Damage Localization of RC Beams via Wavelet Analysis of Noise Contaminated Modal Curvatures

Journal of Soft Computing in Civil Engineering 5-3 (2021) 101-133
How to cite this article: Jahangir H, Hasani H, Esfahani MR. Damage localization of RC beams via wavelet analysis of noise
contaminated modal curvatures. J Soft Comput Civ Eng 2021;5(3):101–133. https://doi.org/10.22115/scce.2021.292279.1340.
2588-2872/ © 2021 The Authors. Published by Pouyan Press.
This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
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Journal of Soft Computing in Civil Engineering
Journal homepage: www.jsoftcivil.com
Damage Localization of RC Beams via Wavelet Analysis of Noise
Contaminated Modal Curvatures
H. Jahangir1
, H. Hasani2
, M.R. Esfahani3*
1. Assistant Professor, Department of Civil Engineering, University of Birjand, Birjand, Iran
2. M.Sc. Student, Department of Civil Engineering, University of Birjand, Birjand, Iran
3. Professor, Department of Civil Engineering, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad,
Iran
Corresponding author: esfahani@um.ac.ir
https://doi.org/10.22115/SCCE.2021.292279.1340
ARTICLE INFO ABSTRACT
Article history:
Received: 26 June 2021
Revised: 20 September 2021
Accepted: 29 October 2021
In this paper, the location of single, double, and triple
damage scenarios in reinforced concrete (RC) beams are
assessed using the Wavelet transform coefficient. To achieve
this goal, the numerical models of RC concrete beams were
conducted based on the experimental specimens. The mode
shapes and corresponding modal curvatures of the RC beam
models in damaged and undamaged status were considered
as input signals in Wavelet transform. By considering the
Wavelet coefficient as damage index, Daubechies,
Biorthogonal, and Reverse Biorthogonal Wavelet families
were compared to select the most proper one to identify
damage locations. Moreover, various sampling distances and
their influence on the damage index were studied. In order to
simulate the practical situations, two kinds of noises were
added to modal data and then denoised by Wavelet analysis
to check the proposed damage index in noisy conditions. The
results revealed that among the wavelet families, rbio2.4 and
rbio2.2 outperform others in detecting damage locations
using mode shapes and modal curvatures, respectively. As
expected, the sensitivity of modal curvatures to different
damage scenarios is more the mode shapes. By increasing
sampling distances from 25 mm to 100 mm, the accuracy of
the proposed damage index reduces. In order to eliminate
boundary effects, it is necessary to use windowing
techniques. Applying Wavelet denoising methods on noise-
contaminated modal curvatures leads to proper damage
localization in both types of noises.
Keywords:
Damage location;
Wavelet transform;
Noisy contaminated condition;
Modal curvature;
RC beams.

102 H. Jahangir et al./ Journal of Soft Computing in Civil Engineering 5-3 (2021) 101-133
1. Introduction
Reinforced concrete (RC) structures [1–3] and consequently their elements such as RC columns,
slabs and beam are classified as important in maintenance point of view [3,4]. Therefore,
detection of initial cracks and damages in these structural elements cause future maintenance and
repair costs [5,6]. Early structural damage detection is one of the first steps in structural
rehabilitation [7–11] and strengthening [12–15]. Among different damage detection techniques,
vibrational based damage detection is a regular and prominent one which consists of detecting
how changes in the structure, such as stiffness loss, cause changes in its dynamics characteristics,
such as natural frequencies, mode shapes, and modal damping, named modal data [16–18]. The
literature recommends a variety of approaches for identifying structural damage using modal
data [19–22]. In his analysis of damage detection methods, Salawu [23] found that frequency
changes were effective in most circumstances, although mode shapes were also necessary. It has
been shown by Pandey et al. [24] that the absolute change in mode shape curving can be used as
an effective damage indicator since it is much more sensitive to damage than mode shapes.
According to Dawari and Vesmawala [25], the method of modal curvature (second derivative of
mode shape) and the modal flexibility method were used to detect cracking in reinforced
concrete beams. In particular, mode shapes are often used to detect damages, since they contain
local data about the damage and are highly resistant to environmental influences [26,27]. But,
noise can easily contaminate damage induced uniqueness in mode shapes, and it only gets shown
at unreasonably high damage levels [28]. Elimination of noise from mode shapes needs effective
soft computing methods [29–32] or signal processing techniques such as wavelet transforms. The
simultaneous ability of reducing noises and detecting minor singularities in mode shapes, caused
many researcher to utilize different Wavelet mothers such as Mexican hat [33]. Gabor [34], Haar
[35] and Debauchies [36] in structural damage detection field. Zhu et al. [37] utilized Wavelet
transform in order to identify damages in functionally graded beams. An index calculating the
location of damage is calculated based on the maximum wavelet coefficients in the scale field;
this index is known as the damage index. As described in Table 1, several researchers used
wavelets as damage detection techniques and consider modal data of damaged and undamaged
structure as input signals.
Table 1
The tabulated damage detection related studies.
Reference Year Type of studied structure Method
[38] 2012 Steel Beams Wavelet / Frequencies
[39] 2013 Steel Plates Wavelet / Deflections
[40] 2014 General Analytical Beam Wavelet Finite Element / Support Vector Regression
[41] 2016 General Analytical Beam Complex Wavelet / Modal Curvature
[42] 2021 Concrete Beam Wavelet / Time History Domain
[43] 2021 Concrete Beam Wavelet / Vibrational Based Responses
[44] 2021 Pre-stressed Concrete Slab Contourlet / Mode Shapes

