# Mechanical Behavior of a Double-Column Self-Centering Pier Fused with Shear Links

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## Abstract

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## Featured Application

**A double-column self-centering pier fused with shear links is proposed as a novel structure, which can improve the seismic resilience of bridges. Corresponding simplified FE model and theoretical model are developed. The research outcome of this work can guide the design of the innovated pier.**

## Abstract

## 1. Introduction

## 2. FE Model and Validation

#### 2.1. Modeling Method

#### 2.2. Validation of Modeling Method

#### 2.2.1. Self-Centering Pier

_{c}= 52.7 MPa, measured elastic modulus E

_{c}= 3.8 × 10

^{4}MPa) is adopted, while 20 mm diameter hot rolled ribbed steel bars (measured yield strength f

_{y}= 349 MPa, measured elastic modulus E

_{s}= 202 GPa) and 15.2 mm diameter strands (ultimate strength f

_{PT}= 1860 MPa, elastic modulus E

_{s}= 195 GPa) are used for the internal ED bars and unbonded prestress steel strands, respectively. The average prestress of each strand is 78.26 kN.

#### 2.2.2. Shear Links

_{k}, a parameter defining unloading stiffness; (5) K

_{0}, initial stiffness; (6) F

_{y}, yield strength; (7) η, hardening ratio; (8) η

_{soft}, softening ratio; and (9) α, the ratio of peak strength to yield strength. These parameters can be divided into two categories: parameters (1)–(4), namely, the backbone curve parameters, describing the force-displacement backbone curve under the cyclic lateral load; and parameters (5)–(9), namely, the hysteretic parameters, specifying the hysteretic rules. The experiment conducted by Ji et al. [44] is employed to calibrate the hysteretic parameter values.

## 3. Performance of Double-Column Self-Centering Piers with Shear Links under Quasi-Static Cyclic Loading

^{2}and an initial stress of 1395 MPa. Further, a vertical concentrated load of 3.08 × 10

^{4}kN is applied at the bent cap to simulate the superstructure gravity. Grade C60 concrete (Chinese concrete grade, standard uniaxial unconfined compressive strength f

_{c}= 38.5 MPa, standard elastic modulus E

_{c}= 3.6 × 10

^{4}MPa) is adopted, while 15.2 mm diameter strands (ultimate strength f

_{PT}= 1860 MPa, elastic modulus E

_{s}= 195 GPa) are used for unbonded prestress steel strands. The initial stiffness and yield strength of the adopted shear link are 6.0×105 kN/m and 1.4 × 106 kN, respectively. The responses of the piers under quasi-static cyclic loading are discussed in this section.

#### 3.1. Rocking Process

#### 3.2. Influence of Shear Link Arrangement

_{ISM}and F

_{yISM}represent the initial stiffness and yield strength of the spring in the integrated spring model, respectively, while k

_{UDSMi}and F

_{yUDSMi}represent the initial stiffness and yield strength of each spring in the uniformly distributed spring model, respectively; n is the number of springs.

## 4. Theoretical Analysis of Double-Column Self-Centering Piers with Shear Links

#### 4.1. Theoretical Model Parameters

- Geometrical dimensions: h and A
_{pier}are the height and cross section of the columns, respectively; d is the distance between columns and A_{PT}is the area of steel strands. - Material properties: E
_{c}and f_{c}are the elastic modulus and axial compressive strength of concrete, respectively; E_{PT}and f_{PT}are the elastic modulus and the ultimate strength of steel strands, respectively. - Gravitational forces: W
_{beam}and W_{pier}are superstructure gravity and column gravity, respectively. - Shear link parameters: k
_{SL}and F_{ySL}are the initial stiffness and yield strength of shear links, respectively. - Other: F
_{PT0}is the initial prestress of the steel strands. All of these parameters are illustrated in Figure 12.

#### 4.2. Piers with Shear Links

#### 4.2.1. Pre-Rocking State

_{3}) and E(u

_{6}); the horizontal linear displacements of nodes B(u

_{1}), C(u

_{4}), D(u

_{4}), and E(u

_{5}); and the vertical linear displacement of node B(u

_{2}). Obviously, u

_{4}can be considered the lateral displacement at the top of the pier u

_{x}, i.e., u

_{x}= u

_{4}. Since elements ① and ② in Figure 13b refer to the column, A

_{1}, I

_{1}, and E

_{1}are the area, inertia moment, and elastic modulus of each column, which are expressed as

_{c}is the elastic modulus of concrete.

