# Bayesian Calibration of Hysteretic Parameters with Consideration of the Model Discrepancy for Use in Seismic Structural Health Monitoring

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Discrepancy Model

#### 2.2. Bayesian Calibration Procedure

- Phase 1: definition of the seismic input excitation and of the computational model. First, a ground earthquake acceleration record must be selected as seismic input excitation of the system. Then, one specifies general options for the physical model and its governing laws. In this work the system of Ordinary Differential Equations (ODEs) governing the Bouc–Wen hysteretic oscillator (Section 3) are implemented and solved numerically with the explicit Runge–Kutta method.
- Phase 2: definition of the probabilistic prior information on the model parameters. Once the computational model $\mathcal{M}\left(\mathit{x}\right)$ is defined, one has to select carefully which model parameters ${\mathit{x}}_{\mathcal{M}}$ to include in the calibration in order to get a reliable set of physical values from the resulting posteriors estimates after the Bayesian updating. Prior information on possible values of the hyperparameters are set by setting their prior probability distributions $\pi \left({\mathit{x}}_{\mathcal{M}}\right)$, defining for each hyperparameter the type of univariate distribution (i.e., uniform, Gaussian, lognormal distributions, etc.) and its statistical moments. The prior information is obtained making some considerations about the amount of the dissipated energy during the hysteresis.
- Phase 3: Bayesian model updating. At this stage, the Bayesian model updating can be carried out using the experimental data $\mathit{Y}$ inferring the posterior distributions of the hyperparameters. However, in many practical applications, a closed form of Equation (7) does not exist. For this reason, Markov Chain Monte Carlo (MCMC) simulations have been conducted herein, allowing for an approximate expectation in Equation (8).

## 3. Results

#### 3.1. Numerical Benchmark: Calibration of a SDoF Bouc–Wen Type Hysteretic System

_{0}, ${\nu}_{0},{\eta}_{0}$ are usually set to unity [38]. Whereas the values ${\delta}_{A},{\delta}_{\nu},{\delta}_{\eta}$ are constant terms which specify the amount of stiffness and strength degradation [39].

^{®}[6] with 100 chains, 700 steps and AIES solver algorithm [28]. The number of the BWBN forward model ${\mathcal{M}}^{BW}\left(\mathit{x}\right)$ calls in MCMC was 70′000. The system of ordinary differential equation (ODEs; Equations (10) and (12)) was solved with the MATLAB solver ode45 (explicit Runge–Kutta method with relative error tolerance 1 × 10

^{−3}). The Latin hypercube sampling (LHS) method was used to get the parameters prior distributions (Table 2) of the BWBN model, which are shown in Figure 4.

^{−4}(i.e., half of the average of the MSE computed on all the post-predictive model runs, MSE = 4.0415 × 10

^{−4}). In fact, we should not dwell on this specific prediction; rather, we should consider its confidence intervals. The latter tells us how uncertainties on model input propagate through the model.

#### 3.2. Demonstration on a Case Study

^{2}while the volume is about 5000 m

^{3}. Figure 8 reports the schematizations of the analyzed building. The Town Hall of Pizzoli belongs to the network of buildings monitored by OSS [26]. The OSS monitors the structure of Pizzoli thanks to a dynamic monitoring system composed of eight accelerometers installed on the building and one placed in the basement. More specifically, a tri-axial accelerometer to record the seismic input in the three space directions is fixed in the basement. On the raised ground floor, no accelerometers are present. On the first floor, three biaxial accelerometers and one monoaxial accelerometer (in X direction) are present to capture the acceleration responses of the first floor in the two horizontal directions. Finally, the same scheme of sensors installed on the first floor is present on the second floor. It is worth noting that the acquisition system and the sensors setup was designed by OSS and represents an input data for our analyses. In this regard, to obtain an optimal positioning of the sensors, a finite element model should be used. In particular, the optimal sensor placement is done recurring to a linear finite element model (using eigen-analysis, singular value decomposition, etc.). The model is often linear to perform this task because for nonlinear analysis the linear elastic component commonly brings the predominant amount of the output variance.

#### 3.2.1. Reference Model

- The adopted procedure allowed verification of the consistency of the assumed nonlinear model. This was done by checking the stability of the values of the model parameters over time.
- The procedure allowed for the collection of timely information on the health of the structure immediately after the occurrence of the earthquake.

