# Bearing Capacity Studies on Square Steel Tube Confined Steel Reinforced Concrete Column under Eccentric Load

Authors: Tongfeng Zhao
Journal: Advances in Civil Engineering
 (2020)

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## Abstract

In the present study, an experimental research was conducted on square steel tube confined steel reinforced concrete column under eccentric load. The major parameters of the specimens included slenderness ratio, eccentricity ratio, and structural steel reinforced ratio. According to the tested results, the eccentricity ratio, from 0 to 0.55, significantly affects the structural bearing capacity. The slenderness ratio, from 3 to 8, and steel reinforced ratio, from 0.3 to 0.41, slightly affect the capacity. Furthermore, a numerical analysis program was developed, and the calculated results are well consistent with the experimental results. Also, the theoretical formula for eccentrically loaded columns was proposed based on numerical results.


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## Body

## 1. Introduction

The composite steel-concrete columns have been used in high-rise and long-span buildings extensively. With the booming of urban construction, columns should bear higher load owing to the height and span extension of buildings. They should exhibit the characteristics, e.g., high strength, small cross section, high ductility, and convenience of construction. Also, they must be economical and practical to fit the safety and utilization of buildings.There are now two traditional types of composite columns, namely, the steel tube columns filled with concrete and steel reinforced concrete.For the steel tube filled with concrete, effective lateral confinement has been provided to concrete by steel tube to improve the overall ductility and strength. In the meantime, steel tube can serve as a formwork, thereby accelerating the construction process. The concrete around the steel tube can defer the local buckling of the steel tube. However, steel tube filled with concrete has a lower fire resistance than concrete-encased steel columns. On the other hand, the existence of steel in reinforced concrete can improve the shear resistance and antiseismic performance of the column. The encasement of concrete makes the steel more effective against both local and overall buckling, which enhances the stiffness of the specimen. Also, the concrete can provide protection for the steel against chemical corrosion and fire. The major disadvantage of steel reinforced concrete is that reinforced cage and formwork should be set up, thereby making the construction process complicated and construction speed slow.Based on the studies of steel tube filled with concrete and steel reinforced concrete, a novel composite column of steel tube confined steel reinforced concrete (STSRC) has been studied in recent years [1–5]. The composite column is formed by inserting structural steel reinforced into the steel tube. Subsequently, the whole case is filled with concrete. It maintains the advantages of both steel tubes confined concrete and steel reinforced concrete, e.g., high capacity, high strength, good fire resistance, smaller section, and good shear resistance as well as antiseismic property.According to the sectional shape of the steel tube, STSRC can be separated into cylinder, square, and rectangular types. I-shape and cross shape are divided according to reinforced steel (Figure1).Figure 1
The section types of STSRC.
(a)
(b)
(c)
(d)
(e)
(f)The major confinement of square steel tube occurs at the corners and core position. Thus, its effect of confinement and bearing capacity is lower than that of the circular steel tube. But it is found that confinement to core concrete provided by square tube can improve the ductility of the specimen. In this way, square steel tube-confined concrete embodies almost all the advantages of circular steel tube-confined concrete to a great extent. Furthermore, square tube-confined concrete has the advantages of large stiffness of bending, simplicity of node domain, convenience of connection, and easiness for indoor layout. It can be confirmed that economic and social benefits are achieved in practical engineering.Patel et al. studied the verification of a multiscale numerical model described in a companion paper and an extensive parametric study on the performance of high-strength thin-walled rectangular CFST slender beam-columns under biaxial loads. Comparisons of computer solutions with existing experimental results were made to examine the accuracy of the multiscale numerical model developed [6]. Zhao et al. experimentally studied the effect of axial compressive load and eccentric load. According to the results, the propagation of cracks in core concrete can be put off or avoided by the inserted structural steel, so the ultimate bearing capacity and ductility of the composite column are effectively improved [1, 2]. Jung and Mahmoud developed a new direct tension test setup for cylindrical specimens similar to those used in the compression test. The experimental and numerical results of the cracking location showed that gravity effect on fresh concrete during setting time might affect the distribution of concrete cracking strength along the height of the structural elements [7]. Zhao and Wang studied the calculation on compression-bending, axial loading, and pure bending of square tube confined steel reinforced high-strength concrete. Formula for compression-bending was proposed based on the test results [3–5]. Han et al. carried out a series of tests on concrete-filled SHS (square hollow section) stub columns, columns, and beam-columns. A unified theory is described where a confinement factor is introduced to describe the composite action between the steel tube and filled concrete. Simplified models are derived for section capacities and modulus in different stages of the composite sections [8, 9]. Mossahebi et al. carried out an investigation in an effort to comprehend the behavior of a bridge girder consisting of a steel tube confined reinforced concrete. Results of this investigation demonstrated the potential for using a concrete-filled tube as a bridge girder [10]. In addition, there are a large number of research studies on the behavior of steel tube-confined concrete columns under eccentric compression, and also numerical modeling program was developed to study different characteristics of columns [11–19].Though scholars carry out some research on STSRC, Zhao and Wang [3] deduced the formulas for calculating eccentric bearing capacity using superposition method, but it is too complicated to use in practice. Zhao et al. [1] only studied the formulas for circular columns under eccentric loading. It indicates further research on STSRC is needed, such as more experiments to study its properties and simplified calculation. In this study, an experimental research was conducted on square steel tube column confined steel reinforced concrete subjected to eccentric load. Furthermore, numerical analysis program was developed. Also, theoretical formula for eccentrically loaded columns was proposed by analyzing primary parameters and numerical results. Thus, the safety requirements for engineering structure under eccentricity can be met.


