## 5 Verification of the Numerical Model-Examples
### Reinforced Concrete Girder  
The second example is the analysis of a reinforced concrete girder taken from Ref. [33]. According to this reference, this girder with all of the ambient influences was monitored for two years. The geometry, the position of the load, the measuring line and the material characteristics are presented in Figure 22 and Table 2, all of the data were taken from Ref [33]. Figure 22: Geometry of the reinforced concrete girder. Figure 24: (**a–d**) Analysis of the dominant shear stress state \(\tau_{xx}\) in the concrete. Figure 25: Stresses in the reinforcement along the girder (\(-\)compression, \(+\)tension). Figure 23a-d shows the long-term analysis of dominant normal stresses in the x-axis (\(\sigma_{xx}\)) for the indicated time points. Figure 23a shows the state of stresses resulting from the shrinkage of concrete before the application of the external load (day 29). From the figure, it can be seen that the tensile stresses appear in the zone where reinforcement is placed, while the compressive stresses appear in the zone without reinforcement. It also can be seen that the tensile stresses are at the level of the tensile strength of concrete and that the first crack was formed even earlier and at this point, new cracks are opening. The compressive stresses are much smaller than the compressive strength of concrete. Figure 23b shows the stress conditions after the external load is applied (day 30), and a significant increase in the compressive stresses is visible. On the other hand, the tensile stresses remain at the level of the tensile strength of concrete which indicates the opening of new cracks and the widening of old. The shown crack condition (c) corresponds to the maximum size of the external load (day 30). Here the compressive stresses have reached their maximum value of 13.42 MPa, while the cracking of concrete in the tensile zone is considerable. Thus, the concrete loses its load-bearing capacity under tension, resulting in tensile softening of the concrete and a decrease of the tensile stresses in concrete. Figure 23d shows the stress distributions for a permanent external load, i.e., the changes in stresses only due to the long-term effects: the creep and the shrinkage of concrete. It can be seen that both the tensile and the compressive stresses in concrete are reduced. The decrease in the compressive stresses is due to the lowering of the neutral axis towards the tensile zone caused by the shrinkage of concrete, while the decrease in the tensile stresses is due to the increase in the cracking of concrete in the tensile zone. This is caused by the redistribution of stresses in the girder as a consequence of long-term deformations of concrete. The reduction of the compressive stresses at the end of the observation (day 730), relative to the stresses at the time of the application of the full external load (day 30), is about 21%, while the reduction of the tensile stresses is about 56%. The dominant stresses in the x-axis are analysed in more detail, while the normal stresses in the y-axis (\(\sigma_{yy}\)) are caused by the triaxial stress state and are negligible compared to the normal stresses in the x-axis, and their change over time is also negligible. The normal stresses in the z-axis (\(\sigma_{zz}\)) are also a consequence of the triaxial stress state, and are also negligible relative to the stresses in the x-axis, although they are slightly larger than the stresses in the y-axis, which is clearly due to the external load in the z-axis. These stresses decrease over time due to the creep and the shrinkage of concrete. The compressive stresses at the end of the observation period (day 730) are about 14% lower than at the time of the application of the full external load (day 30), while the tensile stresses on day 730 are about 12% lower compared to day 30. Figure 24a-d shows the redistribution of dominant shear stresses. The magnitudes of shear stresses \(\tau_{xy}\) and \(\tau_{yz}\) are negligible compared to the magnitudes of normal stresses.