## 4 Numerical Procedure for the Time-Dependent Analysis
### Numerical Procedure for Concrete Creep
#### 4.1.2 The Correction of the Elasticity Modulus According to Creep  
The numerical model performs an iterative calculation of the problem by time increments and load phases. At the end of each phase/time increment, stresses and strains in all of the Gauss points of the structure are known so that the principal strains and stresses (obtained from global stresses) and their directions can be easily calculated.  
Based on the stress/strain state, the appropriate procedure is applied depending on whether the concrete is in the elastic region, the plastic (yielding) area, the region of tensile softening, or if the cracks have appeared in certain areas.  
When the principal stress in a certain direction is known and the corresponding current deformation is obtained from the constitutive diagram of concrete, the current secant modulus of elasticity of concrete in that direction can be determined. For established directions of principal stresses (\(\sigma_{1}\), \(\sigma_{2}\) and \(\sigma_{3}\)), and their corresponding strains (\(\varepsilon_{1}\), \(\varepsilon_{2}\) and \(\varepsilon_{3}\)), it is now possible to determine the secant current modulus of elasticity in the direction of principal stresses for each Gauss point with the following expressions:  
\[\begin{array}{llll}E_{1}^{t}(t_{0})=\frac{\sigma_{1}(t_{0})}{\varepsilon_{1 }(t_{0})},&E_{2}^{t}(t_{0})=\frac{\sigma_{2}(t_{0})}{\varepsilon_{2}(t_{0})}, &E_{3}^{t}(t_{0})=\frac{\sigma_{3}(t_{0})}{\varepsilon_{3}(t_{0})},\\ E_{1}^{t}(t_{1})=\frac{\sigma_{1}(t_{1})}{\varepsilon_{1}(t_{1})},&E_{2}^{t}(t_ {1})=\frac{\sigma_{2}(t_{1})}{\varepsilon_{2}(t_{1})},&E_{3}^{t}(t_{1})=\frac {\sigma_{3}(t_{1})}{\varepsilon_{3}(t_{1})},\\ &&\cdot&\cdot\\ E_{1}^{t}(t_{n})=\frac{\sigma_{1}(t_{n})}{\varepsilon_{1}(t_{n})},&E_{2}^{t}(t_ {n})=\frac{\sigma_{2}(t_{n})}{\varepsilon_{2}(t_{n})},&E_{3}^{t}(t_{n})=\frac{ \sigma_{3}(t_{n})}{\varepsilon_{3}(t_{n})}.\end{array} \tag{22}\]  
By determining the current effective (corrected) modulus according to expressions (10) and (22), their corrections are made concerning the development of the creep coefficient.  
Based on expressions (10) and (22), the corrected modulus of elasticity for a single main direction and a single Gauss point with the compressive stresses and the tensile stresses smaller than the tensile strength of concrete at any given time can now be calculated.  
This way, the corrected modulus of elasticity of concrete enters the matrix of the material, which now, for the spatial state of stress and a certain moment in time, takes the following form:  
\[D_{\text{ort}}=\left[\begin{array}{cccccc}\frac{E_{1}^{t}(t)\cdot(1-\
u_{23 }\cdot\
u_{32})}{\Delta}&\frac{E_{1}^{t}(t)\cdot(\
u_{12}+\
u_{32}\cdot\
u_{ 13})}{\Delta}&\frac{E_{1}^{t}(t)\cdot(\
u_{13}+\
u_{12}\cdot\
u_{23})}{\Delta} &0&0&0\\ \frac{E_{1}^{t}(t)\cdot(\
u_{21}+\
u_{31}\cdot\
u_{23})}{\Delta}&\frac{E_{2}^ {t}(t)\cdot(1-\
u_{13}\cdot\
u_{31})}{\Delta}&\frac{E_{3}^{t}(t)\cdot(\
u_{2 3}+\
u_{21}\cdot\
u_{13})}{\Delta}&0&0&0\\ \frac{E_{1}^{t}(t)\cdot(\
u_{31}+\
u_{21}\cdot\
u_{32})}{\Delta}&\frac{E_{1}^ {t}(t)\cdot(\
u_{32}+\
u_{12}\cdot\
u_{31})}{\Delta}&\frac{E_{1}^{t}(t)\cdot( 1-\
u_{12}\cdot\
u_{21})}{\Delta}&0&0&0\\ 0&0&0&G_{12}&0&0\\ 0&0&0&0&G_{23}&0\\ 0&0&0&0&0&G_{31}\end{array}\right] \tag{23}\]  
Furthermore, the matrix of the orthotropic material should be transformed from the material coordinate system into the global coordinate system using the transformation matrix [20]. The orthotropic matrix of the material for the global coordinate system enters the finite element stiffness matrix.  
Now the creep deformations in principal directions can be calculated as follows:  
\[\begin{array}{llll}\varepsilon_{1}(t_{1})=\frac{\sigma_{1}(t_{1})}{E_{1}^{ t}(t_{1})}\cdot\varphi(t_{1},t_{0}),\varepsilon_{2}(t_{1})=\frac{\sigma_{2}(t_ {1})}{E_{2}^{t}(t_{1})}\cdot\varphi(t_{1},t_{0}),\varepsilon_{3}(t_{1})= \frac{\sigma_{3}(t_{1})}{E_{3}^{t}(t_{1})}\cdot\varphi(t_{1},t_{0})\\ \varepsilon_{1}(t_{2})=\frac{\sigma_{1}(t_{2})}{E_{1}^{t}(t_