H. Jahangir et al./ Journal of Soft Computing in Civil Engineering 5-3 (2021) 101-133 103
2. Research significance
Reviewing previous studies in wavelet based damage detection field reveals that, although there
are various kinds of mother wavelets with different properties, up to now, none of them is
introduces as the most proper and unique one in detecting structural damages. On the other hand,
RC structural elements such as RC beams play an important role in construction industry and
their failure cause catastrophic financial and life detriments. Moreover, just limited research
works investigate the influence of noises in modal based damage detection methods. Therefore,
in this paper, an experimental RC beam reported in previous study of one of the authors of this
paper is considered as the basis of conducting numerical finite element model. Then, single,
double and triple damage scenarios were created by reducing the stiffness in some specific areas
of the numerical model. The mode shape and modal curvatures of the damaged and undamaged
models were introduced to wavelet transform as input signals and the resulted wavelet coefficient
were utilized to identify damage scenarios. After comparison of different mother wavelets, the
most suitable wavelet for detection of locations of the damage scenarios is selected. Moreover,
the effect of sampling distances on the results of wavelet transform is investigated. After
selection of suitable wavelet and considering the degrees of freedom with the distance of 100
mm, the hypothesis testing method was used for locating the damage scenarios. In addition, in
order to have more consistency with the data and environmental conditions, which happen in
practice, two kind of noises were added to mode shapes of numerical specimens. Then the noise
was eliminated from the modal data by adopting the deniosing techniques using wavelets; and
ability of the index resulted from hypothesis testing was tested for locating the damages in noisy
data. As defined in Ibrahim [45], the soft computing techniques are computational methods deal
with approximate models and gives solutions to complex real-life problems, which is completely
related to the core goal of the paper in deniosing process, as a complex problem by
approximately dividing the high and low frequency parts of the signals, which could not be done
without using computers.
3. Method
3.1. Wavelet transform
In order to conduct signal processing, it is possible to use wavelets, as a kind of transforms, to
generate variable size windows. Such transformation has been enhanced with regard to local
changes by focusing on short time intervals for high frequency components and focusing on
lengthy periods for low-frequency components [46]. Instead of using frequency, the wavelets
transform uses an inverse proportional scale. As a result of wavelet transformation, an input
collection of functions is divided into waves, each associated with a wavelet called Mother
Wavelet, 𝜓(𝑥) [47]. The function 𝜓(𝑡, 𝑎, 𝑏) is the scaled and translated mother wavelet in
wavelet transformation:
1
( , , )
t b
t s b
s
s
 

 
  
 
(1)

Within Eq. (1), s and b represent scale and translation, respectively. Parameterizing the wavelet
transformation is scaling and translating its coefficients. The scaling of a wavelet refers to its
compression or stretching. However, translating a wavelet means that its start has been delayed
[47].
Continuous wavelet transform (CWT) means the whole amount of the signal multiplied by the
wavelet function scaled and translated 𝜓(𝑡, 𝑎, 𝑏):
( , ) ( ) ( , , )
CWT s b f t t s b dt



 
 (2)
The wavelet coefficients of the scale and position functions are provided by Eq. (2). The
coefficients illustrate how the wavelet function relates to the original signal and how similar it is.
As the coefficients increase, the wavelet's form and result will become more similar. Coefficients
of calculation generate a large amount of data at any scale. Using dyadic scales and positions, the
analysis data is reduced by making use of the power of two (known as dyadic scales and
positions). This kind of analysis is known as Discrete Wavelet Transform (DWT). As a result of
Mallat's study [46], filter analysis is called fast transformation wavelets today. The wavelet
coefficient could be the output of this toolbox in which the input is the original signal. The
wavelet function is defined as follows during this transformation:
/2
,
1 2
( ) ( ) 2 (2 )
2
2
j
j j
j k j
j
t k
t t k
  
 

   (3)
Eq. (3) gives j as the decomposition level, t as the time, and 2j as the scale. According to Eq. (4),
the discrete wavelet transformation coefficients are calculated by multiplying the f(t) signal by
the translated and scaled wavelet 𝜓𝑗,𝑘(𝑡):
,
( , ) ( ) ( )
j k
DWT j k f t t dt



