# Numerical Simulation on Seismic Behavior of Steel Fiber Reinforced Concrete Beam—Column Joints

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Experimental Introduction

_{f}/d

_{f}= 35 mm/0.55 mm = 64, and the nominal yield tensile strength was 1345 MPa. The mixed proportion of materials is summarized in Table 3. More information on the SFRC–BCJs could be referred in Shi [47].

## 3. Finite Element Model

#### 3.1. Element Model

#### 3.1.1. Nonlinear Fiber Beam–Column Element

#### 3.1.2. Beam–Column Joint Element

#### 3.1.3. Constitutive Model of Concrete

_{c}′ is compressive strength; ${\epsilon}_{0}$ is strain at maximum strength; f

_{cu}is crushing strength; ${\epsilon}_{cu}$ is strain at crushing strength; λ is ratio between unloading slope at crushing strength and initial slope (default value 0.01); f

_{t}′ is tensile strength; E

_{0}is elasticity modulus; and E

_{ts}tension softening stiffness. These variables were determined based on the cyclic compressive test for SFRC, which can be briefly described as follows

#### 3.1.4. Constitutive Model of Reinforcement

_{y}, elasticity modulus E, and strain hardening rate B when reinforcing bars enter the strengthening stage after yielding. The curve transition shape is determined by the parameter R, which is called the curvature coefficient of the transition curve, and the curvature radius of the curve increases with the decrease in R, and its specific value can be calculated according to the following formula:

_{0}is the curvature coefficient of the curve under initial loading. The a

_{1}and a

_{2}are curvature degradation coefficients under reciprocating loading. The parameter considering isotropic strain hardening adopts the default value of Opensees.

#### 3.1.5. Analysis Module

#### 3.2. Applicability Analysis of Beam–Column Joint Element Model

#### 3.3. Improvement of Beam–Column Joint Element Mode

#### 3.3.1. Constitutive Model of Reinforced Bond–Slip Spring

_{y}and F

_{u}represent the yield strength and ultimate strength of reinforcing bars, S

_{y}is rebar slip at member interface under yield stress, and S

_{u}is rebar slip at the loaded end at the bar fracture strength. The hardening rate b of the hardening starting point on the slip stress relation curve of monotonically loaded reinforcing bars is 0.4. The pinching coefficient r of the slip stress hysteretic curve of cyclically loaded reinforcing bars is 0.6.

_{sp}is yield bond stress, u

_{m}is the maximum bond stress, c is the minimum concrete cover thickness, f

_{c}is the compressive strength of concrete, V

_{f}is the volume ratio of steel fiber, L

_{f}/D

_{f}is the aspect ratio of steel fiber, and d

_{b}is the diameter of longitudinal reinforcement.

_{u}can be computed as follows

#### 3.3.2. Constitutive Model of Joint Shear Block

_{θ}is the average crack spacing in the core area of SFRC–BCJs.

_{b}represents section area of the concrete beam.

## 4. Numerical Result Analysis

#### 4.1. Hysteretic Curve

#### 4.2. Skeleton Curve

_{y}, P

_{m}, and P

_{u}are the yield load, ultimate load, and failure load, respectively. As can be shown from Table 4, the discrepancies of the yield load, ultimate load, and failure load between experimental and corresponding simulate results are less than 9.1% for all the specimens. Besides, the average ratios on P

_{y}, P

_{m}, and P

_{u}were 1.04, 0.98, and 0.94, and the corresponding COV (coefficients of variation) were 0.002, 0.0022, and 0.0002, respectively.

#### 4.3. Energy Dissipation and Stiffness Degradation

## 5. Parameter Expansion Analysis

#### 5.1. Steel Fiber Volume Ratio

#### 5.2. Stirrup Amount of Joint Core Area

#### 5.3. Axial Compression Ratio

## 6. Ultimate Shear Capacity of SFRC–BCJs

_{j}, which can be expressed as

_{cf}can be expressed by

_{0}is the effective height of the core area of the joint; ${a}_{\mathrm{s}}^{\prime}$ is the distance from the resultant force point of longitudinal reinforcement to the edge of concrete.

_{j}can be calculated as follows

**and the calculated value V**

_{jt}_{jc}is 0.997 with the COV of 0.094. Figure 19 shows the comparisons of regression analysis curve between the simulated and experimental results. There is no obvious dispersion between the calculated values and the measured values. The dispersion degree is low, proving the feasibility of using this formula to calculate the ultimate shear capacity of the SFRC beam–column joint core area.