## 2. Test Method

### 2.1. Survey of Test

A total of nine specimens were constructed and tested under eccentric load. A summary of the specimens is presented in Table1. The width of the stub specimens (B) is 200 mm, and the thickness (t) is 5 mm, respectively. Tension tests were carried out to determine the material properties of the steel tubes and steel section. The cross sections of steel tube are cold-formed. The tensile coupons are from the sidewalls of the steel tubes. The yield strength (fsy) and ultimate tensile strength (fsy) of the steel tube are 350 MPa and 425 MPa, respectively. The tensile coupons are from the flanges of the steel section. The yield strength (fsy) and ultimate tensile strength (fsu) of the steel section are listed in Table 1. The compressive cube strength (λ) of concrete is 48.4 MPa. Cross section of the steel section is I-shape. The major parameters of the specimens included slenderness ratio λ (λ=L/B), from 3 to 8, eccentricity ratio e/r (e is initial eccentricity r=B/2), from 0 to 0.55, and steel section index ρ (ρ=Asfsy/Acfck, where As and Ac are the area of concrete and steel cross section, respectively), from 0.3 to 0.41. Confinement index θ (θ=Atfty/Acfck, where At is the area of steel tube) is 0.87.Table 1
Tested parameters and results.
Specimen no.
L (mm)
A
s mm2
e
/
r
f
sy (MPa)
N
ue (kN)
N
us (kN)
(kN)
N
ue
/
N
us
PY10I-0-3
600
1433
0
355
420
0.3
3740
—
—
—
—
PY10I-25-3
600
1433
0.25
355
420
0.3
3216
2888
2870
1.114
1.121
PY10I-40-3
600
1433
0.4
355
420
0.3
2693
2503
2490
1.076
1.082
PY10I-55-3
600
1433
0.55
355
420
0.3
2365
2209
2205
1.071
1.073
PY10I-40-4
800
1433
0.4
355
420
0.3
2670
2468
2457
1.082
1.087
PY10I-40-6
1200
1433
0.4
355
420
0.3
2608
2375
2370
1.103
1.110
PY10I-40-8
1600
1433
0.4
355
420
0.3
2512
2297
2268
1.094
1.108
PY12I-40-3
600
1810
0.4
320
425
0.35
2725
2555
2500
1.067
1.090
PY14I-40-3
600
2150
0.4
420
545
0.41
2833
2686
2651
1.055
1.069In this table, 10I, 12I, and 14I denote the height of steel sections which are 100 mm, 126 mm, and 140 mm, respectively. Detailed information is available in [20]. Nue is the tested bearing capacity; Nus is the numerical bearing capacity; Nuc is the calculated bearing capacity of equation (3) in Section 3.The manufacturing process of the specimens that experienced several phases is as follows.First, the steel tube and steel section were accurately cut to size, and then the end of the steel section was welded to an end plate (thickness is 25 mm), focusing on keeping the steel section perpendicular to the end plate. Subsequently, strain gauges were glued to the estimated area. Moisture-proof processing was performed using epoxy resin. Steel tube was welded carefully to the end plate to assure that the steel reinforced can lie in the center of the steel tube.Second, concrete was poured into the tube gradually, and vibrating stick was used to compact the concrete. A very small amount of longitudinal shrinkage occurred at the top of the specimens during the curing process. Thus, before the test, the top surface of the specimens was covered with cement to ensure that steel tube, concrete, and steel reinforced could bear the load simultaneously.Finally, ribbed slab was welded to solidify the ends of the specimen (Figure2) to avoid its breaking prior to other components.Figure 2
The layout of test.


