# Coupling Effect of Eccentricity and Slenderness Ratios on RCFST Column Instability Modes

Authors: Chao Guo; Zhengran Lu
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

The instability damage modes of rectangle concrete-filled steel tube (RCFST) columns that are subjected to eccentric compression can be divided into two types based on the modified Jezek analytical procedure, namely, the eccentricity ratio (γ) and the slenderness ratio (λ) coupling effect. The RCFST columns have unilateral compression yield failure mode when γ is small. However, it has compressive and tensile mode on both sides when the value of γ is large. In this work, parametric analyses were performed to test 32 RCFST long columns by varying γ with different λ combinations. From the analysis, it was found that the results of the theoretical analysis of the load-tension strain (P-ε) responses and the instability modes of the RCFST long columns are similar to the experimental results. Further, the proposed analytical method aids in better understanding the effects of γ and λ coupling on the stability behavior of the RCFST columns.


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## Body

## 1. Introduction

The rectangle concrete-filled steel tube (RCFST) significantly improves the strength of long columns and reduces the section area of the columns [1]. However, it does not provide a full restraint effect of the steel tubes on the concrete core columns (CCC) as seen in the circular steel tube-confined concrete [2]. Further, under the action of compression and bending loads, the rectangular steel tube is prone to have local buckling in the middle part which transforms CCC into a plain concrete [3]. Moreover, the compression and bending stability of the RCFST long columns is a nonlinear problem under the cross control of the eccentricity ratio (γ), the slenderness ratio (λ), and the column end constraints [4]. Under these conditions of local buckling of a steel tube and a loss of constraint of CCC, the strength of long columns decreases sharply [5]. Therefore, it is necessary to conduct an in-depth analysis of the instability modes of RCFST to explore the influence of γ and λ on RCFST [6]. Further, the plane static instability modes of the RCFST long column should be analyzed emphatically [7]. According to the Euler critical load principle, the critical stability conditions of the RCFST long columns were determined by the elastic modulus, the moment of inertia, and the length [8]. However, the peak-load values of the codes EC4 [9], AISC 360 [10], and GB 50936 [11] were calculated assuming that the elastic modulus and the moment of inertia of the concrete and the steel were unified values by ignoring the difference between these two materials. Even though the composite theory takes into account the contributions of the steel tubes and the concrete to the critical load of the RCFST long columns separately, it is also required that the steel tubes and CCC shall bear the load simultaneously [12]. However, in the actual instability process of the RCFST long columns, the steel and the concrete cannot bear the load jointly. Further, the yield strain (εp) of the steel is 0.002 when the compressive strain (εcc) of the concrete is 0.0033. However, the tensile strain (εct) of the concrete is only 0.0007 [13]. When the RCFST long columns undergo bending deformation, the CCC debond from the steel tubes and enter the failure stage before the steel tubes and then lose the bearing function [14]. However, the instability modes of the RCFST columns are more complicated with various defects [7, 15]. The abovementioned instability modes of the RCFST long columns were verified by the many tests and numerical analysis.Li et al. [16, 17] conducted an experimental and numerical parametric analysis according to 32 sets of high-strength RCFST long columns with γ = 0.13, 0.23, 0.33, and 0.43, respectively, with different λ coupling effects on the RCFST column load-lateral displacement (P-Δ) responses. The instability damage processes on these samples and the RCFST columns instability modes were deduced. Many researchers conducted a lot of research on the instability modes of CFST columns in various methods [18–20]. Fei et al. [21] studied the behavior of multicell concrete-filled steel tube columns under eccentric loading using seven groups of model tests and inversed the compressive and tensile instability mode of RCFST columns. According to the different contribution of steel tube on each side to the bearing capacity of RCFST columns in the process of in-plane instability damage mode, 12 groups of parametric tests and numerical simulation analysis of RCFST columns with unequal wall thickness were carried out by Guangyuan et al. [22], and it was proposed that this kind of RCFST is more suitable when λ is relatively large and when γ is greater than 0.2. Lei et al. [23] utilized eleven specimens of compression concrete-filled narrow rectangular steel tube (CFNRST) members, considering the variation of γ, and λ of the cross section was subjected to combined compression and uniaxial bending on the weak axis. The transition process of the internal stress state of the RCFST long columns was very complex. Therefore, a theoretical analysis theory was needed to reveal the contribution of γ and λ of RCFST long columns in the instability modes [24].To address this issue, in this work, the 32 RCFST specimens’ load-tension strain (P-ε) relationships and the compressive and tensile strain of the steel tube were analyzed. Initially, the two main instability modes, namely, the unilateral compression and bilateral compression tension of RCFST long columns with various γ and λ were obtained. Subsequently, the implicit function equations of two main instability modes were established based on the modified Jezek analytical method, and the regularization slenderness ratio (λ¯) was introduced to conduct the elastoplastic theoretical analysis so that the two main instability modes could be expressed scientifically and reasonably in a dimensionless normalized form. In this way, the great difference of column length and section size of RCFST long columns was avoided, and the application scope of the analytical method was extended. Finally, a parametric analysis of the coupling effects of γ and λ on the instability modes of the RCFST long column system was conducted on 32 specimens of the RCFST long columns using the Maple program to verify the reliability of the theoretical analysis method.