 
 (4)
During the discrete wavelet transform, the original signal passes through high-pass and low-pass
filters. Filters are time-independent linear operators. Using a low-pass filter, bumps in a signal
are smoothed out. With a high-pass filter, bumps in the signal are displayed and smooth regions
are eliminated or reduced [19]. Approximation coefficients are considered high-scale low-
frequency components of a signal and high-scale components are referred to as detail
coefficients. Using the wavelet, the original signal can be filtered in order to identify such
components. Decomposition of the approximation coefficients in sequence might continue the
process. As a result of this method, a large number of components are separated into the original
signal, called the decomposition tree. The decomposing wavelet tree has vital data. This tree
provides access to the signal approximation and detail coefficients at each level. The input signal
used to identify damage consists of abrupt and high frequency damage to a building.
The discrete wavelet transform (DWT) uses both the wavelet function ψ(t) and the scaling
function ∅(x), as opposed to the continuous wavelet transform (CWT) method that only employs
the wavelet function. The low-pass filter is associated with wavelet functions, and the high-pass

filter with scaling functions. The scaling feature works in a similar way to wavelet. In fact,
wavelet coefficients are detailed, and scaling coefficients produce an approximation of the
original signal. Among the wavelet functions, only the orthogonal wavelet has a scaling function
[48].
There are multiple families of wavelets that have been introduced for signal processing and they
may be categorized based on their various properties [46]. Among the wavelet families Gaussian,
Mexican, Shannon, and Morlet, there are no wavelet properties specific to them 𝜓, Their
squatting function is not present ∅, They are therefore incapable of converting discrete wavelets,
and therefore cannot be rebuilt using discretion. Wavelet analysis consists of transforming a
constant wavelet. In summary, Haar, Daubechies, and Coiflet waves do not exhibit a strong
relationship 𝜓. Due to the presence of the scaling function ∅, the discrete wavelet function can
also be used within these wavelets. They are slightly regular and not symmetrical. Another type
of wavelets that are symmetri quely and precisely reconstructable of wavelets is the Biorthogonal
wavelets and the opposite way around, the Reverse Biorthogonal wavelets. In addition, they
feature two types of wavelet and scaling functions.
Wavelets and signal properties determine which wavelet is best suited to process the signal of a
certain phenomenon. In most cases, it can be difficult to know for sure which wavelet is
appropriate for use and that was determined by tried and error. In this study, many mother waves
are used to identify damage using the family of Daubechies, Biorthogonals, and Reverse
Biorthogonals wavelets.
3.2. Experimental specimens
As an experimental reference to assess the ability of proposed damage detection method in
practical applications, as presented in Fig. 1, a reinforced concrete (RC) beams named B1 with
150 mm width, 200 mm height and 2200 mm length which vibrational modal tests are conducted
on them by Baghiee et al. [49] were selected. Table 1 describes the geometrical and mechanical
properties of the RC beam [49]. As presented in Fig. 2, In order to obtain vibrational data, modal
tests were conducted in a suspended position to reduce the effect of supports and the noise
resulted by random vibrations of the floor. In these tests, the degrees of freedom were considered
on central axis on the upper face of the sample with a 100 mm distance from each other [49].
Fig. 1. The cross-section of the experimental specimens (B1) [49].
Table 1
The geometrical and mechanical properties of experimental specimens [49].
Specimen f'cu (MPa) A’s (mm2
) A’s (mm2
) stirrup distance (mm) d (mm) d' (mm)
B1 20 226 226 100 175 25
B1 specimen B2 specimen

Fig. 2. Suspended beams in vibration test.
In conducted modal tests, the frequency response functions (FRFs) were resulted by measuring
the excitation force and responses of specimens in time field and their projection in frequency
field. In order to determine the fundamental modal parameters including natural frequencies and
mode shapes, the mathematical curve fitting techniques were analyzed using STAR Program.
Fig. 3 shows the mode shapes of B1 specimen. The experimental mode shapes were normalized
relative to their greatest vibration amplitude [49].
Fig. 3. Mode shapes of the experimental specimen B1: a) mode 1; b) mode 2 and c) mode 3 [49].
3.3. Verification of numerical model
As the damage scenarios are studied in numerical models, the 2D numerical model of the
undamaged experimental B1 specimen were developed by ABAQUS [50] finite element
software. This model includes 528 plane tension 2D elements (CPE4R) for concrete and 308
truss 2D elements (T2D2) for steel bars. Fig. 4 shows the numerical model of the concrete beam
specimen. The degrees of freedom were considered with distances of 100 mm from each other to
be in accordance with experimental specimens. The properties of concrete and steel used in this
model are provided in Table 2.
Fig. 4. Numerical model of B1 specimen by ABASQUS software.
Table 2
Mechanical properties of the materials in the numerical model.
Material Elasticity Coefficient (GPa) Density (kg/m3
) Poisson Ratio fcu (Mpa) fy (Mpa)
Concrete 17 2400 0.17 20 ---
Steel 200 7850 0.3 --- 400
1 5 10 15 20 23
-1
0
1
degree of freedom
1 5 10 15 20 23
-1
0
1
degree of freedom
1 5 10 15 20 23
-1
0
1
degree of freedom

In this numerical analysis the boundary conditions were considered same as the test of suspended
status and frequencies of numerical analysis and experimental method were so close together;
which can be seen in Table 3. Fig. 5 show the comparison of experimental and numerical mode
shapes. According to Fig. 5, the experimental data and numerical data are in good agreement
with each other.
Table 3
Experimental and numerical natural frequencies of B1 specimen.
Mode Number
Special Frequencies (Hz)
FEM Analyze Experimental Results
First Mode 114.28 114.28
Second Mode 298.42 304.89
Third Mode 560.24 563.19
Fig. 5. Mode shapes of numerical and experimental of B1 specimen: a) First Mode; b) Second Mode and
c) Third Mode.
3.4. Damage scenarios
In this paper, some damage scenarios were developed in the numerical model to test abilities of
the damage detecting methods. These single, double and triple damage scenarios were as
reduction in elastic modulus of concrete in some parts of the beam. The location and percentage
of concrete elastic modulus reduction for these damages are represented in Table 4 and Fig. 6.
(a)
(b)
(c)

