Design Model of Rectangular Concrete-Filled Steel Tubular Stub Columns under Axial Compression

: This research collected and summarized a total of 455 experimental tests of axially loaded square and rectangular concrete-ﬁlled steel tubular (CFST) stub columns. The recently published papers were used to evaluate the current design equations from four international standards, namely the American Concrete Institute (ACI) code, British Standard (BS5400), Chinese standard (BDJ13-51), and Eurocode 4 (EC4). It was found that the results obtained from the codes have appreciable differences and could be improved, especially for the specimens fabricated using high-resistance materials. Therefore, new empirical equations were proposed based on the four standard formulas and the wide range of previously available experimental data to provide more accurate estimations. The proposed equations could predict an average sectional capacity of only 0.1% lower than the experimental results, with better data scattering than the existing equation’s results.


Introduction
Concrete-filled steel tubular (CFST) elements were employed in the construction industry in the early 1900s. Nevertheless, the research investigations in CFST elements did not begin until the early 1960s. After that, many research studies were conducted to understand the behavior of CFST elements. Concrete-filled steel tubular (CFST) structures are increasing in popularity in many engineering applications, including high-rise buildings, bridge structures, supporting platforms of offshore structures and piles. As a consequence of its composite effects, the disadvantages of the two materials can be compensated for, and their advantages in CFST can be combined to provide efficient structural systems. CFST elements have the advantage of high mechanical strength, increased fire resistance, appropriate durability, energy absorption during earthquakes, and no formwork is needed compared to other concrete-steel composite structural elements. The core concrete works to prevent the steel tube from buckling inward. Meanwhile, the confinement of core concrete by the steel tube works to strengthen the mechanical properties of concrete. Consequently, adopting CFST columns results in a relatively small cross-sectional area with considerable economic savings [1][2][3].
CFSTs with regular cross-sections, such as circular, square and rectangular shapes, have been widely investigated [4], yet some research has been conducted on other cross-sectional shapes, for example, Chang et al. [5] conducted an experimental and numerical study to investigate the static axial performance of stiffened concrete-filled double-skin steel tubular (CFDST) stub columns with circular and square shapes. In addition, Rahnavard et al. [6,7] conducted experimental and numerical analysis on four different cross-section shapes of 1.
The columns considered must be tested under static axial compression loads. 2.
The columns must have square or rectangular single-skin cross-sections. 3.
The concrete core and the steel tube must be loaded simultaneously, as illustrated in Figure 1. 4.
The column specimens were checked to ensure that they can be classified as short columns according to the EC4 in which L/b ≤ 4; where L is the specimen length and b is the section width [22]. 5.
Irrelevant CFST columns made of different materials such as stainless steel or aluminum were excluded. 6.
The columns should not have internal steel reinforcements, shear connectors, or any types of stiffeners, etc.
and square CFST stub columns were obtained from this database to be used in the study.
The following inclusion and exclusion criteria were used to set the boundaries for the database: 1-The columns considered must be tested under static axial compression loads. 2-The columns must have square or rectangular single-skin cross-sections. 3-The concrete core and the steel tube must be loaded simultaneously, as illustrated in Figure 1. 4-The column specimens were checked to ensure that they can be classified as short columns according to the EC4 in which L/b ≤ 4; where L is the specimen length and b is the section width [22]. 5-Irrelevant CFST columns made of different materials such as stainless steel or aluminum were excluded. 6-The columns should not have internal steel reinforcements, shear connectors, or any types of stiffeners, etc.
