Damage detection in multi-span beams based on the analysis of frequency changes

Crack identification in multi-span beams is performed to determine whether the structure is healthy or not. Among all crack identification methods, these based on measured natural frequency changes present the advantage of simplicity and easy to use in practical engineering. To accurately identify the cracks characteristics for multi-span beam structure, a mathematical model is established, which can predict frequency changes for any boundary conditions, the intermediate supports being hinges. This relation is based on the modal strain energy concept. Since frequency changes are relative small, to obtain natural frequencies with high resolution, a signal processing algorithm based on superposing of numerous spectra is also proposed, which overcomes the disadvantage of Fast Fourier Transform in the aspect of frequency resolution. Based on above-mentioned mathematical model and signal processing algorithm, the method of identifying cracks on multi-span beams is presented. To verify the accuracy of this identification method, experimental examples are conducted on a two-span structure. The results demonstrate that the method proposed in this paper can accurately identify the crack position and depth.


Introduction
Even if beams with multiple supports are frequently used in practice, the research about such structures in the damaged state is not broadly present in the literature. In this research, uniformly spaced spans are commonly considered. An example is the study of a beam with two identical spans traversed by a train moving at a constant velocity, which had the aim to highlight the resonance characteristics [1]. The Euler-Bernoulli beam theory was used to model the two beam segments, and constraints and compatibility conditions were imposed at the beam center. The effect of the random deviation of the span lengths from the ideal design was studied in [2], using the perturbation method. Transverse vibrations of a slender beam with infinite length mounted on discrete elastic supports are studied in [3]. To describe the dynamic behavior of multi-span beams, the supports are replaced by forces acting on the support locations. The deflection is described by a single function, the reactions at the support locations being included in the eigenvalue solution [4].
In the damage detection methods, the multi-span structures with transverse cracks are treated as an assembly of intact sub-segments. The cracks are replaced with massless rotational spring, while instead of the supports, constraints and compatibility conditions are imposed [5]. The  complicated, and, by replacing the intermediate supports or the crack, a new set of equations has to be solved. Several theoretical studies and experiments aiming to study and analyze the changes of the natural frequencies and mode shapes occurring due to damage in general terms [6][7][8][9][10][11][12][13] and due to a crack in particular [14][15][16][17], were presented in the literature. Previous publications, [18][19][20], present a mathematical relation that predicts the frequency changes occurred in single span beams due to cracks with known depth and position. The input data is limited to the curvature achieved by the intact beam in the location where damage is expected, and the absolute damage severity derived from the deflections of the intact and damaged beam. Based on this relation a database was build for a multitude of damage scenarios and differently supported beams, therefore damage assessment becomes an inverse problem.
This paper presents the exact solution for the frequencies of beams with two unequal spans, for both healthy and damaged state. Any support and crack position can be taken into consideration; the crack depth is limited to around 50% by the linearity condition. The results are implemented in a database and used to demonstrate the robustness of the proposed damage assessment method.

Modal parameters for the intact two-span beam
The study is performed on a long slender beam with two spans, as shown in figure 1, having the length L, the width b and the thickness h, and consequently the second moment of area of the cross section I and the cross section area A. At the right end the support is a hinge and the left end is clamped; supplementary, a hinge at distance 1 L is intercalated. The relevant material characteristics are: the Young Modulus E, the volumetric mass density  , and the Poisson ratio  .
If the beam is subjected to a damage, this is an open transversal crack which has the depth a and the width l . Its position is indicated as c . The bending vibrations of this beam are described by the following equation: where z is the vertical beam displacement at distance x measured from the left end. The solution z of equation (1) can be written as a standing wave, ( , ) ( ) sin( ) z x t W x t     , thus spatial and temporal components are separated. For the spatial component the well-known solution is  In the vibration analysis, each segment between two supports is considered as a separate beam. The left segment has associated the space interval   11 0 xL  , and the origin of the reference system is the fixed beam end (point 1 in figure 2). The right segment has associated the space interval xL  , and the origin of the reference system is the hinge placed at the right beam end (point 3 in figure 2). The two solutions of equation (1) get, for the two segments, the form Applying the eight constraints and continuity conditions to the solutions in equation (3), it results a system with eight algebraic equations. After removing the eight coefficients 12 where . By solving equation (4) for the central hinge position removed iteratively along the beam, the evolution of the dimensionless wave number  in form of variation curves is obtained (see figure 3). From this figure one can easily observe that for 1 0 L  the wave numbers achieve very close values to that of a beam clamped at the left end and hinged at the right end (marked with a red square). If the intermediate hinge is close to the hinged end, i.e. 1 1 L  , the wave number values are very similar with that of a beam with both ends clamped (marked with a blue square).
The two-span beam's frequencies can be derived using the well-known frequency relation while for the right segment it is: Note that, to obtain normalized mode shapes, D1 must be determined separately for each location of L1 and vibration mode.
The analytically achieved results are now compared with that from FEM simulations. A beam with following geometrical and physical-mechanical parameters is involved in the analysis: In the first case the central hinge is placed at the normalized distance 1 / 0.33 LL  and in the second case it is located at distance . The FEM analysis, performed by means of the ANSYS 14 software, made use of a hexahedral mesh element with imposed dimensions 2 mm. Table 1 indicates the wave numbers and frequencies derived analytically, as well as the simulation results. The differences between synonym frequencies is less than 5%, except two modes of the beam with central hinge at 1 / 0.33 LL  . These differences occur because certain slenderness is necessary to permit applying the Euler-Bernoulli theory. Thus, if one of the beam segments is short, bigger errors are present, imposing a more complex model. In addition, the FEM model does not exactly reflect the conditions induced by the hinges in the analytical model.  For the hinge located at L1/L = 0.33, the first six bending mode shapes are depicted in figure 4, where those corresponding to the analytic approach are to the left and those resulted from the FEM analysis to the right. One can observe in this figure that curves fit well.