### 2.2. Test Setup and Procedures

All the specimens were tested using a 5000 kN capacity universal testing machine. Eccentric load was imposed on the specimens through the groove hinge and the spherical hinge.Figure2 shows that strain gauges are distributed in the midheight of the specimen. Eight longitudinal strain gauges were placed at the external surface of the steel tube, and four transverse strain gauges were placed at the axial of the steel tube. Another three longitudinal strain gauges were placed at the flanges and webs of steel sections. Three displacement transducers were placed at the midheight and the quarter-height of the specimen along the bending direction. A displacement transducer was used to monitor the axial deformation. All the data were collected through UCAM-70A automatically.Stepped load approach was used in the test. A load of one-tenth of the computed ultimate bearing capacity was applied when the specimen was in elastic range; a load of one-twentieth of the computed ultimate bearing capacity was applied when the load reached 60% of the computed ultimate bearing capacity. Finally, the load was applied continuously until it reached the computed ultimate bearing capacity. The test was completed when the applied load was less than 80% of the ultimate bearing capacity.


### 2.3. Results and Discussions

There were no significant changes before the eccentric load surpassed 60% of the ultimate bearing capacity. Many diagonal tiny wrinkles were observed from the surface of the steel tube after the axial load surpassed 60% of the ultimate bearing capacity. Sounding occurred when load was close to the ultimate bearing capacity due to cracking of the concrete. Local buckling can be observed in the midheight and quarter-height regions along the length of the specimen. Iron scraps dropped from the surface of the steel tube close to the local buckling regions. Figure3 gives a general view of the specimens after the test.Figure 3
The broken specimens under eccentric load.The load (N) versus extreme fiber compressive and tensile strains (ε) are shown in Figure 4 for steel tube and steel section, respectively. When the load was small, the strain at midheight is approximately proportional to the applied load. When the load reaches about 60–80% of the maximum load, the lateral strain at midheight started to increase significantly, due to the steel section is located near the center of the whole section. When the load reaches about 70–90% of the maximum load, the lateral strain at midheight started to increase significantly.Figure 4
Curves of load-longitudinal strain on (a) steel tube and (b) steel section.
(a)
(b)It was found that, in general, the measured strain of typical specimen (PY10I-40-3) conforms to the assumption of plane (Figure5), and the measured deflection curves are in the shape of a half-sine wave (Figure 6).Figure 5
Distribution of longitudinal strain at midsection.Figure 6
Distribution of deflection along length.Curves of Poisson ratio (ν) versus N/Nu are shown in Figure 7. In general, when N/Nu<0.7, the Poisson ratio of concrete is less the Poisson ratio of steel tube (0.3). Above this load level, the steel tube walls at transverse strain gauges begin to confine the concrete (ν>0.3).Figure 7
Curves ofN/Nu−v.Curves ofN versus midheight deformation (um) are plotted according to the tested data (Figure 8). It can be stratified into elastic, elastic-plastic, and plastic phases. Deflection of slender specimens was insignificant before the load reached 0.6-0.7 Nu with a linear relationship of N and um. Above this load level, the deformation of midheight increased gradually. The deflection increased rapidly when the load approached the Nu. This is considered the elastic-plastic phase. The increased lateral displacement caused the effect of the secondary moment to grow stronger. The load began to drop and the curve of N versus um fell into descending phase. This is considered the plastic phase.Figure 8
Comparisons between the calculated and tested results.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)The influences of primary parameters on ultimate bearing capacity (Figure9) show that the bearing capacity decreased when the eccentricity increased. As the eccentricity changes from 0 to 55, the bearing capacity drops to 40%. As the slenderness ratio changes from 3 to 6, the bearing capacity drops to only 7%. It is shown that a slenderness ratio has a negligible effect on the bearing capacity in tested parameters. This is because the STSRC has high bending stiffness. As steel reinforced changes from 10I to 14I, the steel reinforced ratio increased from 0.3 to 0.41 and bearing capacity increased to only 5%. It also shows that the eccentricity ratio has a significant influence on axial deformation Δ. Slenderness ratio and steel section ratio have little influence on axial deformation.Figure 9
Influence of parameters onN-Δ: (a) slenderness ratios, (b) eccentricities, and (c) steel section ratios.
(a)
(b)
(c)


