6. RESEARCH APPROACH
7.4 Multiple Model Analysis using Bayes Theorem
The calibration process using Bayes Theorem described in Section 3 was also used to identify the candidate models. Once Bayes Theorem was completed on the data, the posterior probabilities of each of the samples were analyzed. The candidate models were identified as the samples with the highest probabilities that comprised 95% of the total posterior probability. For the Route 61 Bridge, the number of samples required to meet or exceed 95% total probability was 234 models. Furthermore, the 234 models that were most probable were used as the candidate models.
7.4.2 Predictive Responses
The candidate models were used to analyze the responses of the bridge. Figure 58- Figure 61 show the response distributions from all of the structural models as well as the response distributions from just the candidate models. The values of the measured responses are also shown.
Figure 58: Candidate Model Strain Response of Bottom Flange of Exterior Girders at Abutment 2: a) Girder 1 and b) Girder 6
Figure 59: Model Strain Response of Web of Interior Girders at Abutment 2: a) Girder 2 and b) Girder 5
Figure 60: Model Strain Response of Web of Interior Girders at Pier 2: a) Girder 2 and b) Girder 5
Figure 61: Model Displacement Response of Interior Girders at Abutment 2: a) Girder 2 and b) Girder 5
The response distributions are given numerically in Table 8. Through the calibration process, the response distributions from the structural models were significantly reduced to the response distributions of the candidate models. Each variance of the distribution reduced between 91% and 99%. As expected, the candidate model distributions were also more accurate to the measured responses. The gages with the most percent differences were the exterior girders with errors of 293% and 483% for Girder 1 and Girder 6, respectively. Again, this could be due to the quality of data used for those particular gages since those locations were highly influenced by adverse gradient effects. The remaining gages had percent differences of 44% or less.
Table 8: Candidate Model Response Distributions Gage Measured
Prior Distribution Posterior Distribution Uncert. Red.
Mean % Diff.
Mean Var. Mean Var.
S-A-G1 49 με -890 με 737 με -166 με 11 με 99% 293% S-A-G2 89 με -569 με 446 με -102 με 7 με 98% 15% S-A-G5 69 με -493 με 385 με -100 με 7 με 98% 44% S-A-G6 31 με -721 με 568 με -181 με 12 με 98% 483% S-P-G2 139 με -176 με 104 με -92 με 5 με 95% 34% S-P-G5 108 με -153 με 87 με -92 με 5 με 95% 15% D-G2 28.7 mm (1.1 in.) 8.8 mm (0.3 in.) 11.0 mm (0.4 in.) 29.7 mm (1.2 in.) 0.9 mm (0.03 in.) 92% 3% D-G5 29.7 mm (1.2 in.) 8.5 mm (0.3 in.) 11.1 mm (0.4 in.) 28.6mm (1.1 in.) 1.1 mm (0.04 in.) 91% 4%
7.4.3 Refined Parameters and Simulation Study
The candidate models were also used to refine the parameters of the models and hence provide insight regarding the boundary conditions of the structure. Figure 62 and Figure 63 show the model parameter distributions from all of the structural models as well as the parameter distributions from just the candidate models. As mentioned previously, the parameters for all of the structural models were uniformly distributed from a free to a fixed
condition. The parameters from the candidate models were able to significantly reduce the uncertainty of two of the model parameters.
Figure 62: Refined Model Parameters of Abutments: a) Abutment 1 and b) Abutment 2
Figure 63: Refined Model Parameters of Approaches: a) Approach 1 and b) Approach 2
The parameter distributions are given numerically in Table 9 below. The model parameters pertaining to the southern end of the bridge (Abut1 and Appr1) did not experience a large reduction of uncertainty. The variance of the distributions reduced only 4% and 2% from all of the models to the candidate models for Abut1 and Appr1, respectively. However, the model parameters pertaining to the northern end of the bridge (Abut2 and Appr2) experienced a significant reduction of uncertainty of 87% and 97%, respectively.
Table 9: Candidate Model Parameter Distributions
Par.
Prior Distribution Posterior Distribution
Uncert. Red. (kN/m) 12*10x (kip/ft) (kN/m) 12*10x (kip/ft)
Mean Var. Mean Var. Mean Var. Mean Var.
Abut1 5.54e5 1.84e4 3.50 2.02 1.61e6 1.50e4 3.96 1.93 4% Abut2 5.54e5 1.84e4 3.50 2.02 477 319 0.43 0.26 87% Appr1 5.54e5 1.84e4 3.50 2.02 1.66e6 1.67e4 3.98 1.98 2% Appr2 5.54e5 1.84e4 3.50 2.02 7.02e4 195 2.60 0.05 97%
The refined parameters provided insight into the thermal behavior of the bridge and led to a more comprehensive understanding of why deterioration was occurring. The reason for the damage can be illustrated by three scenarios shown in Figure 64.
Figure 64: Thermal Evaluation Results: a) Original, b) Current, and c) Recommended (Reprinted from Murphy and Yarnold 2018)
Figure 64(a) shows the “original” scenario, which depicts the behavior of the bridge according to specifications from the original drawings and as the designer intended. In this scenario, the approaches do not experience thermal movement and remain stationary. The bridge expands freely but not enough to contact the wall of the abutment due to a sufficiently sized expansion joint.
Figure 64(b) shows the “current” scenario, which depicts the behavior of the bridge incorporating the refined parameters found during the thermal structural evaluation. The parameters at the north end of the bridge were found to have some degree of stiffness which prevented free thermal movement of the bridge. The results suggest that the expansion joint closed prematurely resulting in crushing of the joint and producing significant boundary stiffness. The premature closure could be due to the lack of consideration of the thermal movement from the large approaches. If the approaches expanded, the expansion joint would lessen but the bridge movement would still occur. The bridge movement was more than the space provided by the joint. As a result, the bridge contacted the abutment and continued to expand, causing damage to occur.
The above scenarios led to two primary findings and recommendations. The first was that the expansion joint at Abutment 2 was too small to accommodate both the bridge and approach or abutment movement. Therefore, enlargement of the expansion joint at Abutment 2 is recommended. The second recommendation was to provide a joint behind the south end of the bridge (behind Abutment 1), removing the restraint provided by the approach during the bridge expansion. Therefore, even if the joint did close at the north end (Abutment 2), the bridge would be able to expand toward Abutment 1 and avoid any stress
build-up. Modifying the fixity of select substructures was considered, but the associated costs were relatively significant and a less desirable alternative.
Finally, the “recommended” scenario shown in Figure 64(c) depicts the behavior of the bridge incorporating recommendations resulting from the thermal evaluation. The enlarged expansion joint is sufficient enough to accommodate both the thermal movement from the bridge as well as from the approaches. As a result, the bridge does not contact the abutment and ceases to contribute further damage to the structure.