Numerical study of axial-loaded circular concrete-filled aluminum tubular slender columns

Abstract

Aluminum alloys have been widely applied in coastal and marine structures because of their superior sustainability and corrosion resistance. Concrete-filled double-skin aluminum tubular columns (CFDAT) possess higher strength and better ductility than traditional reinforced concrete structures. However, few studies have been conducted on numerical simulation methods for circular CFDATs. Specifically, there has been no experimental or numerical study on intermediate-to-slender circular CFDATs. Here, a comprehensive numerical study was conducted on a modeling method for the first time to simulate the axial behavior of a slender circular CFDAT. This study outlines the development of numerical modeling techniques and presents a series of comparative studies using various material nonlinearities, confinement effects, and nonlinearity of the initial geometric imperfections for a slender column. The numerical results were compared with more than 80 previously available stub and slender experimental test results for verification. It was confirmed that the proposed numerical technique was reliable and accurate for simulating the axial behavior of intermediate and slender circular CFDAT. Furthermore, a parametric study was conducted to investigate the effects of geometric and material properties on the axial capacity of the CFDAT. Additionally, the slenderness and strength-to-width ratio of CFDAT were compared with those of concrete-filled double-skin steel tubular columns (CFDST). The simulated axial strengths were compared with those predicted using AS 5100 and AISC 360. New design equations for the CFDATs should be proposed based on AS 5100.

Graphical abstract

Keywords

axial behavior / aluminum / concrete-filled / double-skin / slender columns / numerical modelling

1 Introduction

The investigation of composite columns started in the 1960s and has become increasingly popular as the composite columns combine the benefits of various materials and concrete [1]. Concrete-filled steel tubular columns (CFST) consist of an outer steel tube fully filled with concrete, whereas concrete-filled double-skin steel tubular columns (CFDST) have two concentric inner and outer steel tubes, with an annulus between the skins filled with concrete. Fig.1 shows the typical profiles of CFST and CFDST with circular cross-sections. CFST and CFDST are two common types of composite column sections with superior mechanical performance, including high compressive strength, stiffness, and ductility [2]. In addition, CFST and CFDST are more durable, require lower maintenance, and are more fire-resistant than conventional alloy structures [3,4]. CFST and CFDST tubes can also constitute the framework during construction, which is beneficial in terms of cost and time [5].

Several experimental studies have been conducted on circular CFST and CFDST [4,6–9]. Recently, aluminum alloys have become an increasingly popular alternative to carbon steel in the construction industry because of their superior advantages, including corrosion resistance, high strength-to-weight ratio, recyclability, and low maintenance costs, which meet the requirements for bridges, high-build structures, and offshore structures [10]. A series of experimental tests have also been conducted [5,11] to investigate the axial behavior of circular concrete-filled aluminum tubular (CFAT) and concrete-filled double-skin aluminum tubular columns (CFDAT).

Tab.1 summarizes the experimental tests conducted as well as an overview of the numerical studies undertaken by different researchers on axially loaded circular sectional concrete-filled columns. These tests provided a series of data regarding the axial behavior of concrete-filled stub and slender columns with various alloys, including the ultimate capacity, axial load–strain relationships, and failure modes. The confinement effect provided by the outer tube increased the axial capacity. Local and overall buckling could occur in such columns, not to mention that the failure modes varied owing to the material of the tubes and the slenderness ratios (L_e/r).

Sakino et al. [7] investigated the influence of confinement provided by steel tubes on the axial capacity of a CFST to concrete infill. They concluded that a linear relationship exists between the axial capacity provided by the confinement effect and the yield strength of the outer tubes. Zhou and Young [5,11] conducted experimental studies on stub circular CFAT and CFDAT subjected to axial compression. They observed that the failure patterns depend on the diameter of the outer tube to the thickness ratio (D_o/t_o) and the concrete strength. They also observed that local buckling occurred on both, the outer and inner tubes of the CFDAT columns, and that the concrete was crushed. Tao et al. [8] performed a series of tests on a stub circular CFDST under axial loading. The composite action between the outer tube and infilled concrete was justified and described using a parameter of confinement factor (ξ), as shown in Eq. (1):

(1)\xi =\frac{A_{\mathrm{s}}{f}_{\mathrm{y}}}{A_{\mathrm{c}}{f}_{\mathrm{c}\mathrm{k}}},

where A_{\mathrm{s}} and A_{\mathrm{c}} are the cross-sectional areas of the steel and concrete, respectively, and f_y and f_ck are the yield strength of the outer tube and the characteristic concrete strength, respectively. Han [6], Zeghiche and Chaoui [9] conducted a series of experiments to investigate the axial behavior of circular CFST with various L_e/r ratios. Zeghiche and Chaoui [9] investigated columns which have L_e/r of 50–100, and Han [6] investigated those with L_e/r of more than 130. The results revealed that the concrete strength made a partial contribution to the capacity of the CFST. In addition, the slender CFST failed after the steel tube’s yielding at mid-height. Moreover, no local buckling was observed for any of the slender columns. Essopjee and Dundu [4] conducted tests on axially loaded slender CFDST. The infilled concrete was observed crushing in the CFDST with a length of 1000 mm (L_e/r = 19.4–25.5), and outward local buckling occurred only in the upmost region of the outer tubes. Additionally, no signs of local buckling were observed in the inner tubes. In the other tested CFDST (L_e/r > 29.0), the observed failure mode was only due to overall buckling.

A nonlinear static analysis should be conducted considering several factors to accurately predict the behavior of concrete-filled columns. Among these, material nonlinearity and initial geometrical imperfections have significant effects on the modeling results and are investigated through a comparative study.

Tao et al. [1], Liang and Fragomeni [12] developed numerical models for axially loaded stub CFST with normal- or high-strength steels using different constitutive models. Liang and Fragomeni [12] indicated that a confined concrete model which considers a typical-strength material tends to overestimate the strength of a high-strength material column. Tao et al. [1] determined the variables of confined concrete using the concrete damaged plasticity model (CDP), including the dilation angle and strain hardening/softening relationship. Furthermore, Wang et al. [3], Zhou and Young [13], and Patel et al. [14] proposed numerical methods to predict the behavior of stub CFAT subjected to axial loading. These models were verified based on the experimental results from Zhou and Young [5]. Different methods have been used to determine the constitutive model of confined concrete. Zhou and Young [13] and Patel et al. [15] used similar constitutive models for confined concrete based on Mander et al. [16], whereas Wang et al. [3] adopted the constitutive model proposed by Han et al. [17]. Although the three constitutive models can predict the behavior of confined concrete for CFAT with reasonable errors, the formula used by Wang et al. [3] has a simpler form. Huang et al. [18] and Liang [19] proposed numerical methods for axially loaded stub CFDST. Huang et al. [18] indicated that the inner voids of CFDST with small hollow ratios had a partial influence on the behavior of sandwiched concrete; hence, the confined concrete model for CFST can be used. However, Liang [19] suggested that the inner tube has a significant effect on the confinement mechanism, thus, a new model should be proposed for the CFDST. Concerning CFDAT, further studies should be conducted on the confinement mechanism owing to the significant differences in Young’s moduli between aluminum and steel. Zhou and Young [11] and Patel et al. [15] conducted numerical studies on a stub circular CFDAT subjected to axial loading. Zhou and Young [11] adopted the same constitutive model as the CFAT, whereas Patel et al. [15] proposed a new formula for the lateral confining pressure of the CFDAT. However, the database was quite small, and further investigations should be conducted. An et al. [20] developed a series of finite element models of a slender circular CFST. On the other hand, Liang [21] proposed a fiber-based numerical study of a slender CFDST. Based on numerical methods for stub CFST and CFDST, initial imperfections were considered for slender columns. Both authors indicated that the initial stiffness significantly affected the ultimate capacity of the CFST and CFDST, and that the increase or decrease in the axial capacity was consistent with the initial rigidity. An et al. [20] found that the steel yield strength’s impact is not critical for very slender columns. In addition, Liang [21] indicated that the confinement effect weakened as the slenderness ratio increased and that it could be ignored with L_e/r larger than 61.

