Diagnostics of construction defects in a building by using time-frequency analysis/Pastato konstrukciju defektu diagnostika naudojant laiko-daznio analize.
Volkovas, V. ; Petkevicius, K. ; Eidukeviciute, M. 等
1. Introduction
In many countries, the sudden collapse of the buildings has started
to attract the attention of scientists. This research project could be a
prototype for examining the structures of degradation. In this research,
STATIMON (2010-2011) developed under the project [1, 2]. Similar
accidents in the world and in Lithuania in recent years have increased
the interest in the topic. Deformation structures, and research on this
subject, can be analyzed by different mathematical methods [3]. This is
a different mathematical analysis, important findings of the
investigation of causes of damage in buildings are important to detect.
A reliable evaluation of structural integrity becomes especially
important at design, manufacture and service stages in objects of
increased risk [1-3].
The analysis of standard structures and estimation of their
functionality for a resource period usually are performed by regulations
and norms, which are based on the huge theoretical and practical
experience [4-6]. These regulations and norms however cannot be easily
applied to the items of unique structure without additional detailed and
comprehensive analysis [7, 8]. This is why at present it is allocated to
work of development and improvement of structure strength prediction
methodologies and technologies. Such activity takes place in different
areas such as civil engineering, transportation, power industry and
others [9, 10].
The traditional method of processing signals from transducers has
been in the time domain using digital or analog methods. With the advent
of the efficient computation scheme of the Fast Fourier Transform (FFT),
processing in the frequency domain has become a practical reality. The
frequency domain not only provides a viable alternative to time domain
methods, but permits the solution of problems that involve frequency
dependent parameters [11].
Structural deterioration in buildings can be reduced during the
maintenance. Structural defects can be detected with the aid of a
monitoring system in buildings. Deterioration factors for developing
technological methods can be examined using the buildings [9-12]. This
study, in particular distortion that may occurring buildings, emphasizes
the mathematical analysis. In this study, the analysis of deformations
in buildings is presented as a method of using signal processing
techniques. Data obtained from experiments were analyzed by Short Time
Fourier Transform (STFT).
2. Static models of a building structure
Reliable prediction of static structural response is sensitive to
all structural and analytical parameters. Conclusions on stress-strain
states of structures can precisely match the reality, when structural
parameters, loads, damages, deformations and their causes are identified
correctly. The mathematical models of all processes taking place in a
real structure should be well prepared and properly applied to foreseen
structure monitoring procedures and measures. Geometrical form of
structural elements can be very complex and different. Its precise
reproduction in a numerical model sometimes can require very detailed
meshing. It significantly increases the solved task and extends duration
of their solution. Numerical models should be as simple as possible,
however those should accurately enough reflect the simulated object.
Preparation of it is difficult due to numerous parameters and settings
required to define. For instance, loads and material properties are
random values and their estimations should be defined by characteristics
of probability distributions. Consequently, structural integrity
analysis results are also random values; their probability distributions
characterize probability of accident risk.
In numerical solutions, just as in analytical, some
irregular's forms, which do not strong impact on results, better be
avoided and simplified as rods, panels, shells, and solids.
For the analysis of the simplified buildings a model of floor
structure was developed with included defects, the influence of which
was estimated by measuring parameters of a corresponding physical model:
deflections and strains occurring due to effect of static and dynamic
loads. The floor structure is built using panels and columns. Their
dimensions were selected according to assumed scale factors for suitable
measured values of deflection, strains and frequencies [1]. Dimension of
the flooring model are presented Fig. 1.
Floors and columns are made of construction steel with elasticity
modulus E = 200 GPa, Poisson ratio v = 0.3, density [rho] = 7800
kg/[m.sup.3]. The loading of the structure is realized with ceramic
bricks, which dimensions are 65 x 250 x 120 mm, and total weight of unit
brick 4.2 kg. A single layer of bricks results a regular pressure of 5.4
kPa.
