# Structural Damage Identification Based on the Transmissibility Function and Support Vector Machine

Authors: Yansong Diao; Xue Men; Zuofeng Sun; Kongzheng Guo; Yumei Wang
Journal: Shock and Vibration
 (2018)

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## Abstract

A novel damage identification method based on transmissibility function and support vector machine is proposed and outlined in this paper. Basically, the transmissibility function is calculated with the acceleration responses from damaged structure. Then two damage features, namely, wavelet packet energy vector and the low order principal components, are constructed by analyzing the amplitude of the transmissibility function with wavelet packet decomposition and principal component analysis separately. Finally, the classification algorithm and regression algorithm of support vector machine are employed to identify the damage location and damage severity respectively. The numerical simulation and shaking table model test of an offshore platform under white noise excitation are conducted to verify the proposed damage identification method. The results show that the proposed method does not need the information of excitation and the data from undamaged structure, needs only small size samples, and has certain antinoise ability. The detection accuracy of the proposed method with damage feature constructed by principal component analysis is superior to that constructed by wavelet packet decomposition.


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## Body

## 1. Introduction

Vibration-based damage detection and structural health monitoring (SHM) have been developed in recent years [1, 2]. The goal of SHM is to detect damage that may exist before it becomes critical for the structure’s integrity.Among vibration-based damage detection approaches, transmissibility function (TF) analysis attracts extensive interest because TF defined as the ratio between two outputs is independent of excitation [3]. Up to now, methods based on TF have been developed to detect damage or assess the structural damage severity. Steenackers et al. [4] used the TFs instead of frequency response functions (FRFs) in model updating. The authors concluded that the finite element model updated with TF was equivalent to the model updated with FRFs or operational modes. Maia et al. [5] proposed a method to detect and quantify structural damage of a beam, using response TFs measured along the structure. Some numerical simulations were presented and a comparison was made with results using FRFs. Li et al. [6] proposed a damage detection approach based on TF to identify the damage of shear connectors in slab-on-girder bridge structures with or without reference data from the undamaged structure. Kong et al. [7] derived the TF of a vehicle-bridge coupled system consisting of a simply supported beam and a single-degree-of-freedom vehicle, and a numerical study was conducted to investigate the feasibility of detecting bridge damage using TF. Zhou et al. [8] defined a sensitive damage indicator based on TF coherence, which was used to detect and identify damage severity. The proposed approach was validated on data from a physics-based numerical model as well as experimental data from a three-story aluminum frame structure. Chesné and Deraemaeker [9] reviewed the state-of-the-art using TF for damage detection and localization and discussed the extension of these results to more general dispersive systems such as beams or plates. Zhu et al. [3] proposed a decentralized damage detection approach using TF, which was ideal for using limited number of sensors upon a large-scale structure. Zhou et al. [10] proposed a new approach for detecting structural damage using TF together with hierarchical clustering and similarity analysis. Chen et al. [11] tried to predict damage with neural networks (NNs) trained by TF; the test results showed that the NN classifiers clearly delivered the diagnostic indications of the faults introduced into the structural systems. In [12], a TF based damage detection method using artificial intelligence was proposed for detecting the structural damage and predicting its severity.Broadly, the damage detection methodologies can be mainly classified into two categories: model-driven approaches [13, 14] or data-driven approaches [15, 16]. Model-driven approaches concentrate on the understanding of the structure from its numerical model, where finite element analysis is often employed. In contrast, data-driven approaches construct a model by learning from measured data and then make a comparison between the model and measured responses to detect damage.The machine learning techniques have been raising increasing attentions among the data-driven approaches. There are a variety of machine learning techniques, including Bayesian networks [17], artificial NNs [18], and support vector machines (SVMs) [19].SVM is a supervised learning technique with strong theoretical foundations based on the Vapnik-Chervonenkis theory [20]. This method promises to overcome the shortcomings of conditional NNs such as the local minimization and inadequate statistical capabilities. Furthermore, SVM can unify different types of discriminant functions such as linear, polynomial, and radial basic functions in the same framework [21]. SVM is especially suited in the case of small size samples [22]. SVM