## 4 Numerical Examples and Discussion
### Calculation of Long-Term Deflection of Short-Span PSC Beam (Case A)  
The first combined analysis was performed for a short-span PSC beam selected from a series of tests performed by [PERSON] and [PERSON] between 1981 and 1986 at the University of Brussel [23]. The beam had a rectangular section of \(0.34\times 0.4\) m and a length of 8 m. The beam was prestressed with five strands (diameter 12.7 mm) with an effective area of 93 mm\({}^{2}\). A prestressing stress of 1320 MPa was applied at 14 days. The dimensions and reinforcement details are shown in Figure 4.  
In view of the symmetry of the section, a half-span of the beam was modeled to save computation time. [PERSON] and [PERSON] reported that the compressive strength of the concrete at the time of prestressing was 32.4 MPa [23]. Accordingly, based on MC2010 [13], the material properties of concrete were assumed as follows: [PERSON]'s modulus \(E_{c}\) = 32 GPa, mean compressive strength \(f_{cm}\) = 33 MPa, uniaxial tensile strength \(f_{cm}\) = 2.56 MPa, and unit weight \(\rho_{c}\) = 2350 kg/m\({}^{3}\). For the prestressing and non-prestressing reinforcing bars, it was assumed that the elastic modulus \(E_{s}\) was 200 GPa, and the unit weight \(\rho_{s}\) was 7850 kg/m\({}^{3}\). Two quarter-point loads (\(P\)) of 16.5 kN and 63.75 kN were each applied to the beam for 28 days and 84 days, respectively. The tests were performed in a laboratory with an average RH and temperature of 60% and 20 \({}^{\circ}\)C, respectively. As the beam was kept in the laboratory during the test, the same evaporation rate (\(3\times 10^{-9}\) m/s) was assigned to all surfaces. Due to the absence of experimental data, the same evaporation rate with the case in \(4.1,3\times 10^{-9}\), is assumed. The moisture diffusion coefficient of the concrete was also assumed to be constant and was calculated for a constant RH value of 60%. The hydro-shrinkage constant was calibrated to fit the experimental data. The resulting distribution of the RH across the beam section at different times is presented in Figure 5. Because the evaporation rate was assumed to be the same on all surfaces and the section is symmetrical, the RH values at the top and bottom surfaces are similar. This means that the deflection owing to nonuniform shrinkage is induced owing to the presence of a relatively larger area of reinforcement at the bottom section, as shown in Figure 4. The prestressing and reinforcing bars are mostly located under the neutral axis of the beam section, which restricts the shrinkage strain at the bottom of the beam. Hence, the actual unrestricted strain of the top face of the beam becomes larger than that of the bottom face. As a result, a downward deflection is induced. The resulting deflection has a very small value, i.e., only 0.56 mm at the 1700 th day, as shown in Figure 6a. Figure 6b shows the equivalent load calculated for the nonuniform shrinkage deflection in Figure 6a. The equivalent load at 84 days decreases as shown in Figure 6b because it is adjusted by a load adjusting factor \(\gamma_{a}\) of 0.85 on the 84 th day, i.e., the onset of cracking, which can be estimated from the structural analysis using _Model-2_. It should be noted that the negative deflection in Figure 6a indicates a downward deflection, whereas in Figure 6b, the positive equivalent load indicates the downward distributed load. Figure 4: Longitudinal and cross-sectional view of the short-span prestressed concrete (PSC) beam. Figure 7 shows the change in the mid-span deflection with time. Four different approaches were used to calculate the deflection.