# Practical Design Method of Yielding Steel Dampers in Concrete Cable-Stayed Bridges

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## Abstract

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## Featured Application

**A time efficient method for determining the initial yielding strength of the yielding steel damper applied in cable stayed bridges is proposed.**

## Abstract

## 1. Introduction

_{y}) on the seismic responses of the bridge at different locations longitudinally (auxiliary pier, transition pier and tower column). Primarily, a finite element (FE) model should be established based on a real cable-stayed bridge in China. Furthermore, it is necessary to verify the FE model by comparing with the results of shake table test. Then, the FE model would be used to carry out a comprehensive parametric analysis to investigate the influence of F

_{y}on the seismic responses of the bridge. The theoretically appropriate YSD design would be achieved according to a complicated multivariate function analysis or a simplified single variable function analysis in terms of different designed objectives. In order to develop a practical design method, the YSD working mechanism on the cable-stayed bridge is further investigated, and finally a method is proposed to quickly determine the yield strength of YSDs based on the investigation in this paper.

## 2. The Analytical Model and Input Ground Motions

#### 2.1. FE Modelling of the Cable-Stayed Bridge

_{y}, yielding displacement Δ

_{y}, the ratio of post-yielding stiffness to pre-yielding stiffness and ultimate deformation Δ

_{u}, while the ultimate strength F

_{u}is not an independent parameter. Normally, the Δ

_{y}is set to 10 mm and the ratio of post-yielding stiffness to pre-yielding stiffness is set as 0.6% (less than 5% depending on the different shapes of products), which can provide good hysteretic performance according to related studies [19,20,21,24,25,26,30,32]. Therefore, F

_{y}is the key parameter which needs to be determined in this bilinear model. It is notable that the ultimate deformation Δ

_{u}reaches a certain amount in a fabricated YSD product; however, the maximum deformation of the bilinear model is assumed to be unlimited in the analysis so that the influence of F

_{y}on the relative displacement between the super- and sub-structures can be observed.

#### 2.2. FE Model Validation

#### 2.3. Input Ground Motions

## 3. The Influence of the F_{y} of YSD

_{y}of YSD at different locations—the YSD yielding strength installed at the auxiliary pier (F

_{ya}), the transition pier (F

_{yp}) and the tower (F

_{yt})—it is necessary to determine the effect of all these parameters, resulting in the structural response as a ternary function; it is therefore necessary to carry out a detailed three-variable function analysis.

#### 3.1. Determining the Scope of Analysis

_{y}should be considered firstly when a multivariate function is used. In fact, the seismically induced force of the substructure includes two parts: one is transmitted by the inertial force of the girder, and the other is generated by the self-vibration. Therefore, as shown by the above formula, the value of Fy should be determined according to the designed section capacity (M

_{u}) and self-vibration (M

_{sv}) as well.

_{ut}is around 500,000 kN·m, and M

_{up}(or M

_{ua}) is around 300,000 kN·m, while the H is around 40 m. Therefore, F

_{y1t}could be 12,500 kN at maximum, and F

_{y1p}(or F

_{y1a}) could be 7500 kN at maximum. On the other hand, α is around 0.3, which can be verified in the chart of subsequent calculation. Therefore, the upper limits of the YSD yielding strength F

_{ymax}set at the auxiliary pier, the transition pier and the tower are 5000 kN, 5000 kN, and 9000 kN, respectively.

_{ya}, F

_{yp}and F

_{yt}are 0 kN, 0 kN and 1000 kN, respectively, because the girder–pier can be free but the girder–tower must be restrained to meet the service load function. Thus, 441 cases in total are analyzed in order to find the most appropriate design value of F

_{y}at each location; i.e., F

_{ya}varies from 0 kN to 5000 kN, F

_{yp}varies from 0 kN to 5000 kN and F

_{yt}varies from 1000 kN to 9000 kN with an interval of 1000 kN, respectively.