has been widely accepted as an effective tool for characteristic extraction and damage detection. Kim et al. [23] attempted to systematically integrate the wavelet transform, the autoregressive model, the damage-sensitive energy feature, and the SVM into a single SHM framework for damage detection of smart structures equipped with time-varying nonlinear hysteretic control devices; the damage was identified using the SVM based on the differences of AR parameters between the damaged and undamaged structures. Khatibinia et al. [24] employed SVM with a wavelet kernel for seismic reliability assessment of existing reinforced concrete structures with consideration of soil-structure interaction effects in accordance with performance-based design. Chong et al. [25] proposed a nonlinear multiclass SVM based SHM system for smart structures. Based on the wavelet filtered signals, several wavelet based autoregressive models were constructed. Damage-sensitive features were extracted from the wavelet based autoregressive coefficients and then the nonlinear multiclass SVM was employed to estimate the damage severity. Dushyanth et al. [26] developed a new multilevel SVM and stated that the proposed SVM method required much less training data as compared to other methods. Gui et al. [27] presented three optimization algorithm based SVMs for damage detection. The optimization algorithms, including grid search, partial swarm optimization, and genetic algorithm, were used to optimize the penalty parameters and Gaussian kernel function parameters; two types of feature extraction methods in terms of time series data were selected to capture effective damage characteristics.Nevertheless, little research has been reported on damage identification using SVM combined with TF analysis.For damage identification using SVM, the core issue is to seek some features from the structural responses that are sensitive to structural damage. Since features hold a fundamental role in results performance, one should pay special attention to the feature extraction. In this study, wavelet packet decomposition (WPD) and principal component analysis (PCA) are employed to extract damage features from TF; the aim is to compare the effect of the proposed method using damage features extracted from time-frequency domain and time domain.WPD can be considered as an extension of the discrete wavelet transform, which decomposes not only the approximation but also the detail coefficients at each level of decomposition. WPD creates the same frequency bandwidths in every resolution. The wavelet packet component energies extracted from structural dynamic responses have been used as a damage feature to detect damage. Hou et al. [28] presented a brief background of wavelet analysis and its SHM applications. Sun and Chang [29] proposed a WPD-based damage assessment method of structures. Dynamic signals measured from a structure were first decomposed into wavelet packet components. Component energies were then calculated and used as inputs into NN models for damage assessment. Ding et al. [30] put forward a workable realization procedure for damage alarming of frame structures based on energy variations of structural dynamic responses decomposed by WPD. Ravanfar et al. [31] presented a two-step damage identification approach based on wavelet multiresolution analysis and genetic algorithm in beam structures. The location of the crack was identified in the first step by defining the damage index called relative wavelet packet entropy. Then, the damage severities at the identified locations were assessed in the second step using genetic algorithm.TF has high dimension with high redundancy and correlation. This can slow the damage detection procedure and make the underlying patterns more complicated to classify. Also, computation on high dimensional data can be time-consuming and costly. PCA is a popular technique for dimensionality reduction, which accelerates the detection procedure and makes the underlying patterns more easy to classify; so far, some research works have been done which use PCA for dimensionality reduction in damage detection. Sohn et al. [32] used PCA with the statistical process control for damage detection. Also, Worden and Manson [33] applied linear and nonlinear PCA for reducing the data dimension. Then, Mahalanobis distance was utilized to find anomaly in the reduced dimensional data. In [18], PCA was employed to compress FRF data for damage detection.In this study, a novel damage identification method based on TF and SVM is proposed and outlined. Basically, the TF is calculated with the acceleration responses from damaged structure. Then two damage features, namely, wavelet packet energy vector (WPEV) and the low order principal components (PCs), are constructed by analyzing the amplitudes of TF with WPD and PCA separately. Finally, each of the two damage features (WPEV, PC) is used as input for SVM; the classification algorithm and regression algorithm of SVM are employed to identify the damage location and damage severity, respectively. The numerical simulation and shaking table model test of an offshore platform under white noise excitation are conducted to verify the proposed damage identification method. The main contribution of this study is the introduction of SVM in the context of the development of transmissibility-based damage identification methods.


