Research on Carrying Capacity of T-Shaped Stiffened Concrete-Filled Steel Tubular Columns Subjected to Eccentrically Compressive Load

In this paper, in order to further study themechanical behavior of T-shaped stiffened concrete-filled steel tubular (CFST) columns, numerical programs of eccentrically compressive T-shaped stiffened CFST columns were developed to calculate moment M-φ curvature curves of cross section and column.)e calculated curves with the numerical programs agree well with the experimental results. A parametric analysis was carried out to calculate M-N correlation curves of cross section and column to investigate influence of concrete strength fck, steel yielding strength fy, steel tube thickness t, slenderness ratio λ, and loading angle θ. On the basis of parametric analysis, simplified resistance models of T-shaped CFSTsection and column were proposed and verified by the numerical analysis results. )e simplified resistance models are reliable to predict the mechanical behavior for engineering application.


Introduction
e columns in traditional frame structures, with extended corners to indoor space, normally have larger cross-sectional depths than those of adjacent infilled walls, leading to reduction of usable utilization area and disturbance to indoor environment. Recently, special-shaped columns, as an improved architectural approach, have been increasingly introduced into residential and official buildings. Smooth jointing of special-shaped column and adjacent infilled wall guarantees the efficiency of indoor space and availability to place furniture, which is shown in Figure 1.
Systematic research and extensive engineering practice are mainly on the reinforced concrete special-shaped column. An early study focused on the static behavior of T-shaped and L-shaped columns, subjected to axial compressive load and biaxial eccentrically compressive load. Numerical investigations of RC special-shaped columns were undertaken to computerize the analysis procedure for the purpose of providing design aids for structural engineers [1][2][3][4][5][6]. Many efficient and accurate methods for solving the bearing capacity were proposed, which promoted the development of special-shaped column structure [7,8]. To comply with the develop of housing industry in China, a further experimental and theoretical study was carried extensively concentrating on seismic behaviors for RC special-shaped columns in structural systems, directly for engineering promotions and applications [9,10]. However, with respect to applicable building height and seismic fortification intensity, traditional special-shaped reinforced concrete (RC) columns have strict limitations in seismic behavior.
In order to promote further generalization and application of special-shaped columns, a series of experimental studies on seismic behavior of T-shaped and L-shaped concrete-filled steel tubular (CFST) columns under constant axial compression and cyclic lateral load were carried out to investigate the effects of different parameters such as axial compression ratio, thickness of steel tube, concrete strength, inner stiffener, and loading scheme [11][12][13][14][15][16]. e deformation of steel tube at concave corner was almost completely restrained by the vertical stiffener, which acted as the composite dowel connector at the concave corner [17][18][19]. Besides, extensive researches focused on postponing buckling of steel tubes, and some stiffeners were employed in special-shaped CFST columns. Bar stiffeners were introduced in static and pseudostatic experiments to postpone the premature tube's local buckling and improve constraint effect for concrete [20][21][22][23]. Pulled binding bars were adopted at possible plastic hinge locations in the specialshaped CFST columns subjected to concentric and eccentric loads to study their static behavior [24,25]. Tensile bars were proposed in the T-shaped CFST columns to postpone local buckling of the tube and increase constraint effect for concrete [26,27]. e above experimental results showed that, due to the increase in constraint effect for concrete provided by stiffened steel tubes, special-shaped CFST columns behave advantages on the strength, ductility, and seismic behavior over special-shaped RC columns [28,29]. e research of T-shaped stiffened CFST columns mentioned above, nevertheless, is inadequate for engineering application, and systematic research should be complemented. In this paper, numerical programs of eccentrically compressive T-shaped stiffened CFST section and column are developed to analyze the influence of different parameters. Based on numerical analysis, simplified resistance models of eccentrically compressive T-shaped CFST section and column are proposed, respectively, to predict the mechanical behavior.