## 2. Experimental Analysis

The details of the experimental analysis such as specifications of the specimen, test materials used, and the results are presented below.


### 2.1. Specimen Specifications

To study the influences of the slenderness ratios (λ = 23.09–46.19) and the eccentricity ratios (γ = e0/B = 0.13–0.43) on the behaviors of the RCFST columns, a total of 32 (16 groups) experimental specimens with various lengths (L = 450–2000 mm) and eccentric distances (e0 = 20–65 mm) were used for experimental analysis [16, 17]. The detailed parameters are tabulated in Table 1, where λ=23L/B [25, 26]. Further, the outer width (B) and the thickness (t) of the square steel tube were taken as 150 mm and 4 mm, respectively. The test setup and specimen details are shown in Figure 1. The unit of the stiffener in Figure 1(a) is in mm. When L = 450 mm, a smaller value of 30 mm and 80 mm was taken, otherwise a larger value of 70 mm and 150 mm was considered. Figures 1(b) and 1(d) represent the stress state of the small and large eccentric compressions of the RCFST columns, respectively, and Figure 1(c) represents the number of strain gauges (SG) and surfaces (S) at the central section of the RCFST columns. In this work, S2 and S4 denote the compressive (concave) and tensile (convex) sides of the columns, respectively.Table 1
Parameters of 32 RCFST columns.
No.SpecimenL (mm)λγPu1 (kN)Pu2 (kN)Pucal (kN)Err. (%)Ref.1EC1-1/245010.40.132114213023008.4[17]2EC2-1/245010.40.231846171719398.8[17]3EC3-1/245010.40.331561145514901.2[17]4EC4-1/245010.40.431265124413245.5[17]5LEC1-1/2100023.10.132097220021420.3[16]6LEC2-1/2100023.10.231679166017806.6[16]7LEC3-1/2100023.10.331360141313314[16]8LEC4-1/2100023.10.431176112110905.1[16]9LEC5-1/2150034.60.132012211320152.3[16]10LEC6-1/2150034.60.231617154616454[16]11LEC7-1/2150034.60.331319131013011[16]12LEC8-1/2150034.60.431081110510830.9[16]13LEC9-1/2200046.20.131853184818600.5[16]14LEC10-1/2200046.20.231545143915050.9[16]15LEC11-1/2200046.20.331225117511206.7[16]16LEC12-1/2200046.20.4398210049860.7[16]Figure 1
Test setup and stress state: (a) specimen information, (b) smallγ stress state, (c) strain gauges (SG) arrangement, and (d) large γ stress state.
(a)(b)(c)(d)


### 2.2. Test Materials

The trilinearσ-ε model that was developed for the steel tubes with effective strengths in the range of 200–800 MPa by Han et al. [27] was used in this work. The model is shown in Figure 2.Figure 2
Model for steel tube properties (Figure2 is reproduced from Zhengran Lu et al. 2021).In Figure2, fsy and fsu refer to the yield and the ultimate strengths of the steel tube, respectively. Es refers to the elastic modulus at the initial point on the σ-ε line and is taken as 206 GPa. εsp refers to the strain at the strain hardening site, εsu refers to the ultimate strain corresponding to the ultimate strength, εsy refers to the yield strain that is defined as εsy = fsy/Es, and Esp refers to the initial elasticity modulus at the strain hardening point which is considered to be 0.02Es. The steel tensile test curve that was presented by Li et al. [16, 17] conforms to the constitutive model proposed by Han et al. [27]. Therefore, the high-strength concrete cubic compression strength (fck) values determined by Li were considered for the experimental specimens. The material parameters of the RSTCC specimens [16, 17] are presented in Table 2.Table 2
Material parameters of RSTCC long column specimens.
PhaseE (GPa)μfck (MPa)fsy (MPa)fsu (MPa)RefConcrete41.80.2110.5[16, 17]Steel tube2060.3434.6546.2[16, 17]E: elastic model, μ: Poisson’s ratio, and fck: concrete compressive strength.


