*3.1. Objective Function Minimization*

In order to predict the structural performance under hazardous events accurately, a well-tuned baseline model is essential. With limited modeling information due to lack of design drawings and reports, an approximate model may deviate from the actual behavior of the structure. Based on the field observations, mass estimations, and fixed BCs, modal analysis results of the initial non-updated model are 8.98, 14.41, and 22.05 Hz for first, second, and third modes, respectively. Comparing these results with the actual dynamic response obtained from the identification results, one can see there is a significant mismatch in second and third modes. Therefore, such modeling discrepancy should be diminished to improve the accuracy of the baseline model.

For this purpose, the FEM updating procedure explained in the previous section is adopted. The updating procedure is composed of three loops each manipulating one modeling variable to generate multiple FEM instances. These three parameters are related to the support restraints, member thicknesses, and distributed mass over the entire span. Looking at the support restraints of the bridge, there are two di fferent types of BCs. The first type is anchored to the adjacent buildings, and the second type is bolted connections. The support details observed through visual inspection show that the bolted connections are only used for the arch restraints, and the rest of the connections are most likely anchored to the structure. To decrease the number of parameter updates, considering that anchored connections form rigid supports, the bolted connection type is considered as an updating parameter which leads to three di fferent combinations such as fixed-fixed, fixed-pinned, and fixed-roller. For each BC case, 900 FEM instances are created ranging in stiffness (K), and mass (M) parameters. The objective function error between the FEMs and the modal identification results are computed to find the optimal parameter combination. Figure 8 shows the error surfaces of the fixed-fixed, fixed-pinned, and fixed-roller cases.

**Figure 8.** Modal frequency error surfaces for fixed-fixed, fixed-pinned, fixed-roller BCs.

According to Figure 8, the uppermost three figures of each BC case shows the error due to each individual modal frequency, whereas the three-dimensional figures show the combination of these individual components as the objective function product. For visualization purposes, the error between FEM and identification results is demonstrated with colored surfaces. The error surface ranges between red and blue where red corresponds to maximum dispersion from physical reality (modal frequencies from accelerometer data) and blue corresponds to minimal difference between mathematical model and identified modal parameters. Other colors (e.g., orange, yellow, green, turquoise) lay between maximum and minimum error based on the objective function calculations. The magenta spots on each subfigure points out the optimal combination of updating parameters. The overall behavior shows that the model accuracy is very sensitive to the BCs. In other words, combinatory results as well as individual modal frequency errors heavily rely on the modeling of the support restraints.

Stiffness and mass domains are meshed into 30 pieces when candidate models are developed. So, each dimension consisting of 30 individual values represents the uncertainty range within the minimum and maximum values. To explain, Table 1 presents the modal frequency errors obtained from different BC cases and Figure 9 presents the modal parameters of optimal combination cases for each BC case. Stiffness parameters vary from 1.20 × 10−5m4 to 11.5 × 10−5m4 for moment of inertia and 2.1 × 10−3m2 to 36.8 × 10−3m2 for cross-sectional area of a single element. Meanwhile, mass per unit area ranges between 5.7 × 10−2t/m2 and 1.7t/m2. Table 1 implies that for fixed-fixed and fixed-pinned cases, the optimal solutions from each mode varies significantly, and the objective function is either dominated by one of the modes or an irregular combination of them. Fixed-roller case, on the other hand, is contradictory with the first two BC cases. Optimal combinations obtained from first, second, and third modes are evidently similar with each other (ranging around 21th model number), as well as the optimal objective function solution.


**Table 1.** Optimal models for different BCs.

**Figure 9.** Updated FEM modal parameters for fixed-fixed, fixed-pinned, fixed-roller BCs.

To understand the difference between the fixed-roller case and the other cases, the modal frequencies obtained from each case are investigated. Looking at the first modal frequency of the updated models, it can be observed that the fixed-fixed and fixed-pinned cases have very high errors (47%, 55%), although the second (0.3%, 2.8%) and the third (0.5%, 1.1%) modal frequencies are represented well. In contrast, fixed-roller case represents all three modes with a fair and even accuracy such as 5.8%, 0.2%, and 2.5%. These results show that the arch support fixities are decisive to set the proportion between the first modal frequency and the others, and the fixed-roller case performs significantly better than the other BC cases.

According to Figure 9, comparing the ratio between the modal frequencies, it is seen that the BCs qualitatively do not have a significant effect on the updated mode shapes. On the other hand, without the correct proportion between modal frequencies, even if one or two modes are accurately identified, the remaining mode will have a very high error value. This phenomenon can be proven with a sensitivity study, ye<sup>t</sup> it is the beyond of the scope, and therefore is not addressed further in this paper. Specifically, releasing the arch support fixities in the longitudinal direction can tremendously increase the accuracy of the FEM modal frequencies. Conclusively, an accurate FEM is developed with the presented model updating procedure, and such model can be used to simulate the seismic performance of the structure.

#### *3.2. Simulation of Seismic Response and Reliability*

After the optimal modeling parameters are determined and the FEM with limited information is updated, the resultant model can be used as a baseline to predict structural performance under hazardous events. Specifically, in this study, seismic response is scoped, ye<sup>t</sup> similar analysis procedure can be extended to other damaging events. The PEER Strong Motion Database have an extensive set of real earthquake records, therefore, one of these largest sets, 1994 Northridge Earthquake is taken as an exemplary structural demand due to a seismic event [59]. Table 2 shows the overall information about the ground motion dataset features and strong motion parameters.

One hundred and fifty-one earthquake ground motion records are taken from the Northridge Earthquake dataset and used as structural input for time history analyses. With the time history analysis of the baseline model under different earthquake ground motions, the structural response can be probabilistically simulated. Figure 10 shows an example of these analyses illustrating the time and the frequency content of the structural input and outputs.