## 2.1. Survey of Test

A total of nine specimens were constructed and tested under eccentric load. A summary of the specimens is presented in Table1. The width of the stub specimens (B) is 200 mm, and the thickness (t) is 5 mm, respectively. Tension tests were carried out to determine the material properties of the steel tubes and steel section. The cross sections of steel tube are cold-formed. The tensile coupons are from the sidewalls of the steel tubes. The yield strength (fsy) and ultimate tensile strength (fsy) of the steel tube are 350 MPa and 425 MPa, respectively. The tensile coupons are from the flanges of the steel section. The yield strength (fsy) and ultimate tensile strength (fsu) of the steel section are listed in Table 1. The compressive cube strength (λ) of concrete is 48.4 MPa. Cross section of the steel section is I-shape. The major parameters of the specimens included slenderness ratio λ (λ=L/B), from 3 to 8, eccentricity ratio e/r (e is initial eccentricity r=B/2), from 0 to 0.55, and steel section index ρ (ρ=Asfsy/Acfck, where As and Ac are the area of concrete and steel cross section, respectively), from 0.3 to 0.41. Confinement index θ (θ=Atfty/Acfck, where At is the area of steel tube) is 0.87.Table 1
Tested parameters and results.
Specimen no.
L (mm)
A
s mm2
e
/
r
f
sy (MPa)
N
ue (kN)
N
us (kN)
(kN)
N
ue
/
N
us
PY10I-0-3
600
1433
0
355
420
0.3
3740
—
—
—
—
PY10I-25-3
600
1433
0.25
355
420
0.3
3216
2888
2870
1.114
1.121
PY10I-40-3
600
1433
0.4
355
420
0.3
2693
2503
2490
1.076
1.082
PY10I-55-3
600
1433
0.55
355
420
0.3
2365
2209
2205
1.071
1.073
PY10I-40-4
800
1433
0.4
355
420
0.3
2670
2468
2457
1.082
1.087
PY10I-40-6
1200
1433
0.4
355
420
0.3
2608
2375
2370
1.103
1.110
PY10I-40-8
1600
1433
0.4
355
420
0.3
2512
2297
2268
1.094
1.108
PY12I-40-3
600
1810
0.4
320
425
0.35
2725
2555
2500
1.067
1.090
PY14I-40-3
600
2150
0.4
420
545
0.41
2833
2686
2651
1.055
1.069In this table, 10I, 12I, and 14I denote the height of steel sections which are 100 mm, 126 mm, and 140 mm, respectively. Detailed information is available in [20]. Nue is the tested bearing capacity; Nus is the numerical bearing capacity; Nuc is the calculated bearing capacity of equation (3) in Section 3.The manufacturing process of the specimens that experienced several phases is as follows.First, the steel tube and steel section were accurately cut to size, and then the end of the steel section was welded to an end plate (thickness is 25 mm), focusing on keeping the steel section perpendicular to the end plate. Subsequently, strain gauges were glued to the estimated area. Moisture-proof processing was performed using epoxy resin. Steel tube was welded carefully to the end plate to assure that the steel reinforced can lie in the center of the steel tube.Second, concrete was poured into the tube gradually, and vibrating stick was used to compact the concrete. A very small amount of longitudinal shrinkage occurred at the top of the specimens during the curing process. Thus, before the test, the top surface of the specimens was covered with cement to ensure that steel tube, concrete, and steel reinforced could bear the load simultaneously.Finally, ribbed slab was welded to solidify the ends of the specimen (Figure2) to avoid its breaking prior to other components.Figure 2
The layout of test.


