Nonlinear analysis of square steel‐reinforced concrete‐filled steel tubular short columns considering local buckling

This paper presents a fiber element analysis model that simulates the structural responses of square steel‐reinforced concrete‐filled steel tubular (SRCFST) short columns under concentric compression including local buckling effects. The method of effective widths is utilized to model the gradual postlocal buckling of the steel tube walls of a SRCFST column loaded axially to failure. A new confinement model is developed for the concrete based on test results, considering the confinement induced by the embedded steel section. This confinement model is incorporated into the fiber model, and its accuracy is verified by experimental results. The accuracy of various confinement models proposed for concrete‐filled steel tubular (CFST) square columns in predicting the performance of SRCFST columns is evaluated. A parametric study is performed to investigate the performance of SRCFST columns with various parameters. The applicability of the design formulas specified in current standards for CFST columns to the design of SRCFST columns is examined. A formula is proposed to predict the strength of SRCFST short columns. The developed inelastic simulation model, confinement model, and design formula are found to yield performance predictions of SRCFST columns with good accuracy.


| INTRODUCTION
Steel-reinforced concrete (SRC) columns are utilized as primary load-carrying members in tall structures and bridge piers because they can withstand significant loads. 1 In a SRC column, the steel section is encased by concrete, thus providing resistance to corrosion in an acidic environment.Despite their numerous benefits, previous studies have indicated that high-strength SRC columns exhibit poor ductility.2][3] In addition, the construction process of high-strength SRC columns and beam-column connections, which requires the use of formwork, is time-consuming and challenging. 4Moreover, the formwork must be meticulously constructed and maintained to ensure that the concrete is properly placed and cured, which significantly increases construction costs.
Another type of filled composite column is the concrete-filled steel tubular (CFST) column.The CFST column has higher strength and ductility than the equivalent concrete column due to the concrete confinement provided by the steel tube.In the construction of CFST columns, the steel tube acts as permanent formwork.By eliminating the need for additional formwork, CFST columns can be constructed more efficiently and costeffectively compared to other types of composite columns.The CFST columns can have various crosssectional shapes, such as circular, square, or polygonal.It has been found that the circular section offers the most effective confinement to the concrete, while the square or rectangular shape is easier to connect to the adjacent beams. 5hu et al. 6 studied the behavior of square steelreinforced concrete-filled steel tubular (SRCFST) column, which combines the structural and construction advantages of both CFST and SRC columns.A SRCFST column consists of an outer square steel box, an internal steel H section or double H section, as illustrated in Figure 1.By offering extra reinforcement and confinement, this design improves the ductility and strength performance of the column.
The study by Zhu et al. 6 involved testing short square SRCFST columns with high-strength concrete under axial compression.The studied parameters were the ratio of width-to-thickness B=t ð Þ of the outer steel box, slenderness ratio, concrete strength, and the reinforcement ratio ρ s ð Þof the embedded steel section.The primary failure modes of the columns were identified as local buckling developed in the outer tube and the crushing of the internal concrete.The experiments exhibited that the ultimate strength of the columns improved with an increase in the concrete strength and ρ s ratio.Wang et al. 7 conducted a numerical study utilizing a finite element (FE) model to determine the responses of SRCFST columns loaded eccentrically.Their findings revealed an inverse relationship between the loading eccentricity or slenderness ratio and the strength of the column.Ding et al. 4 presented a FE model for SRCFST columns where the behavior of concrete was simulated by the material model of concrete suggested for CFST columns.However, neither Wang et al. 7 nor Ding et al. 4 included the effect of localized instability of the outer steel box in their modeling.More recently, Wang et al. 8 undertook experiments on highstrength SRCFST columns to explore the effects of the ρ s ratio, yield strength, and thickness of the square box on the behavior of SRCFST columns.It was concluded that using thicker steel tube and higher ρ s ratio improved the capacity of SRCFST columns.
The literature review shows that only limited research works on SRCFST columns have been undertaken.In addition, existing numerical models for SRCFST columns do not consider the confinement induced by the embedded steel section and the unique local buckling of the outer tube.To address these limitations, a new confinement model is proposed for concrete confined by the steel box and internal steel section and incorporated into the fiber analysis model of SRCFST columns.The proposed computational model accounting for the interaction local buckling of steel tube walls is validated by experimental results.A parametric study is conducted to explore the effects of key parameters on the structural behavior of SRCFST columns.Finally, the results are used to propose a design model for computing the ultimate strength of these columns.

