# An Investigation of Compression Bearing Capacity of Concrete-Filled Rectangular Stainless Steel Tubular Columns under Axial Load and Eccentric Axial Load

^{*}

## Abstract

**:**

_{uf}

_{em}to the test N

_{uexp}is 0.985, and the variance is 0.000621. The slenderness ratio and relative eccentricity have a great influence on the load–displacement curves. The thickness of the stainless steel tube has little influence on the load–displacement curves. With the increase in slenderness ratio and relative eccentricity, the compression bearing capacity decreases. With the increase in the slenderness ratio, the failure model of the specimen gradually changes from plastic failure to elastoplastic failure and then elastic failure. When the slenderness ratio is the same, if the relative eccentricity is larger, increasing the thickness of the stainless steel tube will be more effective in improving the compression bearing capacity. When the relative eccentricity is the same, if the slenderness ratio is smaller, increasing the thickness of the stainless steel tube will be more effective for improving the compression bearing capacity. The slenderness ratio and relative eccentricity have a great influence on the longitudinal stress distribution in the cross-section. When the slenderness ratio and relative eccentricity are greater, the longitudinal compressive stress in parts of the cross-section gradually becomes longitudinal tensile stress. The proposed formula can effectively predict the compression bearing capacity of concrete-filled rectangular stainless steel tubular columns. The mean of theoretical calculations to the test and the finite element is 1.054, and the variance is 0.0247.

## 1. Introduction

## 2. Experiment Overview

_{y}was 534.3 MPa, 572.3 MPa, and 598.0 MPa, respectively. The core concrete in specimens was C40, the average cubic compressive strength f

_{cu}was 43.96 MPa, and the axial compressive strength f

_{c}was 29.48 MPa. The Eurocode 4, Eurocode 3, and Chinese code [25,26,27,28,29,30] were followed.

## 3. Finite Element Analysis

#### 3.1. Finite Element Model

#### 3.2. Finite Element Model Verification

_{u}

_{fem}to the test N

_{uexp}is 0.985, and the variance is 0.000621. Then it also can be seen from Figure 7 that there is a variance between the displacements with peak load by the finite element method and the test, and in the straight ascending segment period, the load by FEM is always greater than the test. It is because the steel tube and concrete poured in the test specimen are uneven, the compactness of the concrete is less than that of the finite element model, and the stiffness of the finite element model is greater than that of the test specimen. Overall, it is effective to use the FEM to calculate the compression bearing capacity of concrete-filled rectangular stainless steel tubular columns.

#### 3.3. Analysis of Finite Element Calculation

#### 3.3.1. Analysis of Load–Displacement Curves

#### 3.3.2. Analysis of Compression Bearing Capacity

#### 3.3.3. Analysis of Longitudinal Stress Distribution in the Central Cross-Section

- Analysis of longitudinal stress distribution of the stainless steel tubes in the central cross-section

_{y}, which is the yield strength of the stainless steel tube, at up to 1.020 f

_{y}. When the specimen is under an eccentric compression load, the longitudinal stress of the stainless steel tube does not have a uniform distribution in the cross-section. When the relative eccentricity is one, the maximum longitudinal compressive stress and the minimum longitudinal compressive stress of the stainless steel tube are 0.961 f

_{y}and 0.430 f

_{y}, respectively. With the increase in relative eccentricity, the neutralization axis gradually moves upward along the y-axis, and the longitudinal stress in parts of the cross-section gradually changes from compressive stress to tensile stress. When the relative eccentricity is 2.67, the maximum longitudinal compressive stress and the maximum longitudinal tensile stress do not exceed f

_{y}, and the maximum longitudinal compressive stress and the maximum longitudinal tensile stress are 0.72 f

_{y}and 0.463 f

_{y}, respectively.

_{y}and are 0.819 f

_{y}and 0.602 f

_{y}, respectively. When the relative eccentricity is one, the longitudinal stress of the stainless steel tube is divided into compressive stress and tensile stress on the cross-section, and the maximum longitudinal compressive stress and the maximum longitudinal tensile stress do not exceed f

_{y}, at 0.810 f

_{y}and 0.281 f

_{y}, respectively. When the relative eccentricity is 2.67, the maximum longitudinal compressive stress and the maximum longitudinal tensile stress of the stainless steel tube are 0.727 f

_{y}and 0.497 f

_{y}, respectively.

