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                                                       I. N. Shardakov et alii, Frattura ed Integrità Strutturale, 38 (2016) 339-350; DOI: 10.3221/IGF-ESIS.38.44 
 

339 
 

 
 
  
Process of cracking in reinforced concrete beams  
(simulation and experiment) 
 
 
I. N. Shardakov, A. P. Shestakov, I.O. Glot 
Institute of Continuous Media Mechanics of the Ural Branch of Russian Academy of Science (ICMM UB RAS), Korolev str., 1, 
Perm, 614013, Russia. 
shardakov@icmm.ru, shap@icmm.ru, glot@icmm.ru 
 
A.A. Bykov 
Perm National Research Polytechnic University, Komsomolsky ave., 29, Perm, 614990, Russia  
violentharpy@ya.ru 
 
 
ABSTRACT. The paper presents the results of experimental and theoretical 
investigations of the mechanisms of crack formation in reinforced concrete 
beams subjected to quasi-static bending. The boundary-value problem has 
been formulated in the framework of brittle fracture mechanics and solved 
using the finite-element method. Numerical simulation of the vibrations of an 
uncracked beam and a beam with cracks of different size serves to determine 
the pattern of changes in the spectrum of eigenfrequencies observed during 
crack evolution. A series of sequential quasi-static 4-point bend tests leading 
to the formation of cracks in a reinforced concrete beam were performed. At 
each loading step, the beam was subjected to an impulse load to induce 
vibrations. Two stages of cracking were detected. During the first stage the 
nonconservative process of deformation begins to develope, but has not 
visible signs. The second stage is an active cracking, which is marked by a 
sharp change in eingenfrequencies. The boundary of a transition from one 
stage to another is well registered. The vibration behavior was examined for 
the ordinary concrete beams and the beams strengthened with a carbon-fiber 
polymer. The obtained results show that the vibrodiagnostic approach is an 
effective tool for monitoring crack formation and assessing the quality of 
measures aimed at strengthening concrete structures. 
  
KEYWORDS. Reinforced concrete; Crack nucleation; Vibrodiagnostics; 
Experiment; Mathematical modeling. 
 

 

 
 

Citation: Shardakov, I. N., Bykov, A.A., 
Shestakov, A. P., Glot, I.O., Process of 
cracking in reinforced concrete beams 
(simulation and experiment), Frattura ed 
Integrità Strutturale, 38 (2016) 339-350. 
 
Received: 02.06.2016  
Accepted: 30.08.2016  
Published: 01.10.2016  
 
Copyright: © 2016 This is an open access 
article under the terms of the CC-BY 4.0, 
which permits unrestricted use, distribution, 
and reproduction in any medium, provided 
the original author and source are credited. 

 

 
 
INTRODUCTION 
 

einforced concrete structures have found wide application in the construction industry. Like other construction 
materials, reinforced concrete has limited strength characteristics. When these limiting values are exceeded, the 
structure may fail. Fracture of concrete usually occurs as the process of nucleation and growth of cracks. Initially R 



 

I. N. Shardakov et alii, Frattura ed Integrità Strutturale, 38 (2016) 339-350; DOI: 10.3221/IGF-ESIS.38.44                                                          
 

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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. 
 
 
MATHEMATICAL 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 V2 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 



 

                                                       I. N. Shardakov et alii, Frattura ed Integrità Strutturale, 38 (2016) 339-350; DOI: 10.3221/IGF-ESIS.38.44 
 

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Figure 1: Four-point bending of reinforced concrete beam (a) and the calculation scheme of the beam with a crack (b). 

 
 
According to Fig. 1, the developed mathematical model describes all structural elements of the beam: concrete (1), steel 
reinforcement (2), and supporting elements (3). 
The problem is formulated by using the virtual displacement principle [23]. 

 
 b o a b o af f fA A A A A A            .        (1) 

 
Boundary and initial conditions are specified as 

 
u u u L u u L1 2 3 1 2 3 20, ; 0,      x x   

i it t
u u t i x V V1 20 00, 0, 1, 3;              (2) 

n pp t x S( ),   n n  

 
Boundary conditions imposed on free surfaces, contact lines between the beam and supporting elements and boundary 
between the steel reinforcement and concrete follow from the variational Eq. (1). 
The variation of the work of internal and external forces for the structural elements of the beam can be written as: 