## 2.2. Test Setup and Procedures

All the specimens were tested using a 5000 kN capacity universal testing machine. Eccentric load was imposed on the specimens through the groove hinge and the spherical hinge.Figure2 shows that strain gauges are distributed in the midheight of the specimen. Eight longitudinal strain gauges were placed at the external surface of the steel tube, and four transverse strain gauges were placed at the axial of the steel tube. Another three longitudinal strain gauges were placed at the flanges and webs of steel sections. Three displacement transducers were placed at the midheight and the quarter-height of the specimen along the bending direction. A displacement transducer was used to monitor the axial deformation. All the data were collected through UCAM-70A automatically.Stepped load approach was used in the test. A load of one-tenth of the computed ultimate bearing capacity was applied when the specimen was in elastic range; a load of one-twentieth of the computed ultimate bearing capacity was applied when the load reached 60% of the computed ultimate bearing capacity. Finally, the load was applied continuously until it reached the computed ultimate bearing capacity. The test was completed when the applied load was less than 80% of the ultimate bearing capacity.


## 2.3. Results and Discussions

There were no significant changes before the eccentric load surpassed 60% of the ultimate bearing capacity. Many diagonal tiny wrinkles were observed from the surface of the steel tube after the axial load surpassed 60% of the ultimate bearing capacity. Sounding occurred when load was close to the ultimate bearing capacity due to cracking of the concrete. Local buckling can be observed in the midheight and quarter-height regions along the length of the specimen. Iron scraps dropped from the surface of the steel tube close to the local buckling regions. Figure3 gives a general view of the specimens after the test.Figure 3
The broken specimens under eccentric load.The load (N) versus extreme fiber compressive and tensile strains (ε) are shown in Figure 4 for steel tube and steel section, respectively. When the load was small, the strain at midheight is approximately proportional to the applied load. When the load reaches about 60–80% of the maximum load, the lateral strain at midheight started to increase significantly, due to the steel section is located near the center of the whole section. When the load reaches about 70–90% of the maximum load, the lateral strain at midheight started to increase significantly.Figure 4
Curves of load-longitudinal strain on (a) steel tube and (b) steel section.
(a)
(b)It was found that, in general, the measured strain of typical specimen (PY10I-40-3) conforms to the assumption of plane (Figure5), and the measured deflection curves are in the shape of a half-sine wave (Figure 6).Figure 5
Distribution of longitudinal strain at midsection.Figure 6
Distribution of deflection along length.Curves of Poisson ratio (ν) versus N/Nu are shown in Figure 7. In general, when N/Nu<0.7, the Poisson ratio of concrete is less the Poisson ratio of steel tube (0.3). Above this load level, the steel tube walls at transverse strain gauges begin to confine the concrete (ν>0.3).Figure 7
Curves ofN/Nu−v.Curves ofN versus midheight deformation (um) are plotted according to the tested data (Figure 8). It can be stratified into elastic, elastic-plastic, and plastic phases. Deflection of slender specimens was insignificant before the load reached 0.6-0.7 Nu with a linear relationship of N and um. Above this load level, the deformation of midheight increased gradually. The deflection increased rapidly when the load approached the Nu. This is considered the elastic-plastic phase. The increased lateral displacement caused the effect of the secondary moment to grow stronger. The load began to drop and the curve of N versus um fell into descending phase. This is considered the plastic phase.Figure 8
Comparisons between the calculated and tested results.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)The influences of primary parameters on ultimate bearing capacity (Figure9) show that the bearing capacity decreased when the eccentricity increased. As the eccentricity changes from 0 to 55, the bearing capacity drops to 40%. As the slenderness ratio changes from 3 to 6, the bearing capacity drops to only 7%. It is shown that a slenderness ratio has a negligible effect on the bearing capacity in tested parameters. This is because the STSRC has high bending stiffness. As steel reinforced changes from 10I to 14I, the steel reinforced ratio increased from 0.3 to 0.41 and bearing capacity increased to only 5%. It also shows that the eccentricity ratio has a significant influence on axial deformation Δ. Slenderness ratio and steel section ratio have little influence on axial deformation.Figure 9
Influence of parameters onN-Δ: (a) slenderness ratios, (b) eccentricities, and (c) steel section ratios.
(a)
(b)
(c)


## 3. Theoretical Analysis

### 3.1. Basic Assumption and Constitutive Model

(1) The strain distribution at midheight conforms to plane cross section assumption. (2) There existed no interfacial slip between the steel and the concrete. (3) The effect of shear on the deformation of the specimen was neglected. (4) Internal and external forces were at equilibrium at midheight only. (5) Hinged connection existed at the ends of the specimen with deflection curve conforming to half-sinusoid.The constitutive laws of steel and confined concrete were referenced from the study of Zhao and Wang [3].