## 7. Conclusions

- A numerical simulation method on investigating the seismic behavior of SFRC–BCJs was proposed by modifying the calculation method of shear deformation in the core area of joint and bond–slip deformation of longitudinal reinforcement of beam. The numerical modeling approach can accurately reflect the development of SFRC–BCJs, and the numerical results agreed well with the experimental results.
- Adding the steel fiber volume ratio can effectively improve the seismic behavior of SFRC–BCJs, in terms of the initial stiffness, yield load, ultimate load, and ductility. Besides, increasing the stirrup amount contribute to enhance the yield load, ultimate load, and ductility. However, the axial compression ratio has no obvious influence on the seismic behavior of SFRC–BCJs.
- Based on the numerical simulation results, the formula for calculating the shear capacity of joints is established. Furthermore, the results show that the proposed formula can reflect the influence of steel fibers and stirrups, which is in good agreement with the numerical simulation results.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Geometry and reinforcement configuration of specimens: (

**a**) BCJ1–0, BCJ1–1, BCJ1–2, BCJ3–2 and BCJ3–3; (

**b**) BCJ2–2 and BCJ3–1; (

**c**) BCJ5–1; (

**d**) Reinforcement of specimen (units: mm).

**Figure 11.**Selection of key points of shear block components in the joint core area (note: 1–X includes BCJ1–0, BCJ1–1, and BCJ1–2).

**Figure 16.**(

**a**) Skeleton curves and (

**b**) ductility coefficient under different steel fiber volume ratios.

**Figure 17.**(

**a**) Skeleton curves and (

**b**) ductility coefficient under different stirrup ratios in the core area of joints.

**Figure 18.**(

**a**) Skeleton curves and (

**b**) ductility coefficient under different axial compression ratios.

Category | d (mm) | f_{y} (MPa) | f_{u} (MPa) | δ (%) | E_{s} (MPa) |
---|---|---|---|---|---|

HRB335 | 22 | 418.2 | 652.1 | 27 | 1.95 × 10^{5} |

HRB335 | 16 | 360.5 | 594.9 | 23 | 2.01 × 10^{5} |

HPB235 | 8 | 306.9 | 472.7 | 30 | 2.09 × 10^{5} |

_{y}is yield strength; f

_{u}is ultimate strength; δ is elongation at fracture; and E

_{s}is elasticity modulus of reinforcing bars.

Joint Number | Concrete Strength (Mpa) | Volume Ratio of Steel Fiber V_{f} (%) | Axial Compression Ratio n | Core Area Hoop Reinforcement | Cubic Compressive Strength (MPa) | Split Tensile Strength (MPa) | Elasticity Modulus (MPa) |
---|---|---|---|---|---|---|---|

BCJ1–0 | CF60 | 1.0 | 0.3 | 0 | 81.7 | 7.3 | 45,300 |

BCJ1–1 | CF60 | 1.0 | 0.2 | 0 | 79.1 | 7.4 | 43,700 |

BCJ1–2 | CF60 | 1.0 | 0.4 | 0 | 78.1 | 7.3 | 44,400 |

BCJ2–2 | CF80 | 1.0 | 0.3 | 2ϕ8 | 89.5 | 7.1 | 44,500 |

BCJ3–1 | CF60 | 0.5 | 0.3 | 2ϕ8 | 82.1 | 7.5 | 46,600 |

BCJ3–2 | CF60 | 1.5 | 0.3 | 0 | 86.6 | 8.9 | 40,900 |

BCJ3–3 | CF60 | 2.0 | 0.3 | 0 | 87.4 | 9.1 | 44,100 |

BCJ5–1 | C60 | 0 | 0.3 | 5ϕ8 | 68.6 | 4.9 | 42,500 |

Number | Water (L) | Cement (kg) | Sand (kg) | Stone (kg) | Steel Fiber (kg) | Superplasticizer (kg) |
---|---|---|---|---|---|---|