### 2.3. Test Results and Discussions

In this work, the statistical analysis of the strain gauge 15 (SG15), on the surface 4 (S4) of the steel tube on the tension side, and the strain gauge 7 (SG7) on the surface 2 (S2) of the steel tube on the compression side are focused as shown in Figure1(c). The outer envelopes of the load-strain (P-ε) relationships of the RCFST column under the eccentric compressive load are shown in Figure 3 for different values of γ and λ. From the analysis, it was found that, as the value of γ increases, the peak value of P decreases rapidly, and the P-ε relationship of the RCFST systems showed a linear to nonlinear material behavior up to the peak response. Thereafter, there was a slight parabolic softening phase. During the follow-up process, the value of P decreased slowly showing the characteristics of high ductility. However, when the values of γ were very high (0.43) corresponding to the extreme case assumed in this study, the P-ε linear relationship no longer existed and this RCFST long column became similar to a pure bending member.Figure 3
Steel tubeP-ε curve: (a) γ = 0.13, (b) γ = 0.23, (c) γ = 0.33, and (d) γ = 0.43.
(a)(b)(c)(d)Figure3 shows the curves of C_up/C_low (compressive state up/low limit line) and T_up/T_low (tensile state up/low limit line).Further, based on the SG15 variations, the tension (S4) regional working states and the compression (SG7) working states under the peak-load conditions were categorized into unilateral compression instability and compressions and yield states, although there were four types in [16, 17]. In the case of unilateral compression instability of the RCFST specimens with a relatively small γ and a low λ as shown in Figures 3(a) and 3(b), the value of SG7 at Pu was found to be negative and smaller than εy. Meanwhile, the value of SG15 was found to be larger than εy. This shows that the steel tube and the CCC are in a state of collaborative stress, and the RCFST column is in unilateral compression unstable state. In the case of compression and tension yield states, it was found that, for large γ, the respective SG15 values corresponding to Pu were greater than or equal to εy (Figures 3(c) and 3(d)). Hence, it was concluded that the tension regional steel tubes were in the plastic tension state. During this time, the S2 of the steel tube yielded under compression and S4 yielded under tension. Further, the RCFST was in both sides yield instability mode. Therefore, the steel tube and concrete were unable to work simultaneously.


## 2.1. Specimen Specifications

To study the influences of the slenderness ratios (λ = 23.09–46.19) and the eccentricity ratios (γ = e0/B = 0.13–0.43) on the behaviors of the RCFST columns, a total of 32 (16 groups) experimental specimens with various lengths (L = 450–2000 mm) and eccentric distances (e0 = 20–65 mm) were used for experimental analysis [16, 17]. The detailed parameters are tabulated in Table 1, where λ=23L/B [25, 26]. Further, the outer width (B) and the thickness (t) of the square steel tube were taken as 150 mm and 4 mm, respectively. The test setup and specimen details are shown in Figure 1. The unit of the stiffener in Figure 1(a) is in mm. When L = 450 mm, a smaller value of 30 mm and 80 mm was taken, otherwise a larger value of 70 mm and 150 mm was considered. Figures 1(b) and 1(d) represent the stress state of the small and large eccentric compressions of the RCFST columns, respectively, and Figure 1(c) represents the number of strain gauges (SG) and surfaces (S) at the central section of the RCFST columns. In this work, S2 and S4 denote the compressive (concave) and tensile (convex) sides of the columns, respectively.Table 1
Parameters of 32 RCFST columns.
No.SpecimenL (mm)λγPu1 (kN)Pu2 (kN)Pucal (kN)Err. (%)Ref.1EC1-1/245010.40.132114213023008.4[17]2EC2-1/245010.40.231846171719398.8[17]3EC3-1/245010.40.331561145514901.2[17]4EC4-1/245010.40.431265124413245.5[17]5LEC1-1/2100023.10.132097220021420.3[16]6LEC2-1/2100023.10.231679166017806.6[16]7LEC3-1/2100023.10.331360141313314[16]8LEC4-1/2100023.10.431176112110905.1[16]9LEC5-1/2150034.60.132012211320152.3[16]10LEC6-1/2150034.60.231617154616454[16]11LEC7-1/2150034.60.331319131013011[16]12LEC8-1/2150034.60.431081110510830.9[16]13LEC9-1/2200046.20.131853184818600.5[16]14LEC10-1/2200046.20.231545143915050.9[16]15LEC11-1/2200046.20.331225117511206.7[16]16LEC12-1/2200046.20.4398210049860.7[16]Figure 1
Test setup and stress state: (a) specimen information, (b) smallγ stress state, (c) strain gauges (SG) arrangement, and (d) large γ stress state.
(a)(b)(c)(d)


## 2.2. Test Materials

The trilinearσ-ε model that was developed for the steel tubes with effective strengths in the range of 200–800 MPa by Han et al. [27] was used in this work. The model is shown in Figure 2.Figure 2
Model for steel tube properties (Figure2 is reproduced from Zhengran Lu et al. 2021).In Figure2, fsy and fsu refer to the yield and the ultimate strengths of the steel tube, respectively. Es refers to the elastic modulus at the initial point on the σ-ε line and is taken as 206 GPa. εsp refers to the strain at the strain hardening site, εsu refers to the ultimate strain corresponding to the ultimate strength, εsy refers to the yield strain that is defined as εsy = fsy/Es, and Esp refers to the initial elasticity modulus at the strain hardening point which is considered to be 0.02Es. The steel tensile test curve that was presented by Li et al. [16, 17] conforms to the constitutive model proposed by Han et al. [27]. Therefore, the high-strength concrete cubic compression strength (fck) values determined by Li were considered for the experimental specimens. The material parameters of the RSTCC specimens [16, 17] are presented in Table 2.Table 2
Material parameters of RSTCC long column specimens.
PhaseE (GPa)μfck (MPa)fsy (MPa)fsu (MPa)RefConcrete41.80.2110.5[16, 17]Steel tube2060.3434.6546.2[16, 17]E: elastic model, μ: Poisson’s ratio, and fck: concrete compressive strength.


