# The Influence of Different Dynamic Material Constitutive Models on the Impact Performance of Circular CFST Columns

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## Abstract

**:**

## 1. Introduction

## 2. Finite Element Modeling

#### 2.1. General

#### 2.2. Material Model of Structural Steel

_{s}and f

_{sy}represent the modulus of elasticity and yield strength of the steel used, respectively, and E

_{s}is equal to 2.0 × 10

^{5}MPa; A = 0.2 f

_{sy}/(ε

_{2}− ε

_{1})

^{2}, B = 0.2 A ε

_{2}, C = 0.8 f

_{sy}+ A ε

_{1}

^{2}− B ε

_{1}, ε

_{1}= 0.8 f

_{sy}/E

_{s}, ε

_{2}= 1.5 ε

_{1}, ε

_{3}= 10 ε

_{2}, ε

_{4}= 100 ε

_{2}.

_{eng}and ε

_{eng}are the engineering stress and strain, respectively; σ

_{true}and ε

_{true}are the actual stress and strain, respectively.

_{c}of the element, as the damage criteria were met, the equivalent plastic displacement ${\overline{u}}_{\mathrm{f}}^{\mathrm{pl}}$ at complete cracking was obtained and expressed as ${\overline{u}}_{\mathrm{f}}^{\mathrm{pl}}={L}_{\mathrm{c}}{\overline{\epsilon}}_{\mathrm{f}}^{\mathrm{pl}}$. L

_{c}herein is the specific value of the volume of the minimum element divided by its largest area, and the value of L

_{c}adopts 10 mm in line with the mesh dimension of the FE model.

#### 2.3. Material Model of Concrete

_{0}, y = f/f

_{c}, f represents the axial stress of concrete when the axial strain reaches ε, f

_{c}represents the cylinder concrete strength and the corresponding strain is ε

_{0}; η and β

_{0}denote the parameters that are decided by the form of the column cross-section. H is equal to 2 for the circular cross-section and the corresponding parameter β

_{0}is determined by the following equation:

_{c}and A

_{s}represent the cross-section areas of infilled concrete and steel tube, respectively; f

_{ck}is the characteristic concrete strength.

_{c}, the eccentricity e, the ratio of the biaxial compressive strength to the uniaxial compressive strength ${f}_{\mathrm{bo}}/{f}_{\mathrm{c}}^{\prime}$ and the viscosity parameter v needed to be determined. Based on previous studies [6,14], in this study, the values of ψ, K

_{c}, ${f}_{\mathrm{bo}}/{f}_{\mathrm{c}}^{\prime}$, e and v were taken as 30°, 1.16, 0.1, 2/3 and 0.0005, respectively. According to ACI 318 [25], the Young′s modulus E

_{0}and Poisson′s ratio υ

_{c}of the concrete used were $4730\sqrt{{f}_{\mathrm{c}}^{\prime}}$ and 0.2, respectively.

_{c}and tension damage recovery factor w

_{t}are 1 and 0, respectively.

#### 2.4. Material Strain Rate Effect

^{−1}and 3.91, respectively.

^{3}and 2450 kg/m

^{3}, respectively. This study is in the range of low-speed impact. Researchers have concluded that low-velocity impact has a negligible effect on the elastic modulus and Poisson’s ratio [11,12,13,14,15,16]. In this paper, Poisson’s ratio was adopted as 0.3. The elastic modulus of steel was 200 GPa if it was not given in the literature. In addition, this study mainly focuses on the low-velocity impact loads. Therefore, the volumetric strain as a function of the bulk modulus of material under pressure was not taken into account.

#### 2.5. Element Mesh, Boundary and Contact Conditions

## 3. Verification of Numerical Model

_{0}is the drop velocity at the moment of impact, m

_{0}is the drop mass, E

_{0}is the impact energy and n is the axial load ratio of the specimens.

_{p}) and maximum mid-span deflection (u

_{m}) between the test and FEA results are depicted in Figure 6. From Figure 6, it can be found that the mean values of F

_{p,FEA}/F

_{p,test}and u

_{m,FEA}/u

_{m,test}are 1.01 and 0.99, respectively, and the corresponding standard deviations of are 0.02 and 0.01, respectively. Overall, the established FE model can accurately predict the lateral impact performance of circular CFST columns and can be used for impact response analysis of CFST members.

## 4. Parametric Study

_{sy}was taken as 235 MPa, 355 MPa, 460 MPa and 690 MPa and the concrete strength f

_{c}was taken as 30 MPa, 50 MPa and 70 MPa. In the FE analysis, the section size of the drop hammer with a rigid flat head was 500 mm, with reference to the FAW Jiefang light truck J6F truck. The impact mass was 5000 kg and the corresponding impact speed was 16.67 m/s (60 km/h), where 60 km/h is the general speed limit for urban roads in the city and the minimum speed limit for expressways. The corresponding impact energy was 694 kJ.

