# Influencing Factors and Shear Capacity Formula of Single-Keyed Dry Joints in Segmental Precast Bridges under Direct Shear Loading

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## Abstract

**:**

## 1. Introduction

## 2. Nonlinear Simulation Analysis of a Single-Keyed Dry Joint

#### 2.1. Overview of the Simulation Example

#### 2.2. Constitutive Relationship of Materials

#### 2.2.1. Concrete Uniaxial Stress-Strain Relationship

^{4}MPa, the Poisson’s ratio was 0.2, the uniaxial compressive and tensile strength was 41.5 Mpa and 3.56 Mpa, respectively—corresponding uniaxial stress-strain curves are illustrated in Figure 2a,b.

#### 2.2.2. Plastic Damage Model of Concrete

#### 2.2.3. Stress-Strain Relationship of Rebar

^{5}MPa, and the ultimate strength

**f**is 600 MPa. The ideal elastoplastic constitutive relationship is adopted for the rebar, shown as in Figure 4.

_{u}#### 2.3. Finite Element Model

#### 2.4. Verification of the Simulation Method

#### 2.5. Nonlinear Behavior under Direct Shear

## 3. Discussion on the Existing Shear Capacity Formula

#### 3.1. Existing Shear Capacity Formula

- (1)
- Buyukozturk’s formula [22]

- (2)
- AASHTO (2003) provisions [15]

- (3)
- ATEP provisions [25]

- (4)
- Rombach’s formula [23]

- (5)
- Turmo’s formula [17]

#### 3.2. Comparison of the Existing Formulas

#### 3.3. Force Transmission Mechanism

## 4. Influencing Factors Analysis

#### 4.1. Lateral Prestress

#### 4.2. Concrete Tensile Strength

#### 4.3. Dimension Parameters of the Key

- (1)
- Key depth

**d**on the ultimate shear capacity of the keyed dry joint is illustrated in Figure 12. When

**d**increased from 20 mm to 40 mm, ${\mathit{V}}_{\mathit{u}}$ is increased from 103.31 kN to 110.05 kN, the increasement is only about 6.5%. Therefore, key depth has no much influence on the ultimate shear capacity of the keyed dry joint.

- (2)
- Key inclination

- (3)
- Key aspect ratio (
**B/H**)

**d**and

**B/H**. But in general, dimension parameters of the key have little effect on the ultimate shear bearing capacity of the keyed dry joint.

## 5. Proposal and Evaluation of New Shear Capacity Formula

#### 5.1. New Shear Capacity Formula

**t**. That is, cotα is proportional to ${\sigma}_{n}/\zeta {f}_{t}$, while α is inverse proportional to ${\sigma}_{n}/\zeta {f}_{t}$. Through fitting analysis of the FE simulation results in Table 4, the reasonable value of $\zeta $ is obtained to be 0.394.

#### 5.2. Evaluation of New Formula

## 6. Conclusions

- (1)
- Considering the plastic damage of concrete, the nonlinear FE model of the single-keyed dry joint was established, the ultimate shear capacity analysis under the direct shear loading was carried out on existed test specimen. Results demonstrated that the simulated results are in good agreement with the existing experimental results. Thus, the feasibility and correctness of the finite element simulation method were verified.
- (2)
- Both simulated and experimental results indicated that tensile damage occurred in on the underside of the key root even in the linear stage, which was then developed 45° upwards or vertically along the key root section. Concrete tensile strength at the key root is critical to the ultimate bearing capacity of the single-keyed dry joint under the direct shear loading.
- (3)
- The shear resistance of the concrete key root, related to lateral prestress and concrete tensile strength, is the only contributor to the ultimate shear capacity ${\mathit{V}}_{\mathit{u}}$ of the keyed dry joints. Friction on the joint interface was demonstrated to be only transferring force between the adjacent joints, ${\mathit{V}}_{\mathit{u}}$ does not increase with the increase of friction coefficient. Neither do dimension parameters of the key (such as key depth, key inclination and key aspect ratio) have much effect on ${\mathit{V}}_{\mathit{u}}$. Meanwhile, a reasonable range of key inclination ($\mathrm{tan}\theta $) would be suggested as 0.7~0.9 to obtain comparatively higher value of ${\mathit{V}}_{\mathit{u}}$.
- (4)
- According to the above investigation and supposing the shear stress distributed evenly along key root width, a new formula of the shear capacity of the single-keyed dry joints was proposed based on the maximum principal stress strength criterion. In comparison with the predicted results obtained by other existed formulas, the proposed formula is demonstrated to be in perfect consistency with both tests and the FE simulation results.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) The tensile stress-strain relationship of concrete; (