#### 3.2.2. Bayesian Calibration of the Reduced Single DoF Reference Model

^{2}) the structure exhibited a low level of damage [44].

#### 3.2.3. Models for Stiffness Degradation

_{W}controls the amount of damage. Secondly, a logistic distribution function was adopted:

_{L}defines the amount of degradation.

#### 3.2.4. Comparison of the Calibrated Models

#### 3.3. Influence of the Degradation Level on Model Parameters Inference

^{2}(medium level of degradation); (ii) one in which the input has been rescaled with a PGA of 9 m/s

^{2}(high level of degradation).

#### 3.3.1. Medium Level of Degradation

^{2}) and keeping the same reference damage, all the models were still able to predict the correct value of the initial stiffness ${k}_{i}$. However, an increase of the uncertainties in the model prediction was registered, this time due to the greater values of the discrepancy variance. The choice of the MAP (or of the mean of the distributions) as point estimate led to a slight overrating of the initial stiffness. As the estimation error was negligible, the adopted discrepancy term still performs as a good function to embody all the model errors. However, increasing the level of the input without accounting for a growth of the damage level appears to be unrealistic. This is because higher seismic excitations generally involve greater dissipation of energy. Therefore, a loss up to 40% of the initial stiffness ${k}_{i}$ was taken into account as well, resulting in Model 1 (i.e., the one consistent with the BWBN degrading model, which is able to predict the correct value of the initial stiffness). On the contrary, all the models inconsistent with the BWBN model (i.e., Models 2, 3 and 4) overrate the initial stiffness value in a non-negligible proportion (Figure 13), leading to higher level of uncertainty in the model response prediction. It is also worth noticing how the estimation error was amplified for non-parametric models. This is a considerable indicator of the inadequacy of the discrepancy function adopted. In order to recover the response records, more accurate and well-suited discrepancy models should be investigated, but this is beyond the scope of this paper.

#### 3.3.2. High Level of Degradation

^{2}). Again, when keeping the same level of reference damage, all models were able to predict the correct value of the initial stiffness although with a visibly greater variance than before. When the level of damage increased, taking into account a stiffness loss of up to the 80% of the initial one, the same conclusions made in Section 3.3.1 were also confirmed, stressing the fact that level of uncertainty is now greater due to both greater excitation and a higher level of damage (Figure 14).

## 4. Discussion

## 5. Conclusions

- (i)
- explicitly define the errors and uncertainties present in the model;
- (ii)
- provide the full multivariate distribution of the calibrated parameters;
- (iii)
- estimate the model discrepancy posterior distribution;
- (iv)
- provide insights on quantities of interest (e.g., maximum a posteriori estimates, time-history response prediction).

- (i)
- for low level of damage, and even for high levels of PGA, accurate predictions can be achieved adopting a Gaussian discrepancy term with null mean and unknown variance, which overcomes the low sensitivity of the term used to model the degradation in the response;
- (ii)
- for high levels of damage, on the contrary, the simple Gaussian discrepancy function is unable to tackle the model inaccuracy rising.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Montenegro (1979) ground motion record with peak ground acceleration (PGA) = 3.59 m/s

^{2}.

**Figure 3.**Simulated reference data: (

**a**) response displacement of the Bouc–Wen–Baber–Noori (BWBN) oscillator; (

**b**) response velocity of the BWBN oscillator; (

**c**) hysteresis loop of the BWBN system; (

**d**) total dissipated energy of the BWBN system (both elastic and hysteretic components of the restoring force are gathered), normalized with respect to the mass of the oscillator.

**Figure 5.**Trace plots of the Markov Chains and corresponding kernel density estimation (KDE) for each BWBN model parameter [6]: (

**a**) ${k}_{i}$; (

**b**) $\beta $; (

**c**) $\gamma \ast $; (

**d**) $\alpha $; (

**e**) ${\sigma}^{2}$.

**Figure 7.**Posterior samples of the BWBN benchmark model parameters. The vertical lines denote the mean of the distribution (in yellow) and the maximum a posteriori estimate (MAP) (in red).

**Figure 10.**Posterior samples of the BWBN model parameters for the case study. The vertical lines denote the mean of the distribution (in yellow) and the maximum a posteriori estimate (MAP) (in red).