 

 

k k k
ij ij

bV

k
k k k ki
f i i i

b bV S p

A dV

u
A u dV p u dS

t

2

2

  

   

 


  





 
       (3) 

 
Expressions (1)-(3) contain the following notation: A  – variation of the work of internal forces; fA  – variation of 
the work of inertia and external forces; the superscripts b o a, ,  denote concrete, supporting elements and a reinforcement 
rods, respectively, the upper index k is used as “b” “a” and “o” for different structural elements, i, j – integer indices taking 

values 1, 2, 3 in accordance with the axes of Cartesian coordinate system ( x x x1 2 3, , ); 
k
iu  – displacement vector 

components; kij , 
k
ij  – strain and stress tensor components; 

k  – material density; kip  – components of the external 
force vector describing the impulse force applied along the normal n to the localized surface areas Sp of the beam. 



 

I. N. Shardakov et alii, Frattura ed Integrità Strutturale, 38 (2016) 339-350; DOI: 10.3221/IGF-ESIS.38.44                                                          
 

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The strain tensor components kij  are determined by the relations: 

 

 
bb uu jk i

ij x xj i

1

2


      
 

          (4) 

 
According to the Hooke’s law, the stress tensor components kij  are written as: 

 

 
k k kE Ek k k

ij ij kk ijb b b1 (1 )(1 2 )


   

  
 

  
       (5) 

 
Tab. 1 summarizes the physical properties of materials of the structural elements of the beam. 

 

Structural element 
Elastic modulus 

E, MPa 
Density 
ρ, kg/m3 

Poisson’s ratio, 
  

Concrete 0.35·105 2400 0.12 
Steel reinforcement and Supporting 

elements 
2·105 7800 0.3 

Carbon fiber sheet 2.52·105 2000 0.28 
 

Table 1: Physical properties of materials of the structural elements of the beam. 
 
Numerical implementation of the model is performed using the FEM package ANSYS. Fig. 2 presents finite-element 
meshes used to model a concrete beam with supporting elements and steel reinforcement. To describe the deformation 
process, we used Solid186 (3-D 20-node solid element having 3 degrees of freedom per node and exhibiting quadratic 
displacement behavior) for concrete, Solid189 (3-node beam element with quadratic approximation of displacements) for 
reinforcement, and shell281 (8-nodes shell element with 6 degrees of freedom per node and quadratic approximation of 
displacements and rotation angles) for supporting plates. 

 

 
Figure 2: Finite-element mesh: concrete and supporting elements (a) and steel reinforcement (b).  

 
The finite-element analogue of the variation equation written in matrix form is the system of ordinary linear differential 
equations: 

 
        M U K U f t( )            (6) 
 



 

                                                       I. N. Shardakov et alii, Frattura ed Integrità Strutturale, 38 (2016) 339-350; DOI: 10.3221/IGF-ESIS.38.44 
 

343 
 

where  M  is the mass matrix,  K is the rigidity matrix  U  and  U are the nodal displacement and the nodal 
acceleration vectors, and  f t( )  is the external force vector. 
 
 
VARIATION OF NATURAL FREQUENCIES DUE TO CRACK INITIATION 
 

he matrix equation for eigenmodes and eigenfrequencies of reinforced concrete beam obtained from (6) has the 
form 
 

     K M2 0             (7) 
 

where    is the vector specifying an eigenmode, and 2   is the eigenvalue equal to the squared circular 
eigenfrequency. The solution of system (7) determines the set of eigenmodes i  and eigenfrequencies i  (i = 1, 2,. . . ., 
N), where N is the order of symmetric positive definite rigidity and mass matrices. 
The crack initiation affects the rigidity of the originally intact beam and, as a consequence, changes eigenfrequencies and 
eigenmodes. For a quantitative description of these changes, we introduce small perturbations of the stiffness matrix, 
eigenfrequencies and eigenmode in the form 
 

      o o o oi i i i i iK K K M M[ ] [ ] [ ], [ ] [ ], ,     
             (8) 

 
All variables with superscript ‘o’ correspond to the beam without the crack, and the variables with superscript ‘*’ determine 
the value of perturbation due to the crack appearance. If we substitute (8) into (7) and linearize the obtained expression 
with respect to the perturbation, then, after simple transformations, we get the formula 
 