#### 4.2.2. Rocking State

_{x}is the lateral displacement at the top of the pier. The relationship between the above displacements and the rotation angle of the column can be directly obtained based on the principle of displacement compatibility:

_{PT}(θ) is

_{y}, the shear force at the ends of the shear link, F

_{SL}(θ), is

#### 4.3. Piers without Shear Links

#### 4.4. Model Validation

## 5. Parametric Analysis

#### 5.1. Influence of Initial Stiffness of Shear Links (k_{SL})

#### 5.2. Influence of Yield Strength of Shear Links (F_{ySL})

#### 5.3. Influence of Initial Prestress of Steel Strands (F_{PT0})

^{4}kN to 3.08 × 10

^{4}kN with an increment of 0.77 × 10

^{4}kN, the hysteretic curves of the double-column self-centering pier both with and without shear links are shown in Figure 19. For both piers, the initial prestress of steel strands mainly affects stiffness during the rocking process. The bigger the initial prestress is, the larger the rocking stiffness of the pier is. Further, the initial prestress of the steel strands can efficiently reduce the residual displacement.

## 6. Conclusions

- The prestressed steel strands provide the proposed pier with a stable self-centering ability, which efficiently reduces the residual displacement. Additionally, the rocking behavior effectively prevents the development of plastic hinges in the columns of conventional reinforced concrete piers. Furthermore, the shear links between columns guarantee the pier has a preferable energy dissipation ability. Therefore, the proposed pier can be used to improve seismic resilience with the remarkable features mentioned above.
- The results of the FE model indicate that, for the short piers, it is reasonable to neglect the deformation of each column when the pier rocks. The integrated spring model and the uniformly distributed spring model show almost the same hysteretic performance. These two observations are the basis of the theoretical model.
- The results derived from the theoretical models show good agreement with the results from the FE models. In this case, the theoretical model based on the matrix displacement method and the virtual work principle can be used for future analysis of the innovated pier.
- According to the parametric analyses, the influence of the initial stiffness of the shear links can be neglected. By enhancing the yield strength of shear links, the energy dissipation capacity of the pier can be improved but the residual displacement will increase. Further, the bigger the initial prestress is, the larger the rocking stiffness of the pier is and the smaller the residual displacement is.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Self-centering pier experiment (redrawn from [17]). (

**a**) Design details of the Specimen (unit: mm); (

**b**) load pattern. ED, energy dissipation.

**Figure 6.**Hysteretic model to simulate shear links [41].

**Figure 10.**Different spring models. (

**a**) Uniformly distributed spring model; (

**b**) integrated spring model.

**Figure 13.**Simplified model of double-column self-centering pier with shear links. (

**a**) Full structure; (

**b**) semi-structure.

**Figure 15.**Results of the theoretical and numerical models of the double-column self-centering pier with shear links.

**Figure 16.**Results of the theoretical and numerical models of the double-column self-centering pier without shear links.

**Figure 17.**Influence of the initial stiffness of shear links on the hysteretic curve of the double-column self-centering pier with shear links.

**Figure 18.**Influence of the yield strength of shear links on the hysteretic curve of the double-column self-centering pier with shear links.

**Figure 19.**Influence of the initial prestress of steel strands on the hysteretic curve. (

**a**) Double-column self-centering pier with shear links; (

**b**) double-column self-centering pier without shear links.

Parameters | Pier with Shear Links | Pier without Shear Links | |
---|---|---|---|

Geometrical dimensions | Height of columns, h | 10.0 m | |

Cross section of columns, A_{col} | 2.0 m × 2.0 m | ||

Distance between columns, d | 3.0 m | ||

Area of steel strands, A_{PT} | 12,737.8 mm^{2} | ||

Material properties | Elastic modulus of concrete, E_{c} | 3.6 × 10^{4} MPa | |

Axial compressive strength of concrete, f_{c} | 38.5 MPa | ||

Elastic modulus of steel strands, E_{PT} | 1.95 × 10^{5} MPa | ||

Ultimate strength of steel strands, f_{PT} | 1860 MPa | ||

Gravitational forces | Gravity of superstructure, W_{ss} | 3.08 × 10^{4} kN | |

Gravity of columns, W_{col} | Ignored | ||

Shear links | Initial stiffness of shear links, k_{SL} | 1.6 × 10^{4} kN/m | / |

Yield strength of shear links, F_{ySL} | 2.0 × 10^{4} kN | / | |

Other | Initial prestress of steel strands, F_{PT0} | 1.54 × 10^{4} kN |

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**MDPI and ACS Style**

Xu, L.; Lu, X.; Zou, Q.; Ye, L.; Di, J.
Mechanical Behavior of a Double-Column Self-Centering Pier Fused with Shear Links. *Appl. Sci.* **2019**, *9*, 2497.
https://doi.org/10.3390/app9122497

**AMA Style**

Xu L, Lu X, Zou Q, Ye L, Di J.
Mechanical Behavior of a Double-Column Self-Centering Pier Fused with Shear Links. *Applied Sciences*. 2019; 9(12):2497.
https://doi.org/10.3390/app9122497

**Chicago/Turabian Style**

Xu, Liangjin, Xinzheng Lu, Qiaoshan Zou, Lieping Ye, and Jin Di.
2019. "Mechanical Behavior of a Double-Column Self-Centering Pier Fused with Shear Links" *Applied Sciences* 9, no. 12: 2497.
https://doi.org/10.3390/app9122497