## 3 Markov Chain Monte Carlo Algorithms
### Transitional Markov Chain Monte Carlo (TMCMC)
#### 3.2.3 Implementation and Algorithmic Steps
For 40,000 function evaluations (sample size), DRAM took around 15 min to sample all parameters, although the results of the damage detection were unacceptable. The time spent for damage detection by TMCMC was about 8 min. DREAM displayed the best performance in damage detection, which took around 40 min of computational time. It is understandable that the time elapsed for damage detection is substantially longer when running multiple Markov chains compared to running a single Markov chain. This indicates that the entire computation

Figure 8: Estimation error by DRAM, TMCMC, and DREAM. Figure 1:0 Marginal distributions of model parameters. cost is dependent on each algorithm mechanism and there is a tradeoff between efficiency and accuracy in terms of damage detection. ### Two-Span Continuous Concrete Beam

4.2.1. Structural Information for Damage Detection

A more complex structure with more DOFs than a shear building, a laboratory-scale two-span continuous concrete beam, shown in Figure 9, was used to further numerically investigate the performance of the three MCMC algorithms [71]. Each beam span has a length of 1.9 m. The entire structure was modeled using 30 Euler-Bernoulli elements with thirty-one nodes and three simply supported boundary conditions in MATLAB. This means there are 58 DOFs in total. The cross-sections are 150 mm \(\times\) 250 mm. The elastic modulus of concrete is 28 GPa. The detailed structural information can be found in [71]. In this example, only transitional DOFs are assumed to be measured due to the difficulty in measuring rotational DOFs. For the 40-story shear building, twenty independent measurements were designed to identify modal data, each containing eight natural frequencies and mode shapes. To achieve element-level damage detection, 30 stiffness parameters corresponding to each beam element, denoted as \(\theta_{1}\sim\theta_{30}\) from left to right in Figure 9b, were chosen to be updated by three algorithms. For each stiffness parameter, the ratio of true elastic modulus to nominal elastic modulus provides the value of unity for parameters at intact elements. Similar to the first example, a three-measurement setup strategy was employed to identify the full modal shape for each independent measurement. As depicted in Figure 10, each setup encompasses 4 reference sensors and 8 rover sensors to cover 28 transitional DOFs. The resulting local mode shapes from each setup were assembled using a least squares method for global mode shapes with 28 components related to 28 DOFs. The eight natural frequencies and mode shapes were added with 1% COV for the purpose of measurement noise consideration. In this example, a multi-damage diagnosis is performed, as shown in Figure 1:1 Multi-damage locations and various damage severities would definitely increase difficulties in damage detection. The damage scenarios in Table 2 are designed to reflect relative changes in the elastic modulus, which could result from various damage sources beyond just cracks. We did not specifically investigate the exact cause of stiffness reduction in these scenarios. Additionally, the design of these damage scenarios is informed by the literature on damage detection in beam structures [71; 72; 73]. Figure 1:2 Two-span continuous concrete beam: (**a**) specimen in lab; (**b**) FE model. #### 4.2.2 Results

For the implementation of the three algorithms, a sample size of 40,000 samples was set to ensure the sufficient convergence of parameter estimation. The settings of DRAM, TMCMC, and DREAM are identical to those in Section 4.1. Figures 11-13 show the trace plots of 30 stiffness parameters by DRAM, TMCMC, and DREAM. It can be observed that the 30-parameter Markov chains by DRAM in Figure 11 appear to be a phenomenon of'sampling stagnation', in which trace plots consist of many smooth segments, leading to a large rejection rate of candidate samples. This phenomenon is mainly caused by large proposal variances.