Abstract
Interface debonding identification is critical for ensuring the safety of steelreinforced concrete structures. Laermeasured operating deflection shapes (ODSs) can be utilized to precisely designate the presence and location of interface debondings. However, using such denselysampled ODSs poses a challenge for overcoming intense noise interference. With this concern, this study proposes a noiserobust dynamics feature relying on ODSs, namely TeagerKaiser energy of multiresolution ODS, to identify interface debondings in steelreinforced concrete structures. Owing to the multiresolution analysis, this feature is capable of suppressing noise; and because of the damage sensitivity of the TeagerKaiser energy, the feature can intensify damage signatures. The TeagerKaiser energy of multiresolution ODS is applied to identify surface debondings on a steelreinforced concrete slab, whose ODSs are acquired using a scanning laser vibrometer. The experimental results show that the dynamics feature is of strong noise robustness and damage sensitivity, capable of designating the presence and location of the interface debondings under a noisy environment. Furthermore, damage identification using the proposed dynamics feature is a nonbaseline method, requiring no structural baseline information such as temperature, materials, geometry, and boundary conditions.
1. Introduction
Steelreinforced concrete structures are commonly used in civil engineering. Debonding damage between interfaces of steel and concrete can accumulate and develop, thus identification of such damage is critical for ensuring the safety of the structures [16]. However, identifying interface debondings in steelreinforced concrete structures is a challenge for traditional nondestructive testing methods as interface debondings can be barely judged from appearance of steels or concretes.
Structural damage identification relying on operating deflection shapes (ODSs) has attracted increasing attentions. An ODS is defined as the vibrational deflection of a structure subject to a harmonic excitation at an arbitrary loading frequency [7, 8]. With the aid of the noncontact laser measurement using a scanning laser vibrometer (SLV), denselysampled ODSs can be measured to precisely identify structural damage. Representative studies using SLVmeasured ODSs to identify damage in beamlike structures are as follows. Pai and Young [9] proposed a boundary effect detection method using ODSs measured by a SLV to detect small damage in beams, with the damage treated as an introduced boundary. Waldron et al. [10] utilized the SLV to measure a pair of ODSs of healthy and damaged states of a beam, and they used the difference between the pair of ODSs to indicate the presence and location of the damage. Xu et al. [11] proposed an approach for identifying interfacial debonding between dissimilar structural components by reconstructing the distribution of interfacial forces and canvassing local perturbance to the structural dynamic equilibrium. Asnaashari and Sinha [12] utilized a SLV to measure ODSs of a cracked beam corresponding to a natural frequency and a higher harmonic frequency caused by the crack, respectively. The ODSs' difference, defined as a residual ODS, was experimentally validated to be capable of identifying a crack in a beam. Representative studies are as follows. Cao et al. [13] proposed the wavelettransform curvature ODS for damage detection in beamlike structures. The experimental results show the method can suppress noise and intensify damage signatures.
Although SLVmeasured ODSs have been increasingly utilized for structural damage identification, intense noise interference is the most concern of the damage identification methods. To precisely locate damage, ODSs need to be densely sampled to produce sampling intervals small enough to match damage sizes; however, for most damage identification methods relying on ODSs, densely sampling can cause intense noise interference [1415]. Addressing this problem, this study proposes a noiserobust dynamics feature relying on ODSs, namely TeagerKaiser energy (TKE) of multiresolution ODS (MRODS), for interface debonding identification in steelreinforced concrete structures. First, a measured ODS is decomposed into the approximation and details by the WTbased multiresolution analysis (MRA). Then, by introducing the TKE operator, the TKE of MRODS is formulated based on the decomposed ODS. Damagecaused changes in the TKE of MRODS can be utilized to designate the presence and location of interface debondings.
The rest of this paper is organized as follows. Section 2 formulates the TKE of MRODS. Section 3 applies the TKE of MRODS to identify interface debondings in a steelreinforced concrete slab, whose ODSs are measured via noncontact measurement using a SLV. Section 4 presents the conclusions of this study.