## 2.3. Test Results and Discussions

In this work, the statistical analysis of the strain gauge 15 (SG15), on the surface 4 (S4) of the steel tube on the tension side, and the strain gauge 7 (SG7) on the surface 2 (S2) of the steel tube on the compression side are focused as shown in Figure1(c). The outer envelopes of the load-strain (P-ε) relationships of the RCFST column under the eccentric compressive load are shown in Figure 3 for different values of γ and λ. From the analysis, it was found that, as the value of γ increases, the peak value of P decreases rapidly, and the P-ε relationship of the RCFST systems showed a linear to nonlinear material behavior up to the peak response. Thereafter, there was a slight parabolic softening phase. During the follow-up process, the value of P decreased slowly showing the characteristics of high ductility. However, when the values of γ were very high (0.43) corresponding to the extreme case assumed in this study, the P-ε linear relationship no longer existed and this RCFST long column became similar to a pure bending member.Figure 3
Steel tubeP-ε curve: (a) γ = 0.13, (b) γ = 0.23, (c) γ = 0.33, and (d) γ = 0.43.
(a)(b)(c)(d)Figure3 shows the curves of C_up/C_low (compressive state up/low limit line) and T_up/T_low (tensile state up/low limit line).Further, based on the SG15 variations, the tension (S4) regional working states and the compression (SG7) working states under the peak-load conditions were categorized into unilateral compression instability and compressions and yield states, although there were four types in [16, 17]. In the case of unilateral compression instability of the RCFST specimens with a relatively small γ and a low λ as shown in Figures 3(a) and 3(b), the value of SG7 at Pu was found to be negative and smaller than εy. Meanwhile, the value of SG15 was found to be larger than εy. This shows that the steel tube and the CCC are in a state of collaborative stress, and the RCFST column is in unilateral compression unstable state. In the case of compression and tension yield states, it was found that, for large γ, the respective SG15 values corresponding to Pu were greater than or equal to εy (Figures 3(c) and 3(d)). Hence, it was concluded that the tension regional steel tubes were in the plastic tension state. During this time, the S2 of the steel tube yielded under compression and S4 yielded under tension. Further, the RCFST was in both sides yield instability mode. Therefore, the steel tube and concrete were unable to work simultaneously.


## 3. Analytical Method of RCFST Column Instability

During the mechanical analysis, the experimental results showed that the strength (Pu) and the stability of the RCFST columns were affected by γ and λ coupling control. According to whether the steel tube on the tension side of the RCFST column was in a yield state during the instability process, the instability modes of the RCFST column were divided into the following two types, as shown in Figure 4(A) and 4(B).Figure 4
Instability modes of the RCFST column: (A) unilateral compression and (B) both sides yield instability modes. (a) Instability phenomenon, (b) deformation curve, (c) cross section parameter, (d) strain state, and (e) stress state.Based on a large number of experiments and numerical analysis [28–31], the instability process of the RCFST columns were deduced as follows: initially, under the coupling action of axial load and bending moment, the fiber at the compression edge of the cross section of the RCFST column began to yield, gradually from elastic to plastic working state. Then, as the coupling effects increased, it was found that the elastic region of the cross section shrank the bending stiffness of the RCFST column decreased, and the buckling deformation accelerated rapidly. Thus, it resulted in a rapid increase of additional bending moments. Further, when the increasing value of the bending capacity of the RCFST column was less than the increasing value of the external coupling action, the internal and external forces in the cross section of the RCFST column could not be balanced, which lead to instability.Moreover, the ultimate loadPu of the RCFST column was calculated according to the instability process. Furthermore, under a stable condition, the RCFST column was found to yield from the edge fiber of the cross section with the largest bending moment and developed along the depth direction.From the results, it was found that bothγ and λ had a significant influence on the RCFST column stability modes. Further, from γ=e0/h and λ=23l/h, it was concluded that λ and γ affected the interaction. Moreover, it was observed that when γ and λ were both small, the plastic yield development range was limited to the shaded part in Figure 4(A.c) and was constrained to the compression side. Further, when either γ or λ was larger, a plastic yield zone was found to appear on both sides of the compression and tension as shown in Figure 4(B.c). To analyze the influence of γ and λ coupling on the stability of the RCFST column, the modified Jezek analytical method was used to calculate the stability of the RCFST column under eccentric compression using the boundary conditions as hinged at both ends which are described in the following section.


