# Simulation of Crack Propagation in Reinforced Concrete Elements
## 2 General Crack Propagation  
If an uncracked reinforced concrete specimen (uncracked stage) is subjected to a steadily increasing centric tensile load, cracks will appear at the point of the lowest concrete tensile strength (initial cracking stage). In the crack, the concrete strain value is zero and the steel strain value is maximal. Due to the bond stresses activated on both sides of the crack, tensile stresses are introduced into the concrete and the steel tensile stresses are reduced until the steel and concrete strains are identical again at the end of the transmission length \(l_{\text{t}}\) (Figure 1). With a further increase of the load, the next crack occurs in the remaining uncracked component zone at the point with the lowest concrete tensile strength (crack formation stage; Figure 1, stage 2). This process is repeated until at no point are the steel and concrete strains identical. At this point the stabilized cracking stage is reached (Figure 1, stage 3), however it must be noted that with further load increase beyond this state, further cracks can form.  
The crack width in the crack formation stage results from the integration of the strain differences between the reinforcement and surrounding concrete over the transmission length on both sides of the observed crack. In the stabilized cracking stage, the crack width is determined by integrating the strain difference over half of the crack spacing on both sides of the considered crack.  
Hence, the concrete tensile strength values (specifically the lower concrete tensile strength values) over the length are significant in terms of initiation of cracks, while the bond-slip behaviour is the major parameter affecting the transmission of stresses between the steel bar and the concrete. Therefore, to simulate the whole crack propagation process with increasing load level (i.e., increasing steel stress \(\sigma_{\text{s}}\)), the crack propagation model (CPM) is configured with the following features:  
Figure 1: Stages of crack propagation in reinforced concrete specimens in uniaxial tensile tests.  
For random cracking, concrete tensile strength scattering along the longitudinal axis of the specimen is required. Therefore, the CPM can map different statistical distributions;
* It must be possible to calculate steel and concrete strains separately over the entire length of the specimen. Thus, both materials shall be represented separately;
* A bond stress-slip relationship is required for the interaction of both materials. This should be definable in a general mathematical form to be able to take into account different loading levels and material configurations.  
Using these features, the crack formation can be considered in a very variable and specific way (e.g., through adaptation of input parameters and calculation steps and isolated consideration of influence parameters with almost any number of calculation runs). Using a random definition for the concrete tensile strength over the specimen length, the random locations of the crack initiation and random crack spacings can be determined. Furthermore, the statistical distribution of the output crack widths and crack spacings can be investigated with several runs and numerous random parameters within a specific range.  
However, to obtain a mechanically reasoned model with very good prediction accuracy, specific considerations concerning the tensile strength distribution are needed to reach element-size-independent and reasonable results. Such aspects of the CPM model will be explained in the following.