 
   
   

T To o

o i i
i i T To o o

i i

K

K

[ ]

[ ]

 
 

 


           (9) 

 

This relation allows assessment of variations in the i-th unperturbed mode  o
i

  and the frequency oi  at small 

perturbation of the rigidity matrix K *   . The quantity i
*  is called here the sensitivity parameter of the i-th 

eigenfrequency to crack formation. A comparative analysis of the values of i
*  shows the eigenfrequencies of the 

available frequency spectrum, which demonstrate the strongest response to the prescribed perturbation of the rigidity 
matrix. 
Information on these most sensitive eigenfrequencies and on the distribution of corresponding vibration forms over the 
surface of the beam allows us to determine the location, direction, and duration of the external action required to 
effectively initiate this oscillation and to find a point on the beam, at which the registration of vibration parameters will be 
most effective. The spectral analysis of measured vibration parameter (displacement, velocity, acceleration) ensures the 
possibility of assessing crack nucleation in the originally projected point or its lack. 
Analysis of the spectral properties of the reinforced concrete beam having the crack of 10 mm depth and 1 mm width in 
its central section, showed that there are four natural frequencies in the range of 0 – 5 kHz exhibiting the greatest 
response to the occurrence of such a crack [21]. They are eigenfrequencies No 14 (2072 kHz) and 23 (3897 kHz), 
corresponding to the bending modes, and eigenfrequencies No 16 (2298 kHz) and 23 (3845 kHz), corresponding to the 
torsion modes. Fig. 3 shows the local areas to which the load is applied in order to excite bending and torsion vibrations, 
and the positions of sensors (accelerometers) capable to register these oscillations. 
The solution of the initial boundary value problem modeling the vibrations in reinforced concrete beam caused by the 
impact impulse applied at the selected point can provide information on displacements and accelerations of arbitrary part 
of the structure.  

T 



 

I. N. Shardakov et alii, Frattura ed Integrità Strutturale, 38 (2016) 339-350; DOI: 10.3221/IGF-ESIS.38.44                                                          
 

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Fig. 4 shows the Fourier vibroacceleration image at the sensor location point in two forms: as a diagram illustrating the 
frequency spectrum of vibroaccelerations (a) and as a grey scale image where different shades of grey show the intensity of 
the signal of prescribed frequency (b). These results were found for the beam having a crack with a depth of 10 mm and 
width of 1 mm. 

 

 
Figure 3: Scheme of local application of pulse load and the location of the accelerometer.  

 
 

 
Figure 4: Fourier image of vibroacceleration: as a graph (а) and as a tone distribution (b). 

 
By solving a series of analogous problems, in which the crack depth varies from 0 to 180 mm (the crack propagating 
through the entire cross-section of the beam), we have obtained a set of grey-scale images, which allowed us to get a 
pattern of changes in eigenfrequences with crack extension. The results are shown in Fig. 5. The dark lines in these graphs 
correspond to the natural frequencies. Lines 1–4 with the greatest intensity correspond to frequencies that give the most 
intense signal in the Fourier image (lines 1 and 3 correspond to eigenfrequences No 14 and 23, lines 2 and 4 – to 
eigenfrequences No 16 and 24). The figure shows that as the crack depth increases, the eigenfrequencies reduce.  
 

 
Figure 5: Changes in eigenfrequencies caused by crack propagation: bending vibration mode (a) and torsion vibration modes (b). 

 



 

                                                       I. N. Shardakov et alii, Frattura ed Integrità Strutturale, 38 (2016) 339-350; DOI: 10.3221/IGF-ESIS.38.44 
 

345 
 

 
EXPERIMENT 
 

n our experiments, we used a set-up specially designed for sequential quasistatic four-point loading of a reinforced 
beam to initiate crack nucleation in concrete. During the experiment the beams were subjected to additional impulse 
loading at each loading stage. 

