Comprehensive experimental database and analysis of circular concrete-filled double-skin tube stub columns: A review

Abstract

A concrete-filled double-skin tube (CFDST) is a new type of composite material. Experimental studies have been conducted to investigate the axial compression behavior of CFDST members for approximately 30 years. This paper provides a review of the status of axial compression bearing capacity tests conducted on circular CFDST stub columns as well as a summary of test data for 165 circular CFDST stub columns reported in 22 papers. A relatively complete high-quality test database is established. Based on this database, the main factors affecting the axial compression bearing capacity of the CFDST stub columns are analyzed. The prediction accuracy and robustness of an existing theoretical prediction model, which is a data-driven model, are evaluated, and a numerical simulation of the axial compression bearing capacity of the CFDST stub columns is conducted. In addition, the differences between the basic theory and experimental results of various models are compared, and the possible sources of prediction errors are analyzed. The current model for predicting the axial compression capacity of CFDST stub columns cannot simultaneously satisfy the requirements of high accuracy and confidence, and the stress independency assumption introduced in the test is not valid. The main error source in the theoretical prediction model is the non-simultaneous consideration of the effects of the void ratio and inner steel tube.

Graphical abstract

Keywords

CFDST / bearing capacity model / confined concrete model / test database

1 Introduction

A concrete-filled double-skin steel tube (CFDST) is a type of composite member composed of outer and inner steel tubes with concrete filling between them (see Fig.1). In 1995, Wei et al. [1] employed a CFDST to solve the local and global instabilities of hollow steel tubes in offshore engineering structures and discovered that the sandwich concrete in the CFDST was in a three-dimensional (3D) compressive state because of the lateral expansion restraint by the outer and inner steel tubes. The bearing capacity of this member is 10%–30% higher than the sum of the bearing capacities of the individual components. Henceforth, the axial compression mechanism of CFDST members has been investigated extensively via experiments, numerical simulations, and theoretical analyses. Fig.2 illustrates the research framework.

Fig.1 shows the typical sections of CFDST members. Although various cross-sections and materials can satisfy various engineering requirements, higher requirements are imposed on the prediction model of the axial compression bearing capacity of CFDST members. Based on the results of axial compression experiments and numerical simulations conducted on CFDST stub columns under axial compression, several bearing capacity prediction models have been proposed. Ayough et al. [2] and Vernardos and Gantes [3] analyzed the mechanical behavior of CFDST members under uniaxial compression. Ayough et al. [2], Yan and Zhao [4], and Ipek and Guneyisi [5] summarized and compared existing formulae. However, the accuracy of the same model reported in different studies differed significantly. For example, Yan and Zhao [4] used the prediction model specified in American Specifications ACI 318-14 [6] to predict the bearing capacity of CFDST components under axial compression; the mean value (Mean) and coefficient of variation (COV) of the predicted result were 0.736 and 0.037, respectively. By contrast, the mean value and COV obtained by Ayough et al. [2] for the predicted result of circular CFDST members specified in ACI 318-14 [6] were 1.37 and 0.18, respectively, whereas the corresponding values for square CFDST members were 1.30 and 0.2, respectively. In a study by Ipek and Guneyisi [5], the mean values and COV of the predicted results for circular CFDSTs were 1.182 and 0.163, respectively. Li et al. [7] conducted a comprehensive study based on the confinement effect of CFDSTs with different shapes and proposed a unified analytical model for axially loaded CFDST columns with different cross-sectional shapes. In the study by Li et al. [7], Mean and COV of the predicted result were 1.006 and 0.097, respectively, whereas Mean and COV of the predicted result were 1.000 and 0.094, respectively.

The confinement effect of CFDST members with other shapes on concrete is weaker than that of circular members [3]. Moreover, when the slenderness ratio increased, the failure mechanism of CFDST columns under axial compression became unstable gradually. The formulae in the existing standards and codes cannot yield accurate predictions for the slenderness ratio of CFDST columns [8]. This is because the ranges and sizes of the data sets used by the researchers are different [4,5,9], which results in different analysis results in the aforementioned studies. In addition, researchers in different countries used different standards to determine the material strength; however, they did not unify the units of the test data prior to performing an analysis [2,4,5] or had only unified them in general [5,9]. This renders it difficult to gain a sufficiently accurate understanding of the current status regarding the bearing capacity prediction model of CFDST members under axial compression based on the literature.

Owing to its simple mechanical behavior, a CFDST stub column with a circular section can be used as a reference for designing CFDST stub columns with other types of cross-sections. Moreover, an overall stability mode for short columns does not exist. Therefore, in this study, experimental results of circular CFDST short columns under axial compression reported over the past 30 years are obtained and organized, and a concrete strength database is calibrated. The calibrated database cannot only be used to analyze and evaluate existing design models, but can also provide convenient and reliable data for subsequent studies.

Ipek and Guneyisi [5] and Tran and Kim [9] attempted to develop data-driven models (which are described in detail in Subsection 4.2) to predict the axial bearing capacity of CFDST stub columns. The goodness-of-fit of these models is much higher than that of conventional theoretical models. In 2021, Yan et al. [10] reported that the axial bearing capacity of a CFDST stub column depended significantly on its outer steel tube. Yang et al. [11] discovered that the failure mode of a CFDST short column with a high hollow ratio differed significantly from that with a low hollow ratio member under axial compression. These novel and important studies are included in the database for analysis.

The main contributions of this study are as follows: (a) experimental and theoretical studies pertaining to CFDST members are reviewed and analyzed, and the overall formation of CFDST members is elaborated; (b) based on previous studies regarding CFDST members, a relatively complete database is established and organized accordingly to obtain a high-quality database; (c) the existing prediction models are evaluated using the established database, and an error analysis is conducted.

2 Experimental study on axial bearing capacity of concrete-filled double-skin tube

2.1 Experimental database

2.1.1 Test parameters

In this study, the axial compression test data of 165 CFDST short columns reported in 22 studies [1,4,11–30] conducted over the past 30 years were collected, sorted, and summarized into a relatively complete test data set of CFDST short columns under axial compression. Because CFDST stub columns with stainless steel tubes (CFDSSTs) do not differ significantly from CFDSTs in terms of their axial compression performance, the test data of 57 CFDSST stub columns are included in the data set and discussed in detail herein (Tab.1 [1,4,11–21,23–30]; the details of the database are summarized in Tables S1 and S2 in Electronic Supplementary Material).