| Fiber analysis
A model of fiber elements is formulated for the inelastic analysis of axially loaded SRCFST short columns.The column cross-section is meshed with fine fiber elements using a meshing technique developed by Persson and Strang, 9 as presented in Figure 2. The fibers of concrete and steel are given appropriate material properties to characterize their behavior.A perfect longitudinal bond is considered between the concrete and steel fibers to ensure that both materials experience the same axial strain.To predict the axial load-strain response, a stepby-step incremental analysis is proposed and described herein.The axial strain is incrementally increased by a small value, such as 0.0001, and the corresponding axial stress in each fiber is calculated by using the stress-strain laws.The axial force in each fiber is obtained by multiplying its stress by its cross-sectional area.The axial force in all the fibers is then summed to calculate the applied axial load at that specific axial strain.The numerical analysis continues until the predefined maximum strain ε cu ð Þ is exceeded or the axial load drops below 0:2P ult .The axial strain-load curve is plotted.
The axial load P ð Þ of the SRCFST column is determined using Equation (1).

| Stress-strain laws for steels
The stress-strain responses of structural steel are illustrated in Figure 3, taking into account the effect of biaxial stresses on the steel box of the SRCFST column owing to the confinement and residual stresses locked in the steel sections.It is assumed that these effects reduce the steel yield stress by 10%. 5,13The typical yield plateau observed for the mild structural steel is simulated using the tri-linear stress-strain diagram. 14For the cold-formed steel, the stress-strain curve is curved and for the highstrength steel, the curved part of the cold-formed steel is replaced with a straight line. 14The curved portion of the curve is predicted by utilizing the equation suggested by Liang, 14 which is expressed as: The expressions proposed by Mander 15 are used to compute the stresses of steel fibers beyond the hardening strain ε st until the maximum strain ε su , which is specified as 0.2.

| Modeling of local buckling of square steel box
7][8] It should be noted that ignoring the influence of localized buckling in the numerical analysis of SRCFST columns results in an overestimation of their structural performance.To address this issue, this study uses expressions proposed by Liang et al. 16 for the steel tube walls of a rectangular CFST column to simulate the local and postlocal buckling of the steel tube in SRCFST columns.During numerical calculations, the steel fiber stress is continuously monitored to identify any potential local buckling in comparison to the initial local buckling stress.Liang et al. 16 proposed formula for calculating the initial local buckling stress of the steel tube walls in a rectangular CFST column subjected to uniform compressive stresses as follows: in which and t is the thickness of the steel tube, f syo and σ cr are the yield stress and initial local buckling stress of the steel tube wall, respectively.The stress redistribution method is used to simulate the progressive postlocal buckling behavior of steel plates under the increasing load.In the simulation, the stresses are redistributed from the heavily buckled region to the unloaded edge strips of the steel tube.The effective width concept is utilized to simulate the progressive postlocal buckling of the outer tube as shown in Figure 4. Liang et al. 16 developed an expression for the effective width of the steel tube walls, which is written as in which b and b e are the clear width and the effective width of the steel tube wall, respectively.The ineffective width b ne can be calculated by linear interpolation depending on the stress level of the steel fibers using the following expression: The maximum ineffective width b ne, max of the steel tube wall is determined as In the numerical modeling, the stresses of the steel fibers are first calculated by using the stress-strain relationships of steel and checked for possible local buckling.When the b=t ratio of the steel tube wall is >30 and steel stress is greater than the critical local buckling stress, the ineffective width of the steel tube is computed.The stresses of fiber in the ineffective region are taken as zero until the maximum b ne, max is attained.

| Concrete material modeling
The concrete model depicted in Figure 5 is utilized in this study to determine the ascending portion of the curve expressed as.
Equation ( 8) was suggested by Mander et al. 17 The concrete modulus of elasticity (E c ) is computed using Equation (10). 18 In Equation ( 10), γ c is the reduction factor, which considers the column size effect, was suggested by Liang 14 as Equation ( 11) is utilized to determine the concrete compressive strength (f 0 cc ) in SRCFST columns. 17F I G U R E 5 Stress-strain constitutive curves for concrete with and without confinement.
The lateral pressure induced by the confinement factor (ξ) of SRCFST columns is calculated using Equation (12).
Based on the experimental works of Zhu et al., 6 Wang et al., 7 and Wang et al., 8 a new expression for calculating the confining pressure (f rp ) is proposed as To derive Equation ( 13), the section capacities of steel tube and embedded steel section of the SRCFST column were first subtracted from the ultimate axial load of the column.Equation ( 7) was then used to evaluate f rp,test for each of the tested columns.Finally, the confinement factor (ξ) of each tested column was plotted against the measured f rp,test and a linear equation was proposed based on the statistical analysis, as demonstrated in Figure 6.
The compressive strain (ε 0 cc ) of confined concrete is estimated by means of employing the expression recommended by Wang et al. 19 as The descending part of the curve is determined by Equation ( 15) was proposed by Lim and Ozbakkaloglu 20 where ε ci stands for the strain at the inflection point as presented in Figure 5, equals to 0.01; f cr represents the residual strength of concrete computed as After analyzing the experimental data reported by Zhu et al. 6 and Xiong et al., 21 β c is proposed as 3 | VALIDATIONS OF NUMERICAL MODEL