Mathematical modeling of the vibration process allowed us to calculate the parameters necessary to provide eigenmodes 
exhibiting the strongest response to nucleation and evolution of cracks. Piezoelectric transducers were employed to 
register the vibrations. Simultaneously, visible cracks on the side surface of the beam were registered.  
Tone images based on the Fourier analysis of vibrorecords were obtained from vibroacceleration measurements for each 
quasistatic loading step. A set of these images corresponding to increasing bending moments was used to get two-
dimensional tone images in coordinates: frequency and number of the loading stage (or an appropriate bending moment). 
Fig. 6(a) presents the diagrams illustrating the changes in eigenfrequencies pertaining to the bending eigenmodes, and Fig. 
6(b) shows patterns of visible cracks observed at appropriate loading stages. In the diagrams, the lines corresponding to 
eigenfrequencies are clearly visible so that one can trace the changes in eigenfrequencies as the load increases. It is seen 
that before the appearance of the first crack, eigenfrequencies remain almost unchanged. The first crack with the opening 
width of 0.05 mm occurs at the bending moment of 4.4 kNm. At that instant, the active nucleation of many cracks began 
in concrete. In the tone image, this stage is witnessed by a sharp reduction in eigenfrequencies. For three identified 
eigenfrequencies, 2069, 3904, 4798 Hz, the measured eigenfrequency changes in the interval from the start of loading until 
the propagation of cracks through the entire cross section of the beam are, respectively, 7.2, 13.7, and 18.7% (Tab. 2). 
By solving a series of problems of vibrations of a reinforced beam with a crack propagating through its central cross-
section with increasing bending moment, we obtained theoretically the changes in eigenfrequences. Tab. 2 shows the 
calculated changes in the eigenfrequencies of a beam having a crack extending through the entire cross-section 
(corresponding bending moment 5.0 kNm). As can be seen from the table, the experimentally determined changes in 
eigenfrequencies associated with cracking in concrete agree well with the results of numerical simulation. 

 

 
Figure 6: Changing in eigenfrequencies, corresponding bending vibration modes (a) and observed pattern of cracks (b). 

 
A detailed study of the eigenfrequency diagram shows that at the initial stage of deformation, when no visible changes are 
present in concrete, the changes in eigenfrequencies have already started, yet constituting no more than 0.5% of their 
original value (Fig. 6(c)). Accordingly, we can identify two characteristic stages in the process of loading. At the first stage, 
the nonconservative deformation process begins. The signs of this process cannot be observed visually, but if we have a 
recording apparatus with sufficient sensitivity, they can be well recorded. The second stage is characterized by active 

I 



 

I. N. Shardakov et alii, Frattura ed Integrità Strutturale, 38 (2016) 339-350; DOI: 10.3221/IGF-ESIS.38.44                                                          
 

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cracking reflecting by a sharp decrease in eingenfrequencies. It is of importance that the moment of transition from one 
stage to another is very well registered, because it suggests that the phase of active cracking starts. 

 

, kHz 
Experiment: 

, Hz (/, %) 
Simulation: 

, Hz (/, %) 
2.069 148 (7.2) 109 (5.3) 
3.904 534 (13.7) 489 (12.5) 
4.798 898 (18.7) 1022 (21.4) 

 

Table 2: Changing in eigenfrequencies at the instant of significant opening of the first crack (step 43, M=5.0 kNm). 
 

The strengthening of reinforced concrete structures with carbon fiber reinforced polymers (CFRP) is currently extensively 
used. The developed method of vibration diagnostics turns out to be a promising tool to analyze the deformation 
behavior of a concrete beam reinforced with carbon fiber plastic and to evaluate the effectiveness of this way of 
strengthening concrete structures. 
We tested the beams strengthened with carbon fiber sheet SikaWrap-230 of 40 mm width and 0.13 mm thickness (See 
also the paper by I N. Shardakov, A.A. Bykov, A P. Shestakov in the present issue). A series of physical experiments were 
carried out to explore the spectrum of eigenfrequencies of the beam strengthened prior loading. During stepwise 
increasing quasistatic loading in combination with additional impulse loading, the grey-scale Fourier images were obtained 
(Fig. 7(a)). It is seen that a sharp change in eigenfrequencies corresponding to the formation of first visible cracks is 
associated with the bending moment M = 5.2 kNm; the limiting state accompanied by the rapture of the CFRP sheet is 
achieved at M* = 10.4 kNm. Both values are essentially higher than the relevant values obtained for the unstrengthened 
beam.  
Tab. 3 presents data showing changes in three selected eigenfrequencies registered at successive loading stages (Fig. 7). 
The bending moment M* refers to the instant when the rapture of the strengthening layer takes place. The loading stages 
prior to the onset of first cracks are accompanied by slight changes in eigenfrequencies (~0,6%). The nucleation of cracks 
causes an abrupt change in eigenfrequencies, which reaches at the instant of the rapture of CFRP about 60% of their 
initial values.  