Data from 222 real axial compression tests were obtained to analyze the correlation between variables. The Pearson correlation coefficient was used to reflect the statistics of the degree of linear correlation between variables, and the coefficient is calculated using the formula shown in Eq. (1).

(1)\rho \left(x,y\right)=\frac{COV\left(x,y\right)}{{\delta }_x{\delta }_y},

where x and y are variables that affect the bearing capacity of the CFDST; an COV(x,y) represents the covariance of x and y. The statistics shown in Tables S1 and S2 in Electronic Supplementary Material were substituted into Eq. (1), and the results are depicted in Fig.3(a).

To prevent overfitting caused by multicollinearity (such as specimen length (H) and outer diameter of outer steel tube (D_{\mathrm{o}}), and D_{\mathrm{o}} and outer diameter of inner steel tube (D_{\mathrm{i}})) during feature engineering, decorrelation processing must be performed on independent variables that present clear autocorrelation such that the results shown in Fig.3(b) can be obtained, where the scale on the right shows the different correlation coefficients based on different color shades. The correlation coefficient can only measure the linear correlation between variables, i.e., the higher the correlation coefficient, the higher is the degree of linear correlation between variables.

Independent variables with clear autocorrelations were correlated. Fig.3 shows the thermal diagram obtained. The greater the effects of the variables on the component performance, the higher is the variable correlation value and the darker is the color represented.

As shown in Fig.3, in CFDST axial compression tests, seven research variables are typically used and have been proven to significantly affect the axial bearing capacity. These seven variables are the outer diameter of the outer steel tube (D_{\mathrm{o}}), wall thickness of the outer steel tube (t_{\mathrm{o}}), yield strength of the outer steel tube f_{\mathrm{s}\mathrm{y}\mathrm{o}} (if the outer steel tube is a stainless steel tube, then the nominal yield stress ({\sigma }_{0.2,\mathrm{o}}) is considered), inner diameter of the sandwich concrete (D_{\mathrm{i}}), wall thickness of the inner steel tube (t_{\mathrm{i}}), yield strength of the inner steel tube (f_{\mathrm{s}\mathrm{y}\mathrm{i}}), and standard cylinder concrete compressive strength (f_{\mathrm{c}}^{\prime }).

The research variables selected for each study (Tab.1 [1,4,11–21,23–30]) were regarded as the basis of the component information in Tables S1 and S2 in Electronic Supplementary Material. In Tab.1 [1,4,11–21,23–30], “CS” and “SS” represent carbon steel and stainless steel, respectively. The results of some studies were not included in this data set because some parameter values were not available.

2.1.2 Data processing and data range

In the existing experimental research database (Tables S1 and S2 in Electronic Supplementary Material), the types of specimens (size) for measuring concrete strength are standard cylindrical specimens (150 mm × 300 mm) [4,13,27], nonstandard cylindrical specimens (100 mm × 200 mm) [12,19,25], standard cube specimens (150 mm × 150 mm) [28,29], nonstandard cube specimens (100 mm × 100 mm) [23], and other nonstandard specimens [22]. In this study, the strength was unified based on the compressive strength of a standard cylinder (150 mm × 300 mm) (f_{\mathrm{c}}^{\prime }). The conversion formulae are as follows [31]:

(2){\left(f_{\text{c}}^{\prime }\right)}_{150\times 300}=\left[{\left(f_{\text{cu}}\right)}_{150}-9.94\right]/1.01,

(3){\left(f_{\text{cu}}\right)}_{150}={\left(f_{\text{c}}^{\prime }\right)}_{100\times 200}+6.41,

(4){\left(f_{\text{cu}}\right)}_{150}=0.91{\left(f_{\text{cu}}\right)}_{100}+3.62.

As the concrete strength increases, the effects of the height-to-diameter ratio and shape of the specimen decrease [31,32]. Therefore, the shape effect coefficient for strength conversion from a nonstandard specimen to a standard cylinder is typically set as 1 [22].

Here, f_{\mathrm{s}\mathrm{y}} represents the yield strength of the carbon steel tube. A plastic strain of 0.2% ({\sigma }_{0.2}) represents the nominal yield strength of the stainless-steel tube. The void ratio of the interlayer is calculated as the original inner diameter of the concrete (D_{\mathrm{i}}). The ultimate axial bearing capacity (N_{\mathrm{u}}) is the peak load of the test when a descending section is present in the load–displacement curve, and the axial load at a compression deformation of 5% is set when no descending section is present in the load–displacement curve. Tab.2 presents the range of the components included in the data set.

2.2 Test results of CFDST stub columns

In the axial compression test on the CFDSTs, both ends of the member were subjected to axial loading with a plane quasi-static force. The test results included the axial load–displacement curve, a curve showing the relationship between the lateral and longitudinal strains of the outer tube, and the failure mode of the member.

2.2.1 Load–displacement curve

Three typical types of load–displacement curves exist under small-displacement loading, as shown in Fig.4. These curves are associated with the confinement stress of the member [33]. When the confinement is large, the curve is type III, and when the confinement is small, it is type I. In the middle and later stages of high-displacement loading, the load–displacement curve may appear wavy [3,30,34], which signifies that the CFDST stub column is a ductile member. This characteristic is particularly suitable for structures with high seismic requirements.

2.2.2 Load–strain curve

Owing to the varying deformations of different sections of a member, predicting the failure and deformation position on the inner steel tube is difficult [23], and the longitudinal and transverse strains of the steel tube may vary with the position of the measuring point [13]. Moreover, fixing strain gauges on the inner steel pipe is difficult, the tube wall is sunken inward, and the strain gauges can detach easily, thus resulting in test error. The variation in strain on the inner and outer steel tubes under axial loading can be summarized as follows: the strain change on the outer steel tube is generally continuous and smooth until the strain gauge fails or unloads. Meanwhile, the strain on the inner steel tube is similar to the strain of the outer steel tube in the elastic stage and may fluctuate or decrease after the elastic stage [23]. However, at the initial stage of loading, the longitudinal strain develops rapidly, and the transverse strain does not increase. During the middle and later stages of loading, the growth rate of the transverse strain increases gradually. The longitudinal strain on the inner and outer steel tubes is greater than the transverse strain, as shown in Fig.5. The variation in the longitudinal and transverse strains on the inner and outer steel tubes under axial loading shows that the confinement provided by the steel tube in the CDFST is passive confinement, which is not significant at the beginning but increases with the development of axial deformation, thus effectively constrains the deformation of the entire component before its final failure.