### 3.2. Principle and Procedure of Calculation

The cross section of steel tube, concrete, and steel reinforced were stratified into many strips with homogeneously distributed stress. Assume that the strain at the center of cross section isεc, the stress of concrete σci, steel reinforced section σsi, and steel tube σti can be calculated based on the plane cross section assumption and the stress-strain relationship. Two iterative equations were given for force and moment, respectively:(1)N=∑i=1nσciΔAci+∑i=1mσsiΔAsi+∑i=1nσtiΔAti,(2)M=∑i=1nσciΔAciyci+∑i=1mσsiΔAsiysi+∑i=1nσtiΔAtiyti,where n is the number of strips of concrete and steel tube, m is the number of strips of steel reinforced; yci,ysi, and yti refer to the distance from the center of concrete strip, steel reinforced, and steel tube to the center of whole cross section, respectively; ΔAci, ΔAsi, and ΔAti are the areas of concrete strip, steel reinforced, and steel tube, respectively.Figure10 presents the algorithm for the whole computational procedure. First, the stepped deformation Δum is applied to specimen, and then curvature ϕ is obtained based on the 5th assumption. Internal force Nin and moment Min are calculated by equation (1) and equation (2). Print the results when differences between internal force and external force are less than error D. Once bearing capacity drops to 80% of the ultimate bearing capacity or when the tensile strain of the steel tube reaches 10000με, the computation stops.Figure 10
The algorithm of calculation.


### 3.3. Comparisons between Calculated and Tested Results

Eccentric load calculation program PYZTF was developed in accordance with the theories above.N versus um relationship curves were yielded by inputting the data to the program. Comparison between the calculated results and tested data is drawn in Figure 8, and it is shown that they are consistent well with each other.


### 3.4. Simplified Calculation Method of Ultimate Bearing Capacity

Though the ultimate bearing capacity can be calculated accurately by program PYZTF, it is not adequate for practical application owing to the relatively complicated procedure. Thus, a simplified method is required to calculate the ultimate bearing capacity.First, parameters including initial eccentricityeo were inputted to the program. Next, the ultimate force Nu and corresponding moment Mu could be obtained. In the coordinate system, Nu/No was set as the perpendicular coordinate axis and Nu/No as horizontal coordinate axis. Thus, coordinate point Nu/No,Nu/No was found. By changing Eo gradually, curves of Nu/No versus Nu/No can be plotted with different parameters. For instance, parameters of specimen in Figure 11 are λ = 6, fck = 30 MPa, θ = 0.87, and ρ = 0.32.Figure 11
Influence of parameters onNu/No−Nu/No.
(a)
(b)
(c)
(d)Simplified curves ofNu/No versus Nu/No can be separated into two lines AB and BC (Figure 12) according to [3]. The accuracy of columns under small eccentricity and the safety under big eccentricity of both results were considered. Dashed line in Figure 12 represents a typical Nu/No versus Nu/No relationship curve.Figure 12
The simplified curve forNu/No–Nu/No.The equation for ultimate bearing capacity was proposed using the regression method while considering the effect of reduction factor due to the slenderness of column (φ) on ultimate bearing capacity:(3)Nu=φ1/N0+ηe0/M0,NN0≥φ−a,Nu=M0ηe0,NN0<φ−a,where η=1+0.003λ1.672e0/B−0.003λ1.5, a=0.375+0.746exp−1.394/0.566θ+ρ⋅φ.According to Zhao and Wang [3], the ultimate bearing capacity of the stub column under axial compression is given by N0=1+γθ+ρAcfck,.γ=1.166+0.912exp−3.57θ as(4)φ=1−0.65λ¯,λ¯≤0.215,2λ¯20.965+0.3λ¯+λ¯2−0.965+0.3λ¯+λ¯22−4λ¯2,λ¯>0.215,where λ¯ denotes the relative slenderness ratio, λ¯=L/πAcfc+1.2Atfty+Asfsy/0.6EcIc+EtIt+EsIs.According to Mossahebiet al. [9], the ultimate moment of the column under pure bending is given by M0=0.5ξ⋅B⋅N0, where ξ denotes the plastic development coefficient, ξ=0.134+0.613θ−0.158θ⋅0.81+0.06ρI/Is−0.048.I, Ic, It, and Is denote the second moment of whole area, concrete, steel tube, and steel cross section, respectively, and Ec, Et, and Es denote elastic modulus of concrete, steel tube, and steel, respectively.The formula can be applied once the following condition is met:λ = 3–30, fck = 20–60 MPa, fty = 235–450 MPa, fsy = 235–450 MPa, θ = 0–3, and ρ = 0-1.The tested parameters were inputted into program PYZTF and equation (3), respectively. The results are listed in Table 1. When the average value of Nue/Nus is 1.083, the mean square deviation is up to 0.020. When the average value of Nue/Nuc is 1.093, the mean square deviation reaches 0.019. Experimental data appear to be higher than the corresponding theoretical value. It was estimated that friction was produced by groove hinge and spherical hinges under high axial compression. Thus, the constraint to both ends of specimens was provided, and freedom of the specimen was restrained. Furthermore, the increased end decreased the geometrical length of columns, causing further increase in the bearing capacity.Data of specimens (λ = 6, fck = 30 MPa, θ = 0.87, and ρ = 0.32) were input into the simplified model and program PYZTF. Simplified N-M integration curves were verified by the fiber theoretical model (Figure 13). It can be found that, generally, good agreement is obtained between the fiber theoretical model and simplified model. Simplified model is acceptable to predict the bearing capacity of STRSC under eccentric load.Figure 13
Comparisons between the fiber theoretical model and the simplified curves ofNu/No−Mu/Mo.
(a)
(b)
(c)
(d)