The details of the collected test specimens in this new database are shown in Table  A1, Appendix A Based on the wide range of specimens presented in the current database, Figure 2 displays a schematic view of the failure mechanism of rectangular CFST stub columns. It is well known that hollow steel columns exhibit both inward and outward buckling when subjected to axial loading, while as illustrated in Figure 2, only outward buckling occurs at the mid-height of concrete-filled steel tubular stub columns. This failure pattern is due to the support by concrete infill to the steel tube. Additionally, the presence of the steel tube around the concrete provides a confinement effect that delays the failure of the concrete core and makes it in a more ductile fashion. This interaction between the steel and concrete enhances the ultimate capacity of such columns compared to their counterparts of hollow steel tubes and reinforced concrete columns. The details of the collected test specimens in this new database are shown in Table A1, Appendix A.
Based on the wide range of specimens presented in the current database, Figure 2 displays a schematic view of the failure mechanism of rectangular CFST stub columns. It is well known that hollow steel columns exhibit both inward and outward buckling when subjected to axial loading, while as illustrated in Figure 2, only outward buckling occurs at the mid-height of concrete-filled steel tubular stub columns. This failure pattern is due to the support by concrete infill to the steel tube. Additionally, the presence of the steel tube around the concrete provides a confinement effect that delays the failure of the concrete core and makes it in a more ductile fashion. This interaction between the steel and concrete enhances the ultimate capacity of such columns compared to their counterparts of hollow steel tubes and reinforced concrete columns.
The range and distribution of the database contents concerning the geometric details, material strengths, and ultimate strengths are presented in Figure 3. It can be seen that the concrete strengths (f c ) of the database specimens range from 15 MPa to 120 MPa, and most of them have a value of f c between 30 and 50 MPa. Additionally, the yield strength of the steel tube (f y ) varies from 180 MPa to 840 MPa and mostly ranged between 180 and 300 MPa. The confinement factor [ξ = (A s f y )/(A c f' c )] is distributed between 0.2 and 10.5. Additionally, most of the included specimens have a steel ratio [ρ = A s /A c ] between 2% and 20%, while just a few specimens have a larger ratio. The column width-to-steel thickness ratio (b/t) ranged from 10 to 135. The ultimate section capacity (N u ) varied from 190 to 15,000 KN.  The range and distribution of the database contents concerning the geometric details, material strengths, and ultimate strengths are presented in Figure 3. It can be seen that the concrete strengths (f′c) of the database specimens range from 15 MPa to 120 MPa, and most of them have a value of f′c between 30 and 50 MPa. Additionally, the yield strength of the steel tube (fy) varies from 180 MPa to 840 MPa and mostly ranged between 180 and 300 MPa. The confinement factor [ξ = (As fy)/(Ac f'c)] is distributed between 0.2 and 10.5. Additionally, most of the included specimens have a steel ratio [ρ = As/Ac] between 2% and 20%, while just a few specimens have a larger ratio. The column width-to-steel thickness ratio (b/t) ranged from 10 to 135. The ultimate section capacity (Nu) varied from 190 to 15,000 KN.  The range and distribution of the database contents concerning the geometric details, material strengths, and ultimate strengths are presented in Figure 3. It can be seen that the concrete strengths (f′c) of the database specimens range from 15 MPa to 120 MPa, and most of them have a value of f′c between 30 and 50 MPa. Additionally, the yield strength of the steel tube (fy) varies from 180 MPa to 840 MPa and mostly ranged between 180 and 300 MPa. The confinement factor [ξ = (As fy)/(Ac f'c)] is distributed between 0.2 and 10.5. Additionally, most of the included specimens have a steel ratio [ρ = As/Ac] between 2% and 20%, while just a few specimens have a larger ratio. The column width-to-steel thickness ratio (b/t) ranged from 10 to 135. The ultimate section capacity (Nu) varied from 190 to 15,000 KN.  Finally, all the specimens were such that the width (b) was from 50 mm to 324 mm, the height was from 100 mm to 2400 mm, and the steel thickness was from 1.2 mm to 9.7 mm. The details of the columns are provided in Table A1, Appendix A.
The compressive strength of concrete can be defined using different test methods   Finally, all the specimens were such that the width (b) was from 50 mm to 324 mm, the height was from 100 mm to 2400 mm, and the steel thickness was from 1.2 mm to 9.7 mm. The details of the columns are provided in Table A1, Appendix A.
The compressive strength of concrete can be defined using different test methods depending on the standards used. Therefore, the Eurocode 2 standards [23] were used to convert between f' c and f cu [23], where f' c is the compressive strength of concrete obtained from 150 × 300 mm cylinder specimens, and f cu is the compressive strength of concrete obtained from a 150 mm cube.

