## 5 Numerical Simulation of the Long-Term Behavior of a Bridge
### Long-Term Variation of Stress and Deformation States  
The results of the finite element analysis are presented in the Figures 14-16. For comparison, the figures show the results obtained with different creep models, i.e., EuroCode2, ACI, and B3 model. All these responses have been computed with the same finite element program and the same step-by-step time integration algorithm as presented in Section 2. It must be remarked that the relaxation of the steel tendons is not considered in the present analysis to emphasize only the effect of creep and shrinkage on the long-term behavior of the bridge. Figure 14 presents the mid-span deflection, both in linear and logarithmic time scales, in which the time is measured from the end of construction stage. It must be remarked that the deflection is also evaluated with reference to the configuration existing at the end of the construction stage, i.e., this deflection is not the absolute value of the mid-span deflection. The bridge deflection is calculated as the difference between two large numbers (both affected by some uncertainties): The downward deflection due to the dead load and the upward deflection due to prestress. The numerical analysis takes into account the effect of the differences in thickness of the slabs and webs on their drying rates through a different effective thickness (\(D=2V/S\)) in the cross section (see Table 3). If the shrinkage and the drying creep compliance are considered to be uniform over the cross section the overall effective thickness, \(D=2V/S\) of the whole cross section as reported in Table 4, can be used. The use of non-uniform creep and shrinkage properties throughout the cross section allows to simulate the initial upward deflection due to differential shrinkage and differential drying creep, as shown by Figure 14 for all \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Section** & **Area [m\({}^{2}\)]** & **Segment Length [m]** & **Volume [m\({}^{3}\)]** & **D [mm]** \\ \hline
6 & 14.259 & 4.5 & 64.169 & 453.102 \\
7 & 12.258 & 4.5 & 55.162 & 403.015 \\
8 & 11.410 & 4.5 & 51.349 & 389.807 \\
9 & 10.654 & 5 & 53.274 & 377.120 \\
10 & 9.922 & 5 & 49.610 & 363.916 \\
11 & 9.302 & 5 & 46.508 & 352.999 \\
12 & 8.794 & 5 & 43.972 & 341.620 \\
13 & 8.482 & 5 & 42.410 & 336.480 \\
14 & 8.136 & 5 & 40.682 & 327.697 \\
15 & 7.949 & 5 & 39.743 & 323.048 \\
16 & 7.880 & 5 & 39.402 & 321.317 \\ B1 & 9.417 & 2 & 36.352 & 370.549 \\ B3 & 8.317 & 1 & 8.3175 & 334.471 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: _Cont._  
Figure 14: Bridge deflection (positive if upward) in normal (**a**) and logarithmic (**b**) scale computed with different creep models. the creep models considered in the first years after the end of the bridge construction. The predicted deflections at about 80 years (\(\sim\)30,000 days after span closing) are quite different for shape and values depending on the adopted model. The Model B3 predicts the greater value of the deflection, \(\approx\)40 mm, while using the ACI model the deflection is of few millimeters upward and for the EC2 model it reaches an asymptotic value of \(\approx\)20 mm. The ACI deflection curve presents a shape that is rather different from those of the other models. It gives a deflection growth during the first years which is not compensated at a later time by the creep deformation and, as a result, the deflection tends to a small positive value. The EC2 deflection curve presents a smaller influence of the shrinkage effect at the beginning and then predicts a deflection which tends to an asymptotic value. On the contrary the B3 model deflection curve shows a large influence of the shrinkage for the first 6/7 years with an upward deflection, but then presents an increasing downward deflection with a reduced rate and with no tendency to an asymptotic value. Figure 15 presents the prestress loss in three different tendons at the main (central) pier predicted by the various models in logarithmic time scales.