Fig.2 shows the typical ratio of stability (φ) versus slenderness ratio, λ (i.e., L_e/r) curve, where φ is calculated by dividing the ultimate capacity by the section capacity of the columns. The critical slenderness ratios of columns with various failure modes, namely λ_o and λ_p, are shown in Fig.2. λ_o is the junction of short and intermediate columns (which is 22 according to AS 3600 [22]). λ_p is used to distinguish elastic–plastic instability failure as well as elastic instability failure of columns, and can be determined for CFST having a value of 115/\sqrt{f_{\mathrm{y}}/235} as indicated by An et al. [20]. However, no method has been found yet for determining the critical slenderness ratios for CFDAT.

Several experimental and numerical studies have investigated the axial behavior of concrete-filled columns, as listed in Tab.1. However, these studies have a limited scope, not to mention that there have been no experimental or numerical studies on slender circular CFDAT. This study is the first to develop a finite element numerical model to predict the behavior of such columns. Material nonlinearity and initial geometrical imperfections, as well as other factors, are discussed and presented. This numerical method was verified to accurately predict the axial behavior of circular concrete-filled stub and slender columns with various alloys. The proposed numerical method was then utilized to investigate the behavior of intermediate and slender circular CFDAT with various geometric and material properties through a parametric study.

2 Numerical modeling

2.1 Material characteristics

As an important part of numerical modeling, the material properties should be defined using a proper model to describe the failure criteria and plastic flow rules of concrete and metals. To simulate columns subjected to axial compression, the infilled concrete is considered to fail when it crushes or cracks. The CDP has been utilized in many previous studies [3,8,18,20,23] and was also applied in this study. It employs a non-associated potential plastic flow using the Drucker–Prager hyperbolic and Lubliner yield functions to describe the plasticity of concrete [24]. The parameters used for the CDP should be carefully determined, including the dilation angle, eccentricity, ratio of the biaxial to uniaxial compressive strength (f_b0/f_c0), compressive meridian (K), and viscosity parameter. Based on previous studies, the values of these parameters were taken as 20°, 0.1, 1.16, 0.667, and 0.0001, respectively [1,25].

For carbon steel and aluminum alloys, Ayough et al. [26] used the elastic–plastic model provided by Abaqus (*ELASTIC and *PLASTIC) to describe the behavior of carbon steel. First, a linear elastic model was used for the elastic part of the metallic material. Subsequently, von Mises yield surfaces with associated plastic flow were used to model plasticity in multiaxial stress states and the material was considered isotropic [24].

Residual stresses generally exist in alloys owing to stresses during manufacturing, which may slightly decrease the initial stiffness of the alloy tubes [27]. However, the influence of residual stress on CFST and CFDST is negligible, because the infilled concrete can significantly increase the stiffness of the columns [1].

2.1.1 Concrete

The confinement effect can significantly improve the strength and ductility of the confined concrete under compression [16], which also exists in axially loaded CFST and CFDST owing to the influence of the tubes. Han et al. [17], Hassanein et al. [25], and Ellobody et al. [28] suggested that the confined concrete stress–strain model is suitable for modeling stub CFST by considering the confinement effect. However, for a slender CFDST, the confinement effect causes a limited difference in the ultimate loading, and can be ignored in this case [21].

The model of the confined concrete may differ as the material properties of steel and aluminum vary, including the Young’s modulus and yield strength (0.2% proof stress). However, current confined concrete models for concrete-filled aluminum columns are very limited [14]. In addition, the current models only consider the yield strength of the outer tube as well as D_o/t_o ratio for stub columns only. Thus, more studies regarding the model of confined concrete aluminum columns are needed.

The effects of using confined and unconfined concrete models to simulate axially loaded concrete-filled columns with various alloys were investigated using a series of numerical models, as listed in Tab.2. The average ratios of the numerical results to the corresponding experimental results (P_NUM/P_EXP) obtained using the confined concrete model were generally larger than those obtained using the unconfined concrete one, except for the slender CFST.

As listed in Tab.2, the differences in P_NUM/P_EXP using the confined and unconfined concrete models were small and only 1% when it came to CFST and CFDST, whereas the differences were clear (6% and 9% for CFAT and CFDAT, respectively). The average values of P_NUM/P_EXP from simulation using the confined concrete model were 0.93 and 0.95 for stub CFAT and CFDAT, respectively; but using a confined concrete model can provide more accurate predictions. When confined and unconfined concrete models were used for the development of slender CFST and CFDST, the differences in P_NUM/P_EXP were less than 3%.

However, as shown in Fig.3, the confinement effect significantly influences the post-peak branch because it can increase the ductility of the concrete and delay the reduction in concrete strength. The axial load–strain curve of the model using unconfined concrete rapidly decreased when the ultimate loading was reached, whereas the model utilizing confined concrete showed that the column could bear loads even at the post-peak section. A simulation using the confined concrete model is expected to provide a more accurate simulation of the slender CFAT and CFDAT. A confined concrete model was employed in the numerical technique for axially loaded concrete-filled columns with various alloys.

For concrete under compression, the stress–strain relationship of the confined concrete model was calculated using the method proposed by Han et al. [17], as shown in Eq. (2):

(2)y=\{\begin{array}{rl}& 2x-x^2,\left(x\le 1\right)\\ & \frac{x}{{\beta }_0{\left(x-1\right)}^{\eta }+x},\left(x>1\right)\end{array}

where x=\epsilon /{\epsilon }_0 and y=\sigma /{\sigma }_0; \eta =2 for circular cross-section; {\beta }_0={\left(2.36\times {10}^{-5}\right)}^{0.25+{\left(\xi -0.5\right)}^7}{\left(f_{\mathrm{c}}^{\prime }\right)}^{0.5}\times 0.5\ge 0.12 for circular cross-section; {\epsilon }_0={\epsilon }_{\mathrm{c}}+800{\xi }^{0.2}\times {10}^{-6}; {\epsilon }_{\mathrm{c}}=(1300+12.5f_{\mathrm{c}}^{\prime })\times {10}^{-6}.

