## 3 Markov Chain Monte Carlo Algorithms
### Transitional Markov Chain Monte Carlo (TMCMC)
#### 3.2.3 Implementation and Algorithmic Steps
For reinforced concrete structures, typical causes of stiffness degradation include cracks, reinforcement corrosion, and localized spalling of concrete, all of which contribute to a reduction in stiffness. In practice, the levels of stiffness degradation used in our model are reasonable and commonly observed in structures that have been in service for extended periods, such as concrete bridges that have been operational for more than a decade. Also, reinforced concrete structures often exhibit cracking during their service life. However, after a prolonged period, these cracks can lead to a degradation in the material's performance, ultimately reducing the structure's stiffness. The stiffness reduction assumptions in our study are based on a baseline model representing the structure in its normal service condition. It is important to note that identifying stiffness degradation in reinforced concrete structures does not necessarily imply an imminent collapse. Rather, it highlights a reduction in stiffness that could shorten the structure's service life or potentially reduce its seismic performance and safety reliability. As for the specific causes of stiffness reduction, they could be due to a variety of factors such as loose joints/bolts, holes, cuts, cracks, material degradation, or the removal of structural components. However, this study does not focus on the specific causes of stiffness change. Instead, our primary objective is to provide a tutorial on Bayesian model updating using single-chain and multi-chain MCMC, and its application in structural damage detection. We acknowledge that further research into the relationship between damage identification and classification is a valuable area of future study. \begin{table}
\begin{tabular}{l c} \hline \hline
**Damage Scenario** & **Damage Location (Damage Severity)** \\ \hline
1 & 1 st (\(-20\%\)), 12 th (\(-10\%\)), 25 th (\(-20\%\)), 38 th (\(-20\%\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Damage location and severity of shear building. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Structural Properties** & \multicolumn{2}{c}{**Nominal Values (GPa)**} & **Stiffness Variation** \\ \hline Elastic modulus of frame members & \(E_{1}\sim E_{6}\) & 200 & \(\theta_{1}\): \(-20\%\); \(\theta_{3}\): \(-10\%\) \\ Elastic modulus of truss members & \(E_{t1}\sim E_{6}\) & 200 & \(\theta_{6}\): \(-30\%\); \(\theta_{10}\): \(-10\%\) \\ Supporting spring stiffness & \(k_{y1}\sim k_{y2}\) & \(7.01\times 10^{4}\) & \(\theta_{15}\): \(20\%\) \\  & \(k_{z1}\sim k_{z2}\) & \(8.76\times 10^{4}\) & \(\theta_{15}\): \(20\%\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Structural properties and damage scenario of the pedestrian bridge. In addition, in practical damage detection, a reference or baseline is needed for comparison, meaning measurements should be taken in both the healthy and damaged conditions to accurately distinguish changes in stiffness. In this study, however, the primary focus is a comparative study and tutorial on Bayesian model updating using different MCMC methods. For simplicity, we assume that the baseline model for all examples represents the healthy condition, where all stiffness parameters \(\theta\) are in unity. As such, any identified \(\theta\) values from the Bayesian model updating (e.g., 0.8 or 0.9) are interpreted as representing the damaged condition. ### A Forty-Story Shear Building

#### 4.1.1 Structural Information for Damage Detection

The 40-DOF shear building shown in Figure 1 is employed as our first example, where the mass per floor and inter-story stiffness are assumed to be uniformly distributed throughout the height. The mass \(M\) at each floor is set to 1.2 metric tons, and inter-story stiffness \(K\) is set as 356 MN/m.