#### 3.2. Multivariate Function Analysis

_{y}on the relative displacement (RD) of the auxiliary pier, the transition pier and the bridge tower with the increase of other two yielding strengths, which share the same characteristics. On one hand, there are distinct layers between the RD surfaces at each location, indicating that the RD decreases rapidly to a stable small value with the increase of the yielding strength of the YSD at its own location. On the other hand, each surface is a subduction surface, which indicates that the yielding strengths of the YSDs at the other two locations have a significant impact on the RD only when their values are small. Therefore, in order to use steel dampers efficiently, the yielding strength of YSDs at each location should not be too large.

_{ya}= 4000 kN, F

_{yp}= 4000 kN and F

_{yt}= 1000 kN (layout 1 in Table 3).

_{ya}= 1000 kN, F

_{yp}= 2000 kN and F

_{yt}= 3000 kN or F

_{ya}= 1000 kN, F

_{yp}= 5000 kN and F

_{yt}= 4000 kN. Recalling Figure 13, Figure 14 and Figure 15, the RD of the latter is smaller, so F

_{ya}= 1000 kN, F

_{yp}= 5000 kN and F

_{yt}= 4000 kN is preferable (layout 2 in Table 3).

#### 3.3. Single Variable Function Analysis

_{yt}and F

_{yp}or F

_{yt}and F

_{ya}or F

_{yp}and F

_{ya}) can remain unchanged to study the influence of the other variable (F

_{yt}or F

_{yp}or F

_{ya}) on the structure’s seismic response. Consequently, three series of cases are analyzed: (1) F

_{ya}varies from 0 kN to 5000 kN, while F

_{yt}and F

_{yp}remain constant; (2) F

_{yp}varies from 0 kN to 5000 kN, while F

_{ya}and F

_{yt}remain constant; and (3) F

_{yt}varies from 1000 kN to 9000 kN, while F

_{ya}and F

_{yp}remain constant. In fact, it is required that the others should remain when analyzing the influence of yielding strength of the specified location. For illustration purposes, 2000 kN was set.

_{y}, the girder–pier/column RD at the auxiliary pier, transition pier and tower column decrease significantly. However, when the yielding strength is relatively large, the F

_{y}only affects the displacement at its own location without a significant change at the other two locations. As shown in Figure 20b, Figure 21b and Figure 22b, with the increase of F

_{y}, there is a significant fluctuation of the RCD curve at its own location, but the RCD curves at the other two locations are approximately flat. In addition, the RCD curve at each location increases first and then decreases, indicating that a peak must exist. Obviously, this result is consistent with that of the multivariate function analysis but much more concise. In addition, according to Figure 20b, Figure 21b and Figure 22b, the following Table 4 can be obtained to verify the correctness of the proportion of the above natural vibration response.

_{y}generally constrains the RDs of piers or towers, and only has a significant impact on its own location compared to the other two. Therefore, it can be considered that the selection of F

_{ya}, F

_{yp}and F

_{yt}is independent and not coupled.

_{yt}should be set to its peak value of 2000 kN as shown in Figure 22b. Then, on the grounds of the intersection of the RCD curves of the auxiliary pier and of the tower in Figure 20b, F

_{ya}should be 4000 kN, and F

_{yp}should be approximately 4000 kN. Therefore, F

_{ya}= 4000 kN, F

_{yp}= 4000 kN and F

_{yt}= 2000 kN (layout 3 in Table 3) is one of the reasonable solutions obtained through single variable function analysis.

_{ya}, F

_{yp}and F

_{yt}is 1000 kN~2000 kN, 1000 kN~4000 kN and 2000 kN~4000 kN, respectively. In order to achieve the other designed objective (i.e., a small and uniform RD), the yielding strengths should be as large as possible; therefore, F

_{ya}= 1000 kN, F

_{yp}= 4000 kN and F

_{yt}= 4000 kN is the other reasonable solution (layout 4 in Table 3).

#### 3.4. Comparison of the Designed Results

## 4. Dynamic Vibration of Bridge Systems with and without YSDs

## 5. Equal Yielding Strength Analysis

_{ya}= F

_{yp}= F

_{yt}may be reasonable for engineering practice. Additionally, the calculated cases were greatly reduced from 441 to 5.

_{yt}= 2000 kN is also the designed result by single-variable analysis in terms of RCD at the tower–girder location, and so layout 6 is preferable.