## 3.1. Basic Assumption and Constitutive Model

(1) The strain distribution at midheight conforms to plane cross section assumption. (2) There existed no interfacial slip between the steel and the concrete. (3) The effect of shear on the deformation of the specimen was neglected. (4) Internal and external forces were at equilibrium at midheight only. (5) Hinged connection existed at the ends of the specimen with deflection curve conforming to half-sinusoid.The constitutive laws of steel and confined concrete were referenced from the study of Zhao and Wang [3].


## 3.2. Principle and Procedure of Calculation

The cross section of steel tube, concrete, and steel reinforced were stratified into many strips with homogeneously distributed stress. Assume that the strain at the center of cross section isεc, the stress of concrete σci, steel reinforced section σsi, and steel tube σti can be calculated based on the plane cross section assumption and the stress-strain relationship. Two iterative equations were given for force and moment, respectively:(1)N=∑i=1nσciΔAci+∑i=1mσsiΔAsi+∑i=1nσtiΔAti,(2)M=∑i=1nσciΔAciyci+∑i=1mσsiΔAsiysi+∑i=1nσtiΔAtiyti,where n is the number of strips of concrete and steel tube, m is the number of strips of steel reinforced; yci,ysi, and yti refer to the distance from the center of concrete strip, steel reinforced, and steel tube to the center of whole cross section, respectively; ΔAci, ΔAsi, and ΔAti are the areas of concrete strip, steel reinforced, and steel tube, respectively.Figure10 presents the algorithm for the whole computational procedure. First, the stepped deformation Δum is applied to specimen, and then curvature ϕ is obtained based on the 5th assumption. Internal force Nin and moment Min are calculated by equation (1) and equation (2). Print the results when differences between internal force and external force are less than error D. Once bearing capacity drops to 80% of the ultimate bearing capacity or when the tensile strain of the steel tube reaches 10000με, the computation stops.Figure 10
The algorithm of calculation.


## 3.3. Comparisons between Calculated and Tested Results

Eccentric load calculation program PYZTF was developed in accordance with the theories above.N versus um relationship curves were yielded by inputting the data to the program. Comparison between the calculated results and tested data is drawn in Figure 8, and it is shown that they are consistent well with each other.