## 2. Transmissibility Function

In this paper, TF will be used as primary data to perform the damage identification. A TF is the ratio of FRF at two spatial locations on a structure, and it can be estimated in a nonparametric preprocessing step.By assuming a single forceFk(ω) that is located in the input degree of freedom (DOF) k, the TF is given by [34](1)Tjiω=AiωAjω=-ω2HikωFkω-ω2HjkωFkω=HikωHjkω,where Aiω and Ajω are the Fourier transform of the responses measured at DOF i and DOFj, respectively, and Hikω and Hjkω are the elements of FRF at DOF i and DOF j, respectively.From (1), it can be seen that when the structure is subjected to a single point excitation, the TF only depends on the location of the force, which is not dependent on either the nature of the force signal or its amplitude. The influence of the input signal (excitation force) is eliminated.The TF is very sensitive to damage because it is a ratio of functions with many peaks and valleys. Since the dynamic responses of structure change when damage occurs, it is natural to expect that the TF will also change if damage occurs. A change in the TF represents a change in the structural properties, and it is inferred that the change of TF indicates the damage of structure.


## 3. Damage Feature Based on WPD

The wavelet analysis is a multiresolution analysis method in the time and frequency domain of a signal, and it has the ability to describe the local characteristic of a signal. WPD can be considered as an extension of the discrete wavelet analysis, which does not require a priori knowledge of the frequency bands, but it filters the signal in adaptive way. Discrete wavelet analysis decomposes just the previous approximation coefficients through high and low pass filters, whereas WPD decomposes both the detail and approximation coefficients, which keeps the important information available in higher frequency components. More details can be found in the textbook by Mallat [35].The original signal can be expressed as a summation of WPD components as(2)ft=∑i=12jfjit,where t is time lag and fji(t) is the WPD component signal that can be represented by a linear combination of wavelet packet functions.To examine the structural health condition, it is essential to achieve an index that is sensitive to structural damage. Usually, the measured vibration signals are decomposed by WPD into component signals and then component energies are calculated. The wavelet packet component energy is a suitable tool for identifying and characterizing a specific phenomenon of signal in the time-frequency domain. It has been shown by Yen and Lin [36] that the energy stored in a specific frequency band at a certain level of WPD provides a greater potential for signal feature than the coefficients alone.The wavelet packet component energy of a signal is defined as(3)Efji=∫-∞+∞fjit2dtThe total signal energy can be expressed as the summation of wavelet packet component energies when the mother wavelet is orthogonal.(4)Ef=∑i2jEfji=∑i=12j∫-∞+∞fjit2dtThen, the energy ratio of each wavelet coefficient can be written as(5)pij=EfjiEfThepij values correspond to a ratio of the energy of a particular coefficient Efji to the total energy.The normalized wavelet packet component energy vector of a signal is defined as(6)WPEV=p1j,p2j,…,p2jjWhen the structure is damaged at different location, the values ofpij and WPEV become different; therefore, WPEV can be used as damage feature to identify damage of the structure in this study.


