research addresses the complex development of confining stress and its impact on the concrete core (CC) behavior within these columns, which are essential for urban infrastructure. Through extensive numerical studies, this study proposes a model to estimate the confining stress in axially loaded CFST short columns. Study findings reveal that the confinement effectiveness is influenced by variables such as compressive strength (CS) of the concrete, cross-sectional shape, and depth-to-wall thickness percentage. Additionally, the confinement is also significantly affected by the yield strain of steel ε_y/ε_c to the peak strain of unconfined concrete ε_c. A three-dimensional finite element model (FEM) was built for the simulation of the columns’ nonlinear behavior and was rigorously validated against experimental data. This model aids in the design and implementation of more efficient and resilient urban structures, supporting the principles of sustainable construction. The study underscores the importance of structural integrity in sustainable urban development and provides valuable insights for improving the design of green building materials.

1. Introduction

Concrete-filled steel tube (ST) columns offer several benefits over traditional steel and reinforced concrete columns [1,2,3]. They possess greater strength, superior flexibility, and enhanced energy absorption capacity, and they result in significant cost savings [4].

The hollow steel tube (HST) used in these columns is filled with concrete [5]. As such, the application of CFST columns in the building sector has the potential to improve structural performance significantly. However, it is clear that even with all of the benefits of CFST columns and the research conducted in this area, the building industry still has a limited appetite for using them. This state of affairs can be ascribed to the persistence of unresolved issues and doubts regarding the CFST columns’ behavior and the significant elements influencing their effectiveness. The progress processing of confining stress complicates the mechanism of CFST columns significantly. Recent numerical and experimental investigations were performed to understand this mechanism more thoroughly [6,7,8,9]. Consequently, equations have been proposed to predict confinement in CFST columns. In 2022, Naghipour et al. investigated the impact of concrete confinement levels on the load-bearing capacity of steel-reinforced concrete (SRC) columns subjected to eccentric loading, utilizing both experimental methods and finite element analysis [10]. Piccardo et al. (2024) explored the carbon effects and primary energy of structural frame(s), focusing on different materials and prefabrication levels [11]. Fărcean et al. (2024) developed a waste management policy for a steel company, contributing to sustainability in construction [12]. Luo et al. (2024) reviewed the application of demolished wastes and aggregates in concrete, providing insights into sustainable practices [13,14,15]. Shariati et al. (2023) researched the partial replacement of sulfuric acid resistance of concrete with coal waste for fine and coarse aggregates [16]. Finally, Jin et al. (2021) focused on meso-scale modeling of the size effect in eccentrically-loaded squared CFST columns, emphasizing the effect of eccentricity and confinement [17]. The investigation conducted by Hu, Huang [18] introduced equations designed for the assessment of confinement. According to the equations, the confinement depends on the yield stress and the depth-to-wall thickness ratio (D/t). In a study presented by Starossek, Falah [19], an investigation was undertaken to explore the circular CFST columns’ behavior. The study resulted in the development of equations aimed at predicting the confinement within such columns. This research work reveals that the confinement is contingent upon CC compressive strength (CS) and the D/t ratio. Tao, Wang [20] undertook numerical investigations involving both circular and rectangular CFST columns. Their efforts resulted in formulating equations to predict confinement within these columns. The findings indicate that confinement is influenced by several key parameters including the cross-sectional shape, the ST yield stress, the D/t ratio, and the compressive strength of the CC. Thai, Uy [21] introduced a set of equations designed for the estimation of confinement in rectangular CFST columns. These equations reveal that the level of confinement is influenced by key parameters, including the depth-to-wall thickness ratio, CS, and ST yield stress. Moreover, in a separate investigation conducted by Lai and Ho [22], the external confinement effect on CFSTs is explored, focusing on their confining stress and the resulting composite action. Novel design equations have been formulated to forecast axial strength based on this analysis. These equations elucidate that the CFST columns’ maximum strength experiences augmentation while increasing the yield strength of ST and concrete cylinder strength or decreasing the ST depth-to-wall thickness ratio.

The study’s objective is to focus on and scrutinize the impact of various factors on CC confinement and to provide a predictive model for the development process of confining stress. Building upon this investigation and previous research, a novel parameter, denoted as the ${\epsilon }_y/{\epsilon }_c$ ratio, has been introduced. Utilizing this parameter allows for the consideration of the impact of the disparity between the steel’s yield strain and the unconfined concrete peak stress–strain on the confinement stress. The simultaneous application of load on the CC and ST emphasizes a substantial effect of their discrepancy on confining stress development. In this regard, an extensive numerical analysis was conducted using ABAQUS software (version 6-14-3, 2015). Subsequently, a comprehensive three-dimensional FEM was formulated in anticipation of nonlinear response in axially loaded circular CFST stub columns.

2. Confining Stress

In axially loaded CFST columns, the ST serves a dual purpose: it provides lateral confinement for the CC and bears axial loads. The behavior of CFST columns is significantly influenced by the difference in Poisson’s ratios between the ST and the CC. Due to the steel’s higher Poisson’s ratio compared to concrete, the ST expands more than the CC during the initial loading phase. This difference leads to a separation between the ST and the CC, disrupting their interaction. As the load increases, the lateral expansion of the CC gradually intensifies until the ST and CC reestablish contact. The interaction between the two components improves as the CC’s lateral expansion becomes more pronounced than that of the ST. This finding highlights that CFST columns exhibit a different confinement mechanism compared to traditional spiral-reinforced concrete columns. This study is to provide a model for predicting how confinement will develop. Because of the complexity of this mechanism, it is frequently not possible to determine confining stress by using analytical research alone.

On the other hand, measuring confining stress throughout the loading process is complex, and studies might need to depict how confining stress develops in CFST stub columns. The finite element approach is helpful for successfully demonstrating the fine details of confining stress generation. Consequently, intensive numerical research was conducted in this study using ABAQUS software (version 6-14-3, 2015) and finite element analysis. The goal was to develop a predictive model for the confining stress development under compressive load in CFST columns. The average value of contact stresses at the middle height of the column is used to approximate confining stress since it is directly correlated with the interaction between the ST and CC. The study ignored the impact of column length by concentrating only on short columns. Additionally, in examining CFST columns featuring a thin-walled ST (D/t ≥ 10), it was assumed, for simplicity, that stress throughout the wall thickness remains constant. Consequently, the impact of wall thickness on confining stress was considered negligible. This study explored the impact of various parameters on the development of confining stress, including cross-sectional shape, depth-to-wall thickness ratio, ${\epsilon }_y/{\epsilon }_c$ ratio, and CC compressive strength. An extensive analysis was performed on numerous circular CFST stub columns using ABAQUS software (version 6-14-3, 2015) to investigate the effects of parameters such as the D/t ratio, compressive strength of the concrete core, and the ${\epsilon }_y/{\epsilon }_c$ ratio. The CC compressive strength ranged from 15 to 250 MPa, while the D/t and ${\epsilon }_y/{\epsilon }_c$ ratios varied from 10 to 150 and 0.5 to 5, respectively.

The parameters L/D and t were held constant at 3 and 4 mm, respectively. Moreover, to assess the impact of these parameters on confinement, the unconfined concrete model offered by Karthik and Mander [23] was employed to replicate the CC behavior. Material parameters, including $\psi$, e, k_c, f_b0/fc′, and $\upsilon$, were set to default values to ensure standardized conditions across all specimens. It is important to emphasize that all specimens were treated as geometrically and loading-wise ideal, thus eliminating any eccentricity in loading or slippage between the CC and ST.

