Issue 38

I. N. Shardakov et alii, Frattura ed Integrità Strutturale, 38 (2016) 339-350; DOI: 10.3221/IGF-ESIS.38.44 340 the crack formation does not lead to complete loss of the carrying capacity of the structure, but it can be considered as a fracture precursor. Knowledge of the crack nucleation process is important to ensure early prediction of emergency situations and prompt use of techniques to restore damaged reinforced concrete structures. Nowadays, there are many ways to monitor cracks in concrete structures. Common visual observation methods are helpful in the evaluation of big visible damages, while the hidden or inaccessible defects cannot be identified. The local methods of inspection are actively used as well. They involve recording various physical quantities, such as ultrasonic waves [1], eddy currents [2], temperature fields [3], acoustic emission [4], X-ray, magnetic particles [5], and so on. A survey of nondestructive testing methods applied to reinforced concrete structures can be found in [6]. These methods also have some limitations: one should know a priori the region with defects or use many transducers; the region where a defect may appear should be accessible for examination. The location of a defect may remain unexamined because only a small part of the structure can be examined with these methods. Most local methods are rather expensive, and it takes much time to investigate the structure using these techniques. The methods for crack detection and pattern recognition based on image processing are described in [7]. Another direction in flaw detection technology includes global methods aimed at discovering damage via the assessment of the mechanical behavior of the entire structure. The main advantage of such methods is that they do not require a great number of transducers to be located in close proximity to defects. Most commonly used and promising global methods of nondestructive testing are vibration approaches [8]. They are based on the fact that the variations of physical properties (mass, rigidity, damping) of an object cause the changes in modal parameters, which qualitatively and quantitatively characterize natural vibrations. A survey of vibration approaches is provided in [9-11], and their applications to reinforced concrete structures are described in [12]. Vibration nondestructive testing methods employ various modal parameters. The analysis of eigenfrequencies allowing early damage detection is given in [13]. Damage localization problems have been solved by using a criterion based on the comparison of two mode shapes in damaged and undamaged states [14]. Information about the curvature of eigenmode shapes serves as an indicator of defects in beam structures [15]. Data on eigenmode shape deformation energy have been used for damage localization in [16]. A method for estimating the state of bridges, which is based on the information regarding a compliance matrix, is described in [17]. In connection with the vibration methods of study of quasi-brittle materials destruction the works [18-20] should be noted were the methods of wavelet analysis are successfully used to identify the fracture process. In the present paper, we propose a new variant of the vibration method to detect the crack location and crack opening degree by analyzing the changes in the eigenfrequencies spectrum associated with the occurrence of cracks. The method allows us to perform monitoring the strain state of structures, where the region of most probable fracture is known, but free access to it is absent. M ATHEMATICAL MODEL OF DEFORMATION PROCESSES IN REINFORCED STRUCTURE he stress-strain state of reinforced concrete beams under quasi-static four-point bending is studied. The beam is reinforced by two plane steel frameworks joined together by vertical elements. The structural scheme and characteristic sizes of the beam are shown in Fig. 1 (a). At a certain stage of the loading, cracking starts in the beam. Works [21, 22] focus on the mathematical modeling of this process and comparing the simulation results and experimental data. The proposed approach is based on the analysis of vibrations of a reinforced concrete beam that occur in response to impact loads applied to a certain part of the beam. Comparison of the vibration response of a solid beam and a beam having different-size cracks makes it possible:  to evaluate the sensitivity of spectral properties of a reinforced concrete beam to crack nucleation and growth;  to find pulse load, causing vibrations with eigenfrequencies most sensitive to cracking and to define load parameters;  to identify patterns of change in eigenfrequencies associated with the emergence and growth of cracks. A calculation scheme for the problem of vibrations of a reinforced beam having a crack is given in Fig. 1 (b). A crack that occurs in concrete is represented as a volume V 2 occupied by the material exhibiting practically zero mechanical properties. This approach is valid for cracks called opening mode cracks. It is precisely these cracks that appear in the beam tested under four-point bending conditions used in our experiment. As regards other types of cracks, having no gap between the edges of the crack, it is necessary to consider the interaction between the opposite surfaces of the crack, taking into account the dissipation of mechanical energy. Our approach can be adapted for these types of cracks. T