### 3.1. Modified Jezek Analytical Method

Based on the theory of the modified Jezek analytical method [32], the following assumptions were made: (1) the elastoplastic constitutive models for the steel tube and the CCC in the RCFST column were adopted in this paper. Further, the steel tube and the CCC bear the compressive stress together under compression, while only the steel tube bears the tensile stress under tensile yield as the CCC could not bear the tensile stress. (2) The deformation curve of the RCFST column under eccentric load was a half-wave of sine curve as shown in Figure 4(b). In this work, 32 groups of eccentrically loaded RCFST column samples [16, 17] were simplified as biaxial symmetric rectangular section compression and bending members, whose two ends were hinged, and the sectional area was A = bh. Further, under the coupling action of an axial pressure P and end bending moment M, two kinds of elastic-plastic instability modes were observed as shown in Figures 4(A) and 4(B).Furthermore, it was observed that, for the RCFST column, samples with smallγ yielded only in the compression zone, and the height of the elastic zone of the central cross section was he, as shown in Figure 4(A.c). Moreover, when the samples had large γ, the steel tube in the tension zone was found to reach the yield strength fsy while the compression zone was found to reach the yield strength, and the elastic zone shrank to the middle of the section, as shown in Figure 4(B.c).According to the above two different instability modes of the RCFST columns, the load and the moment balance equations of the middle height cross section were established, and theP-Δ relationship was obtained. Then, the critical load Pu was obtained by using the extreme value condition as dP/dΔ=0.Further, the critical elastic load (PE) of the RCFST long columns are suggested [33], as given in equations (1)–(9):(1)PE=π2EscAscλ2,(2)Esc=fscpεscp,(3)fscp=0.0717⋅fsyfsc,235+0.4513⋅fck,C30fck+0.4955⋅fscy,(4)εscp=3.8437×10−6⋅fsy,(5)fscp=1.212+B⋅ξ+C⋅ξ2⋅fck,(6)B=0.1521⋅fsyfck,C30+0.8421,(7)C=−0.0915⋅fsyfck,C30+0.0272,(8)fsy,235=235MPa,fck,235=20.318MPa,(9)ξ=AsfsyAcfck.


### 3.2. RCFST Column under Unilateral Compression Instability Mode

Figure4(A.d) shows the strain distribution diagram of the middle height section of the RCFST column under small eccentric compression, and the corresponding stress distribution is shown in Figure 4(A.e). Py refers to the plastic yield-bearing load of the RCFST column under axial compression and is given as shown in the following equation:(10)Py=σyscA.The load balance conditions of the section at the middle height of the column are obtained based on the following equations:(11)P=Py−12σysc+σtscbhe,(12)σysc+σtsc=2Py−Pbhe.The moment balance condition of the section at the middle height is obtained as shown in the following equation:(13)M+PΔ=12σysc+σtscbheh2−he3.From equations (12) and (13), the value of he is obtained as shown in the following equation:(14)he=3h2−3M+PΔPy−P.The curvature of the middle height section (Ф) is obtained as shown in the following equation:(15)Φ=εysc+εtsche=σysc+σtscEsche=2Py−PEscbhe2.According to the second assumption of the modified Jezek analytical method, the deflection curve of the RCFST column was defined as shown in the following equation:(16)y=Δsinπxl.From the definition ofФ, it can be concluded as shown in the following equation:(17)Φ=−y″l2=Δπ2l2=2Py−PEscbhe2.By combining equations (14) with (17), the internal and external stresses of the middle height cross section are balanced, and the deformation compatibility condition is transformed into P-Δ relationships as shown in the following equation:(18)Δh21−PPy−M+PΔPy2=2l2Py9bπ2Esc1−PPy3.From the critical equilibrium conditiondP/dΔ=0, it is found that(19)Δ=Py3Ph21−PPy−MPy.By combining equations (19) with (18), the peak value of eccentric compressive strength (Pu) is obtained as shown in the following equation:(20)Pu=bh3π2Esc12l21−2MhPy1−Pu/Py3.In equation (20), M=Pue0.In the derivation of equations (11) to (20), the calculated value of PE in the RCFST column in equation (1) is not applied; however, the RCFST is analyzed as an elastic-plastic material similar to steel. This assumption leads to the calculation error of deviation between the calculated results and the material properties of the RCFST. To solve this problem, the following corrections are necessary which are suitable for the calculation of the actual RCFST column.In this work, it was assumed thatα=Pu/Py, and PE0=π2Escbh3/12l2 is Euler critical stress of long column. Further, the ideal elastic-plastic constitutive relation of steel was modified by using λ of the RCFST with equation (1) to replace PE0 in equation (20) and Pu was obtained as shown in the following equation:(21)Pu=PE1−2MhPy1−α3.SinceM in equation (21) is also a function of Pu, equation (21) had no explicit solution and had to be solved using the Newton interpolation method.Further,λ was regularized as λ¯2=Py/PE [34], and equation (21) was reduced to the following equation:(22)αλ¯2=1−2αγ1−α3.According to equation (22), λ¯ was expressed as a polynomial of γ after regularization. The above equations were established based on the conditions of σtsc≤fscp and he>1−α.From equations (14) and (21), it was concluded as shown in the following equations:(23)heh=1−2αγ1−α>1−α,(24)γ<1−α2.