Table 4
Properties of damage scenarios in numerical models.
Damages Scenario Types of Damage Percentage Reduction of Concrete Damage Location (x) (mm)
D1 Single Damage 40 1100
D2 Double Damage 20 600 and 1600
D3 Triple Damage 20 600, 1200 and 1600
Fig. 6. Locations of different damage scenarios: L1, L2 and L3.
3.5. Best mother wavelet and sampling distance
Before evaluating the ability of wavelet based damage detecting methods and using various
damage indices that can be defined for them, it is required to choose best mother wavelet for
analyzing the received signals from the structure. The quality of selecting the optimized and
suitable wavelet for signal processing depends on the properties of the wavelet and the signal. In
most cases one wavelet could not be used with absolute certainty and this procedure should be
done through trial and error. In this paper, mother wavelets were chosen by considering the first
mode shape of specimen in single damage scenario state as the input signal. Also the degrees of
freedom were set with distances of 25 mm, 50 mm, 75 mm and 100 mm from each other in order
to evaluate the effectiveness of sampling distances on the proposed method.
The signals of vibrational mode shape and its corresponding modal curvature (second derivation
of mode shape) obtained by using central difference approximation were calculated after curve
fitting with spline functions. Fig. 7 demonstrates the first mode shape and corresponding modal
curvature of specimen in single damage scenario status (D1).
Fig. 7. a) The first mode shape and b) modal curvature of the specimen in single damage scenario (D1).
x
a
b
c
x
x
a) b

As it could be seen in Fig. 7, there is no significant change in mode shape in the location of
damage. This change is negligible in the case of modal strain too. Therefore, identification of
damage location demands more process of modal data via best wavelet transform and choosing
optimal sampling distance.
Initial investigations show that among different wavelets, bior2.4 and bior3.5 from Biorthogonal
wavelet family, db2 and db3 from Daubechies family and rbio2.2, rbio2.4 and rbio3.3 from
Reverse Biorthogonal family are more sensitive to location of damages compared to other
wavelets. Figs. 8, 9 and 10 show wavelet functions (ψ(x)) and scale of (ϕ(x)) of these families.
Fig. 8. Biorthogonal wavelet family: a) bior2.4 and b) bior3.5 [51].
Fig. 9. Daubechies wavelet family: a) db2 and b) db3 [51].
(a)
ϕ(x) ψ(x)
ϕ(x) ψ(x)
(b)
(a)
ϕ(x) ψ(x)
(b)
ϕ(x) ψ(x)

Fig. 10. Reverse Biorthogonal wavelet family: a) rbio 2.2; b) rbio 2.4 and c) rbior 3.3 [51].
Studies on the results of these wavelets for single damage scenario show that details signal
obtained from wavelet transform of the first mode shape and modal curvature by use of bior2.4
and db2 have similar behavior to that of rbio2.2. This behavior similarity in different sampling
distances can be seen in Figs. 11 and 12, respectively.
Fig. 11. Details signal of the first mode shape using bior2.4, db2 and rbio2.2 wavelets at sampling
distance of: a) 25 mm; b) 50 mm; c) 75 mm and d) 100 mm.
ϕ(x) ψ(x)
(a)
ψ(x)
ϕ(x)
(b)
0 1 2 3 4 5 6 7
-0.5
0
0.5
1
x
Scaling
function
0 1 2 3 4 5 6
-2
0
2
Reconstruction scaling function phi
0 1 2 3 4 5 6 7
-1
0
1
x
Wavelet
function
0 1 2 3 4 5 6
-4
-2
0
2
4
Reconstruction wavelet function psi
0 1 2 3 4 5 6 7
-0.5
0
0.5
0 1 2 3 4 5 6 7
-0.5
0
0.5
Reconstruction low-pass filter
0 1 2 3 4 5 6 7
-0.5
0
0.5
0 1 2 3 4 5 6 7
-0.5
0
0.5
Reconstruction high-pass filter
ϕ(x) ψ(x)
(c)
(a) (b)
(d)

Fig. 12. Details signal of the first modal curvature using bior2.4, db2 and rbio2.2 wavelets at sampling
As it shown in Fig. 11, the signals of details calculated using rbio2.2 mother wavelet has a
greater amplitude than db2 and bior2.4. In addition, maximum range of these signals estimate
damage location to some extent and increasing the sampling distance will reduce the ability of
the method. Fig. 12 shows the detail signals resulted from rbio2.2 mother wavelet on the first
modal curvature also have a greater range than bior2.4 and db2. Moreover, location of damage
has appeared better by the detail signals obtained from modal curvature. The error developed in
sampling distance of 75 mm can be assumed to be caused by symmetry of some mother wavelets
and asymmetry of these distances. Comparing detail signals obtained from application of bior2.4,
db2 and rbio2.2 mother wavelets on the first mode shape and modal curvature, it can be
concluded that due to similar behavior and greater amplitudes of wavelet rbio2.2 than the two
other wavelets, rbio2.2 wavelet can be utilized instead of both bior2.4 and db2.
The studies indicate that detail signals resulted from wavelets bior3.5 and db3 have a similar
behavior to that of wavelet rbio3.3. Figs. 13 and 14 show this similar behavior for the first mode
shape and modal curvature in the four sampling distances, respectively. Figs. 13 and 14 show
that for both of first mode shape and modal curvature, the detail signals calculated using mother
wavelet rbio3.3 have greater amplitudes compared to bior3.5 and db3. In different sampling
distances, locating damage scenario by using modal curvature is more accurate in comparison
with mode shape. Hence, rbio3.3wavelet can be employed as a replace for the two other
wavelets.
(c) (d)
(a) (b)