Although the longitudinal and transverse strains of the outer steel tubes may be similar in thick- and thin-walled CFDST members, those of the inner steel tubes may differ significantly [13]. The reported samples, albeit few, are sufficient to show that the sandwich concrete in a CFDST may be confined in a non-uniform passive manner, i.e., the confinement provided by the inner and outer steel tubes to the sandwich concrete may not be equal.

2.2.3 Failure mechanism

The failure modes of CFDST columns under axial compression can be categorized into two types: brittle and ductile. 1) Brittle failure: brittle failure has been clearly observed in some members investigated by Uenaka et al. [15] and others [12,16,19,24]. A typical feature is the evident shear failure and buckling of the inner steel tube [24]. Generally, members with brittle fracture exhibit a type I load–displacement curve [15], which decreases significantly after the peak value. In addition, brittle failure was observed in hollow concrete-filled steel tube (HCFST) short columns [18]. 2) Ductile failure: ductile failure has been reported in many studies, where the material and loading degree of members differ slightly [4,12,13,27–30]. The typical feature of ductile failure is the outward bulging of the external steel tube of the component at a later stage of loading, and the irregular buckling of the inner steel tube at the corresponding position [29]. When a sandwich concrete is composed of sea sand concrete and the outer steel tube is composed of stainless steel [30], one or more annular buckling phenomena may occur. When the displacement is small, the failure characteristics are not evident and the component expands; however, the inner steel tube may not buckle [27]. When ductile failure occurs, the load–displacement curve exhibits a type-I, II, or III shape with only a short decrease period after the peak value, as shown in Fig.4.

Notably, the failure mode of the component differed from that mentioned above [11,18] when the void ratio was high. Owing to the overall thin section of the member, a failure mode with local ring-shaped outward bulging and bending occurred. The inner and outer steel tubes of the CFDST column with a larger void ratio exhibited outward bulging [23], which is different from the circular buckling reported by Li et al. [30]. However, for the CFDST columns with a smaller void ratio, the inner steel tubes exhibited radial-symmetric and axial random inward buckling after yielding [24]. The load–displacement curve of the CFDST with a larger void ratio was a type I curve. In the experiments performed on the CFDST, the bearing capacity did not decrease significantly nor did the failure mode change due to local initial defects. Under long-term sustained loading and corrosion conditions, the bearing capacity of CFDST members may diminish owing to the decrease in the wall thickness of the outer carbon steel tube [25].

3 Theoretical study on bearing capacity of CFDST

3.1 Mechanical behavior analysis of concrete-filled double-skin tube

Fig.6 shows the internal force diagram of the CFDST cross-section under axial compression (where {\sigma }_{\mathrm{c}\text{θ}} is the total confining pressure supplied to the concrete, {\sigma }_{\mathrm{o}\text{θ}} is the circumferential stress in the outer tube and {\sigma }_{\mathrm{i}\mathrm{r}} is the circumferential stress in the inner tube). The interaction history of the CFDST members under axial compression observed in the tests [16,18,35] is as follows. In the initial loading stage, owing to the lower Poisson’s ratio of the concrete compared with that of the steel tube, no interaction occurred between the steel tube and concrete. As axial loading increased, longitudinal microcracks began to appear and then extended gradually in the sandwich concrete; the transverse deformation increased gradually and the steel tube wall was continuously extruded. Within a certain range, the inner and outer steel tubes effectively restrained the development of transverse deformation. Consequently, the sandwich concrete was in a 3D compression state, which facilitated the improvement to its axial bearing capacity until the steel tube buckled or the concrete crack penetrated to form shear failure. This implies that the development path of the transverse confined stress in sandwich concrete is related to the material properties of both the concrete and restrained materials. However, because the magnitude of the axial load on sandwich concrete is not directly provided in existing experimental studies, the transverse deformation of concrete cannot be directly obtained, the number of specimens in the same control variable group is typically small, and the results of different studies cannot be compared because of the inconsistency in variable selection. Therefore, quantifying the transverse confined stress in sandwich concrete is challenging.

3.2 Two types of bearing capacity prediction models

Based on the experimental phenomena and data, two types of bearing capacity prediction models have been established: design- and analysis-oriented bearing capacity prediction models.

3.2.1 Design-oriented bearing capacity prediction model

Design-oriented research. This type of research typically establishes an accurate prediction model for the ultimate bearing capacity of CFDSTs, which can be directly used to guide engineering design by directly calibrating the mathematical function expression for predicting the axial bearing capacity and the experimental or numerical results of CFDSTs. For example, the European Code EN 1994-1-1 [36], Australian Standard AS 5100.6-2004 [37], American Specifications ACI 318-14 [6], and ANSI/AISC 360-16 [38], and those proposed by Uenaka et al. [15], Yan and Zhao [4], Han et al. [28], Hassanein and Kharoob [35], Hassanein et al. [39], and Yan et al. [10] involve the use a self-derived formula to calculate the axial compression bearing capacity of actively constrained concrete [19,20], which is then used in the shape function expression for the bearing capacity prediction of sandwich concrete. The ultimate bearing capacity model of the CFDST was established based on the results of axial compression tests. In addition, the data-driven models established by Tran and Kim [9], Ipek and Guneyisi [5], and Ipek et al. [40] should be included in the first type of research because a data-driven model directly establishes nonlinear relationships between independent variables (material and shape parameters of CFDST members) and dependent variables (axial compressive ultimate bearing capacity of CFDST) [21].

3.2.2 Analysis-oriented bearing capacity prediction model

Analysis-oriented research. Analysis-oriented research generally considers the reaction and interaction between concrete and steel tubes; therefore, the analysis-oriented model can potentially be used to predict members with similar stress mechanisms. An analysis of concrete response shows that the transverse confining pressure on the sandwich concrete in CFDSTs is similar to that on the core concrete in concrete-filled steel tubes (CFSTs) [24]. Therefore, in nonlinear CFDST studies [24–28], the typical theoretical models used for CFDST based on other confined concrete members should be included. For example, Tao et al. [41] corrected the damage plastic model and axial stress–strain model of passively confined concrete based on the axial compression test results of CFST members. Han et al. [42] established a model based on the results of the axial compression test of CFST members; the model was similar to the axial stress–strain model of passively confined concrete established by Popovics [8] involving an actively confined concrete model. The model established by Hu and Su [43] combined the linear Drucker–Prager yield criterion and passively constrained the concrete axial stress–strain model proposed by Saenz [44]. Li et al. [7] proposed a unified model of CFDST columns with different cross-sectional shapes.