2.1. Cross-Sectional Shape

The cross-sectional shape constitutes one of the effective parameters influencing CC confinement. In alignment with the findings of Sheikh and Uzumeri [24], it is posited that the effective application of confining stress is limited to a specific portion of the CC in which the stress is fully raised. For this purpose, the effective confining stress can be calculated from Equation (1) expressed by Mander, Priestley [25].

$${f_l}^{\prime }=f_l . k_e$$

(1)

$f_l$ shows the confining stress and $k_e$ shows the confinement effectiveness coefficient. Due to the symmetric and uniform distribution of the confinement in the circular sections, the confinement effectiveness coefficient can be set equal to one for circular CFST columns ($k_e=1$). So, to investigate the impacts of other factors such as the ${\epsilon }_y/{\epsilon }_c$ ratio, concrete core CS, and D/t ratio, a large number of CFST stub columns with the circular section were studied using ABAQUS software (version 6-14-3, 2015).

2.2. The Ratio of ${\epsilon }_y/{\epsilon }_c$

Given the simultaneous loading of the ST and the concrete core, the discrepancy between the yield strain of steel and the strain at peak stress for unconfined concrete can exert a considerable influence on the evolution of confining stress across the development procedure. In this regard, the ${\epsilon }_y/{\epsilon }_c$ ratio has been defined, and its impact on the confining stress has been studied. For this purpose, an extensive numerical analysis is carried out (Figure 1. Observations indicate the ${\epsilon }_y/{\epsilon }_c$ ratio increment results in an increase in contact stress. The observation highlights that the slope of this incremental rise remains nearly constant up to the yielding point in ST, irrespective of varying ${\epsilon }_y/{\epsilon }_c$ ratios (refer to Figure 1). Also, in the case of ${\epsilon }_y/{\epsilon }_c<1$, especially the ratios of 0.5 to 0.8, it is observed that the contact stress drops after the yielding of ST. As the ST enters the plastic region, its Poisson’s ratio increases. This increase in the Poisson’s ratio, or in other words, the lateral expansion of the ST, results in a reduced interaction between the concrete core and ST. When the concrete core begins to crush, its Poisson’s ratio also rises, leading to a renewed escalation in the contact stress between the concrete core and ST. It is important to note that the yielding of the ST reduces its resistance to the expansion of the CC. Consequently, this reduction in resistance leads to a decrease in the slope of the increasing contact stress.

2.3. CS of Concrete Core

The total weight is split proportionately between the ST and concrete core based on each stiff status. As a result, the concrete core shares a significantly higher overall load ratio. Concrete’s nonlinear behavior becomes increasingly apparent, and the Poisson’s ratio rises as compressive stress increases in the material. As a result, the contact stress rises, and so does the concrete core’s lateral expansion. Studies using numbers are another source of this influence.

Conversely, as previously said, raising the CS of concrete raises the core’s stiffness. As a result, the CC’s lateral expansion within the elastic range is reduced. As a result, it is reasonable to anticipate that the development between the CC and ST will take longer to occur.

2.4. Depth-to-Thickness Ratio of ST (D/t)

The first point that comes to mind about the effect of this parameter is that by increasing D/t, the probability of local buckling of the ST increases, and consequently, the confining effects from the ST decrease. Nevertheless, this effect can be significant, particularly when the local buckling of the ST precedes its yielding. Another influence of this parameter is that, with a rise in the D/t percentage, the A_s/A_c ratio decreases, as illustrated in Equation (2). According to Equation (3), by reducing the ratio of A_s/A_c, the coefficient of confinement ($\xi$) decreases.

$${\displaystyle \frac{A_s}{A_c}}={\displaystyle \frac{D^2-D_c^2}{D_c^2}}={\left(1+{\displaystyle \frac{2}{\frac{D}{t}-2}}\right)}^2-1$$

(2)

$$\xi ={\displaystyle \frac{A_s{f}_y}{A_c{f_c}^{\prime }}}$$

(3)

A_s and A_c represent the cross-sectional areas of the ST and the CC, respectively. D and D_c denote the depths of the ST and the CC, respectively. $f_y$ is the yield stress of the steel, indicating the stress at which the steel deforms plastically, and ${f_c}^{\prime }$ is the compressive strength of the unconfined concrete, which measures the maximum stress the concrete can withstand before failing. Hence, it can be observed that an increase in the D/t ratio (where D is the depth of the ST and t is the thickness) results in a reduction in the confinement provided to the CC. This phenomenon is further validated through extensive numerical analysis using the ABAQUS software (version 6-14-3, 2015. The interaction between the ST and the CC becomes significant when the lateral expansion of the CC exceeds that of the ST. To quantify this interaction, the ratio $\Delta D_c/\Delta D$ can be defined where $\Delta D_c$ and $\Delta D$ represent the cross-sectional deformation of the CC and the ST, respectively. This ratio helps in understanding the relative deformations and interactions between the materials. In fact, by increasing the $\Delta D_c/\Delta D$ ratio, the CC expands faster than the steel tube, and so the interaction happens sooner. This ratio can be simplified according to Equation (4).

$${\displaystyle \frac{\Delta D_c}{\Delta D}}={\displaystyle \frac{{\left({\epsilon }_{\theta }\right)}_c . D_c}{{\left({\epsilon }_{\theta }\right)}_s . D}}={\displaystyle \frac{{\vartheta }_c . {\epsilon }_l . D_c}{{\vartheta }_s . {\epsilon }_l . D}}=\left({\displaystyle \frac{{\vartheta }_c}{{\vartheta }_s}}\right).\left(1-{\displaystyle \frac{2}{\frac{D}{t}}}\right)$$

(4)

where ${\left({\epsilon }_{\theta }\right)}_c$ and ${\left({\epsilon }_{\theta }\right)}_s$ are the lateral strain of the CC and ST, respectively. ${\vartheta }_c$ and ${\vartheta }_s$ are the Poisson’s ratio of the CC and the ST, respectively.

From this expression, it could be found that by increasing the ratio of D/t, the $\Delta D_c/\Delta D$ rate increases, and consequently, the interaction between the ST and CC occurs faster.

3. Proposed Confining Stress–Axial Strain Model for Circular CFST Stub Columns

Based on the investigations conducted in this study, four-stage and three-stage models were proposed for the cases of ${\epsilon }_y/{\epsilon }_c\le 1$ and ${\epsilon }_y/{\epsilon }_c>1$, respectively, to represent the development of the confining stress under compression, as depicted in Figure 2.

In the initial stage (from 0 to ${\epsilon }_0$), there is no confining stress. In fact, since the Poisson’s rate for the CC is lower than that for the ST, no interaction occurs between them. According to the obtained results from the software ABAQUS and the presented explanations, ${\epsilon }_0$ relates to the D/t rate and ${f_c}^{\prime }$, as shown in Figure 3. However, it is observed that the D/t rate has no significant effect on the value of ${\epsilon }_0$. The Equation (5) was proposed for determining ${\epsilon }_0$. Figure 3 is depicted to illustrate this, in which a large number of specimens with D/t rates ranging from 10 to 150 and the CS of CCs ranging from 15 MPa to 250 MPa were studied using the software ABAQUS. According to this figure, it can be observed that this influence is not significant.

$${\epsilon }_0=\left\{\begin{array}{l}[-0.0029 {{f_c}^{\prime }}^2+1.5 {f_c}^{\prime }+30]\times {10}^{-5} {f_c}^{\prime }\le 100\\ \left[6.5\ln \left({f_c}^{\prime }\right)-15\right]\times {10}^{-4} {f_c}^{\prime }>100\end{array}\right.$$

(5)

where ${\epsilon }_0$ is the first axial strain at which the interaction between the CC and ST occurs.