**b**) the compressive stress-strain relationship of concrete.

**Figure 3.**(

**a**) Relationship between tensile damage and inelastic strain; (

**b**) relationship between compressive damage and inelastic strain.

**Figure 7.**Tension damage distribution diagram and crack development process; (

**a**) load to point A; (

**b**) load to point B; (

**c**) load to point C.

**Figure 9.**Vertical force component of tangential force at the contact surface; (

**a**) under lateral prestress; (

**b**) 0.004 ${V}_{u}$ state; (

**c**) under ultimate load ${V}_{u}$; (

**d**) diagram of forces on the interface.

**Figure 18.**Relation between the experimental and predicted joint shear strength of different formulas. (

**a**) results of Buyukozturk’s formula

_{.}; (

**b**) results of AASHTO (2003) provisions; (

**c**) results of Turmo’s formula; (

**d**) results of ATEP provisions; (

**e**) results of Rombach’s formula; (

**f**) results of Equation (20).

**Figure 19.**Comparison of the shear capacity predicted by different formulas and FE simulated values.

Test Specimen | FE Results (kN) | Test Results (kN) |
---|---|---|

F1.0-d35-A0.714-B100-H100 | 82.27 | 89.70 |

F2.0-d35-A0.714-B100-H100 | 105.86 | 113.87 |

F1.0-d25-A1-B100-H100 | 74.86 | 80.79 |

F1.0-d50-A0.5-B100-H100 | 88.90 | 94.47 |

Lateral Preload | 1 | 2 | 3 | 4 | 6 | 8 |
---|---|---|---|---|---|---|

AASHTO | 83.37 | 102.58 | 121.78 | 140.99 | 179.40 | 217.81 |

Turmo | 54.01 | 68.41 | 82.81 | 97.21 | 126.01 | 154.81 |

ATEP | 69.61 | 92.41 | 115.21 | 138.01 | 181.31 | 229.21 |

Rombach | 71.10 | 84.10 | 97.10 | 110.10 | 136.10 | 162.10 |

Bakhoum | 110.56 | 137.76 | 164.96 | 192.16 | 246.26 | 300.96 |

V_{E} | 89.70 | 113.87 | - | - | - | - |

V_{F} | 82.3 | 105.3 | 124.4 | 140.2 | 159.5 | 175.7 |

**Table 3.**The shear bearing capacity of test pieces under different friction coefficients (unit: kN).

Friction Coefficient | 0.4 | 0.6 | 0.8 | 1.0 | |
---|---|---|---|---|---|

Specimen | |||||

F1.0-d35-A25/35 | 83.12 | 82.27 | 82.81 | 83.43 | |

F8.0-d35-A25/35 | 161.11 | 161.91 | 161.83 | 162.67 |

Group | Specimen Number | σn (MPa) | Ft (MPa) | d (mm) | tanθ | B/H |
---|---|---|---|---|---|---|

GF | GF1-1 | 1 | 4.01 | 35 | 0.714 | 1.0 |

GF1-2 | 2 | |||||

GF1-3 | 3 | |||||

GF1-4 | 4 | |||||

GF1-5 | 5 | |||||

GF1-6 | 6 | |||||

GCr | GCr1-1 | 2 | 2.20 | |||

GCr1-2 | 2.51 | |||||

GCr1-3 | 2.74 | |||||

GCr1-4 | 2.93 | |||||

GCr1-5 | 4.01 | |||||

Gd | Gd1-1 | 4.01 | 20 | |||

Gd1-2 | 25 | |||||

Gd1-3 | 30 | |||||

Gd1-4 | 35 | |||||

Gd1-5 | 40 | |||||

GA | GA1-1 | 35 | 0.429 | |||

GA1-2 | 0.571 | |||||

GA1-3 | 0.657 | |||||

GA1-4 | 0.714 | |||||

GA1-5 | 0.771 | |||||

GA1-6 | 1.000 | |||||

GBH | GBH1-1 | 0.714 | 1.0 | |||

GBH1-2 | 1.5 | |||||

GBH1-3 | 2.0 | |||||

GBH1-4 | 2.5 | |||||

GBH1-5 | 3.0 | |||||

GBH1-6 | 4.0 |

Reference | Specimen Number | Specimen Number | Concrete Strength (MPa) |
---|---|---|---|