3.3 Constitutive model of confined concrete

Currently, studies pertaining to the constitutive models of confined concrete can be categorized into two types based on the condition hypothesis: path-independent and path-dependent models.

3.3.1 Path-independent constitutive model of confined concrete

The path-independent model assumes that the peak stress of concrete is independent of the development process of the circumferential strain. For example, Tao et al. [41] established a constitutive model for path-independent confined concrete.

Tao et al. [41] proposed a new three-stage model to represent the strain-hardening or softening law of a CFST. In the initial phase, almost no interaction occurred between the steel tube and concrete. Therefore, the initial phase curve segment can be used to represent the ascending stage of the stress–strain relationship of the unconstrained concrete until the peak strength is reached f_{\mathrm{c}}^{\prime }. After initial phase, the peak strain of the concrete increased with confinement at the platform segment. At this stage, based on the interaction between the steel tube and concrete in the simulation, the concrete strength increased. After second stage, the softening stage of the increase in ductility owing to confinement was defined. This new strain-hardening or strain-softening function of concrete has been widely used to model CFST columns.

3.3.2 Path-dependent constitutive model of confined concrete

The second type of path dependence theory of passively confined concrete originated from an experimental study performed by Lim and Ozbakkaloglu [45] in 2015 on the axial compression bearing capacity of ordinary concrete and cylindrical specimens of high-strength concrete under unconstrained, actively restrained, and fibre reinforced plastics (FRP)-constrained conditions. The experimental results showed that the relationship between the transverse and axial strains of actively confined and FRP-confined concrete depended on the instantaneous confined stress under the corresponding axial strain. Under a specified axial strain, transverse strain, and confined stress, the axial stress–strain relationship depends on the action path of the confined stress. Therefore, in different confined systems, owing to the difference in the stress paths, the axial stress–strain relationship of passively confined concrete may differ [45]. In other words, the conventional path-independent theoretical model of confined concrete can be regarded as a constitutive relationship under a specific path, and the specific constitutive relationship will not be applicable if the path changes.

Yan and Zhao [4] investigated the interaction between concrete and steel tubes and concluded that the sandwich concrete in a CFDST exhibited passive confinement under the applied pressure, which increased with the axial load, and that the stress path of the passive confinement affected the ultimate bearing capacity of the CFDST under axial compression. Lim and Ozbakkaloglu [45], Lai et al. [46], Lin and Tsai [21], Lin et al. [47], Yang et al. [11], and Yang and Feng [48] conducted similar studies, in which a reference value was specified for path-dependent models of CFST members or FRP-confined concrete to analyze the interaction between concrete and steel tubes as well as to establish path-related ultimate bearing capacity prediction models for CFDSTs under axial compression.

In 2021, Yan et al. [10] proposed an index, i.e., SI, to evaluate the difference between a passively confined stress path (CSP) and an actively CSP based on the development process of lateral stress in the inner and outer steel tubes via axial compression tests performed on 28 CFDST specimens. The calculation formula is as follows:

(5)SI\left(P_{\mathrm{i}}\right)=S\left(P_{\mathrm{i}}\right)/S\left(P_{\mathrm{i},\mathrm{u}\mathrm{p}}\right).

In Eq. (5), S(P_i) represents the area of the passively confined path. S(P_i,up) represents the area of the actively confined path. Yan et al. [10] provides a comprehensive graphical representation for each parameter, offering detailed descriptions. Two different CSPs appeared in the CFDST column owing to the different confinement modes of the external and internal tubes [10]. The {\sigma }_{\mathrm{o}\mathrm{r}}-f_{\mathrm{c}\mathrm{z}}/f_{\mathrm{c}}^{\prime } (f_{\mathrm{c}\mathrm{z}} represents the axial concrete compressive stress) curve represents the CSP yielded by the external tube, which is denoted as the “external CSP” herein. Similarly, the {\sigma }_{\mathrm{i}\mathrm{r}}-f_{\mathrm{c}\mathrm{z}}/f_{\mathrm{c}}^{\prime } curve represents the CSP yielded by the internal tube, which is denoted as the “internal CSP” herein. The properties of the external CSP were similar to those of previously reported CSPs of confined concrete in CFST columns [10]. However, the change in the internal CSP was not significant and primarily fluctuated near the horizontal zero axis, unlike the behavior exhibited by the external CSP.

Although the method above is intuitive and effective, it does not include all the data from existing CFDST axial compression tests, including the measured lateral strain of the inner and outer steel tubes and the material parameters (such as the Poisson’s ratio and elastic modulus) required to calculate SI. Therefore, the lateral stress path P_i of a member cannot be determined easily. In addition, a constraint effect index η is proposed, which allows the stress path to be considered when calculating the axial compression bearing capacity by fitting the relationship function between η and SI. In fact, the axial compression bearing capacity of CFDST short columns proposed by Yan et al. [10] considers the factors affecting δ (i.e., the void ratio \chi and the confinement of the outer steel tube to concrete).

Yan et al. [10] discovered that CFDST columns generally exert a weaker confinement effect on concrete than CFST columns, in contrast to the results of axial compression tests conducted on CFSTs by Zhao et al. [49] and Lin et al. [47]. The larger the constraint effect coefficient η, the greater is the difference between the CFDST and CFST. The transverse stress in the inner steel tube transforms from tension to compression, and when the stress in the outer steel tube is high, the stress on the inner steel tube remains low. This indicates that the inner steel tube does not effectively confine the sandwich concrete; therefore, using a thick-walled inner steel tube in the CFDST member is likely unnecessary.

3.4 Numerical simulation accuracy

Although the path-independent hypothesis model yielded favorable verification results when the test components were few, it yielded unsatisfactory verification results on larger and broader data sets. In particular, the path-independent assumption disregards the confinement stress of passively confined concrete, which is due to the progressive development of microcracks and the non-consideration of concrete damage prior to the peak stress. Consequently, in all the studies mentioned above, the brittle failure mode of the component was not effectively simulated, and the verification effect on components with brittle failure [15] was inferior to that on other components.