BCJ1–0 | 164 | 547 | 696 | 1044 | 78 | 8.2 |

BCJ1–1 | 164 | 547 | 696 | 1044 | 78 | 8.2 |

BCJ1–2 | 164 | 547 | 696 | 1044 | 78 | 8.2 |

BCJ2–2 | 164 | 547 | 696 | 1044 | 78 | 8.2 |

BCJ3–1 | 156 | 520 | 710 | 1065 | 39 | 7.8 |

BCJ3–2 | 172 | 573 | 682 | 1023 | 117 | 8.6 |

BCJ3–3 | 181 | 599 | 668 | 1001 | 156 | 8.9 |

BCJ5–1 | 146 | 487 | 623 | 1210 | 0 | 7.3 |

Component | P_{y} (kN) | P_{m} (kN) | P_{u} (kN) | ||||||
---|---|---|---|---|---|---|---|---|---|

T | S | T/S | T | S | T/S | T | S | T/S | |

BCJ1–0 | 27.49 | 28.55 | 0.96 | 35.20 | 37.65 | 0.93 | 32.08 | 33.41 | 0.96 |

BCJ1–1 | 26.18 | 24.32 | 1.08 | 34.12 | 37.37 | 0.91 | 30.63 | 33.42 | 0.92 |

BCJ1–2 | 31.64 | 29.01 | 1.09 | 36.45 | 37.51 | 0.97 | 29.88 | 32.08 | 0.93 |

BCJ2–2 | 32.29 | 29.99 | 1.08 | 39.95 | 37.66 | 1.06 | 33.22 | 35.10 | 0.95 |

BCJ3–1 | 26.01 | 26.29 | 0.99 | 33.86 | 34.58 | 0.98 | 29.41 | 30.75 | 0.96 |

BCJ3–2 | 25.74 | 24.65 | 1.04 | 39.27 | 39.60 | 0.99 | 33.87 | 36.90 | 0.92 |

BCJ3–3 | 27.50 | 26.38 | 1.04 | 40.00 | 39.00 | 1.03 | 33.57 | 36.02 | 0.93 |

BCJ5–1 | 24.41 | 23.83 | 1.02 | 30.76 | 30.75 | 1.00 | 28.12 | 30.16 | 0.93 |

Average | 1.04 | 0.98 | 0.94 | ||||||

COV | 0.0020 | 0.0022 | 0.0002 |

**Table 5.**Comparison between calculated results and experimental results of ultimate shear capacity of SFRC–BCJs.

Joint Number | V_{jt} (kN) | V_{jc} (kN) | V_{jt}/V_{jc} |
---|---|---|---|

SF–7 [76] | 398.6 | 393.561 | 1.013 |

SF–8 [76] | 456.6 | 486.258 | 0.939 |

J3–3 [77] | 467.7 | 454.363 | 1.029 |

J3–4 [77] | 456.0 | 503.451 | 0.906 |

S3 [78] | 1375.5 | 1238.093 | 1.111 |

SF–2 [70] | 1087.5 | 980.296 | 1.109 |

S6 [46] | 34.1 | 38.647 | 0.882 |

BCJ1–0 | 348.4 | 359.264 | 0.970 |

BCJ1–1 | 330.9 | 344.214 | 0.961 |

BCJ1–2 | 360.5 | 347.004 | 1.039 |

BCJ2–2 | 384.3 | 385.718 | 0.996 |

BCJ3–1 | 328.1 | 404.009 | 0.812 |

BCJ3–2 | 375.9 | 323.254 | 1.163 |

BCJ3–3 | 390.0 | 370.185 | 1.054 |

BCJ5–1 | 347.6 | 360.635 | 0.964 |

Average | 0.997 | ||

COV | 0.094 |

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## Share and Cite

**MDPI and ACS Style**

Shi, K.; Zhu, J.; Li, P.; Zhang, M.; Xue, R.; Zhang, T.
Numerical Simulation on Seismic Behavior of Steel Fiber Reinforced Concrete Beam—Column Joints. *Materials* **2021**, *14*, 4883.
https://doi.org/10.3390/ma14174883

**AMA Style**

Shi K, Zhu J, Li P, Zhang M, Xue R, Zhang T.
Numerical Simulation on Seismic Behavior of Steel Fiber Reinforced Concrete Beam—Column Joints. *Materials*. 2021; 14(17):4883.
https://doi.org/10.3390/ma14174883

**Chicago/Turabian Style**

Shi, Ke, Junpeng Zhu, Pengfei Li, Mengyue Zhang, Ru Xue, and Tao Zhang.
2021. "Numerical Simulation on Seismic Behavior of Steel Fiber Reinforced Concrete Beam—Column Joints" *Materials* 14, no. 17: 4883.
https://doi.org/10.3390/ma14174883