The second stage, which extends from ${\epsilon }_0$ to ${\epsilon }_1$, is described by Equation (6).

$$f_l=\left[a{\left({\displaystyle \frac{\epsilon -{\epsilon }_0}{{\epsilon }_c-{\epsilon }_0}}\right)}^2+b\left({\displaystyle \frac{\epsilon -{\epsilon }_0}{{\epsilon }_c-{\epsilon }_0}}\right)\right] . d$$

(6)

where $\epsilon$ is the axial strain and ${\epsilon }_c$ is the strain at peak stress (${f_c}^{\prime }$) of unconfined concrete. The parameters of a, b, and d are calculated by using Equations (7) to (9), respectively.

$$a=\left[0.026{{f_c}^{\prime }}^2+14{f_c}^{\prime }+380\right]\times {10}^{-3}$$

(7)

$$b=a^2. \left[0.13-{\displaystyle \frac{33}{100a}}\right]+1.59$$

(8)

$$d={\left[-0.292\ln \left({\displaystyle \frac{D}{t}}\right)+1.94\right]}^{0.785+{f_c}^{\prime }\left[880-1.72{f_c}^{\prime }\right]\times {10}^{-5}}$$

(9)

As elucidated in the preceding sections, the yielding of the ST outcomes in a diminished resistance against the lateral expansion of the CC. Thus, it is expected that the slope of increasing the confining stress reduces. Also, based on the investigations conducted in this research, it was seen that the increase in the confining stress decreases when the lateral strain of the CC becomes greater than ${\epsilon }_c$. Accordingly, ${\epsilon }_1$ is defined as the yield strain of the ST (${\epsilon }_{sy}$) provided that the lateral strain of the CC is not greater than ${\epsilon }_c$ unless it is defined as the axial strain ${\epsilon }_{cl}$, at which the lateral strain of the CC is equal to ${\epsilon }_c$. ${\epsilon }_1$ can be defined according to Equation (10).

$${\epsilon }_1={\epsilon }_{sy}\le {\epsilon }_{cl}$$

(10)

${\epsilon }_{sy}$ is the yield strain of the ST defined by Equation (25). The axial strain ${\epsilon }_{cl}$ could be obtained on the basis of the fact that the lateral strain of the CC and ST are similar across their interaction. The lateral strain of the ST can be measured from Hooke’s law.

$${\left({\epsilon }_{\theta }\right)}_{concrete}={\left({\epsilon }_{\theta }\right)}_{steel}={\displaystyle \frac{1}{E_s}}\left(f_{\theta }-{\vartheta }_s{f}_v\right)$$

(11)

where $E_s$ is the elastic modulus of steel; $f_v$ is the axial stress in the ST calculated from Hooke’s law, and $f_{\theta }$ is the hoop stress in the ST obtained from the equilibrium of the effective forces in cross-section as shown in Figure 4.

$$f_{\theta }={\displaystyle \frac{f_l . D_c}{2t}}=\left({\displaystyle \frac{D}{2t}}-1\right)f_l$$

(12)

$$f_v=E_s{\epsilon }_{cl}-{\vartheta }_s{f}_{\theta }$$

(13)

By substituting the Equations (12) and (13) into Equation (11), ${\epsilon }_{cl}$ is determined according to Equation (14).

$${\epsilon }_{cl}={\epsilon }_0+{\displaystyle \frac{{\epsilon }_c-{\epsilon }_0}{2ag}}\left(-h+\sqrt{h^2+4ag{E}_s\left({\epsilon }_c-{\vartheta }_s{\epsilon }_0\right)}\right)$$

(14)

$$h=bg+{\vartheta }_s{E}_s\left({\epsilon }_c-{\epsilon }_0\right)$$

(15)

$$g={\displaystyle \frac{d}{2}}\left(1-{\vartheta }_s^2\right)\left({\displaystyle \frac{D}{t}}-2\right)$$

(16)

where the parameters of a, b, and d are given in Equation (7) to Equation (9), respectively.

The preciseness of the provided equation has been confirmed against the results of the Finite Element analysis. The comparative analysis is depicted in Figure 5, where the predictions of the proposed equation are represented by the dashed line, while the finite element results are delineated by the solid line.

To describe the process of confining stress beyond the strain ${\epsilon }_1$, Equation (17) is proposed. It should be noted that in the case of ${\epsilon }_y/{\epsilon }_c<1$, the confinement drops after the yielding of the ST is neglected, and it is assumed that the confining stress is constant from ${\epsilon }_{sy}$ to ${\epsilon }_1$, as shown in Figure 2.

$$f_{l2}\left(\epsilon \right)=f_{l1}+m\left[{\displaystyle \frac{1}{\exp \left(-n{\left(\epsilon -{\epsilon }_t\right)}^p\right)+1}}-0.5\right]$$

(17)

where $f_{l1}$ is the confining stress corresponding to ${\epsilon }_1$ in Equation (10); m, n, p, and ${\epsilon }_{sy}$ are determined from Equation (18) to Equation (21), respectively.

$$m=\left[73.185\ln {f_c}^{\prime }+224.91\right]{\left({\displaystyle \frac{D}{t}}\right)}^{-0.66k}$$

(18)

$$n=7.93\times {\left({\displaystyle \frac{D}{t}}\right)}^{0.81z}$$

(19)

$$p=\left[-16.2{\left({\displaystyle \frac{{\epsilon }_y}{{\epsilon }_c}}\right)}^2+39.3\left({\displaystyle \frac{{\epsilon }_y}{{\epsilon }_c}}\right)+1127\right]{f_c}^{\prime -0.1}$$

(20)

$${\epsilon }_t=\left\{\begin{array}{l}{\epsilon }_c {\displaystyle \frac{{\epsilon }_y}{{\epsilon }_c}}\le 1\\ {\epsilon }_1 {\displaystyle \frac{{\epsilon }_y}{{\epsilon }_c}}>1\end{array}\right.$$

(21)

$$k=\left\{\begin{array}{l}{\left({\displaystyle \frac{D}{t}}\right)}^{0.09}{\left({\displaystyle \frac{{\epsilon }_y}{{\epsilon }_c}}\right)}^{-2.7{\left({\displaystyle \frac{D}{t}}\right)}^{-0.71}} {\displaystyle \frac{{\epsilon }_y}{{\epsilon }_c}}\le 2\\ {\left({\displaystyle \frac{D}{t}}\right)}^{0.145}{\left({\displaystyle \frac{{\epsilon }_y}{{\epsilon }_c}}\right)}^{-0.715{\left({\displaystyle \frac{D}{t}}\right)}^{-0.096}} {\displaystyle \frac{{\epsilon }_y}{{\epsilon }_c}}>2\end{array}\right.$$

(22)

$$z={\left({\displaystyle \frac{{\epsilon }_y}{{\epsilon }_c}}\right)}^{-0.21\ln {\displaystyle \frac{D}{t}}-0.376}$$

(23)

Figure 6 compares the proposed model of confining stress to the outcomes gained from the numerical investigations. The findings of the proposed model and that of the numerical investigations are shown with the dashed line and the solid line, respectively.

The Yield Strain of the ST (${\epsilon }_{sy}$)

The loading process of STs within CFST columns can be delineated into two distinct steps. In the initial phase, characterized by the absence of interaction between the ST and CC, the ST is adjusted to uniaxial compression load. In the second step (Beyond ${\epsilon }_0$), the ST is stressed biaxially. In order to determine the yield strain of the steel tube, it is assumed for simplicity that the compressive stress of the ST in the first step is considered as the residual stress for the second step.

$$f_{s1}=E_s{\epsilon }_0$$

(24)

where $f_{s1}$ is the CS of the ST in the first step.

As explained above, the second step begins from ${\epsilon }_0$; so, the yield strain of the ST can be determined according to Equation (25).

$${\epsilon }_{sy}={\epsilon }_{l2}+{\epsilon }_0$$

(25)

where ${\epsilon }_{sy}$ is the yield strain of the ST and ${\epsilon }_{l2}$ is the axial strain of the steel tube in the second step.