In this study, the constitutive models of concrete used by previous researchers were organized, as listed in Tab.3 [1,5,11–17,20,21,23,27–30,33–35,39,44,50–56]. The same researcher used the same constrained concrete constitutive model to perform finite element analysis on different verification sets, and different results were obtained. For example, Li et al. [53] used the constrained concrete constitutive model proposed by Han et al. [42] to predict the bearing capacity of members tested in previous studies [1,15,17,20,21], and the results indicated different accuracies. In addition, the results of different constitutive models used in the same validation set indicated significant differences, such as the prediction of the validation set [17,20] by Pagoulatou et al. [54] and Li et al. [23]. The model proposed by Li et al. [23] demonstrated higher accuracy and less dispersion. These errors occurred because the model used was not considered completely.

Generally, a model with a path-independent assumption cannot account for the effects of confined materials and the concrete void ratio on the passively constrained loading path; consequently, the finite element calculation result may deviate significantly from the actual value [2].

4 Comparative analysis of bearing capacity prediction models for CFDST

4.1 Analysis of prediction model

To verify the effectiveness of the models listed in Table S3, the test results obtained from a database (Tables S1 and S2) were used in this study for a comparative analysis. The database included 165 CFDST members (Table S1). Because the axial compression performance of CFDSST members does not differ significantly from that of CFDSTs, 57 CFDSST components were included (see Table S2). The prediction accuracy was compared with those of standard formulae and prediction models, and a summary is provided in Fig.7. Here, the ordinate represents the ratio of the predicted limit load (N_{\mathrm{u},\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}) to the measured limit load (N_{\mathrm{u}}) of the sample, i.e., N_{\mathrm{u},\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}/N_{\mathrm{u}} the abscissa represents the amount of data; and n is the verification set for the CFDST. The extreme values of N_{\mathrm{u}} can be obtained from Fig.7.

The calculation model listed in Table S3 was used to predict the bearing capacity of the CFDST specimens in the data set. The test results were compared with the bearing capacity prediction results. Tab.4 [4,6,10,15,28,35–39] summarizes the calculation accuracy of each calculation model for the test data in the data set and factors that were not considered.

An analysis of the existing models shown in Fig.7 and Tab.4 [4,6,10,15,28,35–39] show that the prediction model proposed by Yan et al. [10] yields high prediction accuracy for both CFDST and CFDSST components. To understand the accuracy of this model for various components more effectively, an appropriate calculation formula was used to summarize the previous database separately (Tables S1 and S2). Fig.8 shows the deviation between the predicted limit load (N_{\mathrm{u},\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}) and tested ultimate capacity (N_{\mathrm{u}}) of the sample. Fig.8 shows the mean and standard deviation of the value (N_{\mathrm{u},\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}/N_{\mathrm{u}}) in each prediction model. The results are summarized in Tab.5.

A comprehensive comparative analysis is presented in Fig.8 and Tab.5. The prediction model formula used by Yan et al. [10] for the axial compressive bearing capacity is more suitable for CFDST short columns than for CFDSST short columns. Moreover, for the CFDST short columns, the mean was 0.982, and the dispersion was low. Therefore, the formula used by Yan et al. [10] provides a more accurate prediction of the axial compressive bearing capacity of short CFDST columns.

4.2 Data-driven model

Several experimental tests and theoretical studies have been conducted to analyze the axial compression behavior of CFDSTs. However, a reliable and widely applicable high-precision model could not be inferred owing to the numerous factors affecting the axial compression performance of CFDSTs and the complexity of the nonlinear functional form of the ultimate bearing capacity expression.

Tran and Kim [9] applied a multivariate adaptive regression spline (MARS), artificial neural network (ANN), and adaptive neural fuzzy inference system (ANFIS) to determine the shape function of the ultimate bearing capacity expression for a CFDST. In this method, the computer adaptively learns from the data to obtain the implicit nonlinear relationship between the input and output variables; subsequently, it outputs the same visualization coefficient via regression analysis. This type of model does not consider the theoretical relationship between variables; instead, it uses a significant amount of data for fitting and determining the nonlinear and complex relationship between variables, which is known as the data-driven model.

Tab.6 [6,15,28,35–38] presents the calculation accuracy of a data-driven model established by Tran and Kim [9], where R² represents the goodness-of-fit and a20 is a typical engineering index that represents the ratio of the amount of data with a prediction accuracy within 20% of the total amount of data. Based on Tab.6 [6,15,28,35–38], for the same data set, the data-driven model exhibits significantly higher prediction accuracy and stability than the ordinary theoretical calculation formula. The main advantage of data-driven models is their ability in capturing nonlinear and complex interactions between system variables. These interactions between input and output variables can yield explicit mathematical expressions, which can effectively address complex engineering problems.

Although the prediction accuracy of these models varies with the data set size, proportion of training sets, and algorithms used, data-driven models are generally more effective than conventional calculation formulae, and the fitting accuracy of some of their algorithms approaches 1. Among them, ANNs and improved neural networks exhibit stable and good performance on each data set. However, many types of data-driven models exist, and the performance of each machine-learning model may be affected by several factors. In this study, some models utilized in the axial compression bearing capacity prediction of CFSTs in the past 3 years were summarized and compared (see Tab.7 [5,9,40,57–61]), where the goodness-of-fit R² was used as the evaluation index.

Moreover, the method to develop data-driven models is not limited to these models. Although the data-driven models reported thus far exhibit high accuracy, their interpretability is low, which is attributable to two reasons. First, because of the complexity of machine learning, particularly of the neural network involved, the nonlinear relationship provided by machine learning is high dimensional and cannot be directly observed. Therefore, the effect of overfitting on the prediction results is not negligible. Second, as mentioned above, a data-driven model exhibits high data dependency, i.e., it relies on only high-quality data or a sufficiently large database to reduce the effects of abnormal data on the model. However, a detailed description of the data sets used previously has not been provided, including the basis for selecting the input variables, the unity of the input variable units, and the data screening standard. As such, the prediction accuracy of the standard formula listed in Tab.6 [6,15,28,35–38] is significantly lower than that of the standard formula applied in this study (Tab.4 [4,6,10,15,28,35–39]). Furthermore, other researchers have focused on data-driven models for establishing nonlinear equations, which renders the models more interpretable [62]. However, these models have not been applied to the axial bearing capacity prediction of CFDSTs.