2. TeagerKaiser energy of MRODS
2.1. MRODS
By the theory of the MRA [16, 17], an orthonormal, compactly supported wavelet basis of space ${\mathbf{L}}^{2}\left(R\right)$ of measureable, square integral functions is formed by dilating and translating a mother wavelet function $\psi \left(x\right)$:
where $R$ and $Z$ denote sets of real and integer numbers, respectively. $\psi \left(x\right)$ satisfies the following twoscale equation:
where $\varphi \left(x\right)$ is a scaling function that is dilated and translated as:
And $\varphi \left(x\right)$ satisfies the following twoscale equation:
where ${\left\{{g}_{k}\right\}}_{k=0,\dots ,M1}$ and ${\left\{{h}_{k}\right\}}_{k=0,\dots ,M1}$ denote quadrature mirror filters and have the relationship:
Based on the orthonormal base expressed in Eq. (1), the space ${\mathbf{L}}^{2}\left(R\right)$ can be spanned by:
Eq. (6) implies that the analysis and synthesis of an ODS $U\left(x\right)$ in ${\mathbf{L}}^{2}\left(R\right)$ can be, respectively, realized by:
Such wavelets provide a framework for the MRA as stated in the following.
Derived from Eq. (7), ${\mathbf{W}}_{j}=\text{span}\left({\psi}_{j,k}:j,k\in Z\right)$ forms a subspace of ${\mathbf{L}}^{2}\left(R\right)$, leading to:
For all $j$, ${\mathbf{W}}_{j}$ are orthogonal to each other, from which ${\mathbf{L}}^{2}\left(R\right)$ is expressed as:
where $\oplus $ denotes summing vector spaces. On the other hand, ${\mathbf{W}}_{j}=\text{span}\left({\psi}_{j,k}:i,k\in Z,i>j\right)$. ${\mathbf{V}}_{j}=\text{span}\left({\psi}_{j,k}:i,k\in Z,i>j\right)$ forms a subspace of ${\mathbf{L}}^{2}\left(R\right)$, which leads to ${\mathbf{V}}_{j1}={\mathbf{V}}_{j}\oplus {\mathbf{W}}_{j}$. Substituting ${\mathbf{V}}_{j}$ into Eq. (9) results in:
Thus, a sequence of closed subspaces $\mathbf{V}$ nested as:
Forms the MRA of ${\mathbf{L}}^{2}\left(R\right)$.
Based on the above definitions, an ODS $U\left(x\right)$ in the subspace ${\mathbf{V}}_{0}$ with the finest resolution can be decomposed into the first to the $N$th level:
where ${A}_{N}\left(x\right)$ is the approximation of $U\left(x\right)$ at level $N$ in ${\mathbf{V}}_{N}$, and ${D}_{j}\left(x\right)$ is the detail of $U\left(x\right)$ at level $j$ in ${\mathbf{W}}_{j}$. Eq. (12) can be implemented by the discrete wavelet transform (DWT) [18]; the fundamental discrete wavelet, the Haar wavelet, is utilized for the MRA in this study. For a damaged beamlike structure, damage features carried in its ODS can be retained in the approximation while the noise components can be separated and contained in the details.
2.2. TKE
Kaiser formulated the TKE operator $\mathrm{\Psi}\left(\right)$ to measure the TKE of an discrete oscillating signal ${Y}^{2}\left[p\right]$ [19]:
The TKE is very sensitive to slight changes in signals. To demonstrate such sensitivity, consider a signal $y\left(x\right)={y}_{1}\left(x\right)+{y}_{2}\left(x\right)x\in \left[\mathrm{0,10}\right]$ that is a consine signal with a very slight perturbance at $x=$ 5:
The signal $y\left(x\right)$ is discreted with the sampling interval of 0.01, and the discrete $y\left[x\right]$ is shown in Fig. 1(a); its TKE is calculated by Eq. (13) and shown in Fig. 1(b). In Fig. 1(a), the slight perturbance at $x=$ 5 causes barely visible change and can be negligible, whereas a peak rapidly arises in the TKE of $y\left[x\right]$ at the location of perturbance ($x=$ 5). Thus, the TKE can sensitively reflect local slight perturbance in a signal.