Formulations of Current Design Codes
Different design codes provide different formulae for estimating and designing CFST stub columns under axial compression. Generally, the fundamental difference lies in the methods of determining the compressive strength of concrete. Four approaches from different international standards used in the United States, Japan, Europe, and China were compared to the results of the previously available data.

ACI Approach
The approach used to design the reinforced concrete sections has been adopted to calculate the axial capacity of CFST stub columns. American Concrete Institute ACI code [24] utilizes the same formulae for different cross-sections and does not consider the confinement effect.
The compressive capacity of CFST rectangular stub columns can be determined using Equation (1) according to the ACI code. A s is shown in Equation (1), the design load based on ACI (NACI) is the summation of the ultimate axial sectional capacities of the concrete core and steel tube.
where, A s is the cross-sectional area of RHS steel, Ac is the cross-sectional area of the concrete infill.
The same approach was also adopted by the Architectural Institute of Japan (AIJ) [25] guidelines to predict the CFST stub column strength for square and rectangular hollow sections.

EC4 Approach
Australian standard (AS5100) [26] and Eurocode 4 [22] use the same formula, see Equation (2), to determine the section compressive capacity of CFST RHS stub columns. Similar to the ACI code approach, the EC4 design load capacity (N EC4 ) is calculated by summing up the section capacities of the concrete core and steel tube. However, the long-term effects are not considered.

BS5400 Approach
Equation (3) is used by British standards (BS5400) [27] to predict the capacity of CFST short columns under axial compression. Equation (3) is similar to the EC4 formula. However, the British standard utilizes 15 × 15 × 15 cm concrete cubes to determine the compressive strength of concrete.

DBJ13-51 Approach
A different approach is recommended by the Chinese standard (DBJ13-51) [28] to predict the load-carrying capacity of CFST stub columns. The standard takes concrete confinement by the steel tube into account by introducing the confinement factor (ξ 0 ).
where f ck is the characteristic cube compressive strength of concrete, and Equation (6) was used as a conversion relationship between f ck and f cu [1].

Limitations of Current Design Codes
Despite the growing improvement in the properties of steel and concrete, international standards still limit the possibility of using very high-strength materials in CFST elements. Different limitations of concrete compressive strength were also suggested in the international standards. In both AS5110 and EC4, the characteristic compressive cylinder strength at 28 days (f c ) ranges between 25 to 65 MPa and between 20 to 60 MPa, respectively. In BS5400 and DBJ13-51, the characteristic cube compressive strength at 28 days (f cu ) should not be less than 20 and 30 MPa, respectively. While, as shown in Figure 3, the data of the existing experimental studies used in the proposed database covered a more comprehensive range of concrete strengths (f c ) from 15 MPa to 120 MPa (f cu ranges from 19 to 140).
This paper aims to participate in extending the limits allowed by the current design methods and suggests more comprehensive formulations.

Comparative Studies
In this section, the existing experimental results collected in the proposed database were compared with the predicted results using the previously mentioned international standards. This comparison was conducted to evaluate the applicability and accuracy of the standards. Figure 4 illustrates the relationship between the experimental and the predicted values of CFST short-column load-carrying capacity. Figure 4 also shows the mean, defined as the average ratio of the predicted capacity to the experimental strength, and the standard deviation (SD) of the ratio of the predicted capacity to the experimental strength. In this figure, the orange line shows the ideal results when the empirical results are identical to the experimental results. In other words, the further away the points are from this line means that they are more scattered and far from the correct results.
In general, and as expected, the predicted load-carrying capacities from the four codes have nearly the same trends compared to the experimental results. The BS5400 approach overestimates the capacity by 4% compared to the experimental data, while all other approaches underestimate the capacity. For instance, the DBJ13-51 approach predicts sectional capacities 2.8% lower than the experimental results, which have the most reliable mean value of predicted load-carrying capacity compared to the experimental results. However, the scatter of DBJ13-51 approach estimations is relatively high (SD = 0.142). The most faultless scatter was achieved by the ACI approach with standard deviations (SD) of 0.129, while the mean value was relatively higher at 13% lower than the experimental results. For the BS5400 and EC4 approaches, the standard deviation values of the predicted sectional capacities to the experimental results were 0.165 and 0.141, respectively. In light of these results, it can be seen that adjustments are required to make the standard predictions more accurate. These adjustments are discussed in the following sections. dicted values of CFST short-column load-carrying capacity. Figure 4 also shows the mean, defined as the average ratio of the predicted capacity to the experimental strength, and the standard deviation (SD) of the ratio of the predicted capacity to the experimental strength. In this figure, the orange line shows the ideal results when the empirical results are identical to the experimental results. In other words, the further away the points are from this line means that they are more scattered and far from the correct results.  In general, and as expected, the predicted load-carrying capacities from the four codes have nearly the same trends compared to the experimental results. The BS5400 approach overestimates the capacity by 4% compared to the experimental data, while all other approaches underestimate the capacity. For instance, the DBJ13-51 approach predicts sectional capacities 2.8% lower than the experimental results, which have the most reliable mean value of predicted load-carrying capacity compared to the experimental results. However, the scatter of DBJ13-51 approach estimations is relatively high (SD = 0.142). The most faultless scatter was achieved by the ACI approach with standard deviations (SD) of 0.129, while the mean value was relatively higher at 13% lower than the experimental results. For the BS5400 and EC4 approaches, the standard deviation values of the predicted sectional capacities to the experimental results were 0.165 and 0.141, respectively. In light of these results, it can be seen that adjustments are required to make the standard predictions more accurate. These adjustments are discussed in the following sections.