Han et al. [17] suggested that the plastic behavior of concrete is influenced by the confinement factor (ξ) as shown in Eq. (1). For circular CFDST, Huang et al. [18] suggested that the cross-section area of the sandwiched concrete is calculated using the following formula: A_{\mathrm{c}}={\displaystyle \frac{\text{π}}{4}}(D-2t_{\mathrm{o}}{)}^2.

The value of initial elastic modulus is calculated in MPa following ACI 318 [29], using E_{\mathrm{c}}=4730\sqrt{f_{\mathrm{c}}^{\prime }}. The elastic modulus of concrete under compression was considered to be the same as that under tension.

The stress–strain relationship of concrete under tension was obtained using a modified tension stiffening model provided by Wahalathantri et al. [30], as illustrated in Fig.4. The concrete tensile strength is calculated following ACI-318 [29] using f_{\mathrm{c}\mathrm{t}}=0.6\sqrt{{\gamma }_{\mathrm{c}}{f}_{\mathrm{c}}^{\prime }}.

When considering the influence of concrete damage under axial loading, compressive and tensile damage variables (d_{\mathrm{c}} and d_{\mathrm{t}}) were introduced to calculate the plastic and cracking strains on the descending branch of the stress–strain curve for concrete subjected to compression or tension. The compressive and tensile damage variables are calculated as d_{\mathrm{c}}=1-\sigma /f_{\mathrm{c}}^{\prime } and d_{\mathrm{t}}=1-\sigma /f_{\mathrm{c}\mathrm{t}}, respectively [31], where \sigma is the axial stress of concrete; f_{\mathrm{c}}^{\prime } and f_{\mathrm{c}\mathrm{t}} are concrete cylinder compressive and tensile strength, respectively.

2.1.2 Carbon steel

The currently available stress–strain relationships of carbon steel were summarized and quantitatively compared by Ayough et al. [26]. Moreover, it was shown that the prediction using the bilinear stress–strain relationship is 98% accurate compared to the experimental results, whereas the elastic–perfectly plastic model has a 96% accuracy. However, the strain hardening of steel is not substantial under strain conditions in most structures, and effects of the stress–strain relationship of carbon steel on ultimate loads can be neglected [1]. Fig.5 presents the stress–strain curves used for the carbon steel in this study. The values of elastic modulus (E_{\mathrm{s}}), yield stress (f_{\mathrm{y}}) and yield strain ({\epsilon }_{\mathrm{y}}) can easily be obtained based on experimental data. The hardening modulus (E_{\mathrm{s}\mathrm{h}}) has been suggested as 0.05E_s [1]. In this study, the bilinear stress–strain relationship was first considered. However, the elastic–perfectly plastic model was used when the ultimate stress (f_{\mathrm{u}}) was lacking in the experimental data. In this study, Poisson’s ratio for steel was taken to be 0.3.

2.1.3 Aluminum alloy

The stress–strain relationship of aluminum was initially modeled in two stages by Ramberg and Osgood [32]. The boundary between the two stages is at the 0.2% proof strain ({\epsilon }_{0.2}). Rasmussen [33] suggested that the first stage of the stress–strain curve proposed by Ramberg and Osgood [32] accurately describes the behavior of aluminum alloys and provided a modified equation for the second part. These equations have been recognized and employed by several researchers [14,15]. The full-range equations are as follows:

(3)

where E_{0.2} is the tangent modulus of the stress–strain curve at 0.2% proof stress and is calculated using E_{0.2}={\displaystyle \frac{E_0}{1+0.002\left({\displaystyle \frac{n{E}_0}{{\sigma }_{0.2}}}\right)}}. The 0.2% proof strain can be determined using {\epsilon }_{0.2}={\displaystyle \frac{{\sigma }_{0.2}}{E_0}}+0.002. The constant n and the exponent m are expressed as follows: n={\displaystyle \frac{\mathrm{l}\mathrm{n}\left(20\right)}{\mathrm{l}\mathrm{n}({\sigma }_{0.2}/{\sigma }_{0.01})}} and m=1+3.5{\displaystyle \frac{{\sigma }_{0.2}}{{\sigma }_{\mathrm{u}}}}.

The Poisson’s ratio for the elastic part is generally obtained from the experimental data; however, it was taken as 0.33 in this study due to unavailability of experimental data. Abaqus (*PLASTIC) requires consideration of the true stress–strain relationship to calculate the yield stress and plastic strains. Therefore, the engineering stress–strain relationship calculated using Eq. (3) should be converted into the true stress and plastic strains, as follows [24]:

(4)\sigma ={\sigma }_{\mathrm{n}\mathrm{o}\mathrm{m}}\left(1+{\epsilon }_{\mathrm{n}\mathrm{o}\mathrm{m}}\right),

(5){\epsilon }^{\mathrm{p}\mathrm{l}}=\mathrm{l}\mathrm{n}\left(1+{\epsilon }_{\mathrm{n}\mathrm{o}\mathrm{m}}\right)-\frac{\sigma }{E_0}.

2.2 Geometric initial imperfection

Flexural buckling can occur on axial loaded concrete-filled columns with large length to outer diameter ratio (L/D_o) [5,7,11]. The initial geometric imperfections significantly influenced the accuracy of the numerical models of such columns. Without considering the initial imperfections, flexural buckling does not tend to be observed, and slender columns remain ideally straight under axial loading. Therefore, the simulated ultimate capacity was significantly higher than the experimental value.

The initial geometric imperfections include local and global imperfections that provide local buckling and flexural buckling deformation shapes, respectively. However, for concrete-filled columns, the local buckling of tubes is expected to be avoided because the infilled concrete reduces or prevents the inward deformation of the tubes [34]. In light of previous research regarding local and global imperfections, the decrease in the ultimate axial strength of concrete-filled tubular columns is very limited because the infilled concrete has a delay that affects the local buckling of the tube. Therefore, the effect of imperfections on the axial strength is negligible [35–37]. Typically, a similar conclusion can be obtained by comparing the simulated axial strengths with and without considering local imperfections. The difference in the ultimate strengths was less than 1.5%. However, time consumption can be significantly reduced in numerical studies. In addition, the stub columns do not tend to be affected by flexural buckling. Therefore, in this study, only the initial global imperfection was considered for axially loaded concrete-filled columns with L/D_o values larger than three.