Natural frequencies of the damaged beam
The crack considered in this study is transversal and has the depth a, as shown in figure 1. It is located at distance c from the fixed end. As shown in our previous papers, the energy stored in a slice in the undamaged state can indicate the natural frequency drop if a crack occurs in that slice [19]. The mathematical relation permitting to determine the natural frequencies for the damaged beam, adapted for the particular case of the two-span beam is where i f is the natural frequency for the undamaged beam for the i-th vibration mode and ( ) 1 a   is the damage severity characterizing the effect of a crack with relative depth [20], which is null for the undamaged beam. The normalized modal curvature at location c, denoted in equation (8), is found from equations (9) or (10), depending on which segment the crack is located.
The functions in equations (9) and (10) are proportional with the modal curvatures of the two segments. Normalization is achieved by involving the proper coefficients k1 and k2. Based on equation (8), the frequencies of the damaged beam for any crack location can be calculated. Figure 5  . The results are indicated with black squares in figure 5. Having a look onto this figure, it is easy to remark the fit between analytically and FEM-based attained results, meaning that equation (8) can be used to define damage patterns which can be further used in the damage detection process. For beams with multiple supports the frequency shifts are slight, making the improvement of frequency readability an important issue in experimental modal analysis [21][22].

The damage detection algorithm
The proposed assessment algorithm is based on the particularity that any crack provides a particular behavior change to that beam. As consequence of a crack, frequencies of different bending modes shift differently, in respect to the energy stored in the damaged slice, as shown in previous section. For the i-th bending mode of a two-span beam, the relative frequency shift is derived as: For every damage locations c and relative depths a result the relative frequency shifts 1 ... n ff  which constitute the damage pattern. In optimal conditions two frequency shifts are sufficient to identify location and severity [23], but an increased number of analyzed frequencies, we recommend   Herein the Minkowski distance of second order is uded as a Damage Index, that is is used to find the most similar vectors. The value j for which the objective function DIj achieves the lowest value is searched; it indicates the crack location j c . Briefly, the proposed damage assessment algorithm is detailed in the following steps: If the beam is already damaged in the initial state, the assessment algorithm shows the damage evolution from this stage on. S1 -frequency evaluation for the intact state is performed for the first n weak-axis bending vibration modes. It result a vector . The measurements are replaced by FEM simulations, the evaluated frequencies for the intact as well as for the three damage cases being presented in tables 2 to 4. We worked through steps the first 5 steps of the algorithm and found the RFS and the damage signatures  for the three analyzed cases. These are compared in tables 2 to 4 with the DLCs of the closest damage location indicators. Note that, in steps S6 to S8, just 51 crack locations are considered. The reduced number of DLIs assumed in the database increases the degree of difficulty for the identification process. If crack identification is performed successful, it is demonstrated that the damage signature attained from six vibration modes doubtless indicate the crack position.    Alternatively, the damage position can be found by identifying the index j for which the damage index in equation (14) get the lowest value(s). This process can be automated, no human intervention being necessary. If more DI values qualify as possible solutions, in order to eliminate misinterpretation, again the individual elements of the potential DLIs and the DS can be compared.
In both analysis strategies, the increase of DLIs populating the database leads to an improved precision in locating cracks. This is reflected in damage index values close to null, eliminating doubts.

Conclusion
The problem of crack assessment in multi-span beams was considered via a simple and robust vibration-based method. The proposed method is based on the effect that the energy loss in a slice due to damage has upon the magnitude of frequency change. For the intact beam, the energy distribution in various vibration modes was contrived, and a predictive behavioral model developed. This model is used to define DLIs, which are integrated in a database including any possible damage scenarios for the analyzed beam. Adaption of the model, allowing the use of other end supports, can by simply made by changing the boundary conditions.
The method proved effective in detecting cracks and high accuracy is achieved in estimating the crack position. However, the early observation and precise localization depend on the possibility to quantify slight frequency changes and on the density of DLIs involved in the analysis. The similarity estimation strategy plays also a crucial role.