## 4 Numerical Procedure for the Time-Dependent Analysis
### Numerical Procedure for Concrete Shrinkage  
The numerical procedure for the simulation of concrete shrinkage is explained in detail in [25], so here only the necessary guidelines will be given.  
The concrete shrinkage and temperature loads in concrete have very similar effects, causing deflections and strains but not stresses in elements that can be freely deformed [28]. In elements with constrained deformations only stresses occur, not strains. This fact was used for the development of the shrinkage numerical model based on the thermal analogy.  
The first step in the numerical procedure is to determine the current stress-strain state of the element. After the time loop has been established, the total shrinkage for the observed time can be calculated from Equation (13). The shrinkage strain is a scalar value equal in all points for directions of all principal stresses. The second step is to calculate the temperature change T which formally replaces the total shrinkage strain:  
\[T=-\frac{\epsilon_{cs}(t-t_{s})}{\alpha_{t}} \tag{25}\]  
where \(\epsilon_{cs}(t-t_{s})\) is the shrinkage strain, calculated from Equation (13), in the observed time and \(\alpha_{t}\) is the coefficient of the thermal expansion of concrete.  
This temperature change (25) in unreinforced concrete members causes only deformation \(\{\epsilon_{sh}\}\), but not stresses (Figure 10c). If concrete members have reinforcement, part of the deformation is prevented \(\{\epsilon_{c,sh}\}\), so stresses occur (Figure 10d).  
The shrinkage strains are considered only in the directions of principal stresses (strains), where shear stresses are equal to zero [29]:  
\[\{\epsilon_{sh}\}=[\alpha_{t}\cdot T\quad\alpha_{t}\cdot T\quad\alpha_{t}\cdot T \quad 0\quad 0\quad] \tag{26}\]  
Figure 10: (**a**) Concrete member without reinforcement, (**b**) reinforced concrete member with symmetrical reinforcement, (**c**) shrinkage deformation of a concrete member without reinforcement and (**d**) shrinkage deformation of a symmetrically reinforced concrete member.  
Figure 9: Current moduli of elasticity in the main direction 1.  
For this strain state, the appropriate stress state can be calculated:  
\[\{\sigma_{sh}\}=[D]\{\epsilon_{sh}\} \tag{27}\]  
It should be emphasised that the stresses calculated according to expression (27) do not exist in reality and only serve to calculate the equivalent nodal forces from shrinkage (temperature). These nodal forces are returned to the global system, and the deformation state is calculated for them.  
The newly calculated strain vector \(\{\epsilon\}\) consists of the part representing the shrinkage \(\{\epsilon_{s,sh}\}\) and the part caused by other external forces. The actual deformation state can be calculated by subtracting the previously calculated strains \(\{\epsilon_{sh}\}\) from the newly calculated ones (Figure 10d)  
\[\{\epsilon^{\prime}\}=\{\epsilon\}-\{\epsilon_{sh}\} \tag{28}\]  
where vector \(\epsilon^{\prime}\) represents the corrected strains, i.e., the real strain state of the structure.  
After the total strain vector has been obtained, the stress vector can also be calculated. To obtain the real value of the stresses, it is necessary to subtract the free shrinkage part of the stresses from the calculated value.  
\[\{\sigma^{\prime}\}=\{\sigma\}-\{\sigma_{sh}\} \tag{29}\]  
The corrected strains \(\{\epsilon^{\prime}\}\) and stresses \(\{\sigma^{\prime}\}\) of concrete have to be used to control the occurrence and propagation of cracks in tension and yielding of concrete in compression [25; 28].  
The flowchart of the proposed numerical algorithm, previously described, is shown in the Figure 11.  
Figure 11: Flowchart of the proposed model.