# Uncertainty Quantification in Constitutive Models of Highway Bridge Components: Seismic Bars and Elastomeric Bearings

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## Abstract

**:**

## 1. Introduction

## 2. Experimental Data

#### 2.1. Seismic Bars (SBs)

_{l}) was the distance from the bottom of the bent cap to the slab of the bridges. Four specimens were used in the study: WD1, WD2, WOD1, and WOD2 with different h

_{l}; see Table 1.

#### 2.2. Elastomeric Bearings (EBs)

#### 2.2.1. Unanchored Elastomeric Bearing (UEB)

#### 2.2.2. Anchored Elastomeric Bearing (AEB)

## 3. Constitutive Models

#### 3.1. Seismic Bars (SBs)

#### 3.2. Elastomeric Bearings

#### 3.2.1. Unanchored Elastomeric Bearing (UEB)

#### 3.2.2. Anchored Elastomeric Bearing (AEB)

## 4. Bayesian Parameter Estimation

#### 4.1. Bayesian Inversion

#### 4.2. Markov Chain Monte Carlo (MCMC)

#### 4.3. Tempering

- Sample each parameter $\mathit{\theta}$ from a prior distribution.
- Simulate a dataset ${\mathit{y}}^{\ast}$ using a function that takes the parameters and returns the predicted data (${\mathit{y}}_{\mathbf{0}}$), considering the dimension of the observed data.
- Compare ${\mathit{y}}^{\ast}$ and ${\mathit{y}}_{\mathbf{0}}$ using the distance function and a tolerance threshold value.
- When $\beta =1$, the distance function value is less than the threshold value; if this tolerance value is sufficiently small, the distribution obtained will be a good approximation for the posterior $P\left(\mathit{\theta}|{\mathit{y}}_{\mathbf{0}}\right).$

#### 4.4. Convergence Criteria

#### 4.5. Conflation Procedure

## 5. Results and Discussion

#### 5.1. Calibration

#### 5.1.1. Seismic Bars (SBs)

#### 5.1.2. UEB

^{2}and 0.23, respectively. The predicted mean value of $G$falls within the range 981–1275 kN/m

^{2}, suggested by AASHTO (2012) [33], while $\mu $ shows a predicted mean value similar to those recommend in AASHTO (2017) [20] of 0.2.