## 6. Practical Method of YSD Design

## 7. Conclusions

- 1
- The increase of the yielding strength can effectively reduce the RD at each location, but its efficiency decreases with the increase of the yielding strength. For current engineering practice, the seismic design capacity of the auxiliary pier can essentially be decreased for cost efficiency.
- 2
- Equal yielding strength analysis is much simpler than unequal yielding strength analysis; however, it may not be the most appropriate result theoretically, but it is effective enough for engineering practices.
- 3
- The practical design method using equal yielding strength analysis can greatly reduce the analysis cases and hence improve the efficiency of seismic analysis.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 13.**Responses of the auxiliary pier with varying F

_{ya}and F

_{yt}. (

**a**) Relative displacement, (

**b**) ratio of capacity to demand (RCD).

**Figure 14.**Responses of the transition pier with varying F

_{yp}and F

_{yt}. (

**a**) Relative displacement, (

**b**) ratio of capacity to demand.

**Figure 15.**Responses of the tower column with varying F

_{ya}and F

_{yp}. (

**a**) Relative displacement; (

**b**) ratio of capacity to demand.

**Figure 20.**Seismic responses with varying F

_{ya}. (

**a**) Relative displacement (unit: mm); (

**b**) ratio of capacity to demand.

**Figure 21.**Seismic responses with varying F

_{yp}. (

**a**) Relative displacement; (

**b**) ratio of capacity to demand.

**Figure 22.**Seismic responses with varying F

_{yt}. (

**a**) Relative displacement; (

**b**) Ratio of capacity to demand.

**Figure 24.**Seismic responses with varying F

_{y}. (

**a**) Relative displacement; (

**b**) ratio of capacity to demand.

**Figure 25.**Standard deviation with varying F

_{y}. (

**a**) Standard deviation of RCD; (

**b**) standard deviation of RD. Note: 1.0 mass means the original weight of the main girder, 1.5 mass means 1.5 times the original weight, 2.0 mass means 2.0 times the original weight.

Material and Component | Standard Strength (MPa) | Young’s Modulus (MPa) |
---|---|---|

Concrete (tower) | 32.4 | 3.45 × 10^{4} |

Concrete (transition pier) | 26.8 | 3.25 × 10^{4} |

Concrete (auxiliary pier) | 26.8 | 3.25 × 10^{4} |

Cable | 1770 | 2.05 × 10^{5} |

Rebar | 400 | 2.06 × 10^{5} |

Location | Degree of Freedom (without YSDs) | Degree of Freedom (with YSDs) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

UX | UY | UZ | RX | RY | RZ | UX | UY | UZ | RX | RY | RZ | |

Tower | 0 | 1 | 1 | 0 | 0 | 0 | 0 | YSD (2) | 1 | 0 | 0 | 0 |

Auxiliary pier | 0 | 0 | 1 | 0 | 0 | 0 | 0 | YSD (2) | 1 | 0 | 0 | 0 |

Transition pier | 0 | 0 | 1 | 0 | 0 | 0 | 0 | YSD (2) | 1 | 0 | 0 | 0 |

Bridge System | YSD Layout Plan | RCD at Tower | RCD at TP | RCD at AP | RD at Tower | RD at TP | RD at AP | Tower TD | Mid-Span Lateral Drift |
---|---|---|---|---|---|---|---|---|---|

A | Without YSDs | 0.59 | 2.04 | 1.99 | 0 | 417 | 235 | 173 | 515 |

B | Designed layout 1 (M) | 1.21 | 1.19 | 1.19 | 189 | 108 | 69 | 174 | 312 |

Designed layout 2 (M) | 1.17 | 1.02 | 1.99 | 118 | 125 | 125 | 168 | 227 | |

Designed layout 3 (S) | 1.25 | 1.15 | 1.24 | 126 | 106 | 53 | 169 | 235 | |

Designed layout 4 (S) | 1.18 | 1.14 | 2.02 | 125 | 135 | 131 | 164 | 209 |

_{ya}= 4000 kN, F

_{yp}= 4000 kN and F

_{yt}= 1000 kN (designed for a large and uniform RCD), designed layout 2: F

_{ya}= 1000 kN, F

_{yp}= 5000 kN and F

_{yt}= 4000 kN (designed for a small and uniform RD), designed layout 3: F

_{ya}= 4000 kN, F

_{yp}= 4000 kN and F

_{yt}= 2000 kN (designed for a large and uniform RCD), designed layout 4: F

_{ya}= 1000 kN, F

_{yp}= 4000 kN and F

_{yt}= 4000 kN (designed for a small and uniform RD). 2. The unit of displacement is mm, and RCD is a dimensionless constant.