It is well-established that when steel material experiences multiple stresses, the von Mises yield parameter is used to delineate the elastic limit. Consequently, for the second step, it is formulated based on Equation (26).

$$f_v^2-f_v{f}_{\theta }+f_{\theta }^2=F_y^2$$

(26)

where $f_v$, $f_{\theta }$, and $F_y$ are determined according to Equation (27) and Equation (28).

$${\epsilon }_{l2}={\displaystyle \frac{1}{E_s}}\left(f_v-{\vartheta }_s f_{\theta }\right)\to \left|f_v\right|=E_s {\epsilon }_{l2}-{\vartheta }_s f_{\theta }$$

(27)

$$f_{\theta }=\left({\displaystyle \frac{D}{2t}}-1\right)f_l , f_l=\left[a{\left({\displaystyle \frac{{\epsilon }_{l2}}{{\epsilon }_c-{\epsilon }_0}}\right)}^2+b\left({\displaystyle \frac{{\epsilon }_{l2}}{{\epsilon }_c-{\epsilon }_0}}\right)\right]d$$

(28)

where $f_l$ was determined by substituting Equation (25) into Equation (6).

$$F_y=f_y-E_s{\epsilon }_0$$

(29)

where $f_y$ is the yield stress of steel.

By substituting the Equations (27)–(29) into Equation (26), ${\epsilon }_{l2}$ is determined based on Equation (30).

$$A{\epsilon }_{l2}^4+B{\epsilon }_{l2}^3+C{\epsilon }_{l2}^2-D=0$$

(30)

where the factors of A, B, C, and D are defined according to Equation (31) to Equation (36).

$$A={\displaystyle \frac{a^{2}f}{{\left({\epsilon }_c-{\epsilon }_0\right)}^4}}$$

(31)

$$B={\displaystyle \frac{a}{{\left({\epsilon }_c-{\epsilon }_0\right)}^2}}\left(e+{\displaystyle \frac{2bf}{{\epsilon }_c-{\epsilon }_0}}\right)$$

(32)

$$C=E_s^2+{\displaystyle \frac{b}{{\epsilon }_c-{\epsilon }_0}}\left(e+{\displaystyle \frac{bf}{{\epsilon }_c-{\epsilon }_0}}\right)$$

(33)

$$D={\left(f_y-E_s{\epsilon }_0\right)}^2$$

(34)

$$e={\displaystyle \frac{E_s\left(1-2\vartheta \right)\left(\frac{D}{t}-2\right)d}{2}}$$

(35)

$$f={\displaystyle \frac{\left(1+{\vartheta }^2-\vartheta \right){\left(\frac{D}{t}-2\right)}^2{d}^2}{4}}$$

(36)

4. Finite Element Modeling

In this study, a three-dimensional finite element model was developed to predict the ultimate strength and behavior of CFST stub columns under axial compressive stresses using ABAQUS software (version 6-14-3, 2015). Due to the symmetry in both geometry and loading conditions, only one-eighth of the CFST column was modeled. Figure 7 illustrates that symmetric boundary conditions were applied to the planes of symmetry. The top surface of the stub column was restrained against all degrees of freedom except in the direction of the applied load. Compressive loading was consistently applied to the top surface of the stub column using the displacement control mode.

4.1. Element Type and Mesh

The concrete core was simulated using eight-node solid elements with three translation degrees of freedom (C3D8R), whereas the ST was represented using four-node shell elements with decreased integration (S4R).

It is well known that while prediction accuracy can be increased by reducing the size of the elements, doing so results in an increase in computation volume because there are more elements. Therefore, mesh convergence research must be conducted to choose an acceptable mesh density. It can be observed from Tao, Wang [20] that convergence investigations that the N-ε curves are not significantly affected by an aspect ratio of the elements less than 3. Consequently, the element size was selected as D/20 across its cross-section and D/10 along its longitudinal axis, where D is the diameter of the complete section, to shorten the solving time and by the mesh convergence experiments. Figure 7 depicts the finite element mesh of a representative specimen.

4.2. Interaction between the Concrete Core and the Steel Tube

The interaction between the steel tube and the concrete core was simulated using a surface-based interaction model that included a complicated contact in the anticipated direction and a coulomb friction model in the tangential direction. The study used a coefficient of friction of 0.6 between the ST and the CC, drawing on this and other relevant studies [20,26,27,28,29]. A pair of surfaces called the primary and secondary person surfaces must be specified to characterize the contact between two surfaces. In this investigation, the stiffer component was selected as the master surface. To determine which component, between the steel tube and the concrete core, was stiffer, the parameter ${\displaystyle \frac{AE}{L}}$ was used for comparison. In this parameter, $A$ represents the cross-sectional area, $E$ denotes the material’s modulus of elasticity, and $L$ stands for the member’s length. By evaluating ${\displaystyle \frac{AE}{L}}$, the stiffness of the steel tube and the concrete core could be compared effectively.

4.3. Material Modeling

4.3.1. Steel Property

An ideal stress–strain model for the steel material must be defined to explain the behavior of the ST in the CFST columns. By Tao and Wang’s research Tao, Wang [20], It is evident that selecting any of the steel constitutive models has no appreciable impact on the final strength. Also, the ST’s strain-hardening tendency may be minimal because of the compressive loads and the potential for local buckling under compression. Therefore, the behavior of the steel material was described in this study using an elastic-perfectly-plastic model. If the elastic modulus (E_s) and Poisson’s ratio (${\vartheta }_s$) values for a test specimen were not reported, they were equal to 200,000 and 0.3 MPa, respectively.

4.3.2. Concrete Property

The factors of the elastic modulus and Poisson’s rates must be established to model the elastic behavior of CC. The Poisson’s ratio of concrete in this investigation (${\vartheta }_c$) was taken to be equal to 0.2, and the elastic modulus of concrete (E_c) was determined from the empirical Equation (37) proposed by Mander, Priestley [25] if it was not reported for a test specimen.

$$E_c=5000\sqrt{{f_c}^{\prime }}$$

(37)

The damage plasticity model in the material library of the software ABAQUS (version 6-14-3, 2015) was used to show the plastic behavior of the CC. To accurately model the CFST stub columns, five parameters need to be determined: dilation angle ($\psi$), eccentricity (e), the $f_{b0}/{f_c}^{\prime }$ ratio, the parameter K_c, and the viscosity parameter. The model by Tao, Wang [20] was utilized for this purpose. According to their model, the eccentricity and viscosity parameters were set to their default values of 0.1 and 0.0, respectively. The $f_{b0}/{f_c}^{\prime }$ ratio was calculated using Equation (38) by Papanikolaou and Kappos [30].

In this study, five key parameters were required: the dilation angle $\left(\psi \right)$, eccentricity (e), the $f_{b0}/f_{c0}$ ratio, the parameter Kc, and the viscosity parameter. To determine these parameters, the model proposed by Tao, Wang, and colleagues [20] was employed. According to this model, the eccentricity and viscosity parameters were assigned their default values of 0.1 and 0.0, respectively. The $f_{b0}/f_{c0}$ ratio was derived from Equation (38). The $f_{b0}/{f_c}^{\prime }$ ratio was determined from the proposed equation, Equation (38):

$$f_{b0}/{f_c}^{\prime }=1.5{\left({f_c}^{\prime }\right)}^{-0.075}$$

(38)

The parameters of K_c and dilation angle ($\psi$) were obtained from Equation (39) and Equation (40), respectively, proposed by Tao, Wang [20].

$$K_c={\displaystyle \frac{5.5}{5+2{\left({f_c}^{\prime }\right)}^{0.075}}}{k}_c$$

(39)

$$\psi =\left\{\begin{array}{ll}56.3\left(1-\xi \right)& \xi \le 0.5\\ 6.672e^{{\displaystyle \frac{7.4}{4.64+\xi }}}& \xi \ge 0.5\end{array}\right.$$

(40)

where $\xi$ is the confinement coefficient found in Equation (3).

