## 3 Markov Chain Monte Carlo Algorithms
### Transitional Markov Chain Monte Carlo (TMCMC)
#### 3.2.3 Implementation and Algorithmic Steps
TMCMC takes around 14 min, and DREAM requires the most intensive computational cost of around 50 min for damage detection. The high accuracy of damage detection by DREAM comes with high computational time, as it runs many Markov chains simultaneously in order to seek the best solutions and offer more possibilities and robustness. Table 1.0 Estimation error by DRAM, TMCMC, and DREAM. Table 1.1 Marginal distributions of model parameters. ### Steel Pedestrian Bridge

#### 4.3.1 Structural Information for Damage Detection

A large-scale steel pedestrian bridge is considered a benchmark in the study of validating the performances of DRAM, TMCMC, and DREAM in terms of parameter estimation and damage detection. The bridge is located on the Georgia Institute of Technology campus and has a height, width, and longitudinal length of 2.74 m, 2.13 m, and 30.17 m, respectively. The FE model of the pedestrian bridge was constructed in SAP 2000, consisting of 46 nodes and 274 DOFs. Figure 17b shows the bridge model and sensor configuration. Seven uniaxial and biaxial accelerometers were installed on the bridge to, respectively, measure vertical and vertical-transverse vibration responses at 21 out of 274 DOFs. The entire structure was divided into six substructures, including frame members and truss members. More detailed model information is provided in [74; 75]. Table 3 lists the structural properties of the pedestrian bridge, and the damage scenario considered in this study is represented by a reduction in the elastic modulus. \(E_{1}\sim E_{6}\) and \(E_{t1}\sim E_{t6}\) denote the elastic modulus of the frame members and truss members in each substructure. \(k_{y1}\sim k_{y2}\) and \(k_{z1}\sim k_{z2}\) are four spring stiffnesses representing the boundary conditions. In total, there are 15 stiffness parameters to be updated, denoted as \(\theta_{1}\)-\(\theta_{15}\), which correspond to \(E_{1}\sim E_{6}\), \(E_{t1}\sim E_{t6}\), \(k_{y1}\sim k_{y2}\), and \(k_{z1}\sim k_{z2}\). The damage scenario column in Table 3 lists the relative stiffness variation in the true modulus from its nominal value. For example, \(\theta_{1}=\frac{E_{1}^{true}-E_{1}^{nominal}}{E_{1}^{nominal}}=-0.2\) and \(\theta_{15}=\frac{k_{y2}^{true}-k_{y2}^{nominal}}{k_{y2}^{nominal}}=0.2\). Here, the values of \(-0.2\) and \(0.2\) denote a stiffness reduction and increase, respectively. In this example, twenty independent measurements were conducted, each identifying five natural frequencies and modal shapes to perform damage detection. A 1% COV was added to the modal data to simulate a common noise level. It is worth mentioning that unlike a shear building and concrete beam, which involve multiple measurement setups, each independent measurement in the bridge example only contains a single setup to cover 21 DOFs, as we did not execute element-level damage detection for the pedestrian bridge. In practice, the realization of damage detection at the element level was our ultimate target. However, challenges arose as (1) nontrivial model parameterization was required, that is, assigning a parameter to each element; and (2) the identification of too many parameters was unfeasible due to limited measurement information. Although it is a longstanding problem, the practical method is to separate the entire structure into a few substructures, which are assigned to one parameter. Narrowing down damage to the substructure level is desirable, as inspectors can detect potentially damaged substructures and decide whether they need repair, which is economical and time-saving, rather than checking all individual elements one by one. The damage severity presented in Table 3 is also consistent with values found in other studies on damage detection in steel bridges [37; 76; 77]. For this structure, a more

Table 1.2 Three-dimensional pedestrian bridge: (**a**) pedestrian bridge on the Georgia Tech campus; (**b**) bridge model and sensor layout. appropriate interpretation of the damage scenario would be the reduction in the moment of inertia or stiffness in specific subregions.