New Design Models
In this section, two proposed approaches will be introduced to improve the existing equations to predict the load-carrying capacity of the CFST stub columns. The existing formulae were modified using correction factors to improve the mean, standard deviations, and variation coefficient (COV) of the results. The new proposed formulae were determined through regression analysis based on the 455 experimental tests.

New Design Models
In this section, two proposed approaches will be introduced to improve the existing equations to predict the load-carrying capacity of the CFST stub columns. The existing formulae were modified using correction factors to improve the mean, standard deviations, and variation coefficient (COV) of the results. The new proposed formulae were determined through regression analysis based on the 455 experimental tests.

The First Proposal to Modify the Existing Design Formulas
The same approach adopted by Lu and Zhao [32] and Hanoon et al. [33] has been used in this study. In order to improve the available equations for predicting the load axial carrying capacity of square and rectangular CFST stub columns. Some conditions are considered: 1.
The equations must epitomize the experimental data as much as possible.

2.
The expressions have to be similar to the existing expressions provided by the standards. 3.
The formulae should be as simple as possible.
In the first design proposal, new correction coefficients (C1 and C2) were identified to develop the existing equations by improving the mean, standard deviations, and co-Buildings 2023, 13, 128 8 of 23 efficient of variation. These new correction factors were introduced because the original equations described in Section 3 were originally proposed for the conventional column design. Whereas, these equations do not take into account the additional cross-sectional capacity increase due to the confinement effect provided by the steel tube to the concrete core of the CFST stub columns.
A programming code using Microsoft Excel VBA (Visual Basic for Applications) programing language was developed and used to calculate the correction coefficients of the first proposed equations. Figures 5 and 6 illustrate the flowcharts of the algorithm used to design this programming code. As illustrated, the regression analysis approach was employed to correct the prediction for the stub columns based on the 455 experimental tests.
design. Whereas, these equations do not take into account the additional cross-sectional capacity increase due to the confinement effect provided by the steel tube to the concrete core of the CFST stub columns.
A programming code using Microsoft Excel VBA (Visual Basic for Applications) programing language was developed and used to calculate the correction coefficients of the first proposed equations. Figures 5 and 6 illustrate the flowcharts of the algorithm used to design this programming code. As illustrated, the regression analysis approach was employed to correct the prediction for the stub columns based on the 455 experimental tests.
For the ACI code formula, and as illustrated in Equations (7) and (8), only one coefficient (C1) was targeted to be obtained to adjust the concrete capacity in the formula. At the same time, the steel section capacity (C2) was not modified and was assumed to be 1, as proposed in the code (Equation (1)).
C2 (Steel capacity) + C1 (Concrete capacity) As fy + C1 Ac f'c ACI (8) Figure 5. ACI first proposal flowchart. For the BS5400 and EC4 formulae, two coefficients (C1 and C2) were targeted to be found to adjust the concrete and steel capacity in the formulae.
C1 As fy + C2 Ac fcu BS5400 C1 As fy + C2 Ac f'c EC4 (10) For the DBJ13-51 formula, due to the difference in the structure of the equations, two different coefficients (C1 and C2) have been targeted to adjust the formula, as shown in Equation (11).
For all codes, the initial (ii) and final (if) values of the concrete and steel coefficients (C1 and C2) were taken as 0.1 and 5, respectively, and the accuracy of C1 and C2 (Δ) was taken as 0.01. For the ACI code formula, and as illustrated in Equations (7) and (8), only one coefficient (C1) was targeted to be obtained to adjust the concrete capacity in the formula. At the same time, the steel section capacity (C2) was not modified and was assumed to be 1, as proposed in the code (Equation (1)).
C2 (Steel capacity) + C1 (Concrete capacity) A s f y + C1 A c f' c ACI (8) For the BS5400 and EC4 formulae, two coefficients (C1 and C2) were targeted to be found to adjust the concrete and steel capacity in the formulae.
C1 A s f y + C2 A c f ' c EC4 (10) For the DBJ13-51 formula, due to the difference in the structure of the equations, two different coefficients (C1 and C2) have been targeted to adjust the formula, as shown in Equation (11).
(A s + A c )·(C1 + C2·ξ 0 )· f ck (11) For all codes, the initial (i i ) and final (i f ) values of the concrete and steel coefficients (C1 and C2) were taken as 0.1 and 5, respectively, and the accuracy of C1 and C2 (∆) was taken as 0.01.
The first proposal of the modified formula for the four different codes can be expressed as: Comparative Studies As shown in Figure 7, the comparison between the first modified formula of the CFST load-carrying capacity to the experimental results shows a clear improvement in the performance of the four codes. The first proposal of the modified formula for the four different codes can be expressed as: As shown in Figure 7, the comparison between the first modified formula of the CFST load-carrying capacity to the experimental results shows a clear improvement in the performance of the four codes. The most precise mean value (0.999) was achieved using Equation (13) derived from the BS5400 code. In contrast, the less accurate mean was obtained through the modified ACI approach (Equation (12)) and resulted in the same standard deviations of 0.129 as the existing code formulae . The most precise mean value (0.999) was achieved using Equation (13) derived from the BS5400 code. In contrast, the less accurate mean was obtained through the modified ACI approach (Equation (12)) and resulted in the same standard deviations of 0.129 as the existing code formulae.
It can be observed in Figure 7 that all approaches provide a better prediction for small section capacities, while the scatter increases as the section capacities increase. Therefore, effort needs to be invested in improving the proposed formulae by considering the magnitude of the sectional capacity. That is, the direction followed in the second proposed formula.