Numerical Programs of Eccentrically
Compressive T-Shaped CFST Section and Column. With regard to the eccentrically compressive column, fiber-based model is employed in the following to carry out numerical analysis of the T-shaped CFST cross section and column. Hypothesis of half sine wave was introduced to approximate deformation of the column subjected to eccentrically compressive load. e internal axial force and bending moment are calculated on the weakest cross section at midspan. An iteration process is needed in seeking equilibration of internal and external forces. Due to material nonlinearity, geometric nonlinearity, and cross-sectional nonsymmetry, seeking equilibration needs considering impact of torsional deformation.
e numerical analysis program adopts the following hypotheses: (1) during loading process, the cross section always remains plane, and no relative slip among steel tube, reinforcement, and concrete is taken into consideration; (2) in the analysis of simply supported member, the deformation of longitudinal axis is hypothesized as half sine wave; (3) take no account of the concrete tensile strength and consider concrete quitting work as soon as tensile strain emerges; (4) take no account of the shear deformation of columns.

Flow Chart of Numerical Program of T-Shaped CFST Section Subjected to Eccentrically Compressive Load.
e cross-sectional element division and position of vertical load are illustrated in Figure 2. e coordinate origin is located at the centroid of the cross section. e coordinates of axial load N ext are e x and e y . e direction angle θ of eccentricity is clockwise from positive x-axis.
Numerical analysis of the eccentrically compressive behavior of T-shaped concrete-filled steel tubular section is carried out in the following, with the flow chart of numerical program being depicted in Figure 3.

Flow Chart of Numerical Program of T-Shaped CFST Column Subjected to Eccentrically Compressive Load.
A spatial loading diagram of the T-shaped CFST column is shown in Figure 4. e length of the column is L and the column ends are hinged. It is assumed that the column deforms laterally into a half sine wave. e midspan deflection is denoted by U m . e eccentricity of the axial load N ext is e. In the xy plane, the direction angle θ of eccentricity is clockwise from positive x-axis.
Numerical analysis of the eccentrically compressive behavior of T-shaped CFST column is carried out in the following, with the flow chart of numerical program being depicted in Figure 5.

Comparison of Numerical Results and Test Results.
e numerical analysis program is used to simulate the structural behavior of stiffened T-shaped CFST specimens subjected to eccentrically compressive loads. Axial load next-midspan deflection Um relation curves are calculated to be compared with test results of specimens TE5 and TE6 [26]. To further verify the accuracy of the program, we compared another two specimens B20-180 and B40-180 [30]. It can be seen that good agreement in stiffness, peak resistance, and ductility guarantees the accuracy of the numerical program ( Figure 6). e numerical program can be employed to predict mechanical behavior of stiffened T-shaped CFST specimens.

Parametric Analysis
e influences of concrete strength f ck , steel yielding strength f y , steel tube thickness t, slenderness ratio λ, and loading angle θ on eccentrically compressive behaviors of cross section and column are mainly investigated in parametric analysis. e cross-sectional dimensions of the columns employ the test data [26]. e basic parameters are as follows: concrete strength f ck � 26.8 MPa, steel yielding strength f y � 235 MPa, steel tube thickness t � 2 mm, axial length L � 1500 mm, slenderness ratio λ � 20, axial compression ratio n 0 � 0.4, and eccentricity e � 100 mm. e stiffener is welded in the same way with that in the test [26].

Parametric Analysis of Eccentrically Compressive Behavior
of T-Shaped CFST Section 3.1.1. Cross-Sectional M-ϕ Relation Curves. Figure 7 shows the influence of the axial compression ratio on cross-sectional M-ϕ relation curves. It can be seen that there is no significant decline of resistance in the M-ϕ relation curves in the case of forward loading and 45-degree angle loading.
With the increase of axial compression ratio n, the elastic stiffness decreases gradually, and the peak resistance increases first and then decreases. is is because when the axial compression ratio n is relatively small, concrete in the tension zone of cross section will crack earlier to reduce the resistance; when the axial compression ratio n is relatively large, concrete will be compressed to failure, and steel tube will buckle, which reduces the resistance.