### 3.3. RCFST Column under Compression and Tension Yield Instability Modes on Both Sides

In this analysis, the RCFST column was subjected to both compression and tension yield instability modes on both sides of the specimen. The steel tube tension was expressed as shown in the following equation:(25)T=fsybt.The balance conditions of the internal and external stresses were calculated as shown in the following equations:(26)P−T=Py−12σysc+σtscbhe,(27)σysc+σtsc=2Py−P+Tbhe.From the moment balance condition, it was concluded as shown in the following equation:(28)M+PΔ=Th2+12σysc+σtscbheh2−he3.By substituting equation (27) into (28), the following equation was obtained:(29)M+PΔ=Th2+Py−P+Th2−he3.From equation (29), the expression of he was obtained as shown in the following equation:(30)he=3h2−6M+6PΔ−3Th2Py−P+T.Further, the curvature of the middle height section (Ф) was obtained as shown in the following equation:(31)Φ=−y″l2=Δπ2l2=εysc+εtsche=σysc+σtscEsche=2Py−P+TEsche2.By substituting equation (30) into (31), the relationship of P-Δ was obtained as shown in the following equation:(32)Δh21−PPy−M+PΔ+ThPy2=2l2Py9π2bEsc1−P+TPy3.From the critical equilibrium conditiondP/dΔ=0, ∆ was obtained as shown in the following equation:(33)Δ=Py3Ph21−PPy−M+ThPy.By substituting into equation (32), it can be concluded as shown in the following equation:(34)Pu=PE1−2M+ThhPy1−P/Py−P/Py3.Assuming thatT/Py=η, equation (34) was reduced to the following equation:(35)αλ¯2=1−2αγ+η1−α−η3.The above equations were established under the conditions ofσtsc≥σtsc and he/h≤1−α.From equation (30), we know that(36)heh≤1−α+η.When equations (30) and (33) were substituted into equation (36), the following results were obtained as shown in the following equation:(37)α≥1−1−α+η2−2αγ+2η.Due to the complexity of the above reasoning process, the intermediate calculation results of the process were expressed in the form of a diagram, as shown in Figure5. In Figure 5, the horizontal axis represents the horizontal deflection (Δ) at the middle height section of the RCFST column, and the longitudinal axis represents the strength (P) under various assumptions. The distance from o to o' represents the initial eccentricity e0. The gradient of the line represents the initial elastic modulus (Esc) of the RCFST. The top point (PE0, ΔE0) represents the Euler critical stress of the RCFST column under the assumptions of the steel tube and the CCC as a kind of unified ideal elastic-plastic material.Figure 5
Instability mode calculation process.Further, (PE, ΔE) represents the Euler critical stress corrected by λ. (Py, Δy) represents the compressive strength of the RCFST column at plastic yield. (Pe, Δe) represents the strength of the initial elastic stage. (Pu, Δu) represents the calculated compressive strength of the two instability modes proposed in this paper. The curve represents the experimental curve of [17, 18]. From the position height relationship of the above points, it is observed that the Eulerian critical stress PE0 under the ideal elastic-plastic instability mode was the highest, while PE after λ correction reflecting the characteristics of RCFST composite material was lower than PE0. Furthermore, the compressive strength Py that was calculated by area weighting was smaller than PE according to the plastic yield strength of the steel tube and the concrete. The actual ultimate compressive strength Pu of the RCFST was found to be between Pe and Py.From Figure5, it is also observed that the calculation method proposed in this paper can perfectly and scientifically explain the instability mode of the RCFST columns under eccentric compression by the coupling action of γ and λ.


## 3.1. Modified Jezek Analytical Method

Based on the theory of the modified Jezek analytical method [32], the following assumptions were made: (1) the elastoplastic constitutive models for the steel tube and the CCC in the RCFST column were adopted in this paper. Further, the steel tube and the CCC bear the compressive stress together under compression, while only the steel tube bears the tensile stress under tensile yield as the CCC could not bear the tensile stress. (2) The deformation curve of the RCFST column under eccentric load was a half-wave of sine curve as shown in Figure 4(b). In this work, 32 groups of eccentrically loaded RCFST column samples [16, 17] were simplified as biaxial symmetric rectangular section compression and bending members, whose two ends were hinged, and the sectional area was A = bh. Further, under the coupling action of an axial pressure P and end bending moment M, two kinds of elastic-plastic instability modes were observed as shown in Figures 4(A) and 4(B).Furthermore, it was observed that, for the RCFST column, samples with smallγ yielded only in the compression zone, and the height of the elastic zone of the central cross section was he, as shown in Figure 4(A.c). Moreover, when the samples had large γ, the steel tube in the tension zone was found to reach the yield strength fsy while the compression zone was found to reach the yield strength, and the elastic zone shrank to the middle of the section, as shown in Figure 4(B.c).According to the above two different instability modes of the RCFST columns, the load and the moment balance equations of the middle height cross section were established, and theP-Δ relationship was obtained. Then, the critical load Pu was obtained by using the extreme value condition as dP/dΔ=0.Further, the critical elastic load (PE) of the RCFST long columns are suggested [33], as given in equations (1)–(9):(1)PE=π2EscAscλ2,(2)Esc=fscpεscp,(3)fscp=0.0717⋅fsyfsc,235+0.4513⋅fck,C30fck+0.4955⋅fscy,(4)εscp=3.8437×10−6⋅fsy,(5)fscp=1.212+B⋅ξ+C⋅ξ2⋅fck,(6)B=0.1521⋅fsyfck,C30+0.8421,(7)C=−0.0915⋅fsyfck,C30+0.0272,(8)fsy,235=235MPa,fck,235=20.318MPa,(9)ξ=AsfsyAcfck.