_{y}. When the specimen is under an eccentric compression load, the longitudinal stress of the stainless steel tube in the cross-section is not uniformly distributed. With the increase in relative eccentricity, the longitudinal stress in parts of the cross-section changes from compressive stress to tensile stress. When the relative eccentricity is one, the maximum longitudinal compressive stress and the minimum longitudinal compressive stress of the stainless steel tube are 1.078 f

_{y}and 0.258 f

_{y}, respectively. When the relative eccentricity is 1.33, the maximum longitudinal compressive stress and the maximum longitudinal tensile stress of the stainless steel tube are 0.890 f

_{y}and 0.129 f

_{y}, respectively.

_{y}and 0.321 f

_{y}, respectively. When the relative eccentricity is one, the maximum longitudinal compressive stress and the maximum longitudinal tensile stresses of the stainless steel tube are 0.642 f

_{y}and 0.496 f

_{y}, respectively. When the relative eccentricity is one, the maximum longitudinal compressive stress and the maximum longitudinal tensile stresses of the stainless steel tube are 0.622 f

_{y}and 0.462 f

_{y}, respectively.

- 2.
- Analysis of longitudinal stress distribution of core concrete in the central cross-section

_{c}, which is the axial compressive strength of the core concrete, at up to 2.021 f

_{c}. When the relative eccentricity is one, the maximum longitudinal compressive stress and the minimum longitudinal compressive stress of the core concrete are 1.379 f

_{c}and 0.782 f

_{c}, respectively. When the relative eccentricity is 2.67, the maximum longitudinal compressive stress and the minimum longitudinal compressive stress of the core concrete are 1.349 f

_{c}and 0.307 f

_{c}, respectively.

_{c}. When the relative eccentricity is one, the maximum longitudinal compressive stress and the maximum longitudinal tensile stress of core concrete are 0.89 f

_{c}and 0.138 f

_{t}, respectively. When the relative eccentricity is 2.67, the maximum longitudinal compressive stress and the maximum longitudinal tensile stress of core concrete are 0.865 f

_{c}and 0.951 f

_{t}, respectively.

_{c}. With the increase in relative eccentricity, the longitudinal stress distribution status also changes significantly. When the relative eccentricity is one, the maximum longitudinal compressive stress of the core concrete is 2.061 f

_{c}, and it is mainly located in the two corner areas. When the relative eccentricity is 1.33, the longitudinal stress in more areas of the cross-section of the core concrete is smaller. It can be seen from Figure 12j–l that when the slenderness ratio is 48, the longitudinal stress distribution of the core concrete in the cross-section is similar to that of the specimen 120604C40 under the same slenderness ratio. When the relative eccentricity is 0, the maximum longitudinal compressive stress and the minimum longitudinal compressive stress of the core concrete are 1.529 f

_{c}and 0.653 f

_{c}, respectively. When the relative eccentricity is one, the maximum longitudinal compressive stress and the maximum longitudinal tensile stress of the core concrete are 1.124 f

_{c}and 0.933 f

_{t}, respectively. When the relative eccentricity is 1.33, the maximum longitudinal compression stress and the maximum longitudinal tensile stress of the core concrete are 1.049 f

_{c}and 0.942 f

_{t}, respectively.

## 4. Calculation Formula of Compression Bearing Capacity

#### 4.1. Failure Mode Analysis

#### 4.2. Calculation Method of Compression Bearing Capacity

#### 4.2.1. Calculation Formula of Compression Bearing Capacity of Short Column under Axial Compression Load

_{su}is the compression bearing capacity of the short column; N

_{ss}is the compression bearing capacity of the stainless steel tube; N

_{cc}is the compression bearing capacity of the core concrete; α is the yield strength improvement coefficient of the stainless steel tube; β is the compression strength improvement coefficient of the core concrete; A

_{ss}is the cross-sectional area of the stainless steel tube; A

_{cc}is the cross-sectional area of the core concrete; ξ is the constraint effect coefficient, and the calculation method is shown in [34].

#### 4.2.2. Calculation Formula of Compression Bearing Capacity of Long Column under Axial Compression Load

_{lsu}is the compression bearing capacity of the long column, and φ is the stability coefficient.