To define initial imperfection, two parameters, predicted deformation shape and amplitude coefficients of global imperfection ({\omega }_{\mathrm{g}}), should be determined and calculated. Then, *IMPERFECTION in Abaqus was used to define the initial geometric imperfections in this study. The predicted deformation shape was the same as the lowest number of eigenmodes (eigenmode 1) obtained from the linear buckling analysis, which was a half-wave sine curve along the column length. On the other hand, the initial lateral displacement at the mid-length of the column depends on {\omega }_{\mathrm{g}}. In previous experimental study, the measured value of {\omega }_{\mathrm{g}} usually ranges from L/1000 to L/10000, but it varies randomly among different tests [34]. In this section, the numerical values of {\omega }_{\mathrm{g}} for CFST and CFDST are discussed via a sensitivity study as follows. A series of numerical models with various {\omega }_{\mathrm{g}} values (but with the other parameters the same) were developed to investigate the influence of {\omega }_{\mathrm{g}}.

Tab.3 and Tab.4 compare the ultimate loads obtained from previous experimental studies and a series of corresponding numerical simulations for slender CFST and CFDST, respectively. Various values of {\omega }_{\mathrm{g}} (0, L/10000, L/2000, and L/1000) are employed. When the value of {\omega }_{\mathrm{g}} is zero, the initial lateral displacement at the mid-length is also zero, which means that the initial imperfection will not be considered in this case. Typical axial load–strain curves of a slender column with various values of {\omega }_{\mathrm{g}} are shown in Fig.6. Furthermore, the curves based on a slender CFDST (s139.2-1.5) with various values of {\omega }_{\mathrm{g}} are compared. It can be concluded that the values of {\omega }_{\mathrm{g}} have significant effects on both, the predicted ultimate capacity and the post-peak behavior. For the column with a {\omega }_{\mathrm{g}}value of 1/1000, the decreasing of axial loading can be observed, while the predicted ultimate capacity is more accurate.

As shown in Tab.3, the optimum value of {\omega }_{\mathrm{g}} for the modeling of CFST varies due to the value of L_e/r. For slender CFST (L_e/r less than 100), it is obvious that a {\omega }_{\mathrm{g}}value of L/2000 can provide the optimum prediction, in which case the average and standard deviation of P_NUM/P_EXP ratio are 0.92 and 0.05, respectively. Thus, the simulated results with a {\omega }_{\mathrm{g}}value of L/2000 have a good agreement with experimental data. However, the average values of P_NUM/P_EXP with a {\omega }_{\mathrm{g}} of 0, L/10000, and L/1000 are 1.25, 1.20, and 0.87 respectively, which cannot meet the accuracy requirements for numerical studies. For very slender CFST (L_e/r more than 100), the average values of P_NUM/P_EXP with a {\omega }_{\mathrm{g}} of 0, L/10000, L/2000, and L/1000 are 2.57, 0.93, 0.86, and 0.83, respectively. L/10000 is the most suitable value of {\omega }_{\mathrm{g}} for very slender CFST due to the reasonable average and standard deviation. A {\omega }_{\mathrm{g}} of L/1000 and L/10000 can be employed for the modeling of slender CFST with various L_e/r.

In Tab.4, the average values of P_NUM/P_EXP for the slender CFDST were 1.16, 1.19, 1.13, and 1.03 for models with a {\omega }_{\mathrm{g}} of 0, L/10000, L/2000, and L/1000, respectively; whereas, the corresponding standard deviations were 0.16, 0.15, 0.15, and 0.08. The results of using a {\omega }_{\mathrm{g}}value of less than L/1000 are nearly 10% larger than the results of using a {\omega }_{\mathrm{g}} value of L/1000, which overestimates the ultimate capacity of the slender CFDST. The standard deviations of models using a {\omega }_{\mathrm{g}} of less than L/1000 are more than 0.15, while the standard deviation of using the imperfection coefficient of L/1000 is 0.08. It is obvious that the simulated results of using a {\omega }_{\mathrm{g}} of L/1000 and the experimental results are in good agreement. In this case, using L/1000 as {\omega }_{\mathrm{g}} can predict the behavior of the slender CFDST more accurately.

Initial global imperfections should be considered in the simulation of intermediate and slender concrete-filled columns. However, due to the lack of the experimental study of CFAT and CFDAT, the same value of {\omega }_{\mathrm{g}} is used to simulate these columns since concrete-filled columns with various alloys have similar axial behaviors due to infilled concrete. {\omega }_{\mathrm{g}} values of L/2000 and L/10000 are suggested for CFAT with L_e/r of less than 100 and more than 100, respectively. In addition, the initial imperfection with a {\omega }_{\mathrm{g}} of L/1000 should be used for CFDAT with L/D_o greater than 3 to obtain reliable simulations.

2.3 Elements

Discretization was performed by defining suitable elements. An eight-node linear brick solid element with reduced integration (C3D8R) and four-node doubly curved shell element with reduced integration (S4R) provided by the Abaqus element library [24] were used to model the concrete and alloy tubes, respectively. The use of reduced integration elements can reduce simulation time. However, this may cause hour-glassing, which affects the reliability of the modeling results, particularly if a large bending deformation occurs in a model. Thus, the artificial strain energy-to-internal energy ratio (ALLAE/ALLIE) for each model must be less than 5%. Otherwise, the ‘hourglass stiffness’ method provided by Abaqus is used to add artificial stiffness to the elements.

A reasonable mesh size of a model can balance the accuracy of the numerical results and the calculation time of finite element modeling. Zhou and Young [13] suggested that simulated results are more accurate when the shape of most elements of concrete is close to a cube (length:width:depth = 1:1:1). Using this element shape, a convergence study is conducted to determine the optimal mesh size. Fig.7 illustrates a typical relationship between the axial loading and number of elements for an aluminum hollow section tube. The result of the axial loading is considered accurate when the number of elements is 8640. Therefore, the corresponding mesh size is optimal for modeling circular concrete-filled columns, which is one-fifteenth of the outer tube diameter. Tao et al. [1] reached similar conclusions in their convergence study.

An unreasonable mesh generation of the cross-section may cause contact overclosure between the outer tube and the infilled concrete, resulting in an unreasonable physical phenomenon of the models. To avoid this issue, a typical mesh of the concrete-filled cross-sections was generated, as illustrated in Fig.8. Fig.9(a) and Fig.9(b) show the typical mesh of the models developed in Abaqus.

2.4 Boundary and loading conditions

In the previous experimental studies, stub composite columns tended to have fixed supports at both ends, whereas slender columns tended to have simple supports. To reduce the numerical study’s calculation time, only half of the column was modeled by considering symmetric boundary conditions, as illustrated in Fig.9. The columns were symmetric along the axis direction to ensure that the deformation was also symmetric.

In this study, the end plates were not modeled, because the study by Wang et al. [38] showed that the influence of end plates on the behavior of the structure is negligible; hence, time can be saved by not modeling the latter. As shown in Fig.10, a reference point is drawn in the center of the top cross-section of the column for each model, which has the same motion as that of the whole top end of the column using a ‘tie’ constraint. Therefore, the support of the columns (fixed or simple supports) can be defined at the reference points. The top end of the column was not permitted to deform when it was tied to the reference point.

