# Rapid Seismic Evaluation of Continuous Girder Bridges with Localized Plastic Hinges

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Equation of Motion

**X**

_{s}is the displacement of free nodes;

**X**

_{b}is the displacement of supporting nodes;

**X**

_{l}is the relative displacement of the nodes of the plastic hinges in the piers; D is the indicator matrix of the position of the restoring force in the plastic hinge region;

**F**

_{r}(

**X**

_{l}) is the horizontal restoring force column vector of central plastic hinges;

**P**

_{b}is the force due to ground motions;

**M**,

**C**, and

**K**are the mass, damping, and stiffness matrices, respectively; and subscripts s and b represent free nodes and supporting nodes, respectively.

#### 2.2. Restoring Force Model of Plastic Hinges

_{b}is the generalized viscous damping ratio; α is the ratio of the generalized stiffness of the plastic hinge after it is yielded and the original stiffness; F

_{y}and D

_{y}are the generalized yielding force and generalized yielding displacement(curvature) of the plastic hinge, respectively; u and $\dot{u}$ are the generalized relative displacement and velocity of the plastic hinge; z is the hysteretic variable; and coefficients γ, β, and A are the model parameters depending on the shape of the hysteretic curve. In this research, coefficients γ, β, and A are equal to 0.5, 0.5, and 1.0, respectively.

#### 2.3. Equation of State for Explicit Time-Domain Analysis

**K**

_{se},

**K**

_{h}, and

**C**

_{se}are the equivalent stiffness, hysteretic stiffness, and equivalent damping matrices, respectively; and

**Z**can be expressed as:

**X**) from Equation (5), it is necessary to know

**Z**. To solve for

**Z**from Equation (6), it is necessary to know the velocities (${\dot{\mathit{X}}}_{t}$, ${\dot{\mathit{X}}}_{b}$). It is time consuming to directly solve the coupled equations expressed in the implicit form. This research proposes to convert the coupled equations into the state equation [20,21]:

**A**(i,j) are the coefficient matrices dependent on i, and i = 1, 2, …, p. When i = 1,

**A**

_{1,0}=

**S**

_{1}, and

**A**

_{1,1}=

**S**

_{2}. When i = 2,

**A**

_{2,0}=

**TA**

_{1,0},

**A**

_{2,1}=

**TS**

_{2}+

**S**

_{1}, and

**A**

_{2,2}=

**A**

_{1,1}. When i ≥ 3,

**A**

_{(i,0)}=

**TA**

_{(i−1,0)},

**A**

_{(i,1)}=

**TA**

_{(i−1,1)}, and

**A**

_{(i,j)}=

**A**

_{(i−1,j−1)}.

**T**is the exponential matrix.

**S**and

_{1}**S**are shown as:

_{2}**A**

_{(i,0)}and

**A**

_{(i,0)}are determined, thus solving the state equation. This study incorporates the Runge–Kutta method to improve computation precision in the iterations [22,23]. The recursion formula is expressed as:

_{k}= kh, k = 0, 1, …, n − 1; and h = Δt/n. When k = 0,

**Z**

_{i}(s

_{k}) =

**Z**

_{(i−1)}. When k = n,

**Z**

_{i}(s

_{k}) =

**Z**

_{i}.

**Z**

_{i}, the corresponding state

**V**

_{i}can be solved using Equations (10) and (11) and Equations (12) and (13), in turn obtaining (${\dot{\mathit{X}}}_{t}$) and (${\dot{\mathit{X}}}_{b}$), which are plugged into Equations (14)–(18) to obtain the updated

**Z**

_{i}. This process is continued until the termination criterion is satisfied. In this research, the adopted criterion is expressed as:

**A**

_{(i,i)}is generated, the matrix is retained and used in sequential calculations. In each time step, the calculation only involves ${\mathit{A}}_{i,i}\mathit{F}({t}_{i})$ and $\mathit{g}\left({\dot{\mathit{X}}}_{t},\hspace{0.17em}{\dot{\mathit{X}}}_{b},\hspace{0.17em}{\mathit{Z}}_{i}\right)$. Therefore, the proposed approach is expected to achieve high computational efficiency. The performance of the proposed approach is tested through a case study, as elaborated in Section 3.

#### 2.4. Dimension Reduction

**Z**

_{i}, and the other nodal forces are determined. The hysteretic variable is only related to the velocities of the two ends of each plastic hinge. Based on this understanding,

**V**

_{i}is expressed as:

**Z**

_{i}, which is then plugged into Equation (23) to obtain the structural dynamic response. This solving process is the same as that in Section 2.3, but the number of equations is significantly reduced because the number of nodes involving plastic hinges in the piers is much smaller than the total number of nodes of the bridge.