| Comparison with experimental results
The predicted ultimate strengths of SRCFST columns by the numerical model are compared with test results in Table 1 to validate its accuracy.It should be noted that the strength of the concrete cube stated in the experiments has been converted to the cylindrical strength by multiplying the cube strength by 0.85. 22It can be observed from Table 1 that the agreement between experimental data and numerical results is excellent.The average value of the predicted and experimental ultimate strength is 0.99.The computed axial load-strain diagrams of SRCFST columns are compared with experimental data in Figure 7.It appears that the developed numerical model, which incorporates the newly proposed confinement model, accurately computes the experimentally measured axial strain-load relationships of SRCFST columns.

| ASSESSMENTS OF EXISTING CONFINEMENT MODELS
This section assesses the reliability of existing confinement models in predicting the structural responses of short SRCFST columns.Table 2 presents the confinement models proposed by Lai and Varma, 13 Hu et al., 23 Thai et al., 24 and Tao et al. 25 These confinement models have been implemented in the fiber-based modeling program for SRCST columns.The experimental results and computed ultimate strengths using various confinement models are compared in Table 3.It is shown that these confinement models mentioned above result in an underestimation of the column ultimate loads.This is because these confinement models ignore the effect of confinement induced by the embedded sections on the core concrete.The confinement models proposed by Lai and Varma, 13 Hu et al., 23 Thai et al., 24 and Tao et al. 25 result in an average predicted-to-experimental ultimate strength ratio of 0.84, 0.88, 0.88, and 0.90, respectively.Figure 8 compares the predicted axial load-strain diagrams of SRCFST columns with test results.It appears that there are significant discrepancies between the experiments and predictions when using existing confinement models.However, the proposed concrete confinement model in this study provides better predictions than the existing models, as shown in Table 1 and Figure 8.

| BEHAVIOR OF SRCFST COLUMNS
Various parameters that can affect the structural responses of SRCFST columns were examined using this proposed fiber model.A reference column was considered with an external square tube having a 600-mm width and a 10-mm thickness, embedded steel section area of 3757 mm 2 , concrete strength of 70 MPa, and an external and embedded steel yield stress of 350 MPa.Five different groups (G1-G5) of SRCFST columns were studied to investigate the effects of the widthto-thickness B=t ratio, steel reinforcement ratio of the embedded steel section, concrete strength, yield strength of the outer steel box, and the yield stress of the embedded steel section.Details of the reference and analyzed columns are provided in Table 4.

| Influence of B=t ratio
The fiber-based computer model was used to quantify how the width-to-thickness B=t ratio affects the performance of SRCFST columns.The B=t ratio was calculated by altering the thickness of the tube for the same width.10 shows that the ductility index decreases with increasing the B=t ratio.If the B=t ratio is increased, the local buckling of the outer tube may take place.Therefore, increasing this ratio leads to a reduction in the ductility and strength of the columns.

| Influence of embedded steel reinforcement ratio
The influence of the ρ s ratio of the embedded steel section on the responses of SRCFST columns was assessed by varying the ratio from 1.04% to 2.09%, 3.12%, and 4.18%.As depicted in Figure 11, it is apparent that the ρ s ratio has a minor effect on the performance of the columns, especially on the ductility.However, it is discovered that increasing ρ s ratio from 1.04% to 4.18% results in a 9.3% increase in the ultimate strength of the columns.Figure 12 illustrates that the ductility index slightly increases from 2.68 to 2.98 when increasing the ρ s ratio from 1.04% to 4.18%.The additional steel reinforcement enhances the overall strength of the column by providing additional steel area and confinement to the concrete, thus preventing premature failure.The reinforcement also increases the ductility of the column by enabling it to undergo more significant deformation before failing.

| Influence of concrete strength
The computational simulation was performed to assess the effects of concrete compressive strength on the performance of SRCFST columns.The study involved varying the concrete strength from 30 to 90 MPa. Figure 13 compares the axial load-strain relationships as a function of concrete strength.It is shown that increasing the concrete strength remarkably improves the column's resistance but considerably decreases its ductility due to the brittle behavior of high-strength concrete.It is noted from Figure 14 that the ultimate strength of the columns improves by 61% when changing the concrete strength from 30 to 90 MPa.The column with a concrete strength of 30 MPa has a ductility index of 3.55, while the column with a concrete strength of 90 MPa has a ductility index of 2.49.

| Effects of yield stress
The response of SRCFST columns was studied by varying the yield strength of the external tube and embedded steel section from 250 to 550 MPa.The diagrams of axial load-strain as presented in Figure 15 reveal that the steel yield stress of the external tube has a greater influence on the performance of SRCFST columns than that of the embedded steel section.This is because the cross-sectional area of the external tube is larger than that of the internal steel section.Changing the yield stress of the steel box from 250 to 550 MPa results in a substantial increase of 20.6% in the ultimate load of the SRCFST column.On the other hand, when the yield stress of the embedded steel section is raised from 250 to 550 MPa, the ultimate load of the SRCFST column increases only 3.3%.
T A B L E 2 Existing concrete confinement models for concrete-filled steel tubular (CFST) columns.

Confinement model
Formulas for calculating the compressive concrete strength (f 0 cc ) Formulas for calculating (β c ) Lai and Varma 13 The postpeak behavior of concrete is similar to the unconfined concrete model Tao et al. 25 f

| Influence of concrete confinement
The importance of concrete confinement on the performance of SRCFST columns was studied.For this purpose, the reference column was analyzed by either considering the concrete confinement or ignoring it.The results obtained are presented in Figure 16.It appears that ignoring concrete confinement leads to an T A B L E 3 Ultimate strengths of steel-reinforced concrete-filled steel tubular (SRCFST) columns obtained by different confinement models.

Specimen
P u, exp (kN) Hu et al. 23 Thai et al. 24 Lai and Varma 13 Tao et al. 25 P u,num1 (kN)  underestimation of the ultimate strength of the short SRCFST column by 26.2%.This means that the concrete confinement has a significant contribution to the loadcarrying capacity of short SRCFST columns; therefore, it must be considered in the inelastic analysis and design of such filled composite columns.As shown in Figure 16, the concrete confinement has minor effect on the initial stiffness of the SRCFST column and considerable effect on the postpeak behavior and residual strength of the column.

| RELATIVE SIGNIFICANCE OF PARAMETERS
A sensitivity analysis was conducted to determine the relative significance of the parameters on the strength of SRCFST columns.A correlation matrix is a useful tool in this process as it can reveal which variable has the strongest correlation with the output.This statistical tool can F I G U R E 9 Effects of B=t ratios on P À ε relationships.provide valuable insights into the relations between various parameters and their impact on responses, as demonstrated by Abdel Aleem et al. 26 Figure 17 depicts the correlation plot for the parameters considered in this study.The correlation values range from À1.0 to 1.0, where a value of 1.0 represents a perfectly direct relationship, a zero value denotes no correlation, and a À 1 value indicates a perfect inverse relationship between the pairs of parameters under investigation.It appears that the B=t ratio and the steel ratio are negatively correlated, implying that these two parameters are interdependent.
The analysis indicates a very strong correlation between the ultimate axial load and concrete strength, which suggests that the concrete strength is a critical property in estimating the axial capacity of SRCFST columns.The axial capacity is moderately correlated with the outer tube's steel ratio and yield strength.However, the ultimate axial load is almost uncorrelated with the yield stress of the embedded steel section.This is consistent with the findings discussed in preceding sections.Moreover, the width-to-thickness ratio negatively correlates with the ultimate axial load, as the outer tube may experience local buckling for a high ratio.
The order of the relative significance of the parameters on the ultimate axial load of SRCFST columns is as follows: (1) concrete strengths f 0 c , (2) steel ratio ρ s , (3) yield stress of the outer tube f sy,T , (4) width-to-thickness ratio B=t ð Þ, and (5) yield stress of the embedded section f sy,s .

| Current design codes
In this section, the effectiveness of current standards, including AISC 360-16, 27 Eurocode 4, 28 ACI 318-11, 29 and 13-51-2010 30 in estimating the ultimate strength of axially loaded SRCFST columns is investigated.Table 5 presents the formulas specified in these standards for determining the ultimate axial strength of SRCFST columns.The accuracy of the design formulas is evaluated by comparisons of calculations with experimental data given in Table 1 and the numerical solutions of columns listed in Table 4.It can be seen from Table 6 that the current design codes specified for square CFST columns are inadequate in predicting the ultimate strengths of SRCFST columns.All design standards significantly underestimate the ultimate loads of SRCFST columns.Therefore, there is a need for developing an accurate design formula for SRCFST columns under axial compression.

| Proposed design formula
A design equation for estimating the ultimate strength of SRCFST columns is proposed herein as in which f rp is calculated using Equation (13).T A B L E 5 Design expressions for steel-reinforced concrete-filled steel tubular columns using the standards of concrete-filled steel tubular columns.

Standard
Evaluating the accuracy of design codes in predicting the maximum strengths of steel-reinforced concrete-filled steel tubular (SRCFST) short columns.To assess the accuracy of the proposed equation, the ultimate strengths of the SRCFST columns calculated by using Equation ( 17) are compared with test results in Table 6.It would appear that the proposed equation yields accurate strength predictions of SRCFST columns, providing a mean predicting-to-experimental strength ratio of 0.98 and a standard deviation of 0.03.

| CONCLUSIONS
This paper has presented a numerical model developed for computational simulation of the inelastic responses of axially loaded SRCFST columns recognizing the localized buckling of the external square steel box.The proposed inelastic simulation model considering concrete confinement accurately predicts the responses of SRCFST columns.The behavior of SRCFST columns is found to be affected by several parameters, including the ratio of width-to-thickness B=t ð Þ, steel ratio ρ s ð Þ, concrete strength, and yield strength of the steel sections.The suggested simplified design formula is demonstrated to give more accurate strength calculations of SRCFST short columns than existing design codes.Therefore, the study provides valuable insights into the nonlinear analysis, behavior, and design of SRCFST columns under axial loading.The proposed confinement model can be implemented in numerical analysis procedures for simulating the responses of SRCFST columns.reduction of concrete strength due to the columns size effect γ c ¼ 1:85 D À0:135 c ε 0:90 axial compressive strain at which the column's axial load reduces to 90% of its maximum load ε 0:75 Axial compressive strain at which the column's axial load reaches 75% of its maximum load ε c concrete strain ε 0 cc concrete compressive strain ε s strain of the steel fibers ε st hardening strain = 0.005 ε su steel tensile strain ε sy steel yield strain ε y yield strain determined as ε 0:75 =0:75 σ c concrete stress σ s,i longitudinal steel stress of element i σ c,j longitudinal concrete of element j σ s stress of the steel fibers (a) H shape (b) Double H shape F I G U R E 1 Cross-sections of steel-reinforced concrete-filled steel tubular columns.(a) H shape.(b) Double H shape.

F I G U R E 3
Stress-strain behavior of steel material.

2
Fiber element discretization of steel-reinforced concrete-filled steel tubular column sections.(a) H shape.(b) Double H shape.

F I G U R E 4
Effective width of the square tube of steelreinforced concrete-filled steel tubular column under axial loading.

7
Comparison of measured and predicted axial load-strain responses of steelreinforced concrete-filled steel tubular columns using the proposed confinement model.

8
Evaluating the accuracy of various concrete confinement models.
Effects of B/t ratios on ductility of steel-reinforced concrete-filled steel tubular columns.

4 1 2 3 F I G U R E 1 5
Effects of concrete strengths on ductility.Effects of ρ s ratio on P À ε relationships.Effects of ρ s ratio on ductility.Influences of concrete strengths on P À ε relationships.The impacts of steel yield stresses of (a) steel tube and (b) internal steel profile on the P À ε curves of SRCFST columns.

6
Confinement effect on the load-strain responses of steel-reinforced concrete-filled steel tubular columns.F I G U R 1 7 Correlation matrix of input parameters.
Validation of the proposed formula for concrete confinement f rp .

Table 4
Ultimate axial loads of steel-reinforced concrete-filled steel tubular (SRCFST) short columns.
and Figure 9 demonstrate that the axial capacity of the columns reduces significantly as the B=t T A B L E 1 Material and geometric properties of columns investigated in the parametric study.
T A B L E 4