Regarding the loading condition, displacement control for concrete-filled columns was used in line with many experimental studies [4,5,11]. In the static analysis, the columns were slowly subjected to axial compression until the displacement of the reference point reached the estimated displacement. Sufficient axial displacement was defined to satisfy the requirements of the columns to reach the ultimate capacity. In addition, post-buckling can be observed using displacement control. In this study, a total displacement of 20 mm in the axial direction satisfied this requirement.

2.5 Contact algorithm

The modeling of interaction pairs involves geometric and mechanical properties [24]. The former serves to avoid penetration between the tubes and infilled concrete in the models, whereas the latter considers the mechanical interactions between different materials.

To define the geometric properties, surface-to-surface contact was used as contact discretization, while considering the form of both the slave and master surfaces in the touching region so as to avoid large penetrations of master nodes into slave nodes [24]. The surfaces of the infilled concrete were selected as the master surfaces because of their higher stiffness, whereas the surfaces of the tubes were selected as the slave surfaces. Many researchers have used the same approach to define geometric properties [3,11,13,34]. Regarding mechanical contact properties, two directions of interaction were considered, namely tangential and normal. Hard contact is defined as a normal direction and prevents overclosure at constraint locations from the slave to the master surface [24]. In addition, the separation between the contact pairs was permitted under tensile forces along the normal direction. A penalty method is used to describe the interactions along the tangential direction. In this case, isotropic friction is simulated between contact surfaces by defining the friction ratio taken as 0.6 concurring with the experimental data [1] for the interface between carbon steel and infilled concrete, and as 0.25 between aluminum alloys and infilled concrete [3].

3 Verification

A series of finite element models were employed using the modeling technique discussed in Section 2 for concrete-filled stub and slender columns. As described in this section, further verification of over 80 previous test data was conducted by comparing the numerical results with those obtained in the available literature in terms of (1) ultimate capacity, (2) axial load–axial strain relationship, and (3) failure patterns.

The ultimate column capacities obtained from the experiments (P_EXP) and numerical simulations (P_NUM) are shown in Fig.11 and Fig.12. (P_EXP,P_NUM) points are located near the 0% deviation line, and most of them are between ±10% error lines. The mean ratios of P_NUM/P_EXP are approximately 1.00, whereas the standard deviations were less than 0.10. Therefore, the ultimate column capacity could be accurately predicted using the employed models.

Fig.13 and Fig.14 illustrate typical comparisons between the axial load and strain in both experimental and numerical situations. The curves proposed in the experiments and those from the numerical simulations exhibit a similar pattern. Both the initial and post-peak portions of the experimental curves were accurately predicted by the simulated curves. However, slight errors were observed in the post-peak portion. This is due to the difference between the experimental data and the values calculated using empirical equations for the material properties.

The failure modes of the outer and/or inner tubes of the composite columns were observed using the von Mises stress (S, Mises) contour, whereas the crack patterns of the concrete were obtained using the maximum principal plastic strain (PE, Max. Principal) contours [24]. The failure modes are compared in Fig.15–Fig.19, which demonstrates good agreement between the numerical models and the corresponding test specimens. Local buckling and material yielding were observed in concrete-filled stub columns, whereas global buckling occurred in slender columns.

In general, the proposed modeling technique is accurate and reliable for predicting the behavior of concrete-filled stub and slender columns, and can also be used for simulating slender circular CFDAT in a parametric study.

4 Parametric study

The modeling method developed in Section 2 was utilized to conduct a parametric study on the behavior of intermediate and slender circular CFDAT. These columns were divided into five groups. Tab.5 lists the details of their dimensions and material properties. A total of 64 CFDAT were simulated to explore the axial behavior as influenced by L_e/r, D_o/t_o, D_i/t_i ratios, and concrete strength (f_{\mathrm{c}}^{\prime }). In this section, intermediate and slender CFDAT with different L_e/r ratios are considered to investigate the effects of the variables on the axial capacity. This section also compares the behaviors of CFDAT and CFDST with the same variables, instead of the material of both the outer and inner tubes. The Young’s moduli of the aluminum and steel tubes were 69 and 200 GPa, respectively. The nonlinearity index (n) of aluminum was proposed to be 5.0 [15].

4.1 Effects of L_e/r ratio

The columns of Group A were analyzed to investigate the effect of the L_e/r ratio on the behavior of the columns. Only the length of the columns in Group A varied, and all other parameters were consistent. In Fig.20, the axial capacity of the columns is shown as a function of the L_e/r ratio. As mentioned in Section 1, the critical slenderness ratios to differ short, intermediate and slender CFST, i.e. λ_o and λ_p, are 22 [22] and 115/\sqrt{f_{\mathrm{y}}/235} [20], respectively. However, regarding the CFDAT of Group A, if the critical slenderness ratios (approximately 11.25 and 90, respectively) were determined using such equations for CFST, λ_o and λ_p would be taken as 22 and 106, which are not suitable for that shown in Fig.20. Therefore, this study suggests modifying λ_o and λ_p for CFDAT in which more research is needed.

As shown in Fig.21, the effects of parameters, such as D_o/t_o, D_i/t_i, f_{\mathrm{c}}^{\prime } and σ_0.2o, are investigated on the axial capacity (P)–L_e/r relationship of CFDAT. Fig.21(a) shows that the linearity of the curve increases with D_o/t_o rising, whereas the curves become flatter with decreasing D_o/t_o. It is obvious that D_o/t_o significantly influences the value of λ_p. λ_p tends to be larger with an increasing D_o/t_o. As illustrated in Fig.21(b) and Fig.21(c), the P–L_e/r curves are approximately parallel with various D_i/t_i and f_{\mathrm{c}}^{\prime }. In this case, the value of λ_p does not tend to change. In other words, D_i/t_i and f_{\mathrm{c}}^{\prime } have little effects on the value of λ_p. Similarly, CFDAT P–L_e/r curves with various proof stress of aluminum (σ_0.2o) are approximately parallel when L_e/r is between 22.5 and 45, as shown in Fig.21 (d). The curves gradually become closer to each other with L_e/r increasing, and almost overlap when L_e/r = 90. Therefore, σ_0.2o has very limited effects on λ_p. Compared to the P–L_e/r curve of CFDST shown in Fig.20, it is obvious that Young’s modulus of tubes can significantly influence λ_o and λ_p. In further research on intermediate and slender columns, the Young’s modulus of the tubes and D_o/t_o ratios should be considered.

A change in L_e/r ratio of the column can significantly affect the load-carrying capacity of the intermediate CFDAT. As shown in Fig.20, as L_e/r increases from 11.25 to 90, the ultimate load decreases by 59.7%. The effect of L_e/r ratios of the column on the behavior of CFDAT is the most pronounced which is consistent with CFDST [21].