**Figure 12.**(

**a**) Kernel density estimations (KDE) of the initial stiffness model parameter ${k}_{i}$ (on the left) and; (

**b**) of the discrepancy variance ${\sigma}^{2}$ for each model (low-level of degradation, PGA $\approx $ 1 m/s

^{2}). (

**c**) Boxplot representations (based on the five-number summary: minimum, first quartile, median, third quartile and maximum) of the posterior samples for the model parameter ${k}_{i}$ and; (

**d**) ${\sigma}^{2}$. The horizontal and vertical dotted lines in the plots represent the calibrated reference value of ${k}_{i}$.

**Figure 13.**(

**a**) Kernel density estimations (KDE) of the initial stiffness model parameter ${k}_{i}$ and; (

**b**) of the discrepancy variance ${\sigma}^{2}$ for each model (medium level of degradation: PGA $\approx $ 6 m/s

^{2}, stiffness reduction up 40% of the initial one). (

**c**) Boxplot representations (minimum, first quartile, median, third quartile and maximum) of the posterior samples for the model parameter ${k}_{i}$ and; (

**d**) ${\sigma}^{2}$. The horizontal and vertical dotted lines in the plots represent the calibrated reference value of the initial stiffness.

**Figure 14.**(

**a**) Kernel density estimations (KDE) of the initial stiffness model parameter ${k}_{i}$ and; (

**b**) of the discrepancy variance ${\sigma}^{2}$ for each model (medium level of degradation: PGA $\approx $ 9 m/s

^{2}, stiffness reduction up to 80% of the initial one). (

**c**) Boxplot representations (minimum, first quartile, median, third quartile and maximum) of the posterior samples for the model parameter ${k}_{i}$ and; (

**d**) ${\sigma}^{2}$. The horizontal and vertical dotted lines in the plots represent the calibrated reference value of the initial stiffness.

**Figure 15.**Velocity response prediction of the BWBN Model 3 for high level of degradation (PGA of 9 m/s

^{2}, stiffness reduction up to 80% of the initial one).

**Figure 16.**Boxplot representations (minimum, first quartile, median, third quartile and maximum) of the mean square error (MSE) computed on all the post-predictive models runs. Specifically: (

**a**) boxplot of MSE for low level of degradation (PGA $\approx $ 1 m/s

^{2}); (

**b**) boxplot of MSE for medium level of degradation (PGA $\approx $ 6 m/s

^{2}); (

**c**) boxplot of MSE for high level of degradation (PGA $\approx $ 9 m/s

^{2}). (

**d**) Comparison between the average MSE computed over all the post-predictive model runs for each model (square markers), versus the MSE computed for the MAP estimate (cross markers). The comparison is differentiated into three colored subsets: (i) in black, MSEs computed for a low level of degradation (PGA $\approx $ 1 m/s

^{2}); (ii) in blue, MSEs computed for a medium level of degradation (PGA $\approx $ 6 m/s

^{2}); (iii) in red, MSEs computed for a high level of degradation (PGA $\approx $ 9 m/s

^{2}).

${\mathit{k}}_{\mathit{i}}\left(\mathbf{N}/\mathbf{m}\right)$ | $\mathit{\beta}\left({\mathbf{m}}^{\mathbf{-}\mathbf{1}}\right)$ | $\mathit{\gamma}$ | $\mathit{N}$ | ${\mathit{\delta}}_{\mathit{A}}$ | ${\mathit{\delta}}_{\mathit{\nu}}$ | ${\mathit{\delta}}_{\mathit{\eta}}$ |
---|---|---|---|---|---|---|

7.6 | $63$ | $63$ | 1 | 0 | 2.43 | 6.5 |

Parameter | Distribution | Support | Mean | Std. Dev. |
---|---|---|---|---|

${k}_{i}\left(\mathrm{N}/\mathrm{m}\right)$ | Gaussian | $\left[4.27,11.85\right]$ | $7.42$ | $0.1$ |

$\beta \left({\mathrm{m}}^{-1}\right)$ | Uniform | $\left[55,65\right]$ | $60$ | $2.8$ |

$\gamma \ast $ | Uniform | $\left[-1,1\right]$ | $0$ | $0.57$ |

$\alpha $ | Lognormal | $\left[0,1\right]$ | $0.063$ | $0.01$ |

Parameter | Mean | Std. dev. | (0.05–0.95) Quant. | MAP |
---|---|---|---|---|

${k}_{i}\left(\mathrm{N}/\mathrm{m}\right)$ | 4.5 | $0.18\times {10}^{-1}$ | (4.4–4.5) | 4.4 |