 
 

 
Figure 7: Changing in eigenfrequencies for the beam preliminary strengthened with the CFRP sheet (a) and observed pattern of cracks 
(b). 

 
 



 

                                                       I. N. Shardakov et alii, Frattura ed Integrità Strutturale, 38 (2016) 339-350; DOI: 10.3221/IGF-ESIS.38.44 
 

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n , kHz 
before 1-st 

crack 
M<0.56M* 

1-st crack 
M=0.56 M* 

cracks 
M=0.8 M* 

numerous 
cracks 
M=M* 

rapture of  
CFRP sheet 

1 0.751 0.6 2.7 45.8 55.8 60.5 
2 1.413 0.7 8.0 41.5 54.0 60.5 
3 2.173 0.6 3.3 35.0 x x 

 

Table 3: Relative change in eigenfrequencies / (%) in the preliminary reinforced beam. 
 

The diagram of changes in eigenfrequencies reflects the processes taking place in the beam during its repair and restoring 
at some intermediate loading stage (Fig. 8). The restoration procedure carried out directly under the load and consisted of 
the injection of cracks with the low-viscous epoxy material and gluing carbon fiber sheet to the stretched beam surface. 
In the loading range from 0 to 4.5 kNm, the eigenfrequencies remained practically the same, and their values reduced 
sharply at the onset of formation of first visible crack (М=5.7kNm). In the diagram, one can observe the "restoring" of 
eigenfrequencies after the procedure of crack treatment: they are recovered almost to their original values, which are kept 
over the load range from 5.7 to 7.2 kNm. A further increase in the bending moment leads to the nucleation of new 
generation cracks (Fig. 8(b), light lines) and therefore to new reducing in eigenfrequencies. A total loss of the bearing 
capacity of the beam occurs at the instant of rapture of the CFRP sheet at the bending moment of 10.4 kNm. 
Thus, the diagram of eigenfrequencies enables one to visualize the sharp changes in eigenfrequencies at the instant of 
nucleation of both primary and secondary cracks in concrete, and to identify the stage characterized by a full loss of the 
bearing capacity of the beam.  

 

 
Figure 8: Changes in eigenfrequencies for the beam strengthened with the CFRP sheet under the load (a) and the observed pattern of 
cracks (b). 

 
 

n , kHz 
before 1-st 

crack  
M<0.56M* 

1-st crack  
(I generation) 

M=0.44M* 

1-mm crack 
M=0.53M* 

after 
treatment 

*0.53M M  

1-st crack 
(II 

generation)  
M=0.67M* 

cracks 
M=0.8M* 

numerous 
cracks 
M=M* 

rapture of 
CFRP sheet 

1 0.746 0.4 6.0 38.6 1.9 3.7 36.9 50.5 58.1 
2 1.367 0.6 3.5 35.8 1.2 6.9 26.8 41.0 48.7 
3 2.109 0.4 6.9 24.3 2.0 3.5 26.7 x x 

 

Table 4: Relative change in eigenfrequencies / (%) of the beam reinforced under the loading. 



 

I. N. Shardakov et alii, Frattura ed Integrità Strutturale, 38 (2016) 339-350; DOI: 10.3221/IGF-ESIS.38.44                                                          
 

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Based on the diagram given Fig. 9, the values of loading at which first cracks occur in concrete are compared with the 
limiting values. As a limiting condition, we assume a state when multiple cracks spread over the entire cross-section of the 
beam, and the bearing capacity of the beam is completely ensured by the steel reinforcement (in the experiment it is the 
bending moment equal to 6.5 kNm). The experiment proves that, from the appearance of the first cracks until the loss of 
the bearing capacity (4.4 kNm to 6.5 kNm), the beam continues to be functional despite of nucleation of new cracks and 
propagation of existing ones. In this case, the beam has the strength reserve constituting ~32% of the limiting load. The 
beam reinforced with the CFRP sheet also retains its workability after the onset of first cracks. The bending moment of 
the preliminary strengthened beam corresponding to the full rapture of the sheet exceeds the load at which the occurrence 
of first cracks has been registered by ~45%, and the bending moment of the beam strengthened during the loading by 
~56%. The analysis of the diagram confirms the validity of procedures aimed at strengthening a beam under the loading. 
The effectiveness of such reinforcement techniques is competitive with that of the procedures for preliminary 
strengthening of beams with a composite material. Summing up, our experiment clearly demonstrates that crack 
formation in concrete is not a critical factor affecting the loss of bearing capacity by a reinforced concrete beam.  