#### 2.5. Simulation of Random Processes

**F**

_{q(t)}, and q = 1, 2, …, N. The nonlinear dynamic response of the bridge under the q-th sample of ground motion can be determined following the process elaborated in Section 2.1, Section 2.2, Section 2.3 and Section 2.4. It should be noted that, once the coefficient matrix is determined in the first ground motion, the coefficient matrix is sustained and directly used in calculations for the other ground motions without making any change, thus improving the computational efficiency.

_{i}, the mean and standard deviation are calculated as:

## 3. Case Study

#### 3.1. Bridge Description

_{y}= 4169 kN/m. The initial stiffness is K

_{0}= 4169 kN/m. After yielding occurs, the stiffness is K

_{y}= 41.69 kN/m.

#### 3.2. Power Spectrum Model of Ground Motions

^{−3}m

^{2}/s

^{3}, corresponding to an earthquake magnitude of M8 and an earthquake level of E2 [33]; ${\omega}_{k}$ and ${\zeta}_{k}$ are the natural frequency and damping ratio of soils, respectively; and ${\omega}_{sk}$ and ${\zeta}_{sk}$ are the frequency and damping parameters of the filter. The values of ${\omega}_{k}$, ${\zeta}_{k}$, ${\omega}_{sk}$, and ${\zeta}_{sk}$ are listed in Table 4.

#### 3.3. Spatially Varying Excitation

_{jk}is the coherence function of the stationary random processes ${x}_{j}(t)$ and ${x}_{k}(t)$ at the support points j and k; ω is the natural frequency; d

_{jk}is the projection of the horizontal distance between support points j and k along the direction of wave propagation; α is the correlation coefficient; and vs. is the propagation velocity of the earthquake wave. In this research, the values of α and vs. are 0.2 and 600 m/s, respectively.

_{u}of the multi-point ground motions has no influence on the proposed method in this article, it is still set as more than 5 times of the fundamental frequency according to Reference [33]. The duration of earthquake T should be several times longer than the fundamental period of the bridge. The frequency interval dω and the time interval dt are selected according to the Shannon sampling law [33]. Therefore, the cut-off frequency of the multi-point ground motions is ${\omega}_{u}=15\pi \hspace{0.17em}\mathrm{rad}/\mathrm{s}$. The duration of the earthquake is T = 10 s, which is the total duration of earthquake excitation required for calculation and analysis. The frequency interval is $d\omega ={\omega}_{u}/N$, where the interval number N is equal to 300. The time interval is $dt=0.01\hspace{0.17em}\mathrm{s}$.

#### 3.4. Results and Discussions

## 4. Conclusions

- (1)
- The proposed approach is accurate. The seismic performance evaluation of continuous girder bridges mainly focuses on the seismic performance of piers with fixed hinges. The seismic performance evaluation index is mainly based on pier top section displacement, pier bottom section bending moment–curvature, etc. The time history diagrams of pier top section displacement and the pier bottom section bending moment are consistent with the response characteristics of general dynamic problems, so the accuracy of the method can be known.
- (2)
- The proposed approach significantly improves the computational efficiency. Once determined in the first round of analysis, the coefficient matrices in the equations are preserved throughout the analysis. Preservation of the coefficient matrices simplifies the computation process, thus improving computational efficiency. Such improvement is particularly relevant for seismic evaluation because the evaluation involves stochastic processes and requires repeated computations. Compared with the conventional nonlinear time history dynamic analysis performed, the computation time of the proposed approach is only 5%, and the maximum error of the displacement of the pier top section and the bending moment of the pier bottom section is within 10%. The high efficiency of the proposed approach is achieved by the combination of multiple techniques such as explicit time domain analysis using the state equations, the precision integration method, and the dimension reduction method.
- (3)
- The proposed approach represents an explicit nonlinear dynamic analysis method. The bending moment–curvature diagram shows an oval shape, which indicates that the central plastic hinge area of the continuous girder bridge pier has significant nonlinear characteristics and can dissipate certain ground motion energy. Under the current Chinese seismic design code, these problems are typical local nonlinear problems (nonlinearities occur near the pier bottom or pier top section with fixed hinges), and the explicit iterative dimension reduction method can be used to ensure the high efficiency of seismic performance evaluation. Compared to the conventional methods, which implicitly solve the equations of motion based on iterative computation involving matrix inversion, the proposed explicit method is expected to have better convergence performance. It is worth noting that the dimension reduction utilizes the unique feature of localized plastic hinges. The dimension reduction method is likely applicable to other nonlinear stochastic dynamic problems involving local plasticity.
- (4)
- The proposed approach has good generality and can be applied to solving similar problems. The kinematic equation of bridge structure is established based on the general dynamic principle. Then, the nonlinear restoring force of the pier column bottom section is described by the Bouc–Wen model, and the nonlinear motion equation of the continuous girder bridge under multi-point seismic excitation is rewritten into a quasilinear equation, which is also established by combining with the Runge–Kutta method and a precise time-history integration method. The explicit dimension reduction iterative method in time domain adopted in this article is essentially a rapid method for solving local nonlinear random vibration of a class of problems, which is applicable as long as the problem can be described as a local nonlinear problem and contains an explicit nonlinear restoring force model. The above methods are not only appropriate to continuous girder bridges, but also applicable to the analysis of similar problems of other bridges.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Flowchart of the solving process for the hysteretic variable through iterative calculations.