$\beta \left({\mathrm{m}}^{-1}\right)$ | $65$ | $0.36$ | $\left(64\u201365\right)$ | $64$ |

$\gamma \ast $ | $-0.38$ | 0.25 | $\left(-0.4\u2013-0.34\right)$ | $-0.40$ |

$\alpha $ | 0.035 | $0.53\times {10}^{-2}$ | (0.027–0.044) | 0.031 |

${\sigma}^{2}$ | $2\times {10}^{-4}$ | $4.7\times {10}^{-6}$ | $\left(1.9\u20132.1\right)\times {10}^{-4}$ | $60\times {10}^{-3}$ |

Parameter | Value |
---|---|

${k}_{i}\left(N/m\right)$ | 5.735 × 10^{8} |

$\beta \left(1/m\right)$ | 35.01 |

$\gamma $ | −16.68 |

N | 1 |

${\delta}_{A}\left(1/J\right)$ | 1.812 × 10^{−6} |

$m\left(kg\right)$ | 573,459 |

Parameter | Distribution | Support | Mean | Std. Dev. |
---|---|---|---|---|

${k}_{i}\left(\mathrm{N}/\mathrm{m}\right)$ | Lognormal | $\left[0,\infty \right]$ | $6.2\times {10}^{2}$ | $0.5\times {10}^{2}$ |

$\beta \left({\mathrm{m}}^{-1}\right)$ | Uniform | $\left[20,65\right]$ | $4.25$ | $1.3$ |

$\gamma \ast $ | Uniform | $\left[-1,1\right]$ | $0$ | $0.57$ |

$N$ | Uniform | $\left[0.1,2\right]$ | $1.05$ | $5.5\times {10}^{-1}$ |

${\delta}_{A}$ | Uniform | $\left[1\times {10}^{-8},1\times {10}^{-4}\right]$ | $5\times {10}^{-5}$ | $2.88\times {10}^{-5}$ |

Parameter | Mean | Std. Dev. | (0.05–0.95) Quant. | MAP |
---|---|---|---|---|

${k}_{i}\left(\mathrm{N}/\mathrm{m}\right)$ | $5.7\times {10}^{2}$ | $0.55\times {10}^{-1}$ | $\left(5.7\u20135.7\right)\times {10}^{2}$ | $5.7\times {10}^{2}$ |

$\beta \left({\mathrm{m}}^{-1}\right)$ | $34$ | $0.26$ | $\left(34\u201335\right)$ | $34$ |

$\gamma \ast $ | $-0.47$ | 0.007 | $\left(-0.49\u2013-0.46\right)$ | $-0.48$ |

$N$ | 0.98 | $0.15$ | (0.96–1) | 1 |

${\delta}_{A}$ | $4.2\times {10}^{-5}$ | $6.8\times {10}^{-6}$ | $\left(2.8\u20135.3\right)\times {10}^{-5}$ | $3.8\times {10}^{-5}$ |

${\sigma}^{2}$ | $1.2\times {10}^{-10}$ | $3.6\times {10}^{-11}$ | $\left(0.52\u20131.7\right)\times {10}^{-10}$ | $3.5\times {10}^{-11}$ |

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**MDPI and ACS Style**

Ceravolo, R.; Faraci, A.; Miraglia, G. Bayesian Calibration of Hysteretic Parameters with Consideration of the Model Discrepancy for Use in Seismic Structural Health Monitoring. *Appl. Sci.* **2020**, *10*, 5813.
https://doi.org/10.3390/app10175813

**AMA Style**

Ceravolo R, Faraci A, Miraglia G. Bayesian Calibration of Hysteretic Parameters with Consideration of the Model Discrepancy for Use in Seismic Structural Health Monitoring. *Applied Sciences*. 2020; 10(17):5813.
https://doi.org/10.3390/app10175813

**Chicago/Turabian Style**

Ceravolo, Rosario, Alessio Faraci, and Gaetano Miraglia. 2020. "Bayesian Calibration of Hysteretic Parameters with Consideration of the Model Discrepancy for Use in Seismic Structural Health Monitoring" *Applied Sciences* 10, no. 17: 5813.
https://doi.org/10.3390/app10175813