## 4 Numerical Examples and Discussion
### Long-Term Deflection of Long-Span Cantilever PSC Bridge (Koror-Babeldaob Bridge) (Case C)  
The third example analysis was conducted for a long-span PSC bridge. The Koror-Babeldaob Bridge, with a main span of 241 m, was built in 1977 using a cantilever method. It collapsed in 1996, three months after remedial prestressing. The final mid-span deflection in design was expected to be in the range of 0.53 to 0.65 m. However, after 18 years, it reached 1.39 m and continued to increase until it collapsed on September 26, 1996 [25]. The main span of the bridge comprised a pair of large cantilevers connected with a sliding hinge to allow relative longitudinal movements [26]. The specified compressive strength of the concrete was 35 MPa. In the design, the allowable tensile stress was set to zero [26]. The 28-day elastic modulus of the concrete in the initial design was assumed as 28.3 GPa according to AASHTO [10]; however, various studies have reported that the actual value  
Figure 15: Variation of strain and stress of concrete bottom and mean prestress at the mid-span section with time in the medium-span PSC box girder. was in the range of 19.6 to 22 GPa [26; 27]. The average unit weight of the concrete was reported as 23.5 kN/m\({}^{3}\). In the model, the elastic moduli of both the non-prestressing and prestressing bars were set as 200 GPa, whereas that of concrete was assumed to be 22 GPa in the user-defined material. The uniaxial tensile strength of the concrete was calculated as 3.2 MPa, according to MC2010 [13]. The detailed dimensions of the bridge girder are shown in Figure 16. In the area where the bridge was built, the annual average temperature was approximately 28 \({}^{\circ}\)C, and the annual average RH was approximately 82% [26]. Using high-strength threaded bars with a diameter of 32 mm and strength of 1050 MPa, the prestress of 735 MPa, 70% of the tensile strength of the bar, was applied. The same bars were used to provide vertical prestress for the webs with spacing varying from 300 to 3000 mm and the horizontal transverse prestress of the top slab, with a typical spacing of 560 mm [25]. The average thickness of the concrete pavement at the top slab was 75 mm. The half-span of the bridge was modeled with half-section using DIANA FEA with the advantage of symmetry. A total of 17,432 quadratic hexahedron elements were used for the girder, and 158, 344, and 121 bar elements were used to model the longitudinal, lateral, and vertical prestressing bars, respectively. In addition, 1373 bar elements were used to model the non-prestressing reinforcing bars for the half-concrete girder section excluding the supports. Figure 17 shows the 3D FE model for the Koror-Babeldaob Bridge. To study the effect of the environmental RH on the deflection owing to the non-uniform shrinkage, in the moisture transport analysis using _Model-1_, the average atmospheric RH was set as 60% and 70%, in addition to the reported value of 82%. The evaporation rate was assumed to be \(3\times 10^{-9}\) m/s, and the  
Figure 16: Longitudinal and cross-sectional view of Koror–Babeldaob Bridge [28]. Figure 17: 3D FE model of Koror–Babeldaob Bridge. hydro-shrinkage constant value was calibrated for the best fit to the measured data. The moisture diffusion coefficient was calculated for an RH value of 82%. The analysis result in Figure 18 shows that the deflection owing to the non-uniform shrinkage is upward owing to the cantilever arrangement of the bridge and rises up to 55.5 m on the 2000 th day when exposed to 82% RH. After that, as the RH inside the concrete approaches equilibrium with the atmospheric RH, the deflection slowly decreases to a final value of 33.5 m at the end of the analysis period. As the atmospheric RH decreases, the resulting deflection (in this case, the camber) caused by the nonuniform shrinkage increases, as shown in Figure 18.  
As the bridge is non-prismatic, the second moment of area of its section is calculated at every meter along the length of the bridge, as shown in Figure 19a.