Also, the equivalent compressive stress–strain model needs to be determined to simulate the compressive behavior of the CC in CFST stub columns. The researchers reported different models to prescribe the confined concrete’s behavior. Based on the investigations conducted on the CFST stub columns in this study, a modified stress–strain model expressed in Equation (41) was used in this study. According to this model, the ascending part of the curve was measured from an equation by Han, Yao [31], and for the descending part of the model, the equation proposed by Binici [32] was used.

$$f_c=\left\{\begin{array}{l}{f}_{cc}^{\prime }\left[2\left({\displaystyle \frac{\epsilon }{{\epsilon }_{cc}}}\right)-{\left({\displaystyle \frac{\epsilon }{{\epsilon }_{cc}}}\right)}^2\right] \epsilon \le {\epsilon }_{cc}\\ f_r+\left(f_{cc}^{\prime }-f_r\right)\exp \left[-{\left({\displaystyle \frac{\epsilon -{\epsilon }_{cc}}{\alpha }}\right)}^{\beta }\right] \epsilon \ge {\epsilon }_{cc}\end{array}\right.$$

(41)

where the residual stress ($f_r$) and the parameter $\alpha$ are calculated by Equation (42) and Equation (43), respectively, and proposed by Tao, Wang [20].

$$f_r=0.7\left(1-e^{-1.38\xi }\right){f_c}^{\prime }\le 0.25{f_c}^{\prime }$$

(42)

$$\alpha =0.04-{\displaystyle \frac{0.036}{1+e^{6.08\xi -3.49}}}$$

(43)

The parameter $\beta$ was taken as 1.2 according to Tao, Wang [20]. The peak stress ($f_{cc}^{\prime }$) and the corresponding strain (${\epsilon }_{cc}$) of the confined concrete were calculated using Equations (44) and (45), respectively, from the model developed by Mander, Priestley [25]. These equations provide a means to estimate the enhanced strength and strain capacity of concrete confined by a steel tube, taking into account the interaction between the concrete core and the confining steel.

$$f_{cc}^{\prime }={f_c}^{\prime }+k_1{f_l}^{\prime }$$

(44)

$${\epsilon }_{cc}={\epsilon }_c\left(1+k_2{\displaystyle \frac{{f_l}^{\prime }}{{f_c}^{\prime }}}\right)$$

(45)

According to Richart, Brandtzæg [33], the parameters $k_1$ and $k_2$ were set to 4.1 and 20.5, respectively. The parameter $f_1$ represents the effective confining stress, calculated using Equation (1). Additionally, ${ϵ}_c$, the strain at peak stress $\left(f_c\right)$ of the unconfined concrete, was determined using Equation (46) proposed by Karthik and Mander [23].

$${\epsilon }_c=0.0015+{\displaystyle \frac{{f_c}^{\prime }}{70000}}$$

(46)

In order to use Equation (44), the constant value of confining stress needs to be defined. In this regard, it is assumed that the confinement ($f_l$) is equal to the confining stress in ${\epsilon }_1$ (Equation (10)). By substituting the strain $\epsilon ={\epsilon }_1$ into Equation (6), the confinement ($f_l$) is computed according to Equation (47).

$$f_l=\left[a{\left({\displaystyle \frac{{\epsilon }_1-{\epsilon }_0}{{\epsilon }_c-{\epsilon }_0}}\right)}^2+b\left({\displaystyle \frac{{\epsilon }_1-{\epsilon }_0}{{\epsilon }_c-{\epsilon }_0}}\right)\right] . d$$

(47)

For defining the tensile behavior of concrete, a kind of fracture energy model was used, in which the tensile strength ($f_t$) was considered equal to $0.1{f_c}^{\prime }$, and the fracture energy was determined according to Equation (42) of CEB-FIP [34].

$$G_f=\left[0.0469d_{\max }^2-0.5d_{\max }+26\right]{\left({\displaystyle \frac{{f_c}^{\prime }}{10}}\right)}^{0.7}$$

(48)

where $d_{\max }$ is the maximum coarse aggregate size (in mm).

5. Results and Discussion

To validate the finite element models, the numerical outcomes were compared with experimental data. The $N-\Delta$ and $N-\epsilon$ curves produced by the current model are juxtaposed against experimental results. Additionally, comparisons between the finite element strength and experimental strength are presented as well.

According to the plots, it is observed that there are no descending branches in the $N-\Delta$ or $N-\epsilon$ curves for some specimens such as CU-040 (Hu, Huang [18]), SFE (Johansson and Gylltoft [35]), CC4-A-4-1 (Sakino, Nakahara [36]), CC8-A-8 (Sakino, Nakahara [36]) and 3HN (Tomii [37]) and even in some cases, a strain-hardening behavior is observed. It can be seen that the specimens C1 (Schneider [38]), C2 (Schneider [38]), and CC8-D-8 (Sakino, Nakahara [36]) exhibited clear signs of strain-hardening behavior. One plausible explanation for this phenomenon is attributed to the interaction between the ST and the CC. But another point significant in the results is that some specimens exhibit clearly strain hardening behavior while their D/t ratios are quite large compared to some specimens investigated in this study. The D/t ratios of the specimens CU-040 (Hu, Huang [18]), C1 (Schneider [38]), 3HN (Tomii [37]) and CC4-A-4-1 (Sakino, Nakahara [36]) are 40, 47, 46.9, and 50.3, respectively. Also, the ${\epsilon }_y/{\epsilon }_c$ ratios for these specimens are 0.7, 0.79, 0.75, and 0.74. In these specimens, the yielding of the ST occurs before the CC reaches the uniaxial compressive strength of concrete (${f_c}^{\prime }$). Based on the D/t rate and ${\epsilon }_y/{\epsilon }_c$ for these specimens, low confining stress is expected in ${\epsilon }_1$ (Equation (10)). A possible explanation for this may be the confinement mechanism in the CFST columns. The confinement is passive in nature, so it is developed by increasing the lateral expansion, and also, the lateral expansion is dependent on the axial strain. In the equivalent stress–strain curve of the concrete core, a constant value for the confinement was considered, while it is highly dependent on the axial strain. This can be a reasonable explanation in the specimens tested by Wang, Chen [39]. In this study, despite the specimens having relatively high D/t ratios, there is a notable increase of approximately 20% in the load-bearing capacity of the CFST (Concrete-Filled Steel Tube) specimens compared to the combined load-bearing capacities of the concrete core and the steel tube. While according to the relatively high ratios of D/t and low ratios of ${\epsilon }_y/{\epsilon }_c$ for these specimens, it is expected that there is low confinement in ${\epsilon }_1$. This statement suggests that incorporating the development of confining stress and its impact on the equivalent stress–strain curve of the concrete core (as depicted in Figure 8) could enhance the accuracy of predicting the descending branch of the stress–strain curves and the ultimate strength. By factoring in how confining stress influences the concrete’s behavior under load, the model can more precisely simulate how the material will perform under various conditions, leading to better predictions of both its declining strength post-peak and its maximum load-bearing capacity.