## 3.2. RCFST Column under Unilateral Compression Instability Mode

Figure4(A.d) shows the strain distribution diagram of the middle height section of the RCFST column under small eccentric compression, and the corresponding stress distribution is shown in Figure 4(A.e). Py refers to the plastic yield-bearing load of the RCFST column under axial compression and is given as shown in the following equation:(10)Py=σyscA.The load balance conditions of the section at the middle height of the column are obtained based on the following equations:(11)P=Py−12σysc+σtscbhe,(12)σysc+σtsc=2Py−Pbhe.The moment balance condition of the section at the middle height is obtained as shown in the following equation:(13)M+PΔ=12σysc+σtscbheh2−he3.From equations (12) and (13), the value of he is obtained as shown in the following equation:(14)he=3h2−3M+PΔPy−P.The curvature of the middle height section (Ф) is obtained as shown in the following equation:(15)Φ=εysc+εtsche=σysc+σtscEsche=2Py−PEscbhe2.According to the second assumption of the modified Jezek analytical method, the deflection curve of the RCFST column was defined as shown in the following equation:(16)y=Δsinπxl.From the definition ofФ, it can be concluded as shown in the following equation:(17)Φ=−y″l2=Δπ2l2=2Py−PEscbhe2.By combining equations (14) with (17), the internal and external stresses of the middle height cross section are balanced, and the deformation compatibility condition is transformed into P-Δ relationships as shown in the following equation:(18)Δh21−PPy−M+PΔPy2=2l2Py9bπ2Esc1−PPy3.From the critical equilibrium conditiondP/dΔ=0, it is found that(19)Δ=Py3Ph21−PPy−MPy.By combining equations (19) with (18), the peak value of eccentric compressive strength (Pu) is obtained as shown in the following equation:(20)Pu=bh3π2Esc12l21−2MhPy1−Pu/Py3.In equation (20), M=Pue0.In the derivation of equations (11) to (20), the calculated value of PE in the RCFST column in equation (1) is not applied; however, the RCFST is analyzed as an elastic-plastic material similar to steel. This assumption leads to the calculation error of deviation between the calculated results and the material properties of the RCFST. To solve this problem, the following corrections are necessary which are suitable for the calculation of the actual RCFST column.In this work, it was assumed thatα=Pu/Py, and PE0=π2Escbh3/12l2 is Euler critical stress of long column. Further, the ideal elastic-plastic constitutive relation of steel was modified by using λ of the RCFST with equation (1) to replace PE0 in equation (20) and Pu was obtained as shown in the following equation:(21)Pu=PE1−2MhPy1−α3.SinceM in equation (21) is also a function of Pu, equation (21) had no explicit solution and had to be solved using the Newton interpolation method.Further,λ was regularized as λ¯2=Py/PE [34], and equation (21) was reduced to the following equation:(22)αλ¯2=1−2αγ1−α3.According to equation (22), λ¯ was expressed as a polynomial of γ after regularization. The above equations were established based on the conditions of σtsc≤fscp and he>1−α.From equations (14) and (21), it was concluded as shown in the following equations:(23)heh=1−2αγ1−α>1−α,(24)γ<1−α2.