Jiang [31] | F1.0-d35-A25/35-B100-H100 | J-1 | f_{cu} = 48.5 MPa |

F2.0-d35-A25/35-B100-H100 | J-2 | f_{cu} = 48.5 MPa | |

F1.0-d25-A25/25-B100-H100 | J-3 | f_{cu} = 48.5 MPa | |

F1.0-d50-A25/50-B100-H100 | J-4 | f_{cu} = 48.5 MPa | |

Zhou [28] | F1.0-D50-A25/50-B250-H100 | Z-1 | f_{c}^{′} = 38.7 MPa |

F1.0-D50-A25/50-B250-H100 | Z-2 | f_{c}^{′} = 50 MPa | |

F2.0-D50-A25/50-B250-H100 | Z-3 | f_{c}^{′} = 56.2 MPa | |

F2.0-D50-A25/50-B250-H100 | Z-4 | f_{c}^{′} = 59.6 MPa | |

F3.0-D50-A25/50-B250-H100 | Z-5 | f_{c}^{′} = 48.8 MPa | |

F4.0-D50-A25/50-B250-H100 | Z-6 | f_{c}^{′} = 37.1 MPa | |

Liu [27] | F10-D50-A25/50-B150-H100 | L-1 | f_{t} = 4.6 MPa |

Buyukozturk [23] | F0.68-D32-A32/32-B76.2-H100 | B-1 | f_{t} = 4.33 MPa |

F2.07-D32-A32/32-B76.2-H100 | B-2 | f_{t} = 3.97 MPa | |

F3.45-D32-A32/32-B76.2-H100 | B-3 | f_{t} = 4.38MPa |

Specimen | V_{Buyuk.} [23] | V_{AASHTO} [15] | V_{Turmo} [17] | V_{ATEP} [26] | V_{Rombach} [24] | V _{Equation (20)} | V_{E} |
---|---|---|---|---|---|---|---|

J-1 | 110.56 (1.23) | 83.37 (0.93) | 53.95 (0.60) | 69.61 (0.78) | 71.10 (0.79) | 95.44 (1.06) | 89.7 |

J-2 | 137.76 (1.21) | 102.58 (0.90) | 68.34 (0.60) | 92.41 (0.81) | 84.10 (0.74) | 114.43 (1.01) | 113.9 |

J-3 | 109.85 (1.36) | 82.71 (1.02) | 53.41 (0.66) | 68.82 (0.85) | 70.12 (0.87) | 91.22 (1.13) | 80.8 |

J-4 | 109.85 (1.16) | 82.71 (0.88) | 53.41 (0.57) | 68.82 (0.73) | 70.12 (0.74) | 91.22 (0.97) | 94.5 |

Z1 | 269.25 (1.40) | 201.78 (1.05) | 129.43 (0.67) | 166.13 (0.86) | 167.95 (0.87) | 223.86 (1.16) | 193.0 |

Z2 | 378.52 (1.13) | 293.51 (0.88) | 202.42 (0.60) | 272.48 (0.81) | 261.70 (0.78) | 316.07 (0.94) | 335.0 |

Z3 | 296.75 (1.41) | 227.31 (1.08) | 150.74 (0.71) | 198.00 (0.94) | 207.50 (0.98) | 245.74 (1.16) | 211.0 |

Z4 | 385.75 (1.14) | 301.36 (0.89) | 209.31 (0.62) | 282.07 (0.84) | 273.60 (0.81) | 322.25 (0.96) | 337.0 |

Z5 | 429.99 (1.19) | 326.35 (0.91) | 225.30 (0.63) | 308.62 (0.86) | 268.30 (0.75) | 349.46 (0.97) | 360.0 |

Z6 | 469.04 (1.32) | 336.53 (0.95) | 229.66 (0.65) | 332.62 (0.94) | 259.85 (0.73) | 356.51 (1.01) | 354.0 |

L-1 | 544.15 (1.55) | 410.48 (1.17) | 297.47 (0.85) | 425.25 (1.21) | 298.32 (0.85) | 325.68 (0.93) | 351.9 |

B-1 | 66.31 (1.01) | 65.34 (0.99) | 43.33 (0.66) | 42.86 (0.65) | 57.10 (0.87) | 62.69 (0.96) | 65.5 |

B-2 | 84.61 (1.00) | 84.16 (0.99) | 57.94 (0.69) | 56.73 (0.67) | 59.81 (0.71) | 82.01 (0.97) | 84.2 |

B-3 | 112.6 (1.01) | 116.4 (1.05) | 83.83 (0.76) | 81.88 (0.74) | 80.02 (0.72) | 82.01 (0.74) | 111.0 |

R_{Av} | 1.222 | 0.978 | 0.662 | 0.835 | 0.800 | 0.998 | |

R^{2} | 0.956 | 0.973 | 0.968 | 0.937 | 0.976 | 0.982 |

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**MDPI and ACS Style**

Hou, W.; Peng, M.; Jin, B.; Tao, Y.; Guo, W.; Zhou, L.
Influencing Factors and Shear Capacity Formula of Single-Keyed Dry Joints in Segmental Precast Bridges under Direct Shear Loading. *Appl. Sci.* **2020**, *10*, 6304.
https://doi.org/10.3390/app10186304

**AMA Style**

Hou W, Peng M, Jin B, Tao Y, Guo W, Zhou L.
Influencing Factors and Shear Capacity Formula of Single-Keyed Dry Joints in Segmental Precast Bridges under Direct Shear Loading. *Applied Sciences*. 2020; 10(18):6304.
https://doi.org/10.3390/app10186304

**Chicago/Turabian Style**

Hou, Wenqi, Meng Peng, Bo Jin, Yong Tao, Wei Guo, and Lingyu Zhou.
2020. "Influencing Factors and Shear Capacity Formula of Single-Keyed Dry Joints in Segmental Precast Bridges under Direct Shear Loading" *Applied Sciences* 10, no. 18: 6304.
https://doi.org/10.3390/app10186304