# Simulation of Crack Propagation in Reinforced Concrete Elements
## 5 Comparison with Experiments  
The performance of the CPM will be demonstrated in the following using a series of tests from [36]. The test series included three identical large-scale tests. The focus of the tests was on the generation of many cracks and investigation of the crack spacing distribution. For this purpose, a very high reinforcement ratio of \(\rho\) = 8% with a bar diameter of 32 mm was chosen. A tensile force of 900 kN was required to reach a stress level of \(\sigma_{\text{s,max}}\) = 280 N/mm\({}^{2}\). Figure 14 shows the geometrical parameters of the test specimens used in [36]. The material parameters of the concrete were determined in [36] using compression tests and splitting tensile tests (no data available on scattering). The maximum aggregate size was 16 mm. The parameters used in the CPM are listed in Table 3. A correction of the tensile strength according to Section 3.2 was not performed, as splitting tensile tests were conducted (here \(f_{\text{ct,f,m}}\) = \(f_{\text{ct,m}}\)). The model parameters for the test series conducted by [PERSON] [36]. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{2}{c}{**Geometry**} \\ \hline Reinforcement ratio & \(\rho\) & 8\% \\ Diameter of reinforcing bar & \(\emptyset\) & 32 mm \\ Length of model & \(l\) & 2000 mm \\ Element length & \(l_{\text{e}}\) & 10 mm \\ \hline \multicolumn{2}{c}{**Material**} \\ \hline Concrete & & \\ Mean value of compressive strength of concrete & \(f_{\text{cm}}\) & 35 N/mm\({}^{2}\) \\ Mean value of tensile strength of concrete & \(f_{\text{ct,m}}\) & 3.2 N/mm\({}^{2}\) \\ Mean value of modulus of elasticity of concrete & \(E_{\text{cm}}\) & 32,000 N/mm\({}^{2}\) \\ Reinforcement: B500 & & \\ Modulus of elasticity of reinforcing steel & \(E_{\text{s}}\) & 200,000 N/mm\({}^{2}\) \\ \hline \multicolumn{2}{c}{**Bond slip relationship**} \\ \hline Model Code 2010 (good bond conditions) & & \\ Bond stress & \(\tau_{\text{b}}(s)=14.79\cdot(s/1.0)^{0.4}\) \\ Reduction factor & \(\lambda(x)=0.5\cdot x/32\leq 1.0\) \\ \hline \multicolumn{2}{c}{**Scattering**} \\ \hline Tensile strength of concrete & 18 distributions (in each case) \\ Coefficient of variation & \(\
u_{\text{k}}\) & 10\% (15\%) \\ Correlation length & \(L_{\text{x}}\) & 20 mm (10 mm) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: CPM parameters for the test series conducted by [PERSON] [36]. Figure 14: Details of the axial tensile specimen (dimensions shown in mm) [36]. The calculations were first carried out for a coefficient of variation of \(v_{\rm k}=10\%\) and a correlation length of 20 mm (\(\sim\)1.25 \(d_{\rm g}\), cf. Equation (12)). In a second calculation run, a correlation length of 10 mm was applied for comparison and to consider the small crack spacings. To account for the uncommonly high reinforcement ratio and large reinforcement diameter in combination with the relatively small crack spacing, a coefficient of variation of \(v_{\rm k}=15\%\) was also applied. Since in [36] the crack spacing was recorded on 3 sides at the depths of each reinforcement layer (3 specimens \(\times\) 3 surfaces \(\times\) 2 bars = 18 sets), 18 generated distributions were recorded for each calculation run. To take the disturbed stress region at loading points into account, both the values presented in [36] and the CPM did not take into account the last crack spacing on both sides of the test specimens. The distribution of experimental crack spacing and the calculated crack spacing with the CPM is shown in Figure 15a for \(v_{\rm k}=10\%\) and in Figure 15b for \(v_{\rm k}=15\%\). The statistical parameters are summarized in Table 4. A calculation with twice as many elements (\(l_{\rm e}=5\) mm) showed similar results, with the exception of small deviations in the distribution of the crack spacings due to the randomly generated tensile strength distributions.