4.2 Effects of D_o/t_o ratio

Group B was utilized to investigate the effect of D_o/t_o on the load-carrying capacity of intermediate and slender CFDAT of different lengths. As shown in Tab.5, only D_o varied from 400 to 700 for the columns in Group B, whereas the other dimensions and material properties were the same. The ultimate axial loads of the intermediate CFDAT (L = 3524 and 7047 mm) were significantly increased by using greater D_o/t_o ratios. Typically, the numerical results with L = 7047 mm indicated that increasing the D_o/t_o ratio from 40 to 50, 60, and 70 increased the ultimate axial strength in the intermediate column by 73.1%, 162.0%, and 268.0%, respectively. When the D_o/t_o ratio increased linearly, the ultimate axial loading of the CFDAT exhibited an upward trend. This trend was similar for both the intermediate and slender columns, as shown in Fig.22. The main reason for this significant increase was that the cross-sectional area of the concrete increased as D_o increased.

4.3 Effects of D_i/t_i ratio

The columns in Group C were used to investigate the effect of the D_i/t_i ratio on the behavior of the column. The diameter of the inner tube varied to obtain D_i/t_i ratios of 20, 30, 35, and 40 for the numerical analysis. For the intermediate columns (L = 3524 mm), an increase in D_i/t_i significantly affected the axial strength, as shown in Fig.23. When the D_i/t_i ratio increased from 10 to 20, the load-carrying capacity of the CFDAT slightly increased by 2.2%. In this case, the intermediate CFDAT varied from buckling-dominated to material-dominated failure. As the D_i/t_i ratio continued to increase, the ultimate strength of CFDAT started to decrease owing to a reduction in the cross-section of the concrete area. As shown in Fig.23, the ultimate strength of the slender CFDAT decreased by 1.6%, 7.1%, and 20.4% as D_i/t_i of the column increased from 10 to 20, 30, and 40, respectively. D_i/t_i has a significant effect on the ultimate strength of slender CFDAT when the D_i/t_i ratio is very large. However, D_i/t_i ratio has only a limited effect when it increases slightly. For slender columns (L = 21141 mm), the effects of D_i/t_i ratio were very limited, as P–D_i/t_i tended to be flat.

4.4 Effects of σ_0.2o/f_{\mathrm{c}}^{\prime } ratio

As shown in Fig.24, the simulation results for Group D and some columns in Groups A, B, and C were used to investigate the effect of concrete strength on intermediate and slender columns. Some columns from Group A and Group E can be used to investigate the effect of the proof stress (σ_0.2o) on the outer tube. Increasing the strength of the infilled concrete significantly increased the ultimate axial strength of the intermediate CFDAT. For CFDAT with L = 7047 mm, the concrete strength of 32 MPa rises to 40, 50, and 80 MPa when the corresponding σ_0.2o/f_{\mathrm{c}}^{\prime } is 8.63, 6.90, 5.52, 4.25, and 3.45, respectively (Fig.24), whereas the ultimate axial strength of the column increases by approximately 5.5%, 17.9%, 29.5%, and 55.9%. For the slender CFDAT (L = 14094 mm), the increase in concrete strength also had a significant effect on the column load capacity. As f_{\mathrm{c}}^{\prime } increased, the axial capacity increased by 13.3%, 31.5%, 50.0%, and 81.4%, respectively. This case is different from that of CFDST, in which the ultimate axial capacity of CFDST is minimally affected by increases in concrete strength [39]. However, when the CFDAT becomes very slender (L = 21141 mm), the effects of f_{\mathrm{c}}^{\prime } are less significant, as the columns fail owing to overall buckling instead of the material. Thus, the optimum f_{\mathrm{c}}^{\prime } to reach the highest axial strength for CFDAT and CFDST are different.

Fig.25 shows the effect of varying σ_0.2o on the ultimate strength of intermediate and slender CFDAT. For intermediate columns, σ_0.2o increases from 214 to 276, 345, and 503 MPa. The load-carrying capacity for the CFDAT with a 7047 mm of length increases by 13.5%, 22.8%, and 33.3%, respectively. For slender CFDAT, however, the increase in σ_0.2o has only a very limited effect on the axial capacity. This is because the slender columns in this case failed owing to buckling.

4.5 Comparison of slender CFDAT and CFDST

This parametric study also compared CFDST of the same size as the CFDAT. As shown in Fig.20, the CFDST has a smaller load capacity than that of CFDAT of the same size at a relatively small L_e/r ratio, as the σ_0.2 of the aluminum tube is greater than the f_y of the steel tube. As the L_e/r ratio increased, the ultimate capacity of CFDST approached and exceeded that of CFDAT. This is because buckling gradually becomes a critical factor affecting the ultimate capacity of the columns. The flexural stiffness of CFDST was much larger than that of CFDAT when the Young’s modulus of the steel was approximately three times that of the aluminum alloy. Fig.20 shows that, for intermediate and slender columns, a CFDST of the same size has a higher axial capacity than a CFDAT.

Fig.26 shows how the strength-to-weight (STW) ratio varies with the length of CFDAT and CFDST. The STW ratios of both CFDAT and CFDST decreased with an increase in L_e/r ratios owing to buckling behavior. For short and intermediate columns with L_e/r less than 60, the STW ratios of CFDAT remain greater than those of CFDST. For stub columns, the difference between the STW ratios of the two types of columns was 33%. However, with an increase in L_e/r ratio, the STW of the CFDAT is less than that of the CFDST, and the difference can reach 28.9%. For intermediate columns, aluminum performs better than steel at relatively small L_e/r ratios because of its lower density but similar yield stress to steel. With increasing L_e/r ratios, steel performs better than aluminum because it has greater stiffness to bear buckling.

5 Comparisons with the existing design models

In the absence of design codes for CFDAT, the design codes for axially loaded composite steel columns AS 5100 [41] and AISC 360 [42] were deployed to calculate the ultimate strength of the CFDAT, as shown in Tab.5. In this section, the simulated ultimate strength of the CFDAT is compared with that calculated using the design codes. However, the current design codes AS 5100 [41] and AISC 360 [42] cannot predict the ultimate strength of CFDAT.