**Table 2.** Ground motion dataset summary.

**Figure 10.** Exemplary input ground motion and simulated structural response.

According to Figure 10, it can be observed that the frequency content of the input ground motion is dominated in low frequencies (below 5 Hz), whereas the structural response peaks around 8–9 Hz. The mode with the lowest frequency, the first mode, is excited more than the second and third modes, and therefore, the response peaks are observed around the first frequency range. This is due to the fact that the higher structural frequencies (e.g., 8.5, 19, 29 Hz) are very far away from relatively low frequency seismic activity. For these reasons, the seismic response is expected to have less structural damage compared with the low frequency civil infrastructure. As a result, the structure behaves in the linear range, yet, it should still be checked whether the bridge maximum deformations exceed certain regulations. One reason is, the nonstructural earthquake damage losses still compose a significant percentage of overall losses [60]. Likewise, even slight damages following a seismic event might result in functionality losses [61]. Besides, it is seen that the low-frequency sensitive displacement response still includes the effects of seismic input, whereas these effects vanish in case of the acceleration response. Finally, and the most important of all, excessive displacements occurring at façade components can lead to glass failure, which possesses safety threat for campus occupants nearby the bridge during the catastrophic incident [62,63].

To summarize the overall dataset results, Figure 11 shows the maximum acceleration and displacement response values indexed according to the strong motion parameters amplitude, frequency, and duration [64], respectively. The analysis results are obtained considering the excitation in the vertical direction. Location of the output corresponds to the absolute maximum displacement value observed on multiple bridge deck nodes throughout the time history analyses.

Time history analysis results are recorded and the maximum response values from each analysis are collected to form a distribution demand. Figure 12 shows the distributed and the cumulative maximum displacement distribution obtained from 151 analysis results. Assuming that the distribution type is log-normal, if the probability density function (PDF) and cumulative distribution function (CDF) are plotted, one can see that the current dataset is a good representative of such type.

**Figure 11.** Peak responses indexed according to the strong motion parameters.

**Figure 12.** Maximum displacement demands based on Northridge Earthquake records.

The relationship between the arithmetic and logarithmic means can be established with the following relationships,

$$
\sigma\_y^2 = \ln \left( \frac{\sigma\_x^2}{\mu\_x^2} + 1 \right)
$$

$$
\mu\_y = \ln(\mu\_x) - \frac{1}{2}\sigma\_y^2
$$

where μ and σ corresponds to mean and standard deviation, whereas *y* and *x* subscripts correspond to the normal and lognormal distributions. Distribution obtained from the 151 analysis results is treated as log-normal distribution with the specified mean and standard deviation values, rather than following a fragility curve fitting procedure described in [46,47]. Nevertheless, the red plots show that the log-normal distribution assumption is a good representative of the discrete data distribution obtained from time history analyses. Looking at these CDF values of a particular displacement demand, one can determine the structural reliability under that particular threshold.

After the CDF is determined, the bridge performance can be evaluated according to the reference criteria. An example corresponding to an alter load case is that the US pedestrian steel bridges under live loads are limited by a maximum deflection value of L/1000 [65]. Likewise, allowable live load

deflection limit for the bridges in Japan ranges between L/2000 (L shorter than 10 m) and L/500 (L longer than 40 m) depending on the main span length [66]. Considering Mudd-Schapiro Bridge dimensions, L/1000 and L/2000 values correspond to approximately 0.01 and 0.005 m. Static deflection limits for the Ontario highway bridges with pedestrian sidewalks are formulated as a function of the first flexural frequency, and the allowable threshold for 10 Hz is equal to 0.002 m [67]. Although, these parameters are indirectly related to structural damage, extreme relative displacements can possess non-structural threats to the community as well. As mentioned above, Mudd-Schapiro Bridge and its glass facades lay above campus area which has pedestrian access 24 h a day. Quantifying exceedance of certain deflection values is therefore beneficial practice for occupant safety.

Finally, the exceedance probabilities of exemplary reference criteria are investigated according to the CDF values. Considering 0.010, 0.005, and 0.002 m as the performance thresholds, structural reliability values of the data distribution are 0.987, 0.868, and 0.576, respectively. Likewise, log-normal distribution reliability values of the same performance thresholds are within a close range such as 0.981, 0.887, and 0.533, respectively. In general, based on similar reliability values under Northridge Earthquake example, the authorities can take action for pre-event preparation. These can include exemplary decisions such as claiming the structure's safety, service shutdown, initiating a retrofitting process, destruction if the performance thresholds are unachievable and reconstruction needed. Yet, it should be noted that for a di fferent set of earthquake records with di fferent frequency character, the structural performance is likely to be di fferent. In the future, this issue can be further investigated with ground motion simulation using site-specific spectra (theory-driven), utilize location-aware smartphone seismic networks (measurement-driven), or both. Automated, remote, and computer-aided survey approaches will be more and more important for civil infrastructure systems which is in line with advances in measurement techniques and building information modeling. Basically, imagery data such as point clouds obtained from aerial or terrestrial tools can be converted into FEMs [68,69]. In fact, terrestrial laser scanning is recently linked to FEM updating process, and therefore, SHM [70]. In addition, the advent of drone technologies combined with photogrammetry made it possible to collect aerial information for building inspection [71]. Such complementary tools can also take part in the development of future cyber-physical infrastructure and collocated usage of similar systems is likely to happen in the near future. Nevertheless, in summary, with the multilayered and detailed analysis procedure presented in this paper, response distributions to di fferent datasets can autonomously be performed by a well-structured cyber-physical SHM system.