Cross-Sectional N-M and N/N 0 -M/M 0 Resistance
Correlation Curves. e axial force N -moment M resistance correlation curves of the T-shaped CFST cross section are calculated with the numerical program. Accordingly, the normalized N/N 0 -M/M 0 resistance correlation curves are obtained to get rid of the influences of material amount and material strength. N 0 is axial resistance of the cross section, and M 0 is the bending resistance of the cross section. e parametric analysis is carried out to calculate N-M correlation curves of cross section to investigate influence of concrete strength f ck , steel yielding strength f y , steel tube thickness t, and loading angle θ, which are shown in Figures 8-11. Figure 8 shows the influence of concrete strength f ck on cross-sectional N-M and N/N 0 -M/M 0 resistance correlation curves in forward loading. It can be seen that concrete strength f ck has obvious influence on the resistance correlation curves. With the increase of the concrete strength f ck , the area surrounded by the resistance correlation curve increases, which means an increase in the resistance. For the same concrete strength f ck , the bending resistance M increases first and then decreases with the increase in the axial force N. For the normalized N/N 0 -M/M 0 resistance correlation curves, with the increase in the concrete strength f ck , the proportion of concrete resistance in the total resistance increases, and the inflection point of the resistance correlation curve is more significant. While the resistance of concrete is relatively low, the proportion of steel behavior is larger, and the inflection point is relatively insignificant. Figures 9 and 10 show the influence of steel yielding strength f y and steel tube thickness t on cross-sectional N-M and N/N 0 -M/M 0 resistance correlation curves in forward loading. It can be seen that the steel yielding strength f y and steel tube thickness t have obvious influence on the resistance correlation curves. With the increase of the steel yielding strength f y and steel tube thickness t, the area surrounded by the resistance correlation curve increases, which means an increase in resistance. For the same steel yielding strength f y and steel tube thickness t, the bending resistance M increases first and then decreases with an increase in the axial force N. For the normalized N/N 0 -M/M 0 resistance correlation curves, with the increase in steel yielding strength f y and steel tube thickness t, the proportion of steel resistance in the total resistance increases, and the inflection point of the resistance correlation curve is insignificant. While the resistance of steel is relatively low, the proportion of concrete behavior is larger, and the inflection point is relatively significant. Figure 11 shows the influence of loading angle θ on cross-sectional N-M and N/N 0 -M/M 0 resistance correlation curves. e loading angle θ varies from − 90°to 90°, divided into positive angle loading and negative angle loading. For the T-section shown in Figure 2, − 90°angle loading refers to the force state (compressive region in the flange and tensile region in the web), 90°angle loading refers to the force state largest. It can be seen that when the loading angle θ varies from 0 to 90°, the inflection point of the resistance correlation curve is most significant from 30 to 45°; when the loading angle θ varies from 0 to -90°, the inflection point is most significant at − 90°. Figure 12 shows the influence of the axial compression ratio n on cross-sectional M-ϕ relation curves in forward loading. It can be seen that, with the increase of the axial compression ratio n, the elastic stiffness of M-ϕ relation curves with the slenderness ratio of 20 decreases gradually, and the peak resistance increases first and then decreases. When the slenderness ratio is 100, with the increase of axial compression ratio n, the elastic stiffness and the peak resistance are decreased. is is because when the column is slenderer, the cross-sectional height is smaller than the midspan horizontal displacement, and the second-order effect of eccentrically compressive load is larger, so that the column is mainly subjected to bending failure under any axial compression ratio n. at is, concrete in the tension zone of cross section is out of work after cracked, and the resistance is mainly provided by the steel tube. e larger the axial compression ratio n is, the more significant the adverse effect of the second-order effect will be, and the lower the resistance of component will be.

Component-Based M-ϕ Relation Curves.
e resistance and ductility of the component with a slenderness ratio of 20 are better than those with a slenderness ratio of 100. is is because if the adverse effect of the second-order effect of the axial force is larger, the behavior will be worse.

Component-Based N-M and N/N 0 -M/M 0 Resistance Correlation Curves.
e parametric analysis is carried out to calculate N-M correlation curves of cross section to investigate influence of concrete strength f ck , steel yielding strength f y , steel tube thickness t, slenderness ratio λ, and loading angle θ, which are shown in Figures 13-17. Figure 13 shows With the decrease in steel yielding strength f y and steel tube thickness t, the inflection point of the resistance correlation curve is more significant. Figure 16 shows the influence of slenderness ratio λ on component-based N-M and N/N 0 -M/M 0 resistance correlation curves in forward loading. For the different slenderness ratios of 6.7, 13.4, 20, 26.8, 33.5, 40, 60, and 100 are corresponding to the column lengths 500 mm, 1000 mm, 1500 mm, 2000 mm, 2500 mm, 3000 mm, 4500 mm, and 7500 mm, respectively. It can be seen that, with the increase of slenderness ratio λ, the resistance decreases, and the inflection point of the resistance correlation curve gradually disappears, so that the curve approaches the straight line gradually. Figure 17 shows the influence of loading angle θ on component-based N-M and N/N 0 -M/M 0 resistance correlation curves with a slenderness ratio of 20. e loading angle θ varies from − 90°to 90°, divided into positive angle loading and negative angle loading. For the same loading angle θ, the bending resistance M increases first and then decreases with the increase in axial force N. e resistance correlation curve is related to the axial compression ratio n and the T-shaped section size. When the loading angle θ, the vertical force N, and the bending moment M are just to make steel yielding strength f y and concrete strength f ck of the cross section play a maximum role at the same time, the column has large resistance, and the area surrounded by the envelope is largest. It can be seen that when the loading angle θ varies from 0 to 90°, the inflection point of the resistance correlation curve is most significant from 30 to 45°; when the loading angle θ varies from 0 to − 90°, the inflection point is most significant at − 90°.

Basic Hypotheses
(1) Response superposition of axial load and bending moment is adopted.   Advances in Civil Engineering (2) e concrete can develop its compressive strength in compressive region but has no strength in tension region; the steel tube can develop its yielding strength in both compressive and tensile regions. (3) e constraint effect for concrete is considered in the enhancement of concrete compressive strength.

Analysis of Stress Distribution Mode of Cross Section.
Based on the preceding numerical analysis, the crosssectional N-M resistance correlation curve has the form described by the curve ABCD in Figure 18.
where A ci , A sci , and A sti are, respectively, element areas of compressive concrete, compressive steel, and tensile steel; y ci , y sci and y sti are, respectively, distances of compressive concrete, compressive steel, and tensile steel away from neutral axis. If N equals axial load N ext , then the preset y (or x) and corresponding cross-sectional resistance N and M can be accurately determined. If not, the y (or x) should be preset again, and iteration will be carried out until appropriate y (or x) is acquired. For the determination of the location of feature points A, B, C, and D in Figure 18, the specific calculation steps are as follows:   Table 1. e unified theory considers the steel tube and concrete as a new composite material, considering constraint effect for concrete provided by steel tube [31]. e calculated values of numerical program are shown in Figure 20.
e f sc /f ck and ξ are, respectively, taken as the vertical axis and the horizontal axis. e composite strength f sc is the ratio of the calculated axial bearing capacity of stub column to total cross-sectional area A sc (A sc � A s + A c ). rough regression analysis, the composite strength f sc of T-shaped CFST stub column is determined by equation (2) with red line expressed in Figure 19. e coefficients B and C are, respectively, used to consider the influences of steel yield strength and concrete strength (equation (3)). en, the axial compressive bearing capacity of T-shaped CFST stub columns can be calculated with equation (4).

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Advances in Civil Engineering    After determination of feature points A, B, C and D, the simplified model of N-M resistance correlation curves can be obtained by connecting feature points in formulas (5), (6), and (7). AB: CD : In terms of biaxial eccentrically compressive cross sections, each feature point is regressed on the basis of those in the neighboring two feature uniaxial directions. To be mentioned conveniently in the following contents, the coordinates of feature points in positive direction, in negative direction, and in x direction are, respectively, denoted with label (+), (-), and (0). e bending resistance M 0 of point D under biaxial eccentric compression needs considering the impact of loading angle θ. e M 0 is regressed by correcting M 0 (+), with correction factor f 1 expressed in equation (9).
e bending resistance M b of point B under arbitrary loading angle θ (excluding three feature uniaxial directions) is estimated with an elliptical curve determined by the M b in feature loading directions. It can be calculated with the following two formulas, respectively, for positive loading angle (0°-90°) and negative loading angle (− 90°-0°): Advances in Civil Engineering 13 with equations (11) and (12), respectively, for positive loading angle (0°-90°) and negative loading angle (− 90°-0°): Except for loading angle θ, the impact of constraint effect is suggested to be taken into consideration, and a correction expression ΔN b should be subtracted from N b : e bending resistance M c of point C under arbitrary loading angle θ is estimated with expressions (14) and (15), respectively, for positive loading angle (0°-90°) and negative loading angle (− 90°-0°):   14 Advances in Civil Engineering e axial compressive resistance N c of point C under arbitrary loading angle θ is regressed with a straight line determined by the N c in feature loading directions. It can be expressed in the formulas (16) and (17), respectively, for positive loading angle (0°-90°) and negative loading angle (− 90°-0°): After determination of the feature points A, B, C and D under arbitrary loading angle θ, the formulas (5)∼ (7) can be also adopted as simplified resistance model under arbitrary loading angle θ.

Comparison of Simplified Resistance Model and Numerical Results.
e simplified model is used to calculate the cross-sectional resistance correlation curves and compared with the numerical results in Figures 21 and 22. e comparison involves parameters of concrete strength f ck (C30, C50, and C70), steel yielding strength f y (Q235, Q345, and Q420), steel tube thickness t (2 mm, 4 mm, and 6 mm), and loading angle θ (− 90°-90°). e concrete strength C30 (according to a concrete structure code in China GB50010-2002) represents a characteristic value of compressive strength for prismatic standard test blocks. e steel yielding strength Q235 can refer to a steel structure code in China (GB50017-2003). Figures 21 and 22 show that the simplified resistance model has adequate agreement with the numerical results. e correlation curve of simplified resistance model is approximately enveloped by a corresponding numerical correlation curve, which reveals that the simplified model is a little conservative and can be adopted to predict the static behavior of special-shaped stiffened CFST columns subjected to eccentrically compressive load.

Simplified Resistance Model of Eccentrically Compressive
Column. e simplified resistance model for column is introduced in Figure 23. Curve 1 is the numerical result; bilinear curve 2 is the proposed simplified resistance model. N z is the cross-sectional axial compressive resistance considering constraint effect, which is the same as N 0 ; φ is the stability coefficient considering length impact, calculated with the following formula:       . e simplified resistance model has adequate agreement with the numerical results. e correlation curve of simplified resistance model is enveloped by the corresponding numerical correlation curve, which reveals that the simplified model is a little conservative and can be adopted to predict the static behavior of T-shaped stiffened CFST columns subjected to eccentrically compressive load. Figure 25 compares the N-M resistance correlation curves of simplified resistance model (represented by the dotted lines) and numerical method (represented by the continuous lines) under different slenderness ratios λ (λ �13.3 and 60) in a loading angle of 45°. e simplified resistance model has adequate agreement with the numerical results. e correlation curve of simplified resistance model is almost completely enveloped by corresponding numerical correlation curve, which reveals that the simplified model is a little conservative and can be adopted to predict the static behavior of T-shaped stiffened CFST columns under different slenderness ratios λ. Figure 26 compares the N-M resistance correlation curves of simplified resistance model (represented by the dotted lines) and numerical method (represented by the continuous lines) in positive loading angles (0-90°) and negative loading angles (− 90°-0). e simplified resistance model has adequate agreement with the numerical results. e correlation curve of simplified resistance model is

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Advances in Civil Engineering almost completely enveloped by the corresponding numerical correlation curve, which reveals that the simplified model is a little conservative.

Conclusions
(1) e numerical programs of eccentrically compressive T-shaped stiffened CFST columns proposed in this paper are reasonable and reliable, which are in good agreement with the experimental results. (2) Parametric analysis of eccentrically compressive T-shaped stiffened CFST columns shows that the peak resistance increases first and then decreases with the increase in the axial compression ratio n; the resistance increases with the increase of concrete strength f ck , steel yielding strength f y , and steel tube thickness t; the resistance increases with the decrease in slenderness ratio λ. (3) e design formulas for calculating sectional bearing capacity of T-shaped CFST columns are proposed. e constraint effect of steel tube for concrete is considered in the formulas. e calculated sectional bearing capacities are in good agreement with the numerical program results. (4) e simplified resistance models of eccentrically compressive T-shaped stiffened CFST section and column proposed in this paper agree well with the numerical results and can be used to predict the mechanical behavior of T-shaped stiffened CFST section and column.

Data Availability
All the data generated or analyzed during this study are included in this article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.