## 4. Damage Feature Based on PCA

PCA is a multivariate analysis method that can be considered as a linear data compression technique, and it is widely used in the fields of image processing, flow visualization, and the reduced order models. Using an orthogonal projection, the original set of variables in anN-dimensional space is transformed into a new set of uncorrelated variables, the so-called principal components, in aP-dimensional space such that P<N. A summary of the method will be given here for the sake of completeness [37].At the assumption of stationary stochastic process, the amplitude of the TF is used as the analysis signal, and it is divided intop groups, each with n frequency points, forming a matrix Yn×p. Hence the centralized matrix Xn×p can be obtained by using (7)Xk=yk-y-kk=1,2,…,p,where yk is the kth column of matrix Yn×p and y-k is the mean value of yk.The correlation matrix can be defined as(8)R=1p-1Xn×pXTn×pBy definition, the principal components are the eigenvalues and associated eigenvectors of the correlation matrixR(9)Rφi=λiφii=1,2,…,n,where λi and φi are the ith eigenvalue and eigenvectors, respectively. The principal component of the normalized matrix Xn×p is given by (10)Zi=φiTX,where Zi is the ith principal component. Any two principal components are uncorrelated, and the variance of Zi is λi. Because of λ1>λ2>⋯>λn>0, the first principal component, i.e., the highest eigenvalue and its associated eigenvector, represents the direction and amount of maximum variability in the original data. The next principal component, which is orthogonal to the first component, represents the next most significant contribution from the original data, and so on.The contribution rate is given by(11)δi=λi∑i=1nλiThe greater the contribution rateδi value is, the stronger the ability of the corresponding principal component to synthesize the information of original data is.The accumulative contribution rate is given by(12)ψm=∑i=1mλi∑i=1nλi≥0.85In order to increase the reliability of the selected principal component, the accumulative contribution rateψm is more than 85%. The damage feature PC, composed of the firstm-order principal components, could be formed as followed:(13)PC=Z1,Z2,…,ZmTGenerally, the first several principal components contain the most of the information; the entire variation characteristics of the original data can be reflected by analysis of the first several principal components. The damage identification can be implemented in lower dimensional space constituted by the first several principal components. Therefore, the difficulty of the analysis is reduced.


## 5. Support Vector Machines

SVM is initially developed for classification and then it is successfully extended to the regression analysis by Vapnik [38]. By applying kernel function (e.g., linear, polynomial, or radial basis function) as the inner product of functions mapping the data to a high dimensional feature space, SVM can also be applied in a case of nonlinear classification and regression analysis. A brief introduction of SVM is presented here for completeness. Readers are referred to the tutorials on SVM [39, 40] for details.


### 5.1. Classification Algorithm of SVM

The whole process of classification of data by SVM is illustrated in Figure1. Consider a training data set as {(x1,y1),…,(xl,yl)}, where each xi∈Rn denotes the input space of the sample and has a corresponding target value yi∈-1,1 for i=1⋯l, where l corresponds to the size of the training data.Figure 1
Classification of data by SVM.If the training is linearly separable, the boundary can be expressed as follows:(14)w·x+b=0,w∈Rn,b∈R,where the vector w defines the boundary, x is the input vector of dimension n and b is a scalar threshold, and (·) denotes the dot product. At the margins, where the support vectors are located, the equations for classes A and B, respectively, are as follows:(15)w·x+b=1,w·x+b=-1For no-separable data, soft margin of SVM is defined by(16)Margin=2w2and the optimization of the solution will be(17a)min12w2+C∑i=1lξi(17b)Subjecttoyiw·x+b≥1-ξi,ξi≥0,where ξi is slack variable and C is the error penalty. By using Lagrange multipliers algorithm to solve the dual optimization problem as shown in (17a) and (17b), the nonlinear decision function will yield(18)fx=sign∑i=1NαiyiKx,xi+b,αi>0,whereKx,xi=exp-γxi-xj2,γ>0,where the K(x,xi) is the radial basis function and it is defined as the kernel function. By using this kernel function, it can analyze higher dimensional data.


### 5.2. Regression Algorithm of SVM

The main thought behind the regression problem is to determine an approximation which can determine or predict the future target values accurately or at least close to the expected values. In general, a linear estimate of regression function takes the form(19)fx=w·x+b,where w∈Rn, b∈R. The goal is to find the value of w and b such that values of yi can be determined by minimizing the regression risk:(20a)min12w2+C∑i=1lξi+ξi∗(20b)Subjecttoyi-w·xi-b≤ε+ξiw·xi+b-yi≤ε+ξi∗ξi,ξi∗≥0,where ε is the precision, ξi, ξi∗ is slack variable, and C is the constant.By using Lagrange multipliers algorithm to solve the dual optimization problem as shown in (20a) and (20b), the nonlinear decision function will yield(21)fx=∑i=1lαi-αi∗x·xi+b=∑i=1lαi-αi∗Kx,xi+b,where αi,αi∗∈[0,C).In this study, software packageLIBsvm [41] developed by professor Lