# Experimental Study and Modelling on the Structural Response of Fiber Reinforced Concrete Beams
## 4 Modelling  
Based on experimental observations, we accept on the middle span of each beam, subjected to four-point bending (see Figure 4) only one single macro-crack finally opens. This crack grows reaching the neutral axis, opens and finally divides the sample into two pieces. Is possible to suggest that at initial stage micro-cracks are few. Only one of them is forming the macro-crack that is reaching the neutral axis of the beam. Macro-crack growth (increase the length) is characterized by small it's opening (at initial stage non-visible by eye). After reaching the neutral axis, its growth stops (further increase in the crack length is small) and the crack begins to open with an increase in the applied external load. Macro-crack opening is happening by fibers pull-out process. At this stage length of the macro-crack is \"in general\" stable, in our experiments it was 80-85 mm. Each fiber crossing the surface of a macro-crack pulls out the shortest end in the process of its opening. Polymer (polyppropylene) fiber breaks inside the concrete when pulled. Only part of the end length is pulled out.  
Crack opening numerical structural model based on the similar suggestions as in [64; 65] was accepted and used. Basic model's assumptions were: (a) crack's length is stable, it opens, fibers, crossing it's plane under different angles, are pulling out; (b) fibers distribution in the volume is random (according to orientation angles and each fiber geometrical centre spatial location); (c) each fiber is pulling out as a single fiber--is possible to use experimental data set with pull-out curves for fibers oriented under different angles and embedded at different depth in the concrete (single curves and average data) [64]. Number of fibers in every prism with dimensions 100 \(\times\) 100 \(\times\) 400 mm is known (in average) from the experiment. Single prism volume was divided into elementary volumes. N-elementary volumes are on the one side of the crack and the same number on opposite. Number of fibers crossing one particular element grain (which is on the surface of the crack) is easy to calculate.  
For every particular grain is possible to evaluate each fiber tail length and orientation. Using experimental data obtained in pull-out experiments and performing averaging procedure, is possible to calculate force applied to particular grain depending on \"local opening value\". \"Local opening value\" is distance between two grains-neighbours (when macro-crack is closed distance between grains- neighbours is zero. In the model crack mouth opening displacement (CMOD) were numerically changed by steps. At every step, forces applied to every pair of grains-neighbours were calculated, such way all forces on the surface of macro-crack were calculated. From equilibrium conditions applied to the prism external forces were calculated [64]. In the modelling was used commercial software MATLAB. Modelling results were compared with the experimental curves for beams. Predictions generated by the model were validated by 4 PBT of 100 \(\times\) 100 \(\times\) 400 mm prisms.