## 3.4. Simplified Calculation Method of Ultimate Bearing Capacity

Though the ultimate bearing capacity can be calculated accurately by program PYZTF, it is not adequate for practical application owing to the relatively complicated procedure. Thus, a simplified method is required to calculate the ultimate bearing capacity.First, parameters including initial eccentricityeo were inputted to the program. Next, the ultimate force Nu and corresponding moment Mu could be obtained. In the coordinate system, Nu/No was set as the perpendicular coordinate axis and Nu/No as horizontal coordinate axis. Thus, coordinate point Nu/No,Nu/No was found. By changing Eo gradually, curves of Nu/No versus Nu/No can be plotted with different parameters. For instance, parameters of specimen in Figure 11 are λ = 6, fck = 30 MPa, θ = 0.87, and ρ = 0.32.Figure 11
Influence of parameters onNu/No−Nu/No.
(a)
(b)
(c)
(d)Simplified curves ofNu/No versus Nu/No can be separated into two lines AB and BC (Figure 12) according to [3]. The accuracy of columns under small eccentricity and the safety under big eccentricity of both results were considered. Dashed line in Figure 12 represents a typical Nu/No versus Nu/No relationship curve.Figure 12
The simplified curve forNu/No–Nu/No.The equation for ultimate bearing capacity was proposed using the regression method while considering the effect of reduction factor due to the slenderness of column (φ) on ultimate bearing capacity:(3)Nu=φ1/N0+ηe0/M0,NN0≥φ−a,Nu=M0ηe0,NN0<φ−a,where η=1+0.003λ1.672e0/B−0.003λ1.5, a=0.375+0.746exp−1.394/0.566θ+ρ⋅φ.According to Zhao and Wang [3], the ultimate bearing capacity of the stub column under axial compression is given by N0=1+γθ+ρAcfck,.γ=1.166+0.912exp−3.57θ as(4)φ=1−0.65λ¯,λ¯≤0.215,2λ¯20.965+0.3λ¯+λ¯2−0.965+0.3λ¯+λ¯22−4λ¯2,λ¯>0.215,where λ¯ denotes the relative slenderness ratio, λ¯=L/πAcfc+1.2Atfty+Asfsy/0.6EcIc+EtIt+EsIs.According to Mossahebiet al. [9], the ultimate moment of the column under pure bending is given by M0=0.5ξ⋅B⋅N0, where ξ denotes the plastic development coefficient, ξ=0.134+0.613θ−0.158θ⋅0.81+0.06ρI/Is−0.048.I, Ic, It, and Is denote the second moment of whole area, concrete, steel tube, and steel cross section, respectively, and Ec, Et, and Es denote elastic modulus of concrete, steel tube, and steel, respectively.The formula can be applied once the following condition is met:λ = 3–30, fck = 20–60 MPa, fty = 235–450 MPa, fsy = 235–450 MPa, θ = 0–3, and ρ = 0-1.The tested parameters were inputted into program PYZTF and equation (3), respectively. The results are listed in Table 1. When the average value of Nue/Nus is 1.083, the mean square deviation is up to 0.020. When the average value of Nue/Nuc is 1.093, the mean square deviation reaches 0.019. Experimental data appear to be higher than the corresponding theoretical value. It was estimated that friction was produced by groove hinge and spherical hinges under high axial compression. Thus, the constraint to both ends of specimens was provided, and freedom of the specimen was restrained. Furthermore, the increased end decreased the geometrical length of columns, causing further increase in the bearing capacity.Data of specimens (λ = 6, fck = 30 MPa, θ = 0.87, and ρ = 0.32) were input into the simplified model and program PYZTF. Simplified N-M integration curves were verified by the fiber theoretical model (Figure 13). It can be found that, generally, good agreement is obtained between the fiber theoretical model and simplified model. Simplified model is acceptable to predict the bearing capacity of STRSC under eccentric load.Figure 13
Comparisons between the fiber theoretical model and the simplified curves ofNu/No−Mu/Mo.
(a)
(b)
(c)
(d)


## 4. Conclusions

(1)
Eccentricity significantly affects the bearing capacity; big eccentricity should be avoided in practical engineering.
(2)
Slenderness ratio slightly affects capacity in tested parameters, i.e., slenderness ranges from 3 to 8. It is due to the high bending stiffness of STSRC.
(3)
Bearing capacity with adequate precision can be calculated by equation (3) without any repeated computation. This equation can express mechanical properties of columns under the eccentricity.



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*Source: 4212049-2020-04-15.xml*