Fig. 13. Details signal of the first mode shape using bior3.5, db3 and rbio3.3 wavelets at sampling
Fig. 14. Details signal of the first modal curvature using bior3.5, db3 and rbio3.3 wavelets at sampling
(b)
(a)
(c) (d)
(c) (d)
(a) (b)

After choosing rbio2.2 and rbio3.3 as the replace for the similar wavelets, these two mother
wavelets are compared to rbio2.4. Figs. 15 and 16 show the results of evaluation of these three
cognate wavelets on the first mode shape and modal curvature, respectively.
Fig. 15. Details signal of the first mode shape using rbio2.4, rbio3.3 and rbio2.2 wavelets at sampling
Fig. 16. Details signal of the first modal curvature using rbio2.4, rbio3.3 and rbio2.2 wavelets at sampling
(a) (b)
(c) (d)

As it could be found in Fig. 15, for the first mode shape, details signal calculated by using of
rbio2.2 mother wavelet has a greater amplitude than rbio2.4 and rbio3.3; but the locating
accuracy of this wavelet is low. Notwithstanding smaller range, rbio2.4 leads to a better result in
damage locating compared to the two other. According to Fig. 16, for the first modal curvature,
rbio2.2 has a greater amplitude and estimates location of damage better than the two others do.
Investigation of the details signals shows that, generally, modal strains are more sensitive to
damage compared to vibration mode shapes. Moreover, in all cases, the signals calculated by use
of mother wavelet rbio2.2 have greater amplitudes. Wavelets rbio2.4 and rbio2.2 give better
results in locating damage for the first mode shape and for the first modal curvature, respectively.
Regarding the degrees of freedom and sampling distances, as expected, increasing these
distances leads to reduction of ability of the method for identification of accurate location of
damage. Due to asymmetry of basic functions of some of the wavelets used in this section, the
signals resulted from these wavelets in asymmetric sampling distance of 75 mm are unable to
locate damages correctly.
4. Identifying damage locations
4.1. Proposed damage index
This section is focused on evaluation of application of the best mother wavelets on data obtained
from numerical models. Since there are few vibration modes which measured practically, the
first three mode shapes of the specimens were examined. Mode shapes were processed by central
difference method and curve fitting by spline functions. The mode shapes and corresponding
modal curvatures for numerical modes caused by the three damage scenarios are represented in
Fig. 17. The degrees of freedom were considered with 100 mm distances from each other in
order to be in agreement with experimental test.
According to Fig. 17, the affected mode shapes and corresponding modal curvatures by the
damage scenario do not show considerable changes. In order to detect damage scenarios, as
illustrated in in Fig. 18 and 19, respectively, the wavelet transform of mode shapes and
corresponding modal curvatures were conducted by choosing rbio2.4 as the mother wavelet for
mode shapes and rbio2.2 for modal curvatures. The vertical lines indicate damage locations.
As it can be seen in Fig. 18, as the same to many other damage detection methods, the proposed
method has some problems in damage localization at the supports. For single damage scenario,
the second mode is not able to detect damage location but the first and the third mode are able to
detect it. The first and the second modes detect the location of double and triple artificial
damages correctly but the third one is unable to do so. According to Fig. 19, similar to details
signal of mode shapes, the wavelet coefficients obtained from modal curvatures do not have a
correct estimation at supports. The second modal strain does not detect the location of single
damage scenario correctly; also, the third modal curvature does not detect the location of double
and triple artificial damage correctly.

Fig. 17. First (a and b), second (c and d) and third (e and f) mode shapes and corresponding modal
curvatures in the three damage scenarios.
Fig. 18. Details signal of mode shapes using wavelet rbio2.4 for different damage scenarios: a) Single; b)
Double and c) Triple.
(a)
(b)
(c)
(d)
(e)
(f)
(a)
(c)
(b)

Fig. 19. Details signal of modal curvatures using wavelet rbio2.2 for different damage scenarios: a)
Single; b) Double and c) Triple.
Comparing Figs. 18 and 19, it can be concluded that generally the details signal of modal
curvature have greater amplitudes than mode shapes; which indicates their more sensitivity to
damage. Moreover, the details signal obtained from mode shapes and their corresponding modal
curvatures in different modes have different amplitudes. To specify location of damage scenarios
better, and utilize the ability of all the modes, more processes is needed. Some researchers
conduct these processes using statistical methods. One of these methods is hypothesis testing;
which will be used in this paper. In hypothesis testing method the details signal (Dj) as a damage
indicator is assumed a random variables which are normalized and resulted standard variables of
zj as follows [52]:
D
D
j
j
D
z