5.1 AS 5100

Because the CFDATs in the parametric study are all pinned-ended, the effective length of each column is equal to its physical length. The ultimate axial strength of the CFDAT is calculated using Eq. (6) according to AS 5100 [41]:

(6)P_{\mathrm{A}\mathrm{S}5100}={\alpha }_{\mathrm{c}}\{\begin{array}{ll}{\eta }_2\left(A_{\mathrm{o}}{f}_{\mathrm{o}}+A_{\mathrm{i}}{f}_{\mathrm{i}}\right)+A_{\mathrm{c}}{f}_{\mathrm{c}}^{\prime }\left(1+{\displaystyle \frac{{\eta }_1{t}_{\mathrm{o}}{f}_{\mathrm{o}}}{D_{\mathrm{o}}{f}_{\mathrm{c}}^{\prime }}}\right),& {\lambda }_{\mathrm{r}}\le 0.5,\\ A_{\mathrm{o}}{f}_{\mathrm{o}}+A_{\mathrm{i}}{f}_{\mathrm{i}}+A_{\mathrm{c}}{f}_{\mathrm{c}}^{\prime },& {\lambda }_{\mathrm{r}}>0.5,\end{array}

where the subscript “o” and “i” refer to “outer” and “inner”, respectively; A is the cross-sectional area and f is the yield stress (0.2% proof stress for aluminum); D_{\mathrm{o}} and t_{\mathrm{o}} are the diameter and the thickness of the outer tube, respectively; {\eta }_1 and {\eta }_2 are coefficients regarding confinement effect, and are calculated using {\eta }_1=4.9-18.5{\lambda }_{\mathrm{r}}+17{{\lambda }_{\mathrm{r}}}^2\ge 0 and {\eta }_2=0.25\left(3+2{\lambda }_{\mathrm{r}}\right)\le 1, respectively for axially loaded columns. The relative slenderness ({\lambda }_{\mathrm{r}}) is required to be less than 0.5 in order to consider the effects of confinement, and is calculated as shown in Eqs. (7)–(9):

(7){\lambda }_{\mathrm{r}}=\sqrt{\frac{P_{\mathrm{s}}}{P_{\mathrm{c}\mathrm{r}}}},

(8)P_{\mathrm{s}}=A_{\mathrm{o}}{f}_{\mathrm{o}}+A_{\mathrm{i}}{f}_{\mathrm{i}}+A_{\mathrm{c}}{f}_{\mathrm{c}}^{\prime },

(9)P_{\mathrm{c}\mathrm{r}}=\frac{{\text{π}}^2{\left(EI\right)}_{\mathrm{e}}}{L^2},

where P_{\mathrm{s}} is the squash load, and P_{\mathrm{c}\mathrm{r}} is the elastic critical load. The effective stiffness is determined as {\left(EI\right)}_{\mathrm{e}}=E_{\mathrm{o}}{I}_{\mathrm{o}}+E_{\mathrm{i}}{I}_{\mathrm{i}}+E_{\mathrm{c}}{I}_{\mathrm{c}}. No material reduction factor was considered in AS 5100 [41].

The compression member slenderness is taken into account for intermediate and slender columns, and the slenderness reduction factor {\alpha }_{\mathrm{c}} is calculated as shown in Eqs. (10)–(12):

(10){\alpha }_{\mathrm{c}}=\xi \left[1-\sqrt{1-{\left(\frac{90}{\xi \lambda }\right)}^2}\right],

(11)\xi =\frac{{\left({\displaystyle \frac{\lambda }{90}}\right)}^2+1+\eta }{2{\left({\displaystyle \frac{\lambda }{90}}\right)}^2},

(12)\lambda ={\lambda }_{\eta }+{\alpha }_{\mathrm{a}}{\alpha }_{\mathrm{b}},

where \eta =0.00326(\lambda -13.5)\ge 0, {\lambda }_{\eta }=90{\lambda }_{\mathrm{r}}, and {\alpha }_{\mathrm{a}}= {\displaystyle \frac{2100({\lambda }_{\eta }-13.5)}{{\lambda }_{\eta }^2-15.3{\lambda }_{\eta }+2050}}; as recommended in AS 5100 [41], the coefficients {\alpha }_{\mathrm{b}} is taken as −0.5 for {\rho }_{\mathrm{s}}\le 3\mathrm{\%} and 0 for . For concrete-filled columns, {\rho }_{\mathrm{s}} is the ratio of the tube to infilled concrete cross-sectional areas [43].

5.2 AISC 360

Unlike AS 5100 [41], AISC 360 [42] neglects the confinement effect on infilled concrete. The ultimate axial strength of the CFDAT can be determined using Eq. (13) in accordance with AISC 360 [42].

(13)P_{\mathrm{A}\mathrm{I}\mathrm{S}\mathrm{C}360}=\{\begin{array}{ll}\left({0.658}^{\frac{P_{\mathrm{n}\mathrm{o}}}{P_{\mathrm{e}}}}\right)P_{\mathrm{n}\mathrm{o}},& {\displaystyle \frac{P_{\mathrm{n}\mathrm{o}}}{P_{\mathrm{e}}}}\le 2.25,\\ 0.877P_{\mathrm{e}},& {\displaystyle \frac{P_{\mathrm{n}\mathrm{o}}}{P_{\mathrm{e}}}}>2.25,\end{array}

where P_{\mathrm{e}} is the elastic critical buckling load, which can be determined using Eq. (9), P_{\mathrm{n}\mathrm{o}} is given by Eqs. (14)–(17) as follows:

(14)

(15)P_{\mathrm{p}}=f_{\mathrm{o}}{A}_{\mathrm{o}}+f_{\mathrm{i}}{A}_{\mathrm{i}}+0.95f_{\mathrm{c}}^{\prime }{A}_{\mathrm{c}},

(16)P_{\mathrm{y}}=f_{\mathrm{o}}{A}_{\mathrm{o}}+f_{\mathrm{i}}{A}_{\mathrm{i}}+0.7f_{\mathrm{c}}^{\prime }{A}_{\mathrm{c}},

(17)P_{\mathrm{n}\mathrm{o}}=\frac{0.72f_{\mathrm{o}}}{{\left[\left({\displaystyle \frac{D_{\mathrm{o}}}{t_{\mathrm{o}}}}\right){\displaystyle \frac{f_{\mathrm{o}}}{E_{\mathrm{o}}}}\right]}^{0.2}}{A}_{\mathrm{o}}+f_{\mathrm{i}}{A}_{\mathrm{i}}+0.7f_{\mathrm{c}}^{\prime }{A}_{\mathrm{c}},

where \lambda ={\displaystyle \frac{D_{\mathrm{o}}}{t_{\mathrm{o}}}}, {\lambda }_{\mathrm{p}}={\displaystyle \frac{0.15E_{\mathrm{o}}}{f_{\mathrm{o}}}}, and {\lambda }_{\mathrm{r}}={\displaystyle \frac{0.19E_{\mathrm{o}}}{f_{\mathrm{o}}}}.

The effect of the slenderness of the cross sections is calculated using Eq. (14). The effective stiffness is determined using {\left(EI\right)}_{\mathrm{e}}=E_{\mathrm{o}}{I}_{\mathrm{o}}+E_{\mathrm{i}}{I}_{\mathrm{i}}+C_3{E}_{\mathrm{c}}{I}_{\mathrm{c}}, where the coefficient {\mathrm{C}}_3 is taken as C_3=0.45+3\left({\displaystyle \frac{A_{\mathrm{o}}+A_{\mathrm{i}}}{A_{\mathrm{o}}+A_{\mathrm{i}}+A_{\mathrm{c}}}}\right) 0.9.

