## 4 Numerical Examples and Discussion
### Calculation of Long-Term Deflection of Short-Span PSC Beam (Case A)
The first used a built-in MC2010 model for the creep and uniform shrinkage (\(C\&US\)) without considering cracking, as shown by the green line in Figure 7. The second considered creep and uniform shrinkage together with the stiffness reduction of concrete from cracking (\(C\&US+Cracking\)), as shown by the black line in Figure 7. The third considered the effect of non-uniform shrinkage by using an equivalent load in addition to the combined effects of creep, uniform shrinkage, and cracking (\(C\&US+NUS+Cracking\)), as shown by the blue line in Figure 7. The fourth approach was a staggered analysis shown by the purple line in Figure 7. The staggered analysis combined the potential flow and structural analyses provided by DIANA FEA [15, 16]; the moisture analysis was directly linked to the subsequent nonlinear structural analysis. The hydro-shrinkage constant for the best fit to the measured deflection data was found to be \(8\times 10^{-4}\). Figure 5: Relative humidity (RH) distribution along the section of the short-span PSC beam. Figure 6: Mid-span deflection owing to nonuniform shrinkage and equivalent load for the short-span PSC beam. The results in Figure 7 show that the analysis result from using the built-in MC2010 model code without considering cracking and nonuniform shrinkage (_C&US_) starts to diverge from the experimental data and other numerical results at the application of the second load (which induces cracks), and exhibits considerable differences from the measured data at later stages. On the contrary, the analyses considering cracking provide values closer to the measured data (shown with red square dots). The stiffness degradation of the beam owing to the cracking of the concrete significantly increased the long-term deflection. In addition, cracks grow and propagate owing to the creep and nonuniform shrinkage. To demonstrate the effects of creep and nonuniform shrinkage on the crack propagation, an additional crack analysis was performed without considering the creep. Figure 8 shows the crack strain at the bottom of the beam when the second load was applied (84 th day) and at the end of the analysis period (1750 th day). In the analysis considering neither creep nor nonuniform shrinkage, the crack strain does not show any change after the application of the second load. In contrast, under the influence of creep and nonuniform shrinkage, the crack strain not only increases but also extends to a wider region. This further reduces the stiffness of the beam. In addition, the nonuniform shrinkage contributes to the increase in deflection. The final deflection owing to the non-uniform shrinkage alone is only 0.56 mm from the moisture transport analysis as shown in Figure 6a but grows to 2.4 mm on the 1700 th day, when the resulting equivalent load is added to the other loads in _Model-2_. This is also owing to the combined actions of the cracking and creep. Consequently, the analysis using the equivalent load for nonuniform shrinkage in combination with creep, uniform shrinkage, and cracking (_C&US+NUS+Cracking_) provides the results closest to the measured data. Figure 5.0 Variation of mid-span deflection of the short-span PSC beam. As depicted in Figure 7, the staggered analysis also gives results close to the measured data (and the results of the equivalent load analysis (_C&US+NUS+Cracking_)). However, the stress distributions in the staggered analysis and equivalent load analysis are not identical. In the staggered approach, additional tensile stress is developed because the differential drying shrinkage is restrained by the surrounding concrete and reinforcements; i.e., the stress is in the opposite direction of the shrinkage strain. In contrast, in the equivalent load analysis, stress is induced in the same direction as the flexural strain induced by the equivalent load. In this case, however, the reinforcement ratio is larger in the lower part of the section. Hence the shrinkage strain at the bottom is more restrained than that at the top.