^{−1}. The material strain rate effect caused by this medium strain rate is also relatively low, making the effect of the dynamic material model on the components not as good as that of high-speed impact. The following is mainly based on ABAQUS to analyze and compare different dynamic constitutive models of materials to quantify the effects of dynamic material constitutive models on the impact response of CFST members. The dynamic constitutive models of several materials analyzed in this paper are shown in Table 2 and Table 3.

#### 4.1. Comparison of Different Dynamic Constitutive Models of Steel

#### 4.2. Comparison of Different Dynamic Constitutive Models of Concrete

## 5. Conclusions

- (1)
- The established FE model considering the progressive damage of steel and concrete can accurately estimate the dynamic mechanical properties and failure modes of CFST columns, and thus can be used for impact response analysis of CFST members.
- (2)
- The different dynamic constitutive models of steel have different effects on the impact force and mid-span displacement curves at the shock section of CFST columns.
- (3)
- The FE models not considering the strain rate effect of steel have no strain rate hardening effect on the impact process. As a result, the platform value of the impact force of CFST columns is reduced and the maximum mid-span deflection is increased.
- (4)
- The predicted result of the FE model ignoring the strain rate effect of concrete but considering the strain rate effect of steel is in good agreement with that considering the CEB–FIP model for the impact process. This is because the largest proportion of the impact energy of CFST members is mainly assimilated by the outer steel tube.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 7.**Comparison of Q235 steel with different dynamic constitutive models. (

**a**) Comparison of impact force time-history curves of Q235 steel. (

**b**) Comparison of mid-span deflection time-history curves of Q235 steel. (

**c**) Platform value of impact force and maximum mid-span deflection of Q235 steel.

**Figure 8.**Comparison of Q460 steel with different dynamic constitutive models. (

**a**) Comparison of impact force time-history curves of Q460 steel. (

**b**) Comparison of mid-span deflection time-history curves of Q460 steel. (

**c**) Platform value of impact force and maximum mid-span deflection of Q460 steel.

**Figure 9.**Comparison of C30 concrete with different dynamic constitutive models. (

**a**) Comparison of impact force time-history curves of C30 concrete. (

**b**) Comparison of mid-span deflection time-history curves of C30 concrete. (

**c**) Platform value of impact force and maximum mid-span deflection of C30 concrete.

**Figure 10.**Comparison of C70 concrete with different dynamic constitutive models. (

**a**) Comparison of impact force time-history curves of C70 concrete. (

**b**) Comparison of mid-span deflection time-history curves of C70 concrete. (

**c**) Platform value of impact force and maximum mid-span deflection of C70 concrete.

Specimen No. | Boundary Condition | D × t (mm) | L (mm) | H (m) | V_{o} (m/s) | m_{o} (kg) | E_{o} (m) | n | Ref. |
---|---|---|---|---|---|---|---|---|---|

CC1 | Fixed-fixed | 180 × 3.65 | 1940 | 5.5 | 9.21 | 465 | 19.72 | 0 | [30] |

CC2 | Fixed-fixed | 180 × 3.65 | 1940 | 2.5 | 6.40 | 920 | 18.84 | 0 | |

CC3 | Fixed-fixed | 180 × 3.65 | 1940 | 8.0 | 9.67 | 465 | 21.73 | 0 | |

SS1 | Simply-simply | 180 × 3.65 | 2800 | 4.0 | 8.05 | 465 | 15.07 | 0 | |

SS2 | Simply-simply | 180 × 3.65 | 2800 | 2.0 | 5.69 | 920 | 14.89 | 0 | |

SS3 | Simply-simply | 180 × 3.65 | 2800 | 5.0 | 8.93 | 465 | 18.54 | 0 | |

DBF13 | Fixed-fixed | 114 × 1.7 | 1200 | 1.2 | 4.85 | 229.8 | 2.76 | 0 | [31] |

DBF17 | Fixed-fixed | 114 × 1.7 | 1200 | 1.0 | 4.43 | 229.8 | 2.30 | 0 | |

DBF19 | Fixed-fixed | 114 × 1.7 | 1200 | 1.2 | 4.85 | 229.8 | 2.76 | 0.3 | |

DBF21 | Fixed-fixed | 114 × 1.7 | 1200 | 1.0 | 4.43 | 229.8 | 2.30 | 0.6 |

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**MDPI and ACS Style**

Yan, X.-F.; Lin, S.; Ahmed, M.
The Influence of Different Dynamic Material Constitutive Models on the Impact Performance of Circular CFST Columns. *Buildings* **2023**, *13*, 1634.
https://doi.org/10.3390/buildings13071634

**AMA Style**

Yan X-F, Lin S, Ahmed M.
The Influence of Different Dynamic Material Constitutive Models on the Impact Performance of Circular CFST Columns. *Buildings*. 2023; 13(7):1634.
https://doi.org/10.3390/buildings13071634

**Chicago/Turabian Style**

Yan, Xi-Feng, Siqi Lin, and Mizan Ahmed.
2023. "The Influence of Different Dynamic Material Constitutive Models on the Impact Performance of Circular CFST Columns" *Buildings* 13, no. 7: 1634.
https://doi.org/10.3390/buildings13071634