#### 4.2.3. Calculation Formula of Compression Bearing Capacity of Eccentric Column

_{psu}is the compression bearing capacity of the eccentric column; e

_{x}is the relative eccentricity in the x-axis; e

_{y}is the relative eccentricity in the y-axis; η is the relative eccentricity influence coefficient’s correlation with the slenderness ratio, and the calculation formula is shown in Table 4, where e is the relative eccentricity.

#### 4.2.4. Verification of the Proposed Formula of Compression Bearing Capacity

_{ucal}are in good agreement with the test N

_{uexp}and finite element N

_{ufem}, and the mean of theoretical calculations to the test and the finite element is 1.054, while the variance is 0.0247. It is shown that the proposed formula of compression bearing capacity can effectively predict the compression bearing capacity of concrete-filled rectangular stainless steel tubular columns. The application range of the proposed formula for the compression bearing capacity is for concrete-filled rectangular stainless steel tubular columns, and the constraint effect coefficient ξ is from 0.65 to 6.06, the slenderness ratio λ is from 3 to 96, the relative eccentricity e is from 0 to 2.667.

## 5. Conclusions

- (1)
- The finite element model can effectively simulate the compression bearing capacity; the mean of finite element calculations N
_{u}_{fem}to the test N_{uexp}is 0.985, and the variance is 0.000621. - (2)
- The slenderness ratio and relative eccentricity have a great influence on the load–displacement curves. The thickness of the stainless steel tube has little influence on the load–displacement curves. With the increase in slenderness ratio and relative eccentricity, the compression bearing capacity decreases.
- (3)
- With the increase in the slenderness ratio, the failure model of the specimen gradually changes from plastic failure to elastoplastic failure and then elastic failure.
- (4)
- When the slenderness ratio is the same, if the relative eccentricity is larger, increasing the thickness of the stainless steel tube will be more effective in improving the compression bearing capacity. When the relative eccentricity is the same, if the slenderness ratio is smaller, increasing the thickness of the stainless steel tube will be more effective in improving the compression bearing capacity.
- (5)
- The slenderness ratio and relative eccentricity have a great influence on the longitudinal stress distribution in the cross-section. When the slenderness ratio and relative eccentricity are larger, the longitudinal compressive stress in parts of the cross-section gradually becomes the longitudinal tensile stress.
- (6)
- The proposed formula can effectively predict the compression bearing capacity of concrete-filled rectangular stainless steel tubular columns. The mean of theoretical calculations to the test and the finite element is 1.054, and the variance is 0.0247.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 4.**Constitutive models [32]. (

**a**) Stress-strain relation of stainless steel. (

**b**) Compression stress-strain relation of concrete. (

**c**) Tensile stress-strain relation of concrete.

**Figure 6.**Failure mode comparison between finite element and test. (

**a**) Rectangular specimen S2. (

**b**) Square specimen S7.

**Figure 7.**Comparison of load–displacement curves for the finite element model and test. (

**a**) S1. (

**b**) S2. (

**c**) S3. (

**d**) S4. (

**e**) S5. (

**f**) S6. (

**g**) S7.

**Figure 8.**Comparison of load–displacement curves subject to different parameters. (

**a**) 120604C40720. (

**b**) 120604C402880. (

**c**) 120605C40720. (

**d**) 120605C402880. (

**e**) 1201206C40720. (

**f**) 1201206C405760.

**Figure 9.**Comparison of variation of the compression bearing capacity with relative eccentricity under different slenderness ratios. (

**a**) 120604C40. (

**b**) 120605C40. (

**c**) 120804C40. (

**d**) 120805C40. (

**e**) 1201204C40. (

**f**) 1201205C40. (

**g**) 1201206C40.

**Figure 10.**Variation of the compression bearing capacity with slenderness ratio under axial compression load and eccentric axial compression load.