#### 5.1.3. AEB

#### 5.2. Proposed PDFs

## 6. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- SEAOC. A Framework of Performance-Based Seismic Engineering of Buildings; Structural Engineers Association of California: Sacramento, CA, USA, 1995. [Google Scholar]
- ASCE. Seismic Design of Piers and Wharves; ASCE: Reston, VA, USA, 2014; p. 61. [Google Scholar] [CrossRef]
- Tall Building Initiative (TBI). Guidelines for Performance-Based Seismic Design of Tall Buildings; TBI: Berkeley, CA, USA, 2017. [Google Scholar]
- Zhang, P.; Restrepo, J.I.; Conte, J.P.; Ou, J. Nonlinear Finite Element Modeling and Response Analysis of the Collapsed Alto Rio Building in the 2010 Chile Maule Earthquake. Struct. Tall Spec. Build.
**2017**, 26, e1364. [Google Scholar] [CrossRef] - Hube, M.; Santa María, H.; Villalobos, F. Preliminary Analysis of the Seismic Response of Bridges during the Chilean 27 February 2010 Earthquake. Obras Proy. Rev. Ing. Civ.
**2010**, 8, 48–57. [Google Scholar] - Elnashai, A.S.; Gencturk, B.; Kwon, O.S.; Hashash, Y.M.A.; Kim, S.J.; Jeong, S.H.; Dukes, J. The Maule (Chile) Earthquake of February 27, 2010: Development of Hazard, Site Specific Ground Motions and Back-Analysis of Structures. Soil Dyn. Earthq. Eng.
**2012**, 42, 229–245. [Google Scholar] [CrossRef] - Wilches, J.; Santa María, H.; Riddell, R.; Arrate, C. Effects of Changes in Seismic Design Criteria in the Transverse and Vertical Response of Chilean Highway Bridges. Eng. Struct.
**2019**, 191, 370–385. [Google Scholar] [CrossRef] - Aldea, S.; Bazaez, R.; Astroza, R.; Hernandez, F. Seismic Fragility Assessment of Chilean Skewed Highway Bridges. Eng. Struct.
**2021**, 249, 113300. [Google Scholar] [CrossRef] - Martínez, A.; Hube, M.A.; Rollins, K.M. Analytical Fragility Curves for Non-Skewed Highway Bridges in Chile. Eng. Struct.
**2017**, 141, 530–542. [Google Scholar] [CrossRef] - Xiang, N.; Goto, Y.; Alam, M.S.; Li, J. Effect of Bonding or Unbonding on Seismic Behavior of Bridge Elastomeric Bearings: Lessons Learned from Past Earthquakes in China and Japan and Inspirations for Future Design. Adv. Bridge Eng.
**2021**, 2, 1–17. [Google Scholar] [CrossRef] - Aviram, A.; Mackie, K.R.; Stojadinovic, B. Nonlinear Modeling of Bridge Structures in California. ACI Symp. Publ.
**2010**, 271, 1–26. [Google Scholar] [CrossRef] - Steelman, J.S.; Fahnestock, L.A.; Filipov, E.T.; LaFave, J.M.; Hajjar, J.F.; Foutch, D.A. Shear and Friction Response of Nonseismic Laminated Elastomeric Bridge Bearings Subject to Seismic Demands. J. Bridge Eng.
**2013**, 18, 612–623. [Google Scholar] [CrossRef] - Filipov, E.T.; Fahnestock, L.A.; Steelman, J.S.; Hajjar, J.F.; LaFave, J.M.; Foutch, D.A. Evaluation of Quasi-Isolated Seismic Bridge Behavior Using Nonlinear Bearing Models. Eng. Struct.
**2013**, 49, 168–181. [Google Scholar] [CrossRef] - Konstantinidis, D.; Kelly, J.M.; Makris, N. Experimental Investigation on the Seismic Response of Bridge Bearings; Earthquake Engineering Research Center, University of California: Berkeley, CA, USA, 2008. [Google Scholar]
- Rubilar, F. Modelo no Lineal Para Predecir la Respuesta Sísmica de Pasos Superiores; Pontificia Universidad Católica de Chile: Santiago, Chile, 2015. [Google Scholar]
- Astroza, R.; Alessandri, A.; Conte, J.P. A Dual Adaptive Filtering Approach for Nonlinear Finite Element Model Updating Accounting for Modeling Uncertainty. Mech. Syst. Signal Process.
**2019**, 115, 782–800. [Google Scholar] [CrossRef] - Ebrahimian, H.; Astroza, R.; Conte, J.P.; de Callafon, R.A. Nonlinear Finite Element Model Updating for Damage Identification of Civil Structures Using Batch Bayesian Estimation. Mech. Syst. Signal Process.
**2017**, 84, 194–222. [Google Scholar] [CrossRef] - Birrell, M.; Astroza, R.; Carreño, R.; Restrepo, J.I.; Araya-Letelier, G. Bayesian Parameter and Joint Probability Distribution Estimation for a Hysteretic Constitutive Model of Reinforcing Steel. Struct. Saf.
**2021**, 90, 102062. [Google Scholar] [CrossRef] - McKenna, F.; Fenves, G.L.; Scott, M.H. Open System for Earthquake Engineering Simulation; Pacific Earthquake Engineering Research Center, University of California: Berkeley, CA, USA, 2003. [Google Scholar]
- AASHTO. LRFD. Bridge Design Specifications, 8th ed.; American Association of State Highway and Transportation Officials: Washington, DC, USA, 2017. [Google Scholar]
- Wen, Y.-K. Method for Random Vibration of Hysteretic Systems. J. Eng. Mech. Div.
**1976**, 102, 249–263. [Google Scholar] [CrossRef] - Yi, J.; Yang, H.; Li, J. Experimental and Numerical Study on Isolated Simply-Supported Bridges Subjected to a Fault Rupture. Soil Dyn. Earthq. Eng.
**2019**, 127, 105819. [Google Scholar] [CrossRef] - Toni, T.; Welch, D.; Strelkowa, N.; Ipsen, A.; Stumpf, M.P.H. Approximate Bayesian Computation Scheme for Parameter Inference and Model Selection in Dynamical Systems. J. R. Soc. Interface
**2009**, 6, 187–202. [Google Scholar] [CrossRef] [Green Version] - Ramancha, M.K.; Astroza, R.; Madarshahian, R.; Conte, J.P. Bayesian Updating and Identifiability Assessment of Nonlinear Finite Element Models. Mech. Syst. Signal Process.
**2022**, 167, 108517. [Google Scholar] [CrossRef] - Chib, S.; Greenberg, E. Understanding the Metropolis-Hastings Algorithm. Am. Stat.
**1995**, 49, 327–335. [Google Scholar] [CrossRef] [Green Version] - Kroese, D.P.; Taimre, T.; Botev, Z.I. Handbook of Monte Carlo Methods; John Wiley & Sons: Hoboken, NJ, USA, 2013. [Google Scholar]
- Wagner, P.R.; Nagel, J.; Marelli, S.; Sudret, B. UQLab User Manual—Bayesian Inference for Model Calibration and Inverse Problems; Report No. UQLab-V1. In Chair of Risk, Safety & Uncertainty Quantification; ETH Zurich: Zürich, Switzerland, 2019; pp. 3–113. [Google Scholar]
- Lintusaari, J.; Gutmann, M.U.; Kaski, S.; Corander, J. On the Identifiability of Transmission Dynamic Models for Infectious Diseases. Genetics
**2016**, 202, 911–918. [Google Scholar] [CrossRef] [Green Version] - Vats, D.; Flegal, J.M.; Jones, G.L. Multivariate Output Analysis for Markov Chain Monte Carlo. Biometrika
**2019**, 106, 321–337. [Google Scholar] [CrossRef] [Green Version] - Hill, T.P.; Miller, J. How to Combine Independent Data Sets for the Same Quantity. Chaos: Interdiscip. J. Nonlinear Sci.
**2011**, 21, 033102. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Neal, R.M. Slice Sampling. Ann. Stat.
**2003**, 31, 705–767. [Google Scholar] [CrossRef] - Gelman, A.; Rubin, D.B. Inference from Iterative Simulation Using Multiple Sequences. Stat. Sci.
**1992**, 7, 457–472. [Google Scholar] [CrossRef] - AASHTO. LRFD Bridge Design Specifications, 6th ed.; American Association of State Highway and Transportation Officials: Washington, DC, USA, 2012. [Google Scholar]