[FIGURE 1 OMITTED]
[FIGURE 2 OMITTED]
3. FEM modeling of the building structure
The flooring structure stress and strain states were calculated
using finite element models created in Solid Works Simulation system.
The floors were meshed by triangle elements, whereas columns were
modeled using 3D framework elements and also triangles. The columns are
set perpendicularly to the base and the floors. Both types of meshes
show a good accordance of the results. The FE model of the structure is
shown in Fig. 3.
[FIGURE 3 OMITTED]
Some modeling results and the 5 lowest eigenfrequencies of the
flooring structure model without and with defects are presented in Fig.
4, Fig. 5 and Table 1 accordingly.
[FIGURE 4 OMITTED]
[FIGURE 5 OMITTED]
4. Mathematical methods
In terms of the mathematical methods, the Fourier transform based
approaches are considered as follows.
4.1. Power spectral density
A common approach for extracting the information about the
frequency features of a random signal is to transform the signal to the
frequency domain by computing the d (DFT). For a block of data of length
N samples, the transform at frequency m[DELTA]f is given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
where [increment of f] is the frequency resolution and [increment
of t] is the data-sampling interval. The auto-power spectral density
(APSD) of x(t) is estimated as
[S.sub.xx] (f) = [1/N] [[absolute value of X (m[DELTA]f)].sup.2], f
= m[DELTA]f (2)
The cross power spectral density (CPSD) between x(t) and y(t) is
similarly estimated. The statistical accuracy of the estimate in Eq. (2)
increases as the number of data points or the number of blocks of data
increases [13-15].
4.2. Short Time Fourier Transform and spectrogram
The Short Time Fourier Transform (STFT) introduced by Gabor 1946 is
useful in presenting the time localization of frequency components of
signals. The STFT spectrum is obtained by windowing the signal through a
fixed dimension window. The signal may be considered approximately
stationary in this window. The window dimension fixed both time and
frequency resolutions. To define the STFT, let us consider a signal x(t)
with an assumption that it is stationary when it is windowed through a
fixed dimension window g(t), centered at time location [tau]. The
Fourier transform of the windowed signal yields the STFT [13-17].
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
The equation maps the signal into a twodimensional function in the
time-frequency (t, f) plane. Time-frequency analysis identifies the time
at which various signal frequencies are present, usually by calculating
a spectrum at regular intervals of time. Examination of the time domain
for each of the signals clearly demonstrates their differences and their
time varying nature, however the spectral content of these signals
remains predominantly concealed.
The analysis depends on the chosen window g(t). Once the window
g(t) is chosen, the STFT resolution is fixed over the entire
time-frequency plane. In discrete case, it becomes
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
The magnitude squared of the STFT yields the spectrogram of the
function.
Spectrogram {x (t)} = [[absolute value of X ([tau], f)].sup.2]. (5)
5. Data collection system of the building structure
In this experiment, the sampling frequency is 4098 Hz. The analysis
of the data from measurement system was performed using the MATLAB. For
this purpose, considered measurement and data collection system is shown
by Fig. 6.
[FIGURE 6 OMITTED]
5.1. Analysis for the building structure data
In order to analyze the dynamics of the building construction
layers, the experimental structural system was set-up and excited with
modal hammer (Fig. 6). The results were measured using Bruel and Kjear
Pulse 3560. Experimental data: acceleration of different points of floor
(Fig. 2) with damage 1, damage 2 and undamaged floor, and velocity is as
well. All data is first of all, compared to each other.
In this sense, the time domain variations related to the building
structure case are shown by the following figures.
[FIGURE 7 OMITTED]
[FIGURE 8 OMITTED]
Fig. 7 and Fig. 8 shows the dynamics of acceleration at point 1 and
point 5 respectively after excitation using modal hammer. To distinguish
differences in the change of the accelerations are shown on the same
figure. Graphics lets examine the similarities.
