# Simulation of Crack Propagation in Reinforced Concrete Elements
## 6 Discussion and Conclusions  
In the present paper, a simulation model of the random crack propagation in reinforced concrete elements is presented. The model consists of two uniaxial bars (concrete and reinforcing steel) divided into finite elements, which are connected with springs. For the random cracking, random concrete tensile strength values are assigned to the element. Several features and advantages of this model are introduced in the paper, the main objective of which is to discuss the major role of the assumed tensile strength and its distribution, as well as the element size considerations in the CPM. The most important parameter to simulate random cracking is the uniaxial tensile strength along the model axis. It was pointed out that a normal distributed concrete tensile strength can be used for a common range of coefficients of variation (\(v_{\text{k}}<20\%\)). This was confirmed by investigations in [4], according to which the assumed distribution type of the tensile strength has only a minor effect on the calculated crack widths. In CPM, different coefficients of variation can be assigned to the normal distribution. The model also distinguishes between the mean value of the fracture tensile strength \(f_{\text{ct},\text{r}}\) and the mean value of specimen tensile strength \(f_{\text{ct},\text{m}}\). If several tests using a test method where the fracture plane develops randomly at the weakest point of the specimen (e.g., direct tensile test without a notch) are available, then the test results represent only the distribution of the weakest tensile strength values \(f_{\text{ct},\text{f}}\) (here this is the fracture tensile strength). To capture the tensile strength distribution in an entire bar along the longitudinal axis with weak and strong parts (here this is the specimen tensile strength), increasing the mean value of specimen tensile strength \(f_{\text{ct},\text{m}}\) seems logical. For this purpose, an approach was proposed in Section 3.2, setting \(f_{\text{ct},\text{m}}\) as equal to the 95 th percentile of fracture tensile strength values \(f_{\text{ct},\text{f}}\) based on [3]. This approach requires further experimental evidence; however, there are no common alternative approaches available in the literature. Since the approach was applied for direct tensile tests (without a notch), an application for splitting tensile tests (as in experiments in [36]) is not recommended without further research. For numerical models, the results are supposed to be mesh-independent. Since the number of tensile strength values along the model axis in the CPM is dependent on the number of elements (and the respective element lengths), tensile strength values are not supposed to scatter randomly between adjacent elements. For this purpose, a dependency between adjacent elements was defined in the model (autocorrelation). The correlation length--present in the correlation function--specifies the scope of this dependency. The correlation length is a mathematical value, which when combined with mechanical values, enables a mesh-independent and realistic distribution of the tensile strength along the longitudinal axis of a specimen. Since realistic data on the correlation length for tensile strength are scarce [4], for each problem different correlation length values are available in the literature [37]. Therefore, the representative length was introduced in this paper with a mechanically based value considering the aggregate size in order to stay within the macroscale. The comparison with the experimental test in [36] shows a good reproduction of the results with the CPM, although small crack spacings are slightly underestimated by the CPM. The main reason for this underestimation is probably the influence of shrinkage. The investigations in [38] confirm a high influence of shrinkage strains, especially for high reinforcement ratios. However, no information on shrinkage strain is available in [36].