 (5)
Where zj is the normalized damage index of jth component, µD is the mean and σD is standard
deviation of Djs. Figs. 20 and 21 show the result of employing this index for mode shapes and
their corresponding modal curvatures in three damage scenarios.
(a)
(c)
(b)

Fig. 20. Damage index (zj) applied on mode shapes using rbio2.4 in different damage scenarios: a) Single;
b) Double and c) Triple.
Fig. 21. Damage index (zj) applied on modal curvatures using rbio2.2 in different damage scenarios: a)
(a)
(c)
(b)
(a)
(c)
(b)

As it can be seen in Figs. 20 and 21, the wavelet coefficients are normalized by using the zj
index; and it causes the decision making to be easier to determine sensitivity of different modes.
For single damage scenario, all mode shapes and first and third modal curvatures could detect the
location of damage successfully, and the first modal data are more sensitive. The second modal
data are unable to detect single damage scenario. The first and the second modes, with almost the
same sensitivity, have the ability to detect the locations of double and triple damage scenario but
the third one is unable. The triple damage scenario could be detected by suing of the first and the
second modes.
In order to represent damage location better, a threshold can be considered for damage indicator
of zj. This threshold is considered by assuming normal distribution of Dj variable and definition
of zn variable as the starting level of damage. If zj < zη, no damage is occurred, otherwise the
component is damaged. With this definition, the values greater than the threshold remain that are
considered as the estimated locations of damage and the smaller values are zeros. For this reason,
the modified damage index (zrj) has values greater than zero. Fig. 22 shows a schematic of
applying threshold on normalized damage index (zj).
Fig. 22. The schematic of threshold application of zj normalized damage index.
In this paper, three damage scenarios were located by assuming threshold of zη equal to 1.6. Figs.
23 and 24 demonstrate the result of the detection by using of modified damage indicator of zrj for
mode shapes and modal curvatures.
As we see in Figs. 23 and 24, modified damage index of zrj shows the estimated damage
locations are better after applying threshold; so it is more suitable for damage detection. By
investigating modified damage index of zrj, it can be concluded that in order to detect damage
location in concrete beam specimens using the proposed method, a collection of various mode
responses should be considered; and no decision can be made based on one specific mode. In
addition, in order to address the problem of the method at support locations, application of
windowing on the indicator will be useful. In this paper, with respect to the type of details signal,
Turkey window is used for removing their bad results in support locations. As shown in Fig. 25,
this window has a conical-cosine shape and is defined based on the length of the vector and the
ratio of conical part to constant part as below [53]:

Fig. 23. Modified damage index of zrj for mode shapes in different damage scenarios: a) Single; b)
Double and c) Triple.
Fig. 24. Modified damage index of zrj for modal curvature shapes in different damage scenarios: a)
(a)
(c)
(b)
























1
2
1
])}
2
/
1
[
2
cos(
1
{
2
1
2
1
2
1
2
0
])}
2
/
[
2
cos(
1
{
2
1
)
(
x
x
x
x
x
x
w










(6)
Where x is the vector and α is the ratio of the conical part to the constant part; and its value
varies 0 to 1 (i.e. 0 ≤ α ≤ 1). The form of this window is such that it can be coordinated with
vector length by considering different αs. In this research, for removing the support effect in
modified indicator of zrj, biquadratic of the window with =0.2 was applied. Fig. 26 shows this
window.
Fig. 25. Turkey window [53]
Fig. 26. The applied window
After applying the window on modified damage index of zrj to find damaged locations in
numerical model, the results of the index in different modes should be overlapped. Fig. 27 shows
the outcome of overlapping, for each of damage scenarios in numerical model.
α=1
α=0.75
α=0.5
α=0.25

Fig. 27. Overlapping of modified index of zrj in: a) mode shapes; b) modal curvatures
As it seen in Fig. 27, the suggested method detects the location of all damage scenarios for their
corresponding mode shapes and modal strains correctly. So, overlapping of modified damage
indicator of zrj can be used after elimination of support effect for detection of damage locations
in structures.
4.2. Noise contaminated modal data
In most experimental tests, there are many factors involved such as environmental conditions and
noise. Sometimes, the effect of environmental conditions on modal data is more than damage
effect and it led to a wrong decision. Noise affects the frequency response functions and changes
them. The changes in these functions cause the mode shapes to be rough; so it causes data
processing to be problematic. There are so many methods proposed for noise reduction up to
now. Noise can be reduced in signals by modifying testing methods and applying a suitable
isolator. Signal processing knowledge and modern data collecting devices help the researchers to
be closer to this goal. But still, the remaining noise should be removd from collected data to
calculate modal quantities. In this paper, as one of the signal processing methods, wavelet
transforms are used for elimination of noise from modal data.
Donoho and Johnstone [54] showed that small coefficients existed in signals obtained from
wavelet transform includes noise and have a little information of signal. Therefore, the noise-free
version of signal can be achieved by ignoring these coefficients and by reconstruction of other
coefficients of the wavelet. So, for noise elimination, firstly noisy data are considered as input
signal for wavelet transform. Then, the details signal of the data is calculated by selected mother
wavelet for denoising process. In the next stage, the smaller coefficients are eliminated by
applying appropriate threshold; and the main signal is reconstructed by use of remaining wavelet
coefficients. The resulted signal will be noise-free. Threshold application methods for noise
isolation can be divided into two major category of hard and soft. In hard threshold application
method the coefficients of the wavelet that their absolute values are less than a specific value are
considered zero and the remaining coefficients remains with no change. How the hard threshold
is applied is shown in Eq. (7) [54].








