# FE-Study on the Effect of Gradient Concrete on Early Constraint and Crack Risk
## 3 Results  
As described in Section 2.2, four different material distributions were analyzed. These distributions as depicted in Figure 2 are:  
1. homogeneous durable concrete (a)
2. two zones, durable concrete in the surface layer and low hydration energy concrete in the center (b)
3. three zones; gradient concrete in between the surface layer and the core (c)
4. homogeneous low hydration energy concrete (d)  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material Property 1** & **Unit** & \begin{tabular}{c} **Durability Optimized** \\ **Surface Layer Concrete 2** \\ \end{tabular}  & \begin{tabular}{c} **Hydration Energy** \\ **Optimized Core Concrete** \\ \end{tabular}  \\ \hline \ \begin{tabular}{c} Cement type \\ Cement content \\ Initial temperature of the concrete \\ Activation energy \\ Max. thermal hydration energy of 1 kg cement \\ Maximum thermal hydration energy of 1 kg concrete \\ Maximum thermal hydration energy of 1 kg concrete \\ Maximum temperature rise under \\ adiabatic conditions \\ \(E_{c,\mu 28}\) \\ \(f_{c,\mu 28}\) \\ \(f_{c,\mu 28}\) \\ \(f_{c,\mu 28}\) \\ \(f_{c,\mu 28}\) \\ Thermal capacity \\ Thermal conductivity at \(t_{0}\) \\ Thermal conductivity at \(t_{0}\) \\ Thermal conductivity at \(t_{0}\) \\ Poisson’s ratio at \(t_{0}\) \\ Poisson’s ratio at \(t_{0}\) \\ Poisson’s ratio at \(t_{0}\) \\ Thermal expansion coefficient \\ Density \\ kg m\({}^{-3}\) \\ \end{tabular}  &
\begin{tabular}{c} **CEM I** \\ **320** \\ **33.5** \\ **340** \\ **35.1** \\ **36.0** \\ **44.0** \\ **44.4** \\ **45.1** \\ **3** \\ **46.1** \\ **47.5** \\ **48.1** \\ **49.1** \\ **49.2** \\ **49.3** \\ **49.4** \\ **49.5** \\ **49.6** \\ **49.7** \\ **49.8** \\ **49.9** \\ **49. The evolution of the tensile strength and the stresses in the cross-section are calculated in every time step of the calculation. If the tensile strength is lower than the tensile stress cracks are likely. To display the risk of cracking the crack index is calculated, see Equation (13):  
\[I_{cr}=\sigma/f_{ct,m} \tag{13}\]  
\(I_{cr}\)... crack index, -  
\(\sigma\)... stress, N/mm\({}^{2}\)  
\(f_{ct,m}\)... concrete tensile strength, N/mm\({}^{2}\)  
The maximum crack index of the different material distributions is displayed in Figure 6. The calculations show that the critical area for cracking is at the bottom center of the side of the concrete block. For a chosen critical point at the side (Reference Point Figure 6) of the member, the time development of the tensile strength and tensile stress is printed over time in Figure 7. The amplitude of the tensile stress at the concrete surface is caused by temperature change between day and night. To compare the crack tendency of the different mixtures the crack index is shown in Figure 8. The discontinuity in the development at day 1 is caused by the removal of the formwork. Figure 6: Distribution of maximum crack index (\(I_{cr}\)) at a side (mirrored quarter model) for (**a**) homogeneous durability optimized material distribution; (**b**) zoned material property distribution; (**c**) gradient material property distribution and (**d**) homogeneous hydration energy optimized concrete. Figure 7: Time development of the tensile strength (\(f_{ct}\)) and maximum tensile stress (\(\sigma_{max,principal}\)) at the reference point (see Figure 6) for (**a**) homogeneous durability optimized material distribution; (**b**) zoned material property distribution; (**c**) gradient material property distribution and (**d**) homogeneous hydration energy optimized concrete. As can be seen from the resulting curves, the risk of cracking is highest in model (**d**). In this model the whole part consists of hydration energy optimized concrete. The high crack risk is caused by the slow evolution of the tensile strength in this concrete. Tensile stresses in the first five days exceed the tensile strength of the concrete.