4.3 Disadvantages of existing bearing capacity prediction models

Many ultimate bearing capacity prediction models for CFDSTs have been established; however, the existing models exhibit the following disadvantages.

a) In the existing path-independent design-oriented theoretical prediction model [4,11–17], a single model presents a large prediction error for some test data, and the prediction accuracy of all models on the overall data set is not ideal (i.e., either the prediction mean deviation is significant or the prediction accuracy is discrete).

b) The prediction accuracy of the existing path-dependent design-oriented prediction model [18] is not significantly better than that of the path-independent model, and the parameters used to consider the effect of the path fail to achieve the expected goal.

c) Although the existing data-driven models [21–23] can achieve a high level of prediction accuracy, their interpretability and confidence are unsatisfactory. The unit calibration of the data sets used in previous studies is not reported, and the data sets used are relatively small; hence, the reliability of the models used cannot be confirmed [54].

d) The existing prediction models (including numerical simulation models [24–28]) cannot provide a reasonable explanation for some test phenomena. In cases involving different concrete grades, increasing the wall thickness of the outer steel tube in the same manner cannot result in the same bearing capacity improvement to the members [18]. At a certain threshold, improving the concrete grade cannot further increase the axial bearing capacity of CFDSTs [44]. A similar phenomenon has been reported in axial compression tests performed on CFST members [50]. In the axial compression test of a CFST with a square tube, increasing the concrete grade reduced the axial bearing capacity of the entire member [54].

5 Proposed prediction model

5.1 Base model

The results shown in Tab.4 [4,6,10,15,28,35–39] and Fig. 12 indicate that the prediction model of Yan et al. [10] is favorable. Therefore, in this study, the model was optimized based on the prediction model proposed by Yan et al. [10]. Additionally, Yan et al. [10] used the following formula to predict the axial bearing capacity of CFDST:

(6)N_{\mathrm{u}}=0.94f_{\mathrm{s}\mathrm{y}\mathrm{o}}{A}_{\mathrm{o}}+f_{\mathrm{c}\mathrm{c}}{A}_{\mathrm{c}}+f_{\mathrm{s}\mathrm{y}\mathrm{i}}{A}_{\mathrm{i}},

where A_{\mathrm{o}} is the cross-sectional area of the outer tube, A_{\mathrm{c}} the cross-sectional area of the concrete, A_{\mathrm{i}} the cross-sectional area of the inner tube, and f_{\mathrm{c}\mathrm{c}} the compressive strength of confined concrete. The details of these parameters are summarized in Table S3. Based on the table, the method for calculating f_{\mathrm{c}\mathrm{c}} is complicated. To optimize its calculation, Yang and Feng [63] summarized the existing failure surface models of confined concrete and re-expressed them in a generalized form as follows:

(7)\frac{f_{\mathrm{c}\mathrm{c}}^{\prime }}{f_{\mathrm{c}}^{\prime }}=1+A_1{\left(\frac{{\sigma }_{\mathrm{l}}}{f_{\mathrm{c}}^{\prime }}\right)}^{A_2}{\left(f_{\mathrm{c}}^{\prime }\right)}^{A_3}+A_4\sqrt{1+A_5\frac{{\sigma }_{\mathrm{l}}}{f_{\mathrm{c}}^{\prime }}}-A_4,

where A_1\mathrm{a}\mathrm{n}\mathrm{d}{A}_5 are the parameters, respectively, {\sigma }_{\mathrm{l}} is the confining pressure of concrete, and f_{\mathrm{c}}^{\prime } is calculated using Eq. (2).

Generally, in a design-oriented model, the confining pressure of concrete ({\sigma }_{\mathrm{l}}), as expressed in Eq. (7), is calculated by considering the maximum transverse tensile stress yielded by the external restraint material. In the same type of members, because the stress distribution is similar in the stress development path, parameters A_1, A_2, and A_3 are constants.

By referring to Refs. [10,48,63,64] for the values of A_1 and A_5, the following parameters were set in this study: A_4 = A_5 = 0, A_2 = B_1, and 1 + A_3 – A_2 = B_2. Two parameters, i.e., B_3 and B_4=A_1, were added. Equation (7) can be reduced to Eq. (8) as the base model. To enable the use of different constraint materials and concrete cracking stress differences, f_{\mathrm{c}}^{\prime } is multiplied with B_3, as shown in the first term. Parameters B_1 and B_4 simultaneously quantify the effect of the development path of the transverse restraint stress and the inhomogeneity of the restraint on the axial bearing capacity of concrete.

(8)f_{\mathrm{c}\mathrm{c}}^{\prime }=B_3{f}_{\mathrm{c}}^{\prime }+B_4{\left({\sigma }_{\mathrm{l}}\right)}^{B_1}{\left(f_{\mathrm{c}}^{\prime }\right)}^{B_2}.

For thin-walled CFDST members, the stress variation along the wall-thickness direction in the steel tube is negligible. Therefore, when the external steel tube yields, the lateral constraint stress on the outer side of the sandwich concrete is expressed as follows [2]:

(9)p_{\mathrm{o}}=\frac{2f_{\mathrm{s}\mathrm{y}\mathrm{o}}{t}_{\mathrm{o}}}{D_{\mathrm{o}}}.

When the CFDST member does not satisfy the thin-walled assumption, the failure of the sandwich concrete is controlled via the lateral displacement, as mentioned above. In other words, when the outer steel tube enters the plastic limit state completely, it can no longer restrain the transverse deformation of concrete. In this case, the lateral restraint stress on the outer side of the sandwich concrete can be modified as follows:

(10)p_{\mathrm{o}}=\frac{2f_{\mathrm{s}\mathrm{y}\mathrm{o}}}{\sqrt{3}}\ln \frac{D_{\mathrm{o}}}{D_{\mathrm{o}}-2t_{\mathrm{o}}}.

In this study, Eq. (10) was used to calculate the maximum lateral restraint stress on the outer side of concrete.

When the void ratio is high or the inner steel tube is thin, the axial bearing capacity of the CFDST members decreases significantly, although the maximum transverse restraint stress provided by the outer steel tube remains unchanged [12,22].

To compensate for the error caused by disregarding the restraint of the inner steel tube in this study, the transverse restraint effect of the inner steel tube on the sandwich concrete was converted to a certain concrete thickness, and the equivalent void ratio was recalculated. The calculation formula is as follows:

(11)t_{\mathrm{c}}^{\prime }=\frac{E_{\mathrm{i}}}{E_{\mathrm{c}}}{t}_{\mathrm{i}}\le \frac{D_{\mathrm{i}}}{2},

(12)0\le {\chi }^{\prime }=\frac{D_{\mathrm{i}}-2t_{\mathrm{c}}^{\prime }}{D_{\mathrm{o}}-2t_{\mathrm{o}}}⩽D_{\mathrm{i}}.