**Figure 11.**Longitudinal stress distribution of the stainless steel tube in the central cross-section when the ultimate bearing capacity is reached. (

**a**) 120604C40-0-8. (

**b**) 120604C40-1-8. (

**c**) 120604C40-2.67-8. (

**d**) 120604C40-0-48. (

**e**) 120604C40-1-48. (

**f**) 120604C40-2.67-48. (

**g**) 1201206C40-0-4. (

**h**) 1201206C40-1-4. (

**i**) 1201206C40-1.33-4. (

**j**) 1201206C40-0-48. (

**k**) 1201206C40-1-48. (

**l**) 1201206C40-1.33-48.

**Figure 12.**Longitudinal stress distribution of the core concrete in the central cross-section when the ultimate bearing capacity is reached. (

**a**) 120604C40-0-8. (

**b**) 120604C40-1-8. (

**c**) 120604C40-2.67-8. (

**d**) 120604C40-0-48. (

**e**) 120604C40-1-48. (

**f**) 120604C40-2.67-48. (

**g**) 1201206C40-0-4. (

**h**) 1201206C40-1-4. (

**i**) 1201206C40-1.33-4. (

**j**) 1201206C40-0-48. (

**k**) 1201206C40-1-48. (

**l**) 1201206C40-1.33-48.

Specimen Number | Length of Steel Tube a/mm | Width of Steel Tube b/mm | Thickness of Steel Tube t/mm | Length of Specimen L/mm | Compression Bearing Capacity N _{uexp}/kN |
---|---|---|---|---|---|

S1 | 120 | 60 | 4 | 360 | 1261 |

S2 | 120 | 60 | 5 | 360 | 1632 |

S3 | 120 | 80 | 4 | 360 | 1362 |

S4 | 120 | 80 | 5 | 360 | 1732 |

S5 | 120 | 120 | 4 | 360 | 1814 |

S6 | 120 | 120 | 5 | 360 | 2224 |

S7 | 120 | 120 | 6 | 360 | 2913 |

Finite Element Model Type | Length of Steel Tube a/mm | Width of Steel Tube b/mm | Thickness of Steel Tube t/mm | Yield Strength of Steel Tube f_{y}/MPa | Axial Compressive Strength of Concrete f_{c}/MPa | Slenderness Ratio λ | Relative Eccentricity e |
---|---|---|---|---|---|---|---|

FEM 1 | 120 | 60 | 4 | 534.3 | 29.48 | 6~48 | 0~2.667 |

FEM 2 | 120 | 60 | 5 | 572.3 | 29.48 | 6~48 | 0~2.667 |

FEM 3 | 120 | 80 | 4 | 534.3 | 29.48 | 6~48 | 0~2 |

FEM 4 | 120 | 80 | 5 | 572.3 | 29.48 | 6~48 | 0~2 |

FEM 5 | 120 | 120 | 4 | 534.3 | 29.48 | 3~96 | 0~1.333 |

FEM 6 | 120 | 120 | 5 | 572.3 | 29.48 | 3~48 | 0~1.333 |

FEM 7 | 120 | 120 | 6 | 598.0 | 29.48 | 3~48 | 0~1.333 |

Failure Mode | Neutralization Axis | Tensile Area of the Stainless Steel Tube | Compressive Area of the Stainless Steel Tube | Tensile Area of the Core Concrete | Compressive Area of the Core Concrete |
---|---|---|---|---|---|

1 | Not through the cross-section | No tensile area | All are under compression, which is yielding | No tensile area | All are under compression, which has reached the ultimate compressive strength |

2 | Not through the cross-section | No tensile area | All are under compression, which is not yielding | No tensile area | All are under compression, some areas have reached the ultimate compressive strength, while other areas have not reached it |

3 | Not through the cross-section | No tensile area | All are under compression, but some areas are yielding and other areas are not yielding | No tensile area | All are under compression, some areas have reached the ultimate compressive strength, while other areas have not reached it |

4 | Through the cross-section | There are tensile areas, which are not yielding | There are tensile areas, which are not yielding | No tensile area | All are under compression, some areas have reached the ultimate compressive strength, while other areas have not reached it |

5 | Through the cross-section | There are tensile areas, which are not yielding | There are tensile areas, which are not yielding | There are tensile areas, which have reached the ultimate tensile strength | There are compression areas, which have not reached the ultimate compressive strength |

6 | Through the cross-section | There are tensile areas, which are not yielding | There are tensile areas, which are not yielding | There are tensile areas, which have reached the ultimate tensile strength | There are compression areas, which have reached the ultimate compressive strength |

7 | Through the cross-section | There are tensile areas, which are yielding | There are tensile areas, which are yielding | There are tensile areas, which have reached the ultimate tensile strength | There are compression areas, which have reached the ultimate compressive strength |