**Figure 3.**(

**a**) Elastic perfectly plastic load-displacement model for UEB; (

**b**) elastic plastic load-displacement model for UEB; and (

**c**) Bouc–Wen model representing the lateral load-displacement relationship for AEB.

**Figure 4.**Comparison of the responses of (

**a**) SBs-WD (WD2) and (

**b**) SBs-WOD (WOD1) as tested by Martinez et al. (2017) with a calibrated response at posterior means, 5th and 95th percentiles.

**Figure 6.**Conflated PDFs for ${f}_{y}$ (

**a**) and ${g}_{1}$ (

**b**) in the SBs-WD model, and conflated PDF for ${f}_{y}$ (

**c**), ${g}_{1}$ (

**d**), and ${g}_{2}$ (

**e**) on in the SBs-WOD model.

**Figure 7.**Comparison of the response of UEB as tested by Rubilar et al. (2015) with calibrated response at 5th and 95th percentiles.

**Figure 10.**Comparison of the AEB from test series S1 with a calibrated response at 5th and 95th percentiles using (

**a**) AEB-model 1 (

**b**) AEB-model 2.

**Figure 11.**The AEB KDE matrix for test series S1 as a representative case. (

**a**) AEB-model 1 and (

**b**) AEB-model 2.

**Figure 13.**Conflated PDF for ${G}_{eb}$, ${f}_{ya}$, ${a}_{2}$, $\mu $, $\beta $, and $\eta $ on AEB-model 2.

Specimen | h_{l} (cm) |
---|---|

WD1/WD2 | 10 |

WOD1/WOD2 | 72 |

Test Tag | Cyclic Test Velocity (mm/s) |
---|---|

B1 | 25 |

B2 | 25 |

B3 | 10 |

B4 | 50 |

B5 | 75 |

B6 | 100 |

Test Series | N | Width (mm) | Length (mm) | Diameter (mm) | Total Height (mm) |
---|---|---|---|---|---|

S1 | 5 | 650 | 650 | - | 98 |

S2 | 4 | - | - | 500 | 100 |

S3 | 6 | 320 | 320 | - | 53 |

S4 | 4 | 700 | 700 | - | 194 |

4 | 1000 | 1000 | - | 230 | |

S5 | 1 | 585 | 600 | - | 168 |

Type of Seismic Bar | ${\mathit{\gamma}}_{1}$ | ${\mathit{\gamma}}_{2}$ |
---|---|---|