According to Figure 8, by determining the uniaxial compressive stress of concrete (${f_c}^{\prime }$), the factor k, and the confining stress ($f_l$) versus axial strain, the CS of confined CC versus axial strain could be obtained. The axial CS of concrete (${f_c}^{\prime }$) could be obtained from the existing models for unconfined concrete. Also, the contact stress ($f_l$) can be estimated. The problem is that there is no information about the variations of factor k versus the axial strain, bringing more studies to determine this factor during the loading of CFST columns. Figure 9, Figure 10 and Figure 11 show the curves from the model alongside experimental results, while Figure 12 presents the comparison between finite element and experimental strength.

Several statistical indices, including the correlation coefficient (R), coefficient of determination (R2), root mean square error (RMSE), relative root mean square error (RRMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE), were used to evaluate the accuracy of the finite element and test data. Moreover, a performance index ($\xi$) that depends on R and RRMSE was applied to evaluate the performance of the model. Equations (49) through (55) give the mathematical form of these assessment functions.

$$\mathrm{R}={\displaystyle \frac{{\displaystyle \sum _{i=1}^n\left(m_i-\overline{m}\right)\left(t_i-\overline{t}\right)}}{\sqrt{{\displaystyle \sum _{i=1}^n{\left(m_i-\overline{m}\right)}^2}{\displaystyle \sum _{i=1}^n{\left(t_i-\overline{t}\right)}^2}}}}$$

(49)

$${\mathrm{R}}^2={\displaystyle \frac{{\displaystyle \sum _{i=1}^n{\left(m_i-\overline{m}\right)}^2}}{{\displaystyle \sum _{i=1}^n{\left(t_i-\overline{t}\right)}^2}}}$$

(50)

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{{\displaystyle \sum _{i=1}^n{\left(m_i-t_i\right)}^2}}{n}}}$$

(51)

$$\mathrm{RRMSE}={\displaystyle \frac{1}{\left|\overline{t}\right|}}\sqrt{{\displaystyle \frac{{\displaystyle \sum _{i=1}^n{\left(m_i-t_i\right)}^2}}{n}}}$$

(52)

$$\mathrm{MAE}={\displaystyle \frac{{\displaystyle \sum _{i=1}^n\left|m_i-t_i\right|}}{n}}$$

(53)

$$\mathrm{MAPE}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\left|\frac{m_i-t_i}{m_i}\right|\times 100}$$

(54)

$$\xi ={\displaystyle \frac{RRMSE}{1+R}}$$

(55)

where n indicates how many specimens were utilized in the input, with $m_i$ and $t_i$ being the intended experimental value and model prediction linked to the $i^{th}$ specimen. Thus, $\overline{t}$ and $\overline{m}$ are the mean of target experimental values and the mean of model predictions, respectively.

Table 1. Statistical results of the developed FEM.

R	R2	RMSE	RRMSE	MAE	MAPE	$\mathit{\xi }$
0.98	0.97	0.069	0.19	0.045	17.29	0.09

shows the findings related to the statistical analysis of the developed model and the experimental results (normalized predicted and experimental data were used for statistical analysis), showing a good convergence with the testing data.

6. Comparison of the Results from Tests, Finite Element Model and Design Codes

Several popular design codes are used to determine the strength of CFST columns subjected to axial compressive loading. A comparison of several design codes and the finite element model’s forecast accuracy of ultimate strength, specifically, AISC [41], AIJ [42], EC4 [43], and DBJ13-51 [44], is shown in

Table 2. Comparison of the expected and experimental outcomes (the specimens tested by Wang, Chen [39]).

Specimen	D (mm)	t (mm)	$\frac{\mathit{D}}{\mathit{t}}$	$\mathit{L}$ (mm)	${\mathit{f}}_{\mathit{y}}$ (MPa)	${{\mathit{f}}_{\mathit{c}}}^{\mathbf{\prime }}$ (MPa)	$\frac{{\mathit{\epsilon }}_{\mathit{y}}}{{\mathit{\epsilon }}_{\mathit{c}}}$	${\mathit{N}}_{\mathit{E}\mathit{x}\mathit{p}}$ (kN)	$\frac{{\mathit{N}}_{\mathit{A}\mathit{I}\mathit{S}\mathit{C}}}{{\mathit{N}}_{\mathit{E}\mathit{x}\mathit{p}}}$	$\frac{{\mathit{N}}_{\mathit{E}\mathit{C}4}}{{\mathit{N}}_{\mathit{E}\mathit{x}\mathit{p}}}$	$\frac{{\mathit{N}}_{\mathit{D}\mathit{B}\mathit{J}}}{{\mathit{N}}_{\mathit{E}\mathit{x}\mathit{p}}}$	$\frac{{\mathit{N}}_{\mathit{A}\mathit{I}\mathit{J}}}{{\mathit{N}}_{\mathit{E}\mathit{x}\mathit{p}}}$	$\frac{{\mathit{N}}_{\mathit{F}\mathit{E}\mathit{M}}}{{\mathit{N}}_{\mathit{E}\mathit{x}\mathit{p}}}$
CFST-LA-1	153	1.54	99.3	306	345	73.2	0.72	1823	0.69	0.93	0.99	0.78	0.88
CFST-LA-2								1842	0.68	0.92	0.98	0.77	0.87
CFST-LA-3								1872	0.67	0.91	0.96	0.76	0.86
CFST-LB-1	250	2.48	100.8	500	326	73.2	0.67	4871	0.75	0.92	0.98	0.77	0.86
CFST-LB-2								4528	0.81	0.99	1.05	0.82	0.92
CFST-LB-3								4733	0.77	0.95	1.01	0.79	0.88
CFST-LC-1	372	3.64	102.2	744	320	73.2	0.68	9163	0.85	1.07	1.15	0.9	1.01
CFST-LC-2								9996	0.78	0.98	1.05	0.82	0.93
CFST-LC-3								9604	0.81	1.02	1.09	0.86	0.96
CFST-LD-1	469	4.66	100.6	938	291	73.2	0.61	15,827	0.85	0.97	1.04	0.81	0.91
CFST-LD-2								16,670	0.8	0.92	0.99	0.77	0.86
CFST-LD-3								15,239	0.88	1.01	1.08	0.84	0.94
CFST-MA-1	157	2.48	63.3	314	326	73.2	0.67	2117	0.78	0.93	0.96	0.77	0.85
CFST-MA-2								2058	0.8	0.96	0.99	0.79	0.87
CFST-MA-3								2068	0.8	0.95	0.99	0.79	0.87
CFST-MB-1	282	4.36	64.8	564	322	73.2	0.65	6811	0.78	0.92	0.96	0.76	0.84
CFST-MB-2								6380	0.83	0.99	1.03	0.82	0.9
CFST-MB-3								6860	0.77	0.92	0.95	0.76	0.84
CFST-MC-1	358	5.66	63.2	716	290	73.2	0.62	9947	0.84	0.99	1.04	0.82	0.91
CFST-MC-2								10,045	0.83	0.98	1.03	0.81	0.9
CFST-MC-3								9609	0.87	1.03	1.08	0.85	0.94
CFST-MD-1	474	7.42	63.9	948	317	73.2	0.64	17,787	0.84	1	1.04	0.83	0.96
CFST-MD-2								18,306	0.82	0.97	1.01	0.8	0.94
CFST-MD-3								17,885	0.84	0.99	1.03	0.82	0.96
CFST-HA-1	153	3.64	42	306	320	73.2	0.68	2264	0.75	0.93	0.95	0.76	0.82
CFST-HA-2								2274	0.75	0.93	0.94	0.76	0.82
CFST-HA-3								2205	0.77	0.96	0.97	0.79	0.84
CFST-HB-1	235	5.66	41.5	470	290	73.2	0.62	5390	0.73	0.89	0.92	0.73	0.79
CFST-HB-2								5047	0.78	0.95	0.98	0.78	0.84
CFST-HB-3								5096	0.77	0.94	0.97	0.77	0.83
CFST-HC-1	393	9.38	41.9	786	312	73.2	0.63	13,936	0.8	0.99	1.01	0.81	0.87
CFST-HC-2								14,406	0.78	0.96	0.98	0.79	0.84
CFST-HC-3								14,161	0.79	0.97	0.99	0.8	0.86
CFST-HD-1	477	11.36	42	954	310	73.2	0.63	20,237	0.81	1	1.02	0.82	0.88
CFST-HD-2								20,462	0.8	0.99	1.01	0.81	0.87
CFST-HD-3								19,854	0.83	1.02	1.04	0.84	0.9

,

Table 3. Comparison between experimental and predicted results (the specimens tested by Sakino, Nakahara [36]).