If the value of E_{\mathrm{i}} is not reported for conventional carbon steel tubes, then it is set as 200000 MPa, according to Tao et al. [41]. If the value of E_{\mathrm{c}} is not reported for concrete, then it is set as E_{\mathrm{c}}=4730\sqrt{f_{\mathrm{c}}^{\prime }}, based on ACI 318-14 [6].

In the elastic stage, the cross-sectional stresses of the sandwich concrete (as shown in Fig.6) can be expressed as follows:

(13){\sigma }_{\mathrm{c}\mathrm{r}}=-\frac{{\displaystyle \frac{{\left(D_{\mathrm{o}}-2t_{\mathrm{o}}\right)}^2}{4r^2}}-1}{{\displaystyle \frac{{\left(D_{\mathrm{o}}-2t_{\mathrm{o}}\right)}^2}{D_{\mathrm{i}}^2}}-1}{\sigma }_{\mathrm{i}\mathrm{r}}-\frac{1-{\displaystyle \frac{D_{\mathrm{i}}^2}{4r^2}}}{1-{\displaystyle \frac{D_{\mathrm{i}}^2}{{\left(D_{\mathrm{o}}-2t_{\mathrm{o}}\right)}^2}}}{\sigma }_{\mathrm{o}\mathrm{r}},

(14){\sigma }_{\text{cθ}}=\frac{{\displaystyle \frac{{\left(D_{\mathrm{o}}-2t_{\mathrm{o}}\right)}^2}{r^2}}+1}{{\displaystyle \frac{{\left(D_{\mathrm{o}}-2t_{\mathrm{o}}\right)}^2}{D_{\mathrm{i}}^2}}-1}{\sigma }_{\mathrm{i}\mathrm{r}}-\frac{1+{\displaystyle \frac{D_{\mathrm{i}}^2}{4r^2}}}{1-{\displaystyle \frac{D_{\mathrm{i}}^2}{{\left(D_{\mathrm{o}}-2t_{\mathrm{o}}\right)}^2}}}{\sigma }_{\mathrm{o}\mathrm{r}}.

In an experimental study conducted by Jiang et al. [65], the prediction accuracy error of the ultimate strength of non-uniform passively confined concrete under triaxial loading was less than 5% when the weak axis of the uniform passive confinement {\sigma }_{\mathrm{c}\mathrm{r}} < {\sigma }_{\mathrm{c}\text{θ}}. Therefore, in this study, the transverse restraint stress of sandwich concrete was simplified as a uniform stress field, which was measured using the following index:

(15){\sigma }_{\mathrm{c}\overline{\mathrm{r}}}=\frac{{\displaystyle {\int }_{\frac{D_{\mathrm{i}}}{2}}^{\frac{D_{\mathrm{o}}-2t_{\mathrm{o}}}{2}}\text{π}r{\sigma }_{\mathrm{c}\mathrm{r}}\mathrm{d}r}}{\text{π}{\left({\displaystyle \frac{D_{\mathrm{o}}-2t_{\mathrm{o}}}{2}}\right)}^2-\text{π}{\left({\displaystyle \frac{D_{\mathrm{i}}}{2}}\right)}^2},

where {\sigma }_{\mathrm{c}\overline{\mathrm{r}}} represents the equivalent transverse restraint stress of the sandwich concrete; however, it cannot reflect the development path of the stress. An experimental study pertaining to the failure strength conducted by Jiang et al. [65] similarly demonstrated that a path depends more on deformation than on stress. In other words, the ultimate strength of concrete is controlled by its deformation. Damage accumulation during the development of transverse restraint stress reduces the ultimate axial bearing capacity of the sandwich concrete before it reaches the peak load.

When the CFDST member is damaged, the lateral stress of the outer steel tube reaches the peak {\sigma }_{\mathrm{r}\mathrm{o}} = p_{\mathrm{o}} [46]. Setting {\sigma }_{\mathrm{r}\mathrm{i}} = 0 and {\sigma }_{\mathrm{r}\mathrm{o}} = p_{\mathrm{o}}, Eqs. (11) and (12) are substituted into Eq. (13), which yields

(16){\sigma }_{\mathrm{c}\mathrm{r}}=-\frac{1-{\displaystyle \frac{{\left(D_{\mathrm{i}}-2t_{\mathrm{c}}^{\prime }\right)}^2}{4r^2}}}{1-{\displaystyle \frac{{\left(D_{\mathrm{i}}-2t_{\mathrm{c}}^{\prime }\right)}^2}{{\left(D_{\mathrm{o}}-2t_{\mathrm{o}}\right)}^2}}}{p}_{\mathrm{o}}.

The results of Eq. (15) are as follows:

(17){\sigma }_{\mathrm{c}\overline{\mathrm{r}}}=\left[\frac{1}{1-{\chi }^{\prime 2}}+\frac{-{\chi }^{\prime 2}\ln \chi }{\left(1-{\chi }^{\prime 2}\right)\left(1-{\chi }^2\right)}\right]p_{\mathrm{o}}.

The effect of the void ratio on the stress path is represented by the parameter \alpha as follows:

(18)\alpha =\frac{{\sigma }_{\mathrm{o}\mathrm{r}}}{{\sigma }_{\mathrm{c}\overline{\mathrm{r}}}}={\left[\frac{1}{1-{\chi }^{\prime 2}}+\frac{-{\chi }^{\prime 2}\ln \chi }{\left(1-{\chi }^{\prime 2}\right)\left(1-{\chi }^2\right)}\right]}^{-1}.

Solving Eq. (18) is extremely complicated, considering the 3D relationship among the parameter α, void ratio \chi, and equivalent void ratio {\chi }^{\prime }, as shown in Fig.9(a). No clear functional relationship was indicated between \alpha and \chi. However, the relationship between α and {\chi }^{\prime } shows an almost straight line, and a function with a similar shape is used to replace Eq. (14), which yielded a certain calculation accuracy, as shown in Fig.9(b). Hence, the parameter α can be simplified as follows:

(19)\alpha =1.0-{\chi }^{\prime }.

Furthermore, the non-uniform transverse restraint stress exerting on the annular concrete is calculated as follows:

(20){\sigma }_{\mathrm{l}}=\alpha p_{\mathrm{o}}.

Lai et al. [46] investigated the axial stress–strain relationship between a confined system and confined concrete and showed that as the restraint level and concrete strength increased, the difference between passively and actively confined concrete increased.