5.3 Comparisons and discussions

Subsections 5.1 and 5.2 demonstrate the current design formulas from AS 5100 [41] and AISC 360 [42]. It should be noted that AS 5100 [41] considers the confinement effect on infilled concrete, as shown in Eq. (6), whereas AISC 360 [42] uses a relatively more conservative formula and considers only the superposition of each element of the CFDAT. In addition, the calculation of the effective stiffness, {\left(EI\right)}_{\mathrm{e}} differs among the design codes. AISC 360 [42] introduced a stiffness reduction coefficient of less than one (C_3) for infilled concrete, whereas AS 5100 [41] ignored the reduction in the stiffness of the concrete material.

Fig.27 illustrates the comparison of Group A between the simulated ultimate strength of CFDAT and the predicted values obtained from AS 5100 [41] and AISC 360 [42]. For comparison, the dimensionless parameter P_u/P_s was plotted against the L_e/r ratio, which was calculated by dividing the ultimate strength (P_u) by the squash load (P_s). As shown in Fig.27, AS 5100 [41] has a similar trend to the simulated results curve, and the P_u/P_s–L_e/r curve obtained from AS 5100 [41] is almost parallel to that of the simulated results, whereas AISC 360 [42] is in good agreement with the simulated results but only when the value of L_e/r is greater than 60. However, AS 5100 [41] overestimated the ultimate strength of the CFDATs in Group A by an average of 14%, particularly for slender CFDATs. AISC 360 [42] deployed a conservative design strength for stub CFDATs because confinement effects were not considered.

The ratios of the calculated and simulated results (P_AS5100/P_NUM and P_AISC360/P_NUM) are listed in Tab.5. AISC 360 [42] can be utilized to design intermediate and slender CFDAT as the average value of P_AISC360/P_NUM is 0.99 and the standard deviation is 0.12. Although the average P_AISC360/P_NUM ratio is approximately one, AISC 360 [42] tends to provide unsafe results for intermediate and slender CFDATs and can overestimate the ultimate strength by up to 28%, as shown in Fig.28. Among the total of 64 simulated CFDATs, more than 40% of such columns have the P_AISC360/P_NUM ratios beyond the range of 0.90 and 1.10. In other words, the standard deviation of the P_AISC360/P_NUM ratio was too high, and AISC 360 [42] could not provide a reliable design method for all CFDATs.

The value of P_AS5100/P_NUM was 1.13, and the standard deviation was 0.06, indicating that AS 5100 [41] overestimated the average ultimate strength of the CFDAT by 13%. Unlike AISC 360 [42], AS 5100 [41] provides a stable design method for CFDATs, and the discrepancy in the P_AS5100/P_NUM ratios between the various CFDATs is relatively limited, as shown in Fig.29. As both design codes were initially proposed for CFST instead of CFDAT, AS 5100 [41] failed to safely predict the ultimate strength of CFDATs. This is due to the differences between the steel and aluminum tubes, such as the influence of the Young’s modulus and confinement effects. It should be noted that the Young’s modulus of aluminum is approximately one-third that of carbon steel.

None of the design codes can reasonably predict the ultimate strength of CFDATs. However, AS 5100 [41] provided a similar trend to P_u/P_s–L_e/r curve of the CFDATs (Fig.27). In addition, a design method in accordance with AS 5100 [41] is more stable for CFDATs because the standard deviation is acceptable. Consequently, design equations for CFDATs have been proposed based on AS 5100 [41], and the coefficients of the confinement effect of the CFDAT require modification. The slenderness reduction factor according to AS 5100 [41] should be applied to the modified design equations for the CFDATs. Hassanein et al. [43] drew similar conclusions for slender concrete-filled stainless steel columns.

6 Conclusions

This study presents a brief review of the current experimental and numerical studies on concrete-filled alloy tubular stub and slender columns, especially columns with carbon steel and aluminum alloy tubes. The study also pointed to a research gap, i.e., there is no numerical or experimental study to investigate the behavior of slender circular CFDAT, although the experimental data on stub CFDAT is very limited. Further research can be conducted using the modeling technique proposed in this study.

The current numerical modeling technique for concrete-filled stub and slender alloy columns using the commercial software Abaqus is outlined and discussed. The modeling parameters of the material properties of the concrete and alloy, geometric initial imperfection, elements, boundary and loading conditions, and contact algorithm are presented in Section 2. The discussed techniques are summarized as a numerical method for axially loaded concrete-filled composite columns.

1) For the CFST and CFDST, the ultimate strengths simulated using the confined and unconfined concrete models were similar. However, the confinement effect significantly affected the ductility of the columns at the post-peak branch. The simulations using unconfined concrete tended to underestimate the ultimate capacities of CFAT and CFDAT. The confined concrete model is more versatile and can be adopted for modeling.

2) Imperfection coefficients of L/2000 and L/10000 were suggested for CFST with L_e/r values less than 100 and more than 100, respectively. An initial imperfection with an amplitude coefficient of L/1000 should be used for the intermediate and slender CFDST.

3) A series of verification tests of concrete-filled columns with carbon steel and aluminum were conducted using the modeling technique discussed in Section 2. The ultimate load, relationship between loading and axial strain, and failure pattern were obtained from numerical studies using the proposed modeling technique and corresponding experimental studies. After the experimental and numerical results were compared, it was determined that the numerical results could accurately and reliably predict the behavior of concrete-filled columns, and the axial behavior of the slender circular CFDAT could be simulated using the proposed numerical technique.

4) A parametric study of CFDAT was carried out to investigate the effects of L_e/r, D_o/t_o, D_i/t_i and σ_0.2o/f_{\mathrm{c}}^{\prime } ratios on the ultimate axial capacity. L_e/r ratio has significant effects on the axial strength of CFDAT, but the previous method for calculating the critical slenderness ratio of CFDST is not suitable for CFDAT. D_o/t_o ratios can influence the axial capacities of intermediate and slender CFDAT. For an intermediate CFDAT, D_i/t_i can lead to flexural failure instead of material failure. σ_0.2o/f_{\mathrm{c}}^{\prime } ratio also can influence the ultimate capacity of CFDAT greatly by changing f_{\mathrm{c}}^{\prime }, while changing σ_0.2o has very limited effects on slender CFDAT.

5) With the same dimensions and concrete, CFDAT and CFDST were compared to investigate their behaviors at various L_e/r and STW ratios. CFDST tended to have a better axial capacity for the slender columns, whereas CFDAT had higher STW ratios for the intermediate columns.

6) Current codes were used to predict the axial strength of the intermediate and slender CFDAT. AISC 360 can be used to design the axial strength of CFDAT, but it tends to be unstable. However, AS 5100 was unsafe and overestimated it by approximately 13%. Therefore, new design equations for the CFDAT should be proposed based on AS 5100 in further studies.

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