## 3 Stress Estimation Based on the Monitored Strain
### Analysis Results  
Monitoring strains at 14 days (tension of the pre-stress wires at the span-end), 65 days (tension of the pre-stress wires at the mid-span), and 368 days (1 year after concrete pouring) were selected to conduct the stress analysis. Since concrete has a compressive strength and the environmental humidity was measured as \(f_{c}\) = 55 MPa and \(H\) = 60%, they were not considered as model parameters in our inverse analysis. Linear simplified analysis was conducted to determine the prior distribution of stress in each span of the PC beam. The structural stress was approximately calculated through Equation (21). Neglecting the stress relaxation from the creep leads to an overestimation of the stress, and the prior distribution of the stress was assumed to be larger than 0 and smaller than the result from Equation (21). A simplified analysis method was used to determine the range according to the monitored strain:  
\[\sigma_{ul}=E_{c}\epsilon_{M}. \tag{21}\]  
The stress values based on the linear simplified method are shown in Table 4.  
A three-dimensional model space was composed from the prior distribution from the analysis method and two uncertainty factors. The model space was divided into some meshes with equal intervals and the likelihood function of each mesh was calculated. The probability and statistical characteristics of the posterior distribution of the parameters was obtained by introducing the likelihood function into Equations (18)-(20), as shown in Figure 6. In the figure, the height of the columns presents the mean of evaluated stresses, and a range of red lines indicate the standard deviation of the evaluated stresses. It was observed from the figure that the tendencies of the stress at each location changing with the time of the three spans were similar with each other. Some estimated data show significant different tendency with other spans, such as point 0-b in Span #3. This might be because an error in the monitored data leads to the false likelihood estimation of stress, resulting in both the wrong tendency and much larger variance. Due to the decreased quantity of information, Figure 6 indicates that the uncertainty of the stress estimations at 14 days,  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Span**} & \multirow{2}{*}{**Age**} & \multicolumn{3}{c}{**Span-End**} & \multicolumn{3}{c}{**1/4 Span**} & \multicolumn{3}{c}{**Mid-Span**} \\ \cline{3-10}  & & **Top** & **Centroid** & **Bottom** & **Top** & **Centroid** & **Bottom** & **Top** & **Centroid** & **Bottom** \\ \hline \multirow{3}{*}{\#1} & 14-d & 13.53 & 12.30 & 7.00 & 13.05 & 16.33 & 23.43 & 13.68 & 17.28 & 20.87 \\  & 65-d & 21.77 & 17.66 & 7.11 & 19.43 & 22.55 & 32.14 & 15.79 & 20.36 & 30.10 \\  & 368-d & 23.53 & 17.76 & 8.70 & 20.80 & 23.05 &65 days, and 368 days is increasing. Three monitored strains were used as the boundaries to determine stress at 14 days, which resulted in a small variability, while only one strain could be used to estimate the stress at 368 days, resulting in a relatively larger variability. The monitoring point at the top flange of the 1/4 span cross-section of Span #2 was selected as an example. The joint probability density distributions of the stress at 65 days and 368 days are shown in Figure 7. In the figure, the horizontal and vertical axes represent  
Figure 6: Posterior distributions of the stress based on the linear simplified analysis. the stress of concrete at 65 days and 368 days, respectively. The shadow in the figure represents the probability of the stresses. It can be seen from the figure that there are some stress series relative to the monitored strain in the model space. The mean value and standard deviation of the posterior are statistical characteristics of the stress series. As a result, they have a neglected discreteness.