Therefore, a constraint coefficient associated with the constrained material is proposed in this study, namely \beta, to evaluate the effects of constrained materials on the stress path. The calculation formula is as follows:

(21)\beta =\frac{E_{\mathrm{o}}}{E_{\mathrm{c}}}.

According to Lai et al. [46], the parameters of the base function were B_1 = 0.65, B_2 = \beta /20, B_3 = 0.94, and B_4 = 1.05. Notably, parameter \beta cannot completely decouple the effects of constrained materials on the stress path because the transverse strain–axial strain relationship of concrete depends on the instantaneous constraint stress [46].

In conclusion, the bearing capacity model of concrete under non-uniform constraints considering the stress path can be written as follows:

(22)f_{\mathrm{c}\mathrm{c}}^{\prime }=0.94f_{\mathrm{c}}^{\prime }+1.05{\left(\alpha p_{\mathrm{o}}\right)}^{0.65}{\left(f_{\mathrm{c}}^{\prime }\right)}^{\beta /20}.

In CFDST stub columns, the sandwich concrete not only extrudes the steel tube outward, but also delays the local buckling of the steel tube, thus resulting in a more complex stress state. In previous studies, the axial bearing capacity of steel pipes was generally considered to be the product of their area and strength. Meanwhile, Yan et al. [10] used the following the axial bearing capacity prediction formula for CFDST members in their study:

(23)P_{\mathrm{u}}=0.94A_{\mathrm{o}}{f}_{\mathrm{s}\mathrm{y}\mathrm{o}}+A_{\mathrm{c}}{f}_{\mathrm{c}\mathrm{c}}^{\prime }+A_{\mathrm{i}}{f}_{\mathrm{s}\mathrm{y}\mathrm{i}}.

5.2 Model accuracy

The data shown in Table S1 were substituted into Eq. (23) to obtain the result shown in Fig.10(a). For the CFDST data set, 100% of the data were within the error range of [−20%,20%], 97.6% were within the error range of [−15%,15%], and 86.1% were within the error range of [−10%,10%]. Fig.10(b) shows the mean, variance, and extreme values of the ratio of the predicted value to the actual value for each data set. The results show that the proposed prediction model offers high prediction accuracy for the CFDST.

6 Conclusions

The axial bearing capacity of CFDSTs is investigated less compared with that of CFSTs, and CFDSTs under axial compression exhibit more complex bearing mechanisms than CFSTs. Both the experimental and calculation model results showed “fragmentation” as well as better local interpretation and accuracy than those of the entire member. The development of emerging theories and technologies has resulted in new methods and possibilities for investigating CFDSTs. This paper summarizes the existing studies in the following five aspects.

1) Experimental study of CFDSTs

During the design of a test specimen, compared with an actual engineering application, the test piece generally features a smaller diameter. The wall thickness distribution of the outer steel tube is uniform and continuous, and the selected outer steel tube is primarily composed of ordinary steel. The hollow rate distribution is continuous and uniform, and a higher hollow rate is typically selected. A thicker inner steel tube wall is generally preferred, particularly for thin inner steel tubes with high hollow ratios. The strength distribution of the inner steel tube is relatively uniform, and the concrete strength distribution is uniform. The main research variables are the outer steel tube and hollow ratio.

Notably, regardless of the void ratio, researchers generally prefer an inner steel tube with a thicker wall, in which the bearing capacity of some specimens changes slightly because of the void ratio. However, in studies pertaining to hollow CFSTs without an inner steel tube, the strength of the component decreased as the hollow ratio increased gradually. In a study by Miyaki et al. [22], the bearing capacity of a CFST reduced by only 9.3% for a hollow ratio of 18%, whereas it reduced by 47% for a hollow ratio of 55%. Therefore, merely increasing the wall thickness of the inner steel tube to counteract the effect of the hollow ratio on the overall member may reduce the economic efficiency of CFDST members. The current experimental research cannot completely reveal the effects of the hollow ratio and inner steel tube on the overall bearing capacity of CFDST members. In addition, the failure mode of CFDSTs with a large aperture differs significantly from that of CFDSTs with a smaller pore diameter; however, experimental research on this type of member remain scarce.

2) Investigation into CFDST theory

In numerical simulation studies using the conventional path-independent constitutive model of concrete, most researchers validated the finite element method used, and the number and size range of the actual test specimens were relatively small. In a wider range of verification, although the finite element method is generally more accurate than the existing formulae, some deviations are indicated. When the finite element method is used in a parameter study, the application scope of the constitutive model of concrete materials must be prioritized to avoid error accumulation.

Emerging path-dependency studies impose higher requirements on CFDST experimental investigations. Currently, path dependency studies pertaining to CFDSTs is still in the initial stage, and the relevant studies and experimental data are scarce. However, this is consistent with the actual situation theory, and the calculation formula based on this theory indicates a higher accuracy than other path-independent formulae.

3) Non-data-driven axial bearing capacity calculation formula for CFDSTs

The existing formula for calculating the bearing capacity of CFDSTs is generally one-half theoretical and one-half empirical or purely empirical, and verification results based on a test data set typically show large deviations or variances. Synergistic effects affecting the overall bearing capacity of components has not been considered. The formula provided by Yan et al. [10] considers the effect of path dependency and the verification effect is decent; however, the calculation process is complex.

4) Calculation model for load-bearing capacity of CFDSTs

Currently, the models developed to calculate the load-bearing capacity of CFDSTs offer excellent prediction accuracy. However, a few issues are encountered: First, the algorithm codes of these models as well as the data sets used in some studies are not publically available, which renders it difficult to reproduce and verify the models. Second, data-driven models are data dependent. Results of the final model may deviate significantly from the actual situation because of the uneven, extremely small, or low-quality data sets used, which cannot be directly reflected by various loss indicators. Because the current algorithm model cannot identify or assess abnormal data, the current data-driven models must be comprehensively verified experimentally and theoretically. Furthermore, establishing a high-quality large-scale database will significantly benefit the development of data-driven models as well as investigations into the axial compression capacity of CFDSTs.

5) Optimization of load-bearing capacity calculation model of CFDSTs

The axial bearing capacity of CFDST stub columns was modified, and a simple, high-precision (Mean = 0.989), and low-dispersion (SD = 0.069) theoretical calculation model of the axial compression bearing capacity of CFDST stub columns was established. The model exhibited excellent performance in calculating the axial bearing capacity of CFDST members. The prediction errors for 100% and 97.5% of the data were ˂ 20% and ˂ 15%, respectively.

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