## 4 Numerical Examples and Discussion
### Calculation of Long-Term Deflection of Medium-Span PSC Box Girder (Case B)  
The second example analysis was performed for a 40-m span precast PSC box girder bridge for railways located in Osong, South Korea. It was erected at the end of 2017. The girder has a total height of 2.2 m. The thickness of the bottom slab is 0.34 m, and that of the top slab is 0.37 m. The details of the shape and dimensions of the girder are shown in Figure 10.  
The compressive strength of the concrete cylinder was 50 MPa. The modulus of elasticity and the tensile strength of the concrete were calculated as 41.6 GPa and 4.1 MPa, respectively, according to MC2010 [13]. The moduli of both the non-prestressing and prestressing rebar were set to 200 GPa. The prestressing force was applied using ninety 15.2-mm diameter bars in three layers. The yield strength of each bar was 1600 MPa with an allowable stress of 1502 MPa. The design applied stress was assumed to be 87% of the allowable stress, i.e., 1308 MPa. In the model, an effective stress of 1250 MPa was applied while assuming an immediate loss of 4% owing to the combined effects of the elastic shortening, friction, and anchorage slip. The half-span and half-section of the bridge were modeled in DIANA FEA while using the advantages of symmetry. A total of 3817 quadratic hexahedron elements were used for the half concrete girder section. The prestressing and longitudinal reinforcing bars were modeled using 45 and 41 bar elements, respectively. In addition, for the transverse bar and stirrups, 2423 bar elements were used. Figure 11 shows the 3D FE model for the medium-span PSC box girder. Figure 9: Variation of strain and stress of concrete bottom and mean prestress at the mid-span section with time in the short-span PSC beam. The daily average temperature at the site was lowest at -4 \({}^{\circ}\)C in January and highest at 24 \({}^{\circ}\)C in July. The annual average RH was approximately 68.5%. Accordingly, the average atmospheric RH was set to 68.5% in the moisture transport analysis. The moisture diffusion coefficient of the concrete was calculated for a constant RH value of 69%. The evaporation rate was assumed to be \(3\times 10^{-9}\) m/s  
Figure 11: Three-dimensional (3D) finite element (FE) model for the medium-span PSC girder. Figure 10: Longitudinal and cross-sectional view of medium-span PSC box girder. for the top and outer side faces of the girder. To account for the different environmental exposure conditions such as sunlight, the evaporation rate through the bottom and inner faces of the box girder was set to 50% of that of the top and the outer side faces. The hydro-shrinkage constant value was calibrated for the best fit to the measured data. Figure 12a shows the results from the moisture distribution analysis on the 200 th day. The lower web section of the girder, which is thinner, dried out faster. Figure 12b shows the variation in the mid-span deflection owing to the nonuniform shrinkage strain. It rises to a peak value of 5.1 mm after 120 days and then slowly decreases to 4.5 mm on the 1000 th day, as the moisture level gradually equilibrates with the atmosphere. As the girder is almost prismatic, the resulting equivalent load has the same profile as the deflection-time curve (see Figure 12c). In the structural analysis using _Model-2_, the loads were applied according to the construction history of the bridge. The self-weight and prestressing force were applied on the 5 th day. The deck slab was placed on the 150 th day, and other permanent loads (excluding the railway track) were added between the 145 th and 180 th days. The first and second layers of the ballast were installed on the 332 nd and 346 th days, respectively. The load adjusting factor for the equivalent load (0.75) was applied after crack initiation on the 5 th day. The deflection of the girder was measured for 380 days [24].