Specimen	D (mm)	t (mm)	$\frac{\mathit{D}}{\mathit{t}}$	$\mathit{L}$ (mm)	${\mathit{f}}_{\mathit{y}}$ (MPa)	${{\mathit{f}}_{\mathit{c}}}^{\mathbf{\prime }}$ (MPa)	$\frac{{\mathit{\epsilon }}_{\mathit{y}}}{{\mathit{\epsilon }}_{\mathit{c}}}$	${\mathit{N}}_{\mathit{E}\mathit{x}\mathit{p}}$ (kN)	$\frac{{\mathit{N}}_{\mathit{A}\mathit{I}\mathit{S}\mathit{C}}}{{\mathit{N}}_{\mathit{E}\mathit{x}\mathit{p}}}$	$\frac{{\mathit{N}}_{\mathit{E}\mathit{C}4}}{{\mathit{N}}_{\mathit{E}\mathit{x}\mathit{p}}}$	$\frac{{\mathit{N}}_{\mathit{D}\mathit{B}\mathit{J}}}{{\mathit{N}}_{\mathit{E}\mathit{x}\mathit{p}}}$	$\frac{{\mathit{N}}_{\mathit{A}\mathit{I}\mathit{J}}}{{\mathit{N}}_{\mathit{E}\mathit{x}\mathit{p}}}$	$\frac{{\mathit{N}}_{\mathit{F}\mathit{E}\mathit{M}}}{{\mathit{N}}_{\mathit{E}\mathit{x}\mathit{p}}}$
CC4-A-2	149	2.96	50.4	3	308	25.4	0.83	941	0.86	1.14	1.03	0.93	0.98
CC4-A-4-1	149	2.96	50.3	3	308	40.5	0.74	1064	0.97	1.23	1.19	1.02	1.07
CC4-A-4-2			50.4					1080	0.96	1.21	1.17	1	1.06
CC4-A-8	149	2.96	50.5	3	308	77	0.59	1781	0.9	1.05	1.12	0.89	0.97
CC4-C-2	301	2.96	101.5	3	279	25.4	0.75	2382	1.02	1.25	1.21	1.03	1.2
CC4-C-4-1	300	2.96	101.4	3	279	41.1	0.67	3277	1.04	1.22	1.26	1.02	1.17
CC4-C-4-2								3152	1.09	1.27	1.31	1.06	1.22
CC4-C-8	301	2.96	101.5	3	279	80.3	0.53	5540	1.08	1.2	1.32	1.02	1.15
CC4-D-2	450	2.96	152	3	279	25.4	0.75	4415	0.88	1.31	1.32	1.09	1.31
CC4-D-4-1	450	2.96	152	3	279	41.1	0.67	6870	0.81	1.19	1.26	1	1.15
CC4-D-4-2								6985	0.8	1.17	1.24	0.99	1.13
CC4-D-8	450	2.96	152	3	279	85.1	0.51	11,665	0.89	1.28	1.43	1.09	1.25
CC6-A-2	122	4.54	26.9	3	576	25.4	1.55	1509	0.8	1.12	0.99	0.96	0.95
CC6-A-4-1	122	4.54	26.8	3	576	40.5	1.39	1657	0.81	1.11	1.02	0.95	0.95
CC6-A-4-2								1663	0.81	1.1	1.02	0.94	0.95
CC6-A-8	122	4.54	26.8	3	576	77	1.11	2100	0.81	1.04	1.04	0.9	0.91
CC6-C-2	239	4.54	52.5	3	507	25.4	1.36	3035	0.89	1.21	1.04	1	1.11
CC6-C-4-1	238	4.54	52.5	3	507	40.5	1.22	3583	0.91	1.19	1.09	0.99	1.1
CC6-C-4-2			52.4					3647	0.9	1.17	1.07	0.98	1.08
CC6-C-8	238	4.54	52.4	3	507	77	0.98	5578	0.84	1.02	1.03	0.87	0.93
CC6-D-2	361	4.54	79.4	3	525	25.4	1.41	5633	0.77	1.18	1.03	0.97	1.14
CC6-D-4-1	361	4.54	79.4	3	525	41.1	1.26	7260	0.75	1.12	1.06	0.93	1.07
CC6-D-4-2	360		79.3					7045	0.77	1.14	1.08	0.96	1.1
CC6-D-8	360	4.54	79.4	3	525	85.1	0.97	11,505	0.73	1.06	1.11	0.9	1.04
CC8-A-2	108	6.47	16.7	3	853	25.4	2.29	2275	0.85	1.15	1.14	1.05	0.96
CC8-A-4-1	109	6.47	16.8	3	853	40.5	2.05	2446	0.84	1.13	1.13	1.02	0.98
CC8-A-4-2	108		16.7					2402	0.85	1.13	1.14	1.03	0.98
CC8-A-8	108	6.47	16.7	3	853	77	1.64	2713	0.84	1.09	1.15	1	0.97
CC8-C-2	222	6.47	34.3	3	843	25.4	2.26	4964	0.91	1.27	1.08	1.09	1.18
CC8-C-4-1	222	6.47	34.3	3	843	40.5	2.03	5638	0.89	1.2	1.07	1.04	1.14
CC8-C-4-2								5714	0.88	1.19	1.06	1.03	1.13
CC8-C-8	222	6.47	34.4	3	843	77	1.62	7304	0.85	1.09	1.05	0.95	1.04
CC8-D-2	337	6.47	52.1	3	823	25.4	2.21	8475	0.81	1.22	1.02	1.04	1.2
CC8-D-4-1	337	6.47	52	3	823	41.1	1.97	9668	0.81	1.2	1.06	1.02	1.21
CC8-D-4-2								9835	0.79	1.18	1.05	1.01	1.19
CC8-D-8	337	6.47	52	3	823	85.1	1.52	13,776	0.75	1.09	1.07	0.94	1.09

,

Table 4. Comparing the test and predicted results.