6. Time-frequency analysis for the building structure data
In the structures deformation, acceleration, velocity and power
will be examined as a comparison of results of time frequency analysis.
[FIGURE 9 OMITTED]
Fig. 9 depicts the time-frequency spectrum of the acceleration at
point 1, when the floor has defect 1. Here high-amplitude frequencies
around 450 Hz are observed. This situation can be considered as the
fundamental frequency for this point. Other plots are examined in the
different frequency components of each graph are seen to be effective.
Fig. 10 presents the results of the time-frequency analysis of the
point 5, the floor is with defect 1.
The high-amplitude frequencies of the parameters at point 5 are
more dominant.
Time-frequency analysis of the floor without defect is presented in
Fig. 11. Here, there are no frequencies between 400-800 Hz or the
amplitudes are low. However, the analysis of defect in the intensive
frequency components between 400 Hz and 800 Hz are available.
[FIGURE 10 OMITTED]
[FIGURE 11 OMITTED]
The results of the analysis of the structures with defect and
without defect are compared in Fig. 12 where differences between those
areas are shown. In the marked regions, the amplitude of the frequency
cannot be read.
PSD analysis is examined in Fig. 13, it shows that here are two
harmonic components: the first frequency is 380 Hz, and the second 780
Hz near multiples of the frequency of 380 Hz.
[FIGURE 12 OMITTED]
[FIGURE 13 OMITTED]
[FIGURE 14 OMITTED]
Comparing the results of dynamical experiment and analysis of floor
construction the ratio of experimentally and analytically determined
lowest frequency relative difference is not higher than 5%. In
particular the experimental results are consistent with the results of
FEM analysis.
7. Conclusions
In this study, a new analysis method for the detection of damage to
buildings was investigated. The method is applied on real experimental
systems. First, on the experimental systems the sensors are placed on
the designated spots. These sensors are used to measure the pulse. After
excitation the measurements of the velocity, power, and acceleration of
the plate were made. Meaningful analysis of the data has been obtained.
In addition, spectral analysis of all data in the study was made.
Significant findings include the results of spectral analysis. This
method is very useful to diagnosis of the high-frequency regions. The
results of the analysis performed with the data obtained from the areas
with defect and without it allow making comparison of the results of
spectral methods. The analysis of the power spectrum densities was also
performed. These results can be achieved important diagnostically
features. Figure 14 shows the PSD for the given velocity to be
undamaged. Analysis of the PSD analysis of the damaged and undamaged
regions contains important findings of the useful for the determination
of building's structure state. The findings are evaluated; the
results have proved to be consistent and positive for the diagnostics of
building's state.
Received May 20, 2011
Accepted August 21, 2012
Acknowledgments
The presented work was funded by the grant (No. MIP-71/2010) from
the Research Council of Lithuania.
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V. Volkovas, Kaunas University of Technology, Technological Systems
Diagnostics Institute, Kaunas, Lithuania, E-mail:
[email protected]
K. Petkevicius, Kaunas University of Technology, Technological
Systems Diagnostics Institute, Lithuania, E-mail:
[email protected]
M. Eidukeviciute, Kaunas University of Technology, Technological
Systems Diagnostics Institute, Lithuania, E-mail:
[email protected]
T.C. Akinci, Kirklareli University, Engineering Faculty, Department
of Electrical & Electronics Engineering, Kirklareli, Turkey,
E-mail:
[email protected]
[cross.sup.ref] http://dx.doi.org/10.5755/j01.mech.18.4.2340
Table 1
Natural vibration frequencies of flooring
with defects, Hz
Vibration Floor Floor Floor Floor
mode without with with two with a
defects single defects defect in
defect in two the
supports center
1 71.698 71.007 70.125 68.158
2 139.02 136.85 134.54 138.91
3 201.62 200.16 198.15 200.15
4 319.07 315.53 312.32 318.17
5 324.01 322.35 319.49 320.09