## 3.3. RCFST Column under Compression and Tension Yield Instability Modes on Both Sides

In this analysis, the RCFST column was subjected to both compression and tension yield instability modes on both sides of the specimen. The steel tube tension was expressed as shown in the following equation:(25)T=fsybt.The balance conditions of the internal and external stresses were calculated as shown in the following equations:(26)P−T=Py−12σysc+σtscbhe,(27)σysc+σtsc=2Py−P+Tbhe.From the moment balance condition, it was concluded as shown in the following equation:(28)M+PΔ=Th2+12σysc+σtscbheh2−he3.By substituting equation (27) into (28), the following equation was obtained:(29)M+PΔ=Th2+Py−P+Th2−he3.From equation (29), the expression of he was obtained as shown in the following equation:(30)he=3h2−6M+6PΔ−3Th2Py−P+T.Further, the curvature of the middle height section (Ф) was obtained as shown in the following equation:(31)Φ=−y″l2=Δπ2l2=εysc+εtsche=σysc+σtscEsche=2Py−P+TEsche2.By substituting equation (30) into (31), the relationship of P-Δ was obtained as shown in the following equation:(32)Δh21−PPy−M+PΔ+ThPy2=2l2Py9π2bEsc1−P+TPy3.From the critical equilibrium conditiondP/dΔ=0, ∆ was obtained as shown in the following equation:(33)Δ=Py3Ph21−PPy−M+ThPy.By substituting into equation (32), it can be concluded as shown in the following equation:(34)Pu=PE1−2M+ThhPy1−P/Py−P/Py3.Assuming thatT/Py=η, equation (34) was reduced to the following equation:(35)αλ¯2=1−2αγ+η1−α−η3.The above equations were established under the conditions ofσtsc≥σtsc and he/h≤1−α.From equation (30), we know that(36)heh≤1−α+η.When equations (30) and (33) were substituted into equation (36), the following results were obtained as shown in the following equation:(37)α≥1−1−α+η2−2αγ+2η.Due to the complexity of the above reasoning process, the intermediate calculation results of the process were expressed in the form of a diagram, as shown in Figure5. In Figure 5, the horizontal axis represents the horizontal deflection (Δ) at the middle height section of the RCFST column, and the longitudinal axis represents the strength (P) under various assumptions. The distance from o to o' represents the initial eccentricity e0. The gradient of the line represents the initial elastic modulus (Esc) of the RCFST. The top point (PE0, ΔE0) represents the Euler critical stress of the RCFST column under the assumptions of the steel tube and the CCC as a kind of unified ideal elastic-plastic material.Figure 5
Instability mode calculation process.Further, (PE, ΔE) represents the Euler critical stress corrected by λ. (Py, Δy) represents the compressive strength of the RCFST column at plastic yield. (Pe, Δe) represents the strength of the initial elastic stage. (Pu, Δu) represents the calculated compressive strength of the two instability modes proposed in this paper. The curve represents the experimental curve of [17, 18]. From the position height relationship of the above points, it is observed that the Eulerian critical stress PE0 under the ideal elastic-plastic instability mode was the highest, while PE after λ correction reflecting the characteristics of RCFST composite material was lower than PE0. Furthermore, the compressive strength Py that was calculated by area weighting was smaller than PE according to the plastic yield strength of the steel tube and the concrete. The actual ultimate compressive strength Pu of the RCFST was found to be between Pe and Py.From Figure5, it is also observed that the calculation method proposed in this paper can perfectly and scientifically explain the instability mode of the RCFST columns under eccentric compression by the coupling action of γ and λ.


## 4. Comparison of the Test Results

As equations (21) and (34) were both implicit functions of Pu, based on Newton’s interpolation method, the Maple program was used to calculate Pu under two instability modes. The program flow is shown in Figure 6 [35].Figure 6
Maple program flowchart.The physical and mechanical indexes and the geometric parameters of the RCFST columns are shown in Tables1 and 2.The analytical results were compared with the samples of the RCFST columns under the coupling action ofγ and λ in [16, 17]. The results are shown in Figure 7 and Table 1. From the results, it was observed that the calculation formula and the program proposed in this paper ensured that the error between theoretical calculation results and samples is within 10%, and thereby met the needs of the detailed design of the RCFST column.Figure 7
Comparison of sample and calculation results.From equations (21) and (34), it was concluded that Pu is related to the square of λ and the cube of γ. Therefore, the influence of γ on Pu was found to be far greater than the weight of λ. This was verified by the analytical results in Table 1 and Figure 7.


## 5. Conclusions

In this paper, the modified Jezek analytical method was developed using the Maple program to calculate the RCFST columnsPu. The RCFST column instability modes were divided into two types on the yield states of the steel tube tension side. Further, the corresponding P-ε curve relationship was studied for each RCFST system. The following conclusions were drawn from the analytical method performed for the RCFST column systems:(1)
The effect of theγ on the peak value of the P-ε curve relationship of the RCFST long columns was more significant than that of λ. High γ contributed to a more rapid decrease in the peak value of the P-ε curve of the RCFST long columns during the compressive loading processes. This facilitated the RCFST long column systems to reach higher nonlinear characteristics before softening and lower residual strength after softening.(2)
The RCFST column instability modes were divided into two types based on the yielding of the tensile side of the steel tube under variousγ. In the case of lower eccentricity, such as γ = 0.13 and 0.23, the instability modes of the RCFST long columns exhibited unilateral compression. However, at higher eccentricity, such as γ = 0.33, and 0.43, the instability modes of the RCFST long columns were both sides yield instability modes.(3)
Theλ calculated of the RCFST column was converted as a composite material, and the Euler critical stress that was calculated based on the ideal elastic-plastic body of steel was replaced by the composite material, which made the theoretical calculation of instability modes of the RCFST column possible.(4)
The ratio of the yield stress on the tensile side of the steel tabular to the plastic yield strength of the full section of the RCFST column was introduced, so that the two modes of instability of the RCFST column could be transformed into dimensionless normalization formula. This not only expanded the application scope of the Jezek analytical method in the calculation of the RCFST column instability but also ensured that the error of the above theoretical calculation was less than 10% by comparing with many groups of reference samples [16, 17].



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*Source: 6530425-2021-10-27.xml*