F_{yt} | M_{1} | M of Tower | M_{sv} of Tower | α of Tower | F_{yp}/F_{ya} | M_{2} | M of TP | M of AP | M_{sv} of TP | M_{sv} of AP | α of TP | α of AP |
---|---|---|---|---|---|---|---|---|---|---|---|---|

9000 | 360000 | 508542 | 148542 | 0.29 | 5000 | 200000 | 272719 | 256396 | 72719 | 56396 | 0.27 | 0.22 |

_{1}means the moment of the tower due to the YSD, M

_{2}means the moment of the pier due to the YSD, M

_{sv}means the moment due to self-vibration, α means the proportion of the self-vibration response to the total, TP means the transition pier, and AP means the auxiliary pier. 2. The unit of bending moment is kN·m, and α is a dimensionless constant.

Conventional System | YSD System (After Yielding) | ||||||||
---|---|---|---|---|---|---|---|---|---|

Top View | Side View | layout 3 | layout 4 | ||||||

Top View | Side View | MPMR | Period | MPMR | Period | MPMR | Period | ||

10.8% | 3.42 s | 48.0% | 7.30 s | 48.4% | 7.52 s | ||||

53.5% | 1.18 s | 19.0% | 0.95 s | 19.0% | 0.95 s |

Bridge System | YSD Layout Plan | RCD of Tower | RCD of TP | RCD of AP | RD of Tower | RD of TP | RD of AP | Tower TD | Mid-Span Lateral Drift |
---|---|---|---|---|---|---|---|---|---|

A | Without YSDs | 0.59 | 2.04 | 1.99 | 0 | 417 | 235 | 173 | 515 |

B | Designed layout 5 | 1.24 | 1.30 | 1.44 | 124 | 130 | 80 | 164 | 198 |

Designed layout 6 | 1.28 | 1.70 | 1.66 | 189 | 170 | 152 | 165 | 245 |

_{ya}= 3000 kN, F

_{yp}= 3000 kN and F

_{yt}= 3000 kN (designed for a large and uniform RCD), designed layout 6: F

_{ya}= 2000 kN, F

_{yp}= 2000 kN and F

_{yt}= 2000 kN (designed for a small and uniform RD).

**Table 7.**The ratio between the yield strength of YSDs and the weight of the superstructure (unit: kN).

For a Large and Uniform RCD | For a Small and Uniform RD | |||||
---|---|---|---|---|---|---|

Yield Strength | Weight | Ratio | Yield Strength | Weight | Ratio | |

1.0 mass | 12 × 3000 | 280,000 | 13% | 12 × 2000 | 280,000 | 9% |

1.5 mass | 12 × 4000 | 420,000 | 11% | 12 × 3000 | 420,000 | 9% |

2.0 mass | 12 × 4000 | 560,000 | 9% | 12 × 4000 | 560,000 | 9% |

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## Share and Cite

**MDPI and ACS Style**

Xu, Y.; Zeng, Z.; Cui, C.; Zeng, S.
Practical Design Method of Yielding Steel Dampers in Concrete Cable-Stayed Bridges. *Appl. Sci.* **2019**, *9*, 2857.
https://doi.org/10.3390/app9142857

**AMA Style**

Xu Y, Zeng Z, Cui C, Zeng S.
Practical Design Method of Yielding Steel Dampers in Concrete Cable-Stayed Bridges. *Applied Sciences*. 2019; 9(14):2857.
https://doi.org/10.3390/app9142857

**Chicago/Turabian Style**

Xu, Yan, Zeng Zeng, Cunyu Cui, and Shijie Zeng.
2019. "Practical Design Method of Yielding Steel Dampers in Concrete Cable-Stayed Bridges" *Applied Sciences* 9, no. 14: 2857.
https://doi.org/10.3390/app9142857