Slenderness Ratio λ | The Formula of η | Scope of Application |
---|---|---|

λ ≤ 4 | $\eta =0.07487{e}^{3}-0.04403{e}^{2}-0.41727e+1.00409$ | 0 < e ≤ 2.667 e = max{e _{x}, e_{y}} |

4 < λ ≤ 6 | $\eta =-0.03224{e}^{3}+0.24713{e}^{2}-0.63878e+1.00018$ | |

6 < λ ≤ 8 | $\eta =-0.03193{e}^{3}+0.19992{e}^{2}-0.53382e+1.00317$ | |

8 < λ ≤ 13.5 | $\eta =-0.06516{e}^{3}+0.37696{e}^{2}-0.80405e+0.99221$ | |

13.5 < λ ≤ 18 | $\eta =-0.07015{e}^{3}+0.39878{e}^{2}-0.83841e+0.97895$ | |

18 < λ ≤ 24 | $\eta =-0.10133{e}^{3}+0.52723{e}^{2}-0.95313e+0.95699$ | |

24 < λ ≤ 36 | $\eta =-0.12129{e}^{3}+0.60338{e}^{2}-1.01076e+0.83234$ | |

36 < λ ≤ 48 | $\eta =-0.19951{e}^{3}+0.90127{e}^{2}-1.20665e+0.78439$ |

Specimen Number | Compression Bearing Capacity N _{uexp}/kN | Data Sources | Specimen Number | Compression Bearing Capacity N _{uexp}/kN | Data Sources |
---|---|---|---|---|---|

304-t8C50 | 6290 | Reference [20] | 120 × 60 × 4 | 1261 | Reference [21] |

304-t10C50 | 7113 | 120 × 60 × 5 | 1632 | ||

304-t12C50 | 7924 | 120 × 80 × 4 | 1362 | ||

304-t8C70 | 6743 | 120 × 80 × 5 | 1732 | ||

304-t10C70 | 7947 | 120 × 120 × 4 | 1814 | ||

304-t12C70 | 8575 | 120 × 120 × 5 | 2224 | ||

304-t8C80 | 7436 | 120 × 120 × 6 | 2913 | ||

304-t10C80 | 8430 | r-0-0-a | 1542 | Reference [22] | |

304-t12C80 | 9257 | r-0-0-b | 1498 | ||

2205-t8C50 | 8771 | r-0.50-0.50-a | 734 | ||

2205-t10C50 | 10,111 | r-0.50-0.50-b | 716 | ||

2205-t12C50 | 12,472 | r-0.75-0.75-a | 485 | ||

2205-t8C70 | 9686 | r-0.75-0.75-b | 497 | ||

2205-t10C70 | 10,820 | rc1-0.5-0.5-a | 533 | ||

2205-t12C70 | 12,560 | rc1-0.5-0.5-b | 524 | ||

2205-t8C80 | 9962 | rc2-0.5-0.5-a | 824 | ||

2205-t10C80 | 11,728 | rc2-0.5-0.5-b | 814 | ||

2205-t12C80 | 13,272 | rl1-0.5-0.5-a | 795 | ||

– | – | rl1-0.5-0.5-b | 778 | ||

– | – | rl2-0.5-0.5-a | 562 | ||

– | – | rl2-0.5-0.5-b | 564 |

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**MDPI and ACS Style**

Cao, B.; Zhu, L.; Jiang, X.; Wang, C.
An Investigation of Compression Bearing Capacity of Concrete-Filled Rectangular Stainless Steel Tubular Columns under Axial Load and Eccentric Axial Load. *Sustainability* **2022**, *14*, 8946.
https://doi.org/10.3390/su14148946

**AMA Style**

Cao B, Zhu L, Jiang X, Wang C.
An Investigation of Compression Bearing Capacity of Concrete-Filled Rectangular Stainless Steel Tubular Columns under Axial Load and Eccentric Axial Load. *Sustainability*. 2022; 14(14):8946.
https://doi.org/10.3390/su14148946

**Chicago/Turabian Style**

Cao, Bing, Longfei Zhu, Xintong Jiang, and Changsheng Wang.
2022. "An Investigation of Compression Bearing Capacity of Concrete-Filled Rectangular Stainless Steel Tubular Columns under Axial Load and Eccentric Axial Load" *Sustainability* 14, no. 14: 8946.
https://doi.org/10.3390/su14148946