x
x
x
y
0
(7)
(b)
(a)

Where σ is a specific value for threshold and x represents small coefficient. Method of soft
threshold application is the developed form of hard method. In this method, the coefficients that
are smaller than threshold are considered zero and then the remaining coefficients will be
concentrated and shrank toward zero. The method is expressed as Eq. (8) [54].










x
x
x
x
sign
y
0
)
).(
(
(8)
The method of soft threshold application is based on the fact that any wavelet coefficient
includes a part of main signal and noise; and by application of this method, noises will be
isolated from coefficients and the main signal will remain. Donoho [55] defined his general law
for threshold as below:
)
log(
2 n
s

 (9)
Where n is signal length and s is standard deviation of the estimated noise. Noise can be
eliminated from the signal by considering the threshold and using one of the isolation methods;
then the noise-free signal can be used for the next processes [55].
In this paper, in order to evaluate the ability of the wavelets for damage detection using noisy
data, two types of noise called High-frequency Noise and White Gaussian Noise were added in
the form of vibration modes resulted from B1 specimen in three damage scenarios. The
following equation was used for simulation of the noise type I (High-frequency Noise):
)
95
sin(
01
.
0 t
Data
Data free
Noise
Noise 

  (10)
According to this equation, frequency of the noise has been considered 95 Hz. The three first
mode shapes with this noise are shown in Fig. 28.
The effect of noise is recognizable in all the mode shapes showed in Fig. 28. It shows that for a
small damage, the irregularities that have been developed in data due to damage may be
confused with the irregularities caused by noise effect. Also for the relatively large damages,
these irregularities may cause problem in detection of the correct location of damage.
In this paper, the soft isolation method is used for reduction of the effect of these noises. The
threshold for noise isolation has been considered σ = 0.00745; and in order to apply the soft
method, the symmetric wavelet with 8 vanishing moments (sym8) has been used. Fig. 29 shows
the outcome of noise elimination from the three vibration mode shapes.

Fig. 28. The mode shapes contaminated by the noise type I: a) first mode; b) second mode and c) third
mode.
Fig. 29. The primary noise-free mode shapes for noise type I: a) first mode; b) second mode and c) third
mode.
As seen in Fig. 29, after presentation of the mode shapes having noise type I by soft isolation
method, the effect of noises is reduced fairly.
(a)
(b)
(c)

For simulation of noise type II, random number generator of MATLAB was used to develop
White Gaussian Noise. The random numbers were between 0 and 1. The noise is obtained by Eq.
(11) as following:
free
Noise
Noise Data
rand
e
Data 





 ))
1
2
(
1
( (11)
In this equation, e represents the amount of noise. As shown in Fig. 30, the mode shapes having
this type of noise are demonstrated. Fig. 32 shows that the noise type II has an unpredictable
irregular behavior; also it is more similar to the data occurred in real world compared to noise
type I. The outcome of soft isolation of this noise from vibration mode shapes is shown in Fig.
31.
As seen in Fig. 31, isolation of noise from mode shapes having noise type II eliminated the effect
of this noise as much as possible but it was less successful than noise type I. After isolation of
noise from data, the ability of modified damage index zrj in detection of damage in noisy data is
evaluated. For this end, details signals were obtained for three damage scenarios by presenting
noise-removed mode shapes to rbio2.4 wavelet transform. Then, the damage scenarios were
detected by using of hypothesis testing method and applying threshold of zn = 1.15 for noise type
I and zn = 1.6 for noise type II. Figs. 32 and 33 show the results of zrj index for type I and type II
noises, respectively.
Fig. 30. The mode shapes contaminated by noise type II: a) first mode; b) second mode and c) third mode.

Fig. 31. The primary noise-free mode shapes for noise type II: a) first mode; b) second mode and c) third
mode.
Fig. 32. zrj index for vibration mode shapes having noise type I in different damage scenarios: a) single;
b) double and c) triple.