Specimen	D (mm)	t (mm)	$\frac{\mathit{D}}{\mathit{t}}$	$\mathit{L}$ (mm)	${\mathit{f}}_{\mathit{y}}$ (MPa)	${{\mathit{f}}_{\mathit{c}}}^{\mathbf{\prime }}$ (MPa)	$\frac{{\mathit{\epsilon }}_{\mathit{y}}}{{\mathit{\epsilon }}_{\mathit{c}}}$	${\mathit{N}}_{\mathit{e}\mathit{x}\mathit{p}}$ (kN)	$\frac{{\mathit{N}}_{\mathit{A}\mathit{I}\mathit{S}\mathit{C}}}{{\mathit{N}}_{\mathit{E}\mathit{x}\mathit{p}}}$	$\frac{{\mathit{N}}_{\mathit{E}\mathit{C}4}}{{\mathit{N}}_{\mathit{E}\mathit{x}\mathit{p}}}$	$\frac{{\mathit{N}}_{\mathit{D}\mathit{B}\mathit{J}}}{{\mathit{N}}_{\mathit{E}\mathit{x}\mathit{p}}}$	$\frac{{\mathit{N}}_{\mathit{A}\mathit{I}\mathit{J}}}{{\mathit{N}}_{\mathit{E}\mathit{x}\mathit{p}}}$	$\frac{{\mathit{N}}_{\mathit{F}\mathit{E}\mathit{M}}}{{\mathit{N}}_{\mathit{E}\mathit{x}\mathit{p}}}$	Ref.
CU-040	200	5	40	3	265.8	27.15	0.7	2013	0.77	1.1	1.06	0.84	0.86	Hu, Huang [18]
CU-070	280	4	70	3	272.6	31.15	0.7	3025	0.88	1.11	1.1	0.91	0.97
CU-150	300	2	150	3	341.7	27.23	0.9	2608	0.73	1.12	1.04	0.92	1.12
C1	140.8	3	47	4.3	285	28.18	0.79	881	1.07	1.41	1.3	1.15	1.21	Schneider [38]
C2	141.4	6.5	21.7	4.3	313	28.805	0.83	1825	0.89	1.27	1.16	1.05	0.96
C3	140	6.68	21	4.4	537	28.18	1.37	2715	0.92	1.28	1.18	1.1	1.03
3HN	150	3.2	46.9	450	287.4	28.7	0.75	1000	0.87	1.14	1.05	0.93	0.97	Tomii [37]
CS-1	219	6.3	34.8	600	300	163	0.39	6880	0.94	1.06	1.23	0.91	1	Liew, Xiong [40]
SFE	159	4.8	33.1	650	433	64.5	4.09	2150	0.97	1.18	1.22	1.04	1.05	Johansson and Gylltoft [35]

and

Table 5. Comparing the test and predicted results (O’Shea and Bridge [45]).

Specimen	D (mm)	t (mm)	$\frac{\mathit{D}}{\mathit{t}}$	$\mathit{L}$ (mm)	${\mathit{f}}_{\mathit{y}}$ (MPa)	${{\mathit{f}}_{\mathit{c}}}^{\mathbf{\prime }}$ (MPa)	$\frac{{\mathit{\epsilon }}_{\mathit{y}}}{{\mathit{\epsilon }}_{\mathit{c}}}$	${\mathit{N}}_{\mathit{E}\mathit{x}\mathit{p}}$ (kN)	$\frac{{\mathit{N}}_{\mathit{A}\mathit{I}\mathit{S}\mathit{C}}}{{\mathit{N}}_{\mathit{E}\mathit{x}\mathit{p}}}$	$\frac{{\mathit{N}}_{\mathit{E}\mathit{C}4}}{{\mathit{N}}_{\mathit{E}\mathit{x}\mathit{p}}}$	$\frac{{\mathit{N}}_{\mathit{D}\mathit{B}\mathit{J}}}{{\mathit{N}}_{\mathit{E}\mathit{x}\mathit{p}}}$	$\frac{{\mathit{N}}_{\mathit{A}\mathit{I}\mathit{J}}}{{\mathit{N}}_{\mathit{E}\mathit{x}\mathit{p}}}$	$\frac{{\mathit{N}}_{\mathit{F}\mathit{E}\mathit{M}}}{{\mathit{N}}_{\mathit{E}\mathit{x}\mathit{p}}}$
S30CS50B	165	2.82	58.5	580.5	363.3	48.3	0.83	1662	0.86	1.04	1.05	0.89	0.93
S20CS50A	190	1.94	97.9	663.5	256.4	41	0.6	1678	0.81	0.92	0.98	0.79	0.87
S16CS50B	190	1.52	125	664.5	306.1	48.3	0.67	1695	0.76	1.02	1.09	0.87	0.96
S12CS50A	190	1.13	168.1	664.5	185.7	41	0.5	1377	0.79	0.95	1.06	0.82	0.92
S10CS50A	190	0.86	220.9	659	210.7	41	0.57	1350	0.67	0.96	1.06	0.82	0.92
S30CS80A	165	2.82	58.5	580.5	363.3	80.2	0.68	2295	0.89	1.02	1.1	0.88	0.93
S20CS80B	190	1.94	97.9	663.5	256.4	74.7	0.49	2592	0.86	0.94	1.05	0.81	0.9
S16CS80A	190	1.52	125	663.5	306.1	80.2	0.56	2602	0.75	0.99	1.11	0.85	0.95
S12CS80A	190	1.13	168.1	662.5	185.7	80.2	0.39	2295	0.88	1.04	1.19	0.89	1.02
S10CS80B	190	0.86	220.9	663.5	210.7	74.7	0.46	2451	0.64	0.91	1.03	0.78	0.88
S30CS10A	165	2.82	58.5	577.5	363.3	108	0.6	2673	0.96	1.07	1.2	0.93	1
S20CS10A	190	1.94	97.9	660	256.4	108	0.41	3360	0.92	0.99	1.13	0.85	0.96
S16CS10A	190	1.52	125	661.5	306.1	108	0.49	3260	0.78	1.02	1.16	0.88	0.98
S12CS10A	190	1.13	168.1	660	185.7	108	0.34	3058	0.87	1.03	1.18	0.88	1.01
S10CS10A	190	0.86	220.9	662	210.7	108	0.39	3070	0.72	1.02	1.17	0.88	0.99

. Consequently, as observed in

Table 6. Comparison of FEM results with code predictions.

	$\frac{{\mathit{N}}_{\mathit{A}\mathit{I}\mathit{S}\mathit{C}}}{{\mathit{N}}_{\mathit{E}\mathit{x}\mathit{p}}}$	$\frac{{\mathit{N}}_{\mathit{E}\mathit{C}4}}{{\mathit{N}}_{\mathit{E}\mathit{x}\mathit{p}}}$	$\frac{{\mathit{N}}_{\mathit{D}\mathit{B}\mathit{J}}}{{\mathit{N}}_{\mathit{E}\mathit{x}\mathit{p}}}$	$\frac{{\mathit{N}}_{\mathit{A}\mathit{I}\mathit{J}}}{{\mathit{N}}_{\mathit{E}\mathit{x}\mathit{p}}}$	$\frac{{\mathit{N}}_{\mathit{F}\mathit{E}\mathit{M}}}{{\mathit{N}}_{\mathit{E}\mathit{x}\mathit{p}}}$
Mean	0.83	1.06	1.08	0.9	0.98
Min	1.09	1.41	1.43	1.15	1.31
Max	0.64	0.89	0.92	0.73	0.79
STD	0.085	0.116	0.098	0.103	0.117

, AISC [41] and AIJ [42] provide conservative forecasts with a mean value of 17% and 10% less than the test findings. EC4 [43] and DBJ13-51 [44] predict slightly higher capacities than the experimental results. The results show an average of 6% and 8% for EC4 [43] and DBJ13-51 [44], respectively. Also, the mean value of $N_{FEM}/N_{Exp}$ is 0.98 for the finite element model, which shows reasonable accuracy.

7. Conclusions

This study analyzed how confining stress develops in axially loaded CFST (Concrete-Filled Steel Tube) stub columns. It examined various elements that affect material confinement, such as the cross-sectional shape, the ratio of depth to wall thickness, the compressive strength of the concrete core, and the ratio of $e_y/e_c$.

Through detailed numerical investigations in this study, a model was formulated to depict the evolution of confining stress during the compression of the CFST stub columns. An updated finite element model was subsequently introduced to accurately mimic the behavior of these columns. Comparisons between the predictions of this model and experimental outcomes and design code projections were used to verify its accuracy. It is shown that the finite element model can accurately predict the behavior and ultimate strength of the CFST stub columns. Nevertheless, it is anticipated that by considering the development of the compressive stress over the loading of the CFST columns and adding its impact on the equivalent stress–strain relationship of the CC, the predictions’ accuracy may be improved.