The Second Proposal to Modify the Existing Design Formulas
In order to improve the scatter of the results for higher section capacities, the second proposed formulation divides the capacity equations into two groups. Similar to the first modification approach, regression analysis was utilized to propose new equations of the four design equations. However, the second modification for the Chinese Standard was divided based on the confinement factor (ξ).
The same VBA code from the first medication proposal was utilized to perform the second proposal after developing it to divide the data into two groups using an iterative method to get the perfectas COV value of all Standards. The load-carrying capacities from the second proposal can be expressed as follows: ACI Comparative Studies Table 1 and Figure 8 illustrate the comparisons between the second proposed equations and the experimental results. Overall, the second proposed formulations give the predictions of mean, standard deviation, and coefficient of variation compared to the existing and first proposed equations. The mean values of the second approach ranged between 0.940 and 0.999, while the standard deviation ranged between 0.129 and 0.142, as shown in Table 1. The scatter of the second proposed equations for the British and Chinese approaches clearly becomes smaller; meanwhile, the mean values of all approaches have significantly improved compared to the experimental results. The improvement in the second modification approach can be obviously observed in Figure 9. The improvement in the predictions of the second modification equations can be attributed to the fact that the existing code equations were proposed for sections with maximum steel and concrete strengths of 450 MPa and 60 MPa, respectively. However, the behavior of higher grades of such materials is relatively different.

Conclusions
The main objective of this paper is to establish a new comprehensive database consisting of 455 experimental tests of axially loaded square and rectangular concrete-filled steel tubular (CFST) stub columns. The database contains specimens with a concrete core of compressive strengths ranging from normal-to high-strength concrete and steel yield strength ranging from normal to high-strength steel, and with a wide variety of width-tothickness ratios, steel ratios, and confinement factors. The database was first used to assess the accuracy of four international codes for estimating the axial load capacity and then to suggest more accurate prediction approaches based on the regression analysis method of the collected data. The following conclusions made from this study can be drawn:

1.
Despite the considerable number of experimental tests on CFST stub columns, it was found that fewer tests were performed on columns made using high-strength materials. In addition, the tests on large-scale columns were very limited and the majority of these tests were focused on CFST stub columns with small cross-sections.

2.
All four codes provided a conservative prediction of CFST stub columns. The Chinese Standard (DBJ13-51) approach gave the most reliable mean value of the predicted load-carrying capacity to experimental results, while the best scatter was achieved by the ACI approach.

3.
Better predictions have been generally achieved using the first modified equations of the CFST load-carrying capacity than that of the original four codes. The modified formula based on the BS5400 approach gave the most precise results. However, it was found that the first proposed equations mainly provide accurate predictions for small sectional capacities, while the scattering of the results increases with the increase in section resistance.

4.
To improve the scatter of the results, other proposed equations were introduced. It was found that this second proposed approach can provide the most precise results compared to the existing and the first proposed formulas, especially the equation derived from the BS5400 code with a coefficient of variation (COV) of 0.1371. Funding: This research received no external funding.

Data Availability Statement:
The resulted data of this study are available on request from the corresponding author.

Conflicts of Interest:
The authors declare no conflict of interest.