Fig. 33. zrj indicator for vibration mode shapes having noise type II in different damage scenarios: a)
single; b) double and c) triple.
According to Fig. 32, notwithstanding initial noise in mode shapes, zrj damage index was
successful in detection of damage locations. The location of single damage scenario was
correctly detected by the first and third mode shapes; also the location of double and triple
damage scenarios were correctly detected by the first and third mode shapes. The results of Fig.
33, the denoising method is not successful on the mode shapes having noise type II and it is
unable to detect the correct location of damage in any of damage scenarios. The nature of both
artificial Type I and Type II noises were randomly added high frequency low domain signals,
which in most real field and also experimental condition, most of previously proposed damage
detection methods are unable to clarify and make distinction between if the presented fluctuation
in input signal are related to real damage condition or it is just originated from the environmental
noises. Therefore, determining the type of fluctuations and singularities in input signals (if it is
originated from environmental noise or real damage condition) is the most challenging issue in
current damage detection investigations. Therefore, in this paper, to overcome this regular
challenge, instead of using the mode shapes, modal curvatures were utilized as input signals. The
details signal of modal strain of the specimens were examined in order to improve the proposed
method and for comparing ability of noise isolation method in vibration mode shapes and their
modal strains as well. After isolation of noise from mode shapes, their modal curvatures were
obtained by central difference method. Details signal of strains was calculated by using of
rbio2.2 wavelet and zj coordinated indicator was achieved. Then, the damages were assessed by
applying a threshold of zn = 1.3 for noise type I and zn = 1.24 for noise type II. Fig. 34 and 35
show the results of assessment for modal curvatures having these two types of noise.
According to Fig. 34 for noise type I, modal curvatures could detect locations of damage
scenarios as well as mode shapes. Therefore, it can be concluded that for regular high frequency
noises the isolation method can have a good performance for both vibration modes and modal

curvatures. Fig. 35 shows that unlike the result of applying the proposed method on shape modes
having noise type II, which was not able to damage scenarios localization, applying this method
on modal curvatures can lead to a suitable estimation of damage scenarios locations. Hence, as
expected, modal curvatures are more sensitive to damages compared to vibration mode shapes;
and the can be used for damage detection in practice.
Fig. 34. zrj index for modal curvatures having noise type I in different damage scenarios: a) single; b)
double and c) triple.
Fig. 35. zrj index for modal curvatures having noise type II in different damage scenarios: a) single; b)
double and c) triple.

A notable point in Fig. 34 and 35 is extreme sensitivity of the method to supports and wrong
estimation of damage location at support points. Also, just a few mode shapes and modal
curvatures are able to detect damages. In order to reach a single method, overlapping of different
mode responses as well as windowing will be used for elimination of support effects. Figs. 36
and 37 show the results of this equalization for mode shapes and modal curvatures contaminated
with both noise types.
Fig. 36. Accumulation of zrj index for mode shapes contaminated with: a) noise type I and b) noise type II
Fig. 37. Accumulation of zrj index for modal curvatures contaminated with: a) noise type I and b) noise
type II.
Analyzing Figs. 36 and 37 shows that for noise type I, the damage index obtained from vibration
mode shapes and modal curvatures have detected damage location correctly. This success for
noise type II is only seen in damage index calculated by modal curvatures. Noises of
experimental modal data are irregular and are more similar to noise type II. Therefore, the
vibration mode shapes cannot be used for damage detection using real modal data in practice;
and it is necessary to calculate modal curvatures.
5. Conclusions
The goal of this paper is to propose a wavelet based damage index to identify damage locations
in RC beams. To achieve this goal, single, double and triple damage scenarios were applied in
numerical models of RC beams which were simulated based on an experimental specimen.

Considering mode shape and modal curvatures as input signals of Wavelet transform, a damage
index is proposed based on the resulted Wavelet coefficients. Different Wavelets mothers
including bior2.4 and bior3.5 from Biorthogonal wavelet family, db2 and db3 from Daubechies
family and rbio2.2, rbio2.4 and rbio3.3 from Reverse Biorthogonal family were compared to
select the most proper one. In addition, the influence of various sampling distances of 25 mm, 50
mm, 75 mm and 100 mm on damage localization accuracy were investigated. In addition, in
order to simulate the practical conditions, two kind of noises were added to modal data and
deniosed by wavelet denoising techniques, before introducing them as inputs to damage index. It
should be mentioned that the boundary effect and detecting damage locations near the boundaries
is the only limitation of the proposed damage detection method. To overcome this limitation, as
presented in modal tests details, the structural element would be hanged on in a suspended
condition to eliminate the effects of boundaries on obtained modal data. The following results
were obtained in this study:
 Investigation of different wavelets for localization of damage scenarios revealed that
rbio2.4 and rbio2.2 have better performance for wavelet analyzing of mode shapes and
modal curvatures, respectively. Moreover, the detail signals using rbio2.2 mother wavelet
has a greater amplitude and shows more sensitivity to presence of damages.
 Evaluation of details signals show that generally modal curvatures are more sensitive to
damage scenarios in comparison with mode shapes.
 As expected, increasing sampling distances causes a performance reduction in proposed
method in detection of accurate location of damage scenarios. In addition, due to
symmetry of basic functions of some of the wavelets used in this paper, the detail signals
obtained from these wavelets failed to detect damage location correctly in asymmetric
sampling distance of 75 mm.
 The results of application of the damage index obtained from hypothesis testing showed
that it cannot be decided based on one specific mode for detection of damage location in
concrete beam specimen. Moreover, the proposed method has a poor estimation of
damages at support places. For this reason, Turkey window and overlapping of results
were applied and the locations of damages were correctly revealed by introducing
modified damage index.
 Examining the noise contaminated modal data showed that the damage index obtained
from mode shapes and modal strains for noise type I was able to detect damage location
accurately. This success for noise type II is only seen in damage index calculated by
modal curvatures. Because of more similarity of real data to noise type II, modal
curvatures and wavelet rbio2.2 were proposed as the best input signals and most proper
mother wavelet.
Funding
This research received no external funding.

Conflicts of interest
The authors declare no conflict of interest.
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