 

 
 

Figure 9: Critical values of the bending moment.  
 
CONCLUSION 
 

ased on the results of experimental and theoretical studies, we conclude that vibration diagnostics of reinforced 
concrete structures is an effective tool for early detection, assessment, and monitoring of cracking. The real-time 
vibration analysis gives a great deal of information about the level of cracking and the residual life of the concrete 

structure. The developed method of vibration diagnostics enables one to assess the degree of efficiency of measures for 
the restoration and strengthening of the structure. The obtained data provide a general algorithm for creating and 
performing procedures for vibration diagnostics of cracking in reinforced concrete structures. The algorithm consists of 
the following steps: 
1) Based on mathematical modeling, the deformation state of the structure is assessed by using information on actual 

loading conditions. The most probable locations of cracks are predicted. 
2) A series of numerical experiments are carried out to simulate vibrations in the structure. Eigenmodes and 

eigenfrequences are analyzed for the entire structure and the structure having cracks in the predicted areas. 
3) Using numerical simulation results, the parameters of vibrodiagnostic testing are specified, namely: 

 technical characteristics of vibration sensors that provide registration of vibrations in a predetermined 
frequency range with a required precision; 

 vibration sensors location points; 
 position and duration of external impact, ensuring excitation of vibrations with required natural frequencies. 

4) Rational scheme for strengthening a reinforced concrete structure based on numerical simulation data is developed. 
5) Block diagram for the system of vibration diagnostics of cracking in reinforced concrete structures is elaborated. 

 

B 



 

                                                       I. N. Shardakov et alii, Frattura ed Integrità Strutturale, 38 (2016) 339-350; DOI: 10.3221/IGF-ESIS.38.44 
 

349 
 

 
ACKNOWLEDGMENT 
 

his research was supported by the Russian Science Foundation, project No. 14-29- 00172. 
 

 

REFERENCES 
 
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    /NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken die zijn geoptimaliseerd voor prepress-afdrukken van hoge kwaliteit. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
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    /ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing.  Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
  >>
  /Namespace [
    (Adobe)
    (Common)
    (1.0)
  ]
  /OtherNamespaces [
    <<
      /AsReaderSpreads false
      /CropImagesToFrames true
      /ErrorControl /WarnAndContinue
      /FlattenerIgnoreSpreadOverrides false
      /IncludeGuidesGrids false
      /IncludeNonPrinting false
      /IncludeSlug false
      /Namespace [
        (Adobe)
        (InDesign)
        (4.0)
      ]
      /OmitPlacedBitmaps false
      /OmitPlacedEPS false
      /OmitPlacedPDF false
      /SimulateOverprint /Legacy
    >>
    <<
      /AddBleedMarks false
      /AddColorBars false
      /AddCropMarks false
      /AddPageInfo false
      /AddRegMarks false
      /ConvertColors /ConvertToCMYK
      /DestinationProfileName ()
      /DestinationProfileSelector /DocumentCMYK
      /Downsample16BitImages true
      /FlattenerPreset <<
        /PresetSelector /MediumResolution
      >>
      /FormElements false
      /GenerateStructure false
      /IncludeBookmarks false
      /IncludeHyperlinks false
      /IncludeInteractive false
      /IncludeLayers false
      /IncludeProfiles false
      /MultimediaHandling /UseObjectSettings
      /Namespace [
        (Adobe)
        (CreativeSuite)
        (2.0)
      ]
      /PDFXOutputIntentProfileSelector /DocumentCMYK
      /PreserveEditing true
      /UntaggedCMYKHandling /LeaveUntagged
      /UntaggedRGBHandling /UseDocumentProfile
      /UseDocumentBleed false
    >>
  ]
>> setdistillerparams
<<
  /HWResolution [2400 2400]
  /PageSize [612.000 792.000]
>> setpagedevice