**Figure 4.**Illustration of the bridge (unit: cm): (

**a**) longitudinal profile showing spans of bridge; and (

**b**) the transverse profile showing the width of the bridge.

**Figure 6.**Park constitutive model diagram. (

**a**) Stress–strain curve when loading. (

**b**) Disposal strain curve, again when loading.

Area (m^{2}) | Moment of Inertia Iy (m^{4}) | Moment of Inertia Iz (m^{4}) | Elastic Modulus (GPa) | Density (kg/m^{3}) | |
---|---|---|---|---|---|

Girder | 12.49 | 6.70 | 228.91 | 34.5 | 2549 |

Pier | 2.01 | 0.32 | 0.32 | 32.5 | 2549 |

Type (C40) | Elastic Modulus E_{c} (MPa) | f_{c}(MPa) | Yield Strain | Peak Strain | Ultimate Strain |
---|---|---|---|---|---|

non-confined concrete | 31623 | 40.00 | 0.0014 | —— | 0.02 |

confined concrete | 31623 | 41.72 | 0.002429 | 0.002 | 0.013084 |

Type | E_{s} (MPa) | f_{y} (MPa) | f_{u} (MPa) | ε_{y} | ε_{sh} | ε_{su} |
---|---|---|---|---|---|---|

HRB400 | 200,000 | 400 | 540 | 0.002 | 0.01 | 0.1 |

Site Condition | ${\mathit{\omega}}_{\mathit{k}}$ (rad/s) | ${\mathit{\zeta}}_{\mathit{k}}$ | ${\mathit{\omega}}_{\mathit{s}\mathit{k}}$ (rad/s) | ${\mathit{\zeta}}_{\mathit{s}\mathit{k}}$ |
---|---|---|---|---|

Soil 1 | 20.94 | 0.6 | 1.5 | 0.6 |

Soil 2 | 10.0 | 0.4 | 1.0 | 0.6 |

Method | Time (s) |
---|---|

explicit time-domain method | 250 |

Newmark-β method | 5000 |

Average Value × 10^{3} (kN·m) | Average Value of the Standard Deviation × 10^{3} (kN·m) | Average Value of the Absolute Maximum × 10^{3} (kN·m) |
---|---|---|

1.89 | 6.22 | 18.04 |

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**MDPI and ACS Style**

Wei, Z.; Lv, M.; Shen, M.; Wang, H.; You, Q.; Hu, K.; Jia, S. Rapid Seismic Evaluation of Continuous Girder Bridges with Localized Plastic Hinges. *Sensors* **2022**, *22*, 6311.
https://doi.org/10.3390/s22166311

**AMA Style**

Wei Z, Lv M, Shen M, Wang H, You Q, Hu K, Jia S. Rapid Seismic Evaluation of Continuous Girder Bridges with Localized Plastic Hinges. *Sensors*. 2022; 22(16):6311.
https://doi.org/10.3390/s22166311

**Chicago/Turabian Style**

Wei, Zhaolan, Mengting Lv, Minghui Shen, Haijun Wang, Qixuan You, Kai Hu, and Shaomin Jia. 2022. "Rapid Seismic Evaluation of Continuous Girder Bridges with Localized Plastic Hinges" *Sensors* 22, no. 16: 6311.
https://doi.org/10.3390/s22166311