Fig. 1a) Cosine signal with local slight perturbance and b) its TKE
a)
b)
2.3. TKE of MRODS
Damage can cause rapid changes in the TKE of ODSs; in turn, such changes can be utilized to designate the presence and location of damage. However, the TKE operator is very susceptible to noise [19], therefore actual damagecaused changes in the TEK can be masked by the intense noise interference. To this end, noise components need to be separated from ODSs. In accordance with the MRA introduced in Section 2.1, an ODS $U\left[x\right]$ can be decomposed into the $N$th approximation ${A}_{N}\left[x\right]$ and the first to $j$th details ${D}_{j}\left[x\right]$ ($j=1,\dots ,N)$ by Eq. (12), and the TKE of ODS can be expressed as:
By discarding the details ${D}_{j}\left[x\right]$ up to a satisficing level $N$ that contain noise components and retaining approximation ${A}_{N}\left[x\right]$ that contains damage features, the TKE of MRODS is formulated, denoted as ${E}_{N}\left(W\right[x\left]\right)$:
At a satisficing level, noise interference can be basically eliminated in the TKE of MRODS; synchronously, damagecaused changes in the TKE of MRODS can be utilized to designate the presence and location of damage. It is worth mentioning that the damage identification using the TKE of MRODS is a nonbaseline method, requiring no structural baseline information such as temperature, materials, geometry, and boundary conditions.
3. Debonding identification in steelreinforced concrete slabs
3.1. Steelreinforced concrete slabs with interface debonding
A simplysupported steelreinforced concrete slab is taken as a specimen. The concrete slab is reinforced externally with an Ishaped steel beam and internally with rebars. The concrete slab is 2350 mm in length, and seven widththrough debondings denoted as D1D7 are scattered at the interface between the Ishaped steel beam and the concrete slab. The steelreinforced concrete slab is shown in Fig. 2(a) with a zommedin view of the bebonding labeled in a yellow dashed ellipse (Fig. 2(b)). The locations and sizes of the seven debondings are listed in Table 1.
Table 1Debonding locations and relative sizes
Number of debondings  Location (mm)  Relative size (%) 
D1  400430  1.3 
D2  650690  1.7 
D3  890940  2.1 
D4  11601230  3.0 
D5  14501500  2.1 
D6  17051745  1.7 
D7  19501980  1.3 
Fig. 2Steelreinforced concrete slab a) with zoomedin view of debonding labeled in yellow dashed ellipse b)
a)
b)
3.2. Debonding identification
3.2.1. MRODS
Since the span of the steelreinforced concrete slab is too long for laser scanning, the inspection region of the is divided into the left inspection region (3201325 mm) and right inspection region (11452140 mm) with a small overlap covering the fourth debonding D4. An electromechanical shaker was attached to the beam at the location of 480 mm away from the left end of the steel beam to generate a flexural harmonic excitation. A SLV was employed to scan the Ishaped steel beam subject to the harmonic excitation, whereby the corresponding ODS of the steelreinforced concrete slab was measured. The ODSs of the left and right inspection regions of the slab were measured using the SLV under arbitrarily selected excitation frequencies, i.e., 800 and 1200 Hz, respectively. The ODSs, denoted as ${U}_{\omega}$ with $\omega $ the angular frequency, for the left and the right inspection regions at 800 Hz and 2000 Hz are shown in Figs. 3(a) and 3(b), respectively.
Fig. 3ODSs for a) left and b) right inspection regions at 800 Hz and 2000 Hz, respectively
a)
b)
Fig. 4The firstlevel approximations of ODSs for a) left and b) right inspection regions at 800 Hz and 2000 Hz, respectively
a)
b)
Fig. 5The firstlevel details of ODSs for a) left and b) right inspection regions at 800 Hz and 2000 Hz, respectively
a)
b)
By Eq. (12), the firstlevel ($N=$ 1) approximations ${A}_{1}\left[x\right]$ and details ${D}_{1}\left[x\right]$ are extracted from the ODSs of the left and right inspection regions of the slab, shown in Figs. 4 and 5, respectively. By comparing Figs. 3 and 4, it can be seen that the approximations are much smoother that the ODSs. Similarly, the secondlevel ($N=$ 2) approximations ${A}_{2}\left[x\right]$ (Fig. 6) together with details ${D}_{1}\left[x\right]$ and ${D}_{2}\left[x\right]$ (Fig. 7) are extracted from the ODSs. It can be seen that by increasing the decomposition level in the MRA, approximations become smoother (Figs. 4 and 6) because more noise components are separated from the ODSs (Figs. 5 and 7).