SBs-WD | 0.04 | 0.71 |

SBs-WOD | 0.07 | 0.31 |

Parameter | |
---|---|

Post-yield stiffness ratio of non-linear hardening component | ${a}_{2}$ |

Exponent of non-linear hardening component | $\mu $ |

Yield exponent | $\beta $ |

First hysteretic shape parameter | $\eta $ |

Parameter | Distribution | Mean | COV (%) | |
---|---|---|---|---|

SBs-WD (WD2) | ${f}_{y}$ (MPa) | Normal | 235.46 | 5.82 |

${g}_{1}$(-) | 0.11 | 36.36 | ||

SBs-WOD (WOD1) | ${f}_{y}$ (MPa) | Normal | 266.24 | 8.40 |

${g}_{1}$(-) | 0.09 | 22.22 | ||

${g}_{2}$ (-) | 0.39 | 10.26 | ||

UEB (B1-C2) | ${G}_{eb}$ (kN/m^{2}) | Normal | 1083.60 | 3.60 |

$\mu $ (-) | 0.31 | 2.59 | ||

AEB—model 1 (S1) | ${F}_{{y}_{a}}$ (kN) | Normal | 108.00 | 11.10 |

${G}_{eb}$ (kN/m^{2}) | 1033.50 | 2.5 | ||

AEB—model 2 (S1) | ${F}_{{y}_{a}}$ (kN) | Normal | 98.45 | 4.89 |

${G}_{eb}$ (kN/m^{2}) | 994.00 | 3.10 | ||

${a}_{2}$ | 0.50 | 7.98 | ||

$\mu $ | 4.00 | 7.50 | ||

$\beta $ | 0.90 | 4.33 | ||

$\eta $ | 1.05 | 11.41 |

Parameter | |
---|---|

Yield force | ${F}_{ya}$ |

Shear modulus of the bearing | ${G}_{eb}$ |

Post-yield stiffness ratio of non-linear hardening component | ${a}_{2}$ |

Exponent of non-linear hardening component | $\mu $ |

Yield exponent | $\eta $ |

First hysteretic shape parameter | $\beta $ |

Component | Parameter | Distribution | Mean | COV (%) |
---|---|---|---|---|

SBs-WD | ${f}_{y}$ (MPa) | Normal | 206.4 | 6.76 |

${g}_{1}$(-) | 0.104 | 27.6 | ||

SBs-WOD | ${f}_{y}$ (MPa) | Normal | 264.4 | 7.60 |

${g}_{1}$(-) | 0.091 | 13.8 | ||

${g}_{2}$ (-) | 0.395 | 6.25 | ||

UEB | ${G}_{eb}$ (kN/m^{2}) | Normal | 1176 | 1.15 |

$\mu $ (-) | 0.230 | 0.80 | ||

AEB-model 1 | ${f}_{{y}_{a}}$ (kN) | Normal | 167.6 | 1.72 |

${G}_{eb}$ (kN/m^{2}) | 985 | 0.39 | ||

AEB-model 2 | ${f}_{{y}_{a}}$ (kN) | Normal | 157.8 | 1.19 |

${G}_{eb}$ (kN/m^{2}) | 990 | 0.50 | ||

${a}_{2}$ | 0.50 | 2.0 | ||

$\mu $ | 3.765 | 1.74 | ||

$\beta $ | 1.122 | 1.60 | ||

$\eta $ | 0.899 | 1.00 |

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**MDPI and ACS Style**

Pinto, F.J.; Toledo, J.; Birrell, M.; Bazáez, R.; Hernández, F.; Astroza, R.
Uncertainty Quantification in Constitutive Models of Highway Bridge Components: Seismic Bars and Elastomeric Bearings. *Materials* **2023**, *16*, 1792.
https://doi.org/10.3390/ma16051792

**AMA Style**

Pinto FJ, Toledo J, Birrell M, Bazáez R, Hernández F, Astroza R.
Uncertainty Quantification in Constitutive Models of Highway Bridge Components: Seismic Bars and Elastomeric Bearings. *Materials*. 2023; 16(5):1792.
https://doi.org/10.3390/ma16051792

**Chicago/Turabian Style**

Pinto, Francisco J., José Toledo, Matías Birrell, Ramiro Bazáez, Francisco Hernández, and Rodrigo Astroza.
2023. "Uncertainty Quantification in Constitutive Models of Highway Bridge Components: Seismic Bars and Elastomeric Bearings" *Materials* 16, no. 5: 1792.
https://doi.org/10.3390/ma16051792