## 4 Numerical Procedure for the Time-Dependent Analysis
### Numerical Procedure for Concrete Creep
#### 4.1.1 The Spatial State of Stresses and Strains  
In the first step, when stresses, for a given combination of external forces, are within the initial flow surface, the following well-known relationship between strains and stresses applies:  
\[\left\{\begin{array}{c}\sigma_{xx}\\ \sigma_{yy}\\ \sigma_{zz}\\ \tau_{xy}\\ \tau_{yz}\\ \tau_{zx}\end{array}\right\}=\left[\begin{array}{cccccc}\hat{E}\cdot(1-\
u)& \hat{E}\cdot\
u&\hat{E}\cdot\
u&0&0&0\\ \hat{E}\cdot\
u&\hat{E}\cdot(1-\
u)&\hat{E}\cdot\
u&0&0&0\\ \hat{E}\cdot\
u&\hat{E}\cdot\
u&\hat{E}\cdot(1-\
u)&0&0&0\\ 0&0&0&G&0&0\\ 0&0&0&0&G&0\\ 0&0&0&0&0&G\end{array}\right]\cdot\left\{\begin{array}{c}\varepsilon_{xx}\\ \varepsilon_{yy}\\ \varepsilon_{zz}\\ \gamma_{xy}\\ \gamma_{yz}\\ \gamma_{zx}\end{array}\right\}, \tag{14}\]where \(\hat{E}\) is the effective modulus of elasticity modified with the [PERSON]'s coefficient \((\
u)\):  
\[\hat{E}=\frac{E}{(1-2\cdot\
u)\cdot(1+\
u)}, \tag{15}\]  
and \(E\) is the initial modulus of elasticity and \(G\) the modulus of shear, which, according to the linear theory, is calculated by:  
\[G=\frac{E}{2\cdot(1+\
u)}. \tag{16}\]  
As the modulus of elasticity is corrected for principal stress directions in each Gauss point, it is necessary to introduce the orthotropic material model. The orthotropic directions coincide with the principal stress directions and become the major material directions.  
Based on expression (14) for orthotropic materials in the spatial stress state, the relationship can be written as follows [26]:  
\[\left\{\begin{array}{c}\sigma_{11}\\ \sigma_{22}\\ \sigma_{33}\\ \tau_{12}\\ \tau_{23}\\ \tau_{31}\end{array}\right\}=\left[\begin{array}{cccccc}\frac{E_{1}\cdot(1- p_{23}\cdot v_{32})}{\Delta}&\frac{E_{2}\cdot(v_{12}+v_{32}\cdot v_{13})}{\Delta}& \frac{E_{3}\cdot(v_{13}+v_{12}\cdot v_{23})}{\Delta}&0&0&0\\ \frac{E_{1}\cdot(v_{31}+v_{31}\cdot v_{32})}{E_{2}\cdot(1-v_{31}\cdot v_{31}) }&\frac{E_{3}\cdot(v_{23}+v_{31}\cdot v_{13})}{\Delta}&0&0&0\\ \frac{E_{1}\cdot(v_{31}+v_{31}\cdot v_{32})}{\Delta}&\frac{E_{2}\cdot(v_{32}+ v_{31}\cdot v_{31})}{\Delta}&\frac{E_{3}\cdot(1-v_{31}\cdot v_{32})}{\Delta}&0&0&0\\ 0&0&0&G_{12}&0&0\\ 0&0&0&0&G_{23}&0\\ 0&0&0&0&0&G_{31}\end{array}\right].\left\{\begin{array}{c}\varepsilon_{11} \\ \varepsilon_{22}\\ \varepsilon_{33}\\ \tau_{12}\\ \tau_{23}\\ \tau_{31}\end{array}\right\}, \tag{17}\]  
where:  
\(E_{1}\), \(E_{2}\) and \(E_{3}\) --moduli of elasticity in principal directions,  
\(v_{12}\), \(v_{21}\), \(v_{13}\), \(v_{31}\), \(v_{23}\) and \(v_{32}\)--Poisson's coefficients in planes indicated with numbers in indexes,  
\(G_{12}\), \(G_{23}\) and \(G_{31}\) --shear moduli in planes indicated by numbers in indexes  
\[\Delta\ =\ 1-v_{12}\cdot v_{21}-v_{23}\cdot v_{32}-v_{31}\cdot v_{13}-2\cdot v _{21}\cdot v_{32}\cdot v_{13}.\]  
Or shorter:  
\[\sigma=D_{\sigma\sigma}\cdot\varepsilon \tag{18}\]  
The Poisson's coefficients can be calculated from the following expressions [26]:  
\[\
u_{12}=\
u\cdot\sqrt{\frac{E_{1}}{E_{2}}},\ \ \ \ \
u_{23}=\
u\cdot\sqrt{\frac{E_{2}}{E_{ 3}}},\ \ \ \ \
u_{31}=\
u\cdot\sqrt{\frac{E_{3}}{E_{1}}}. \tag{19}\]  
With the following condition, when \(i\
eq j\)[26]:  
\[\frac{\
u_{ij}}{E_{i}}=\frac{\
u_{ji}}{E_{j}} \tag{20}\]  
The shear modulus for the orthotropic material is calculated according to the following expression [26]:  
\[G_{ij}=\frac{\sqrt{E_{i}\cdot E_{j}}}{2\cdot\left(1+\sqrt{\
u_{ij}\cdot\
u_{ ji}}\right)} \tag{21}\]