Fig. 6The secondlevel approximations of ODSs for a) left and b) right inspection regions at 800 Hz and 2000 Hz, respectively
a)
b)
Fig. 7The first and secondlevel details of ODSs for a) left and b) right inspection regions at 800 Hz and 2000 Hz, respectively
a)
b)
3.2.2. TKE of MRODS
The TKE of MRODSs are obtained by Eq. (16) with values in the boundaryeffect interval vanished [20]. For the finest resolution with level $N$ being zero, the approximations of the ODSs are the ODSs themselves, and the corresponding TKE of ODS is denoted as ${E}_{0}\left[x\right]$. ${E}_{0}\left[x\right]$ is shown in Fig. 8 with the actual debonding locations bounded by pairs of red dashed lines. It can be seen from Fig. 8 that intense noise interference dominates the results and debondingcaused peaks are totally masked.
For the firstlevel ($N=$ 1), the corresponding TKE of MRODS, denoted as ${E}_{1}\left[x\right]$ is obtained and shown in Fig. 9, where noise interference is suppressed and debondingcaused peaks to appear at locations of actual debondings. For the firstlevel ($N=$ 2), the corresponding TKE of MRODS, denoted as ${E}_{2}\left[x\right]$ is obtained and shown in Fig. 10, where noise interference is basically eliminated and debondingcaused peaks can be clearly identified, corresponding to the actual debonding locations bounded by pairs of red dashed lines. For this scenario, the second level can be regarded as the satisficing level. It should be noted that the fourth peak for the left inspection region (Fig. 10(a)) and the first peak for the right inspection region (Fig. 10(b)) indicate the same debonding D4 as the two inspection regions have an overlap covering the fourth debonding.
The experimental results show that the TKE of MRODS is of strong noise robustness and damage sensitivity, and can be applied to designate the presence and location of interface debondings of a steelreinforced concrete slab under a noisy environment.
Fig. 8TKE of ODS for a) left and b) right inspection regions at 800 Hz and 2000 Hz, respectively
a)
b)
Fig. 9TKE of MRODS in the first level for a) left and b) right inspection regions at 800 Hz and 2000 Hz, respectively
a)
b)
Fig. 10TKE of MRODS in the second level for a) left and b) right inspection regions at 800 Hz and 2000 Hz, respectively
a)
b)
4. Conclusions
Identifying surface debondings between steel and concrete is critical for ensuring the safety of the steelreinforced concrete structures. Lasermeasured ODSs are of dense sampling and have been increasingly utilized for structural damage identification; however, noise interference is the most concern for such ODSs. This study proposes a noiserobust dynamics feature relying on ODSs, namely TKE of MRODS, to identify surface debondings in steelreinforced concrete structures. The capability of the proposed dynamics feature to identify surface debondings is validated on a steelreinforced concrete slab, whose ODSs are acquired using a SLV. Some conclusions are drawn as follows.
1) Owing to the MRA, a measured ODS can be decomposed into the approximation that contains damage features and details that contain noise components.
2) Owing to the sensitivity of the TKE to slight changes in signals, damagecaused changes in the TKE of MRODS can be utilized to designate the presence and location of damage.
3) Damage identification using the TKE of MRODS is a nonbaseline method, requiring no structural baseline information such as temperature, materials, geometry, and boundary conditions.
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About this article
The authors are grateful for the partial support provided by the Natural Science Foundation of China (No. 11772115). The authors are also grateful for the experimental support from Dr. Hao Xu in The Hong Kong Polytechnic University.