**6. Conclusions**

As a summary of this work, the following general conclusions should be highlighted:

(1) The accuracy of transmission ultrasonic tomography for the detection of brittle damage in concrete can be effectively supported by the graph theory and, in particular, by Dijkstra's algorithm. What is important, it allows the determination of real paths of the fastest ultrasonic wave propagation in concrete containing localized elastically degraded zones at any stage of their evolution. Thanks to the analyses conducted on this basis, the authors developed the method of reducing errors in reconstructions of longitudinal wave speed maps. In this approach, the errors are decreased which are caused by using a simplification of straightness of the fastest wave propagation paths assumed in the typical mathematical apparatus for ultrasonic tomography. The method is based on the appropriate elongation of the measured propagation times of the wave travelling between opposite transmitting-receiving transducers if the actual propagation paths deviate from straight lines.

(2) Transmission ultrasonic tomography allows the estimation of the internal length of concrete defined in accordance with the methodology of damage mechanics. This problem is very important in the case of studies carried within this field of mechanics (e.g., Reference [29–31,49,50]) when predicting the extent and degree of brittle damage evolution in concrete structures. However, this conclusion may be addressed to testing RC beams with certain restrictions regarding issues of bonding between concrete and reinforcing bars and is appropriate for beams of lower reinforcement ratios with arrangement of tomographic rays not along reinforcing bars.

(3) Knowledge of the internal length of concrete allows rational determination of the appropriate resolution in ultrasonic tomography imaging and assessment of evolution of localized elastically degraded zones.

(4) The use of fictitious transmitting-receiving points in ultrasonic tomography, for which wave propagation times are calculated by interpolation of measured times, can contribute to the reduction of the required number of transducers and possible costs in the considered approach to concrete beams assessment while maintaining proper resolution of tomographic images. This outcome was well-grounded in the case of numerical analysis conducted in the work and usefulness of this approach was pre-confirmed in the own experiments. However, due to the limited number of these tests, it needs further justification and experimental studies using different arrangements of distances between sending-receiving points for ultrasonic pulses on the same RC members and confrontation with tests based on, e.g., acoustic emission or X-ray tomography.

**Author Contributions:** Conceptualization Z.P.; Data curation, K.T.; Formal analysis, Z.P.; Investigation, Z.P. and K.T.; Methodology, Z.P.; Software, Z.P.; Validation, Z.P.; Visualization, K.T.; Writing—original draft, Z.P.; Writing—review & editing, K.T. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** Special thanks are dedicated to Bronisław J ˛edraszak for bringing our attention to the possibilities related with testing prefabricated RC beams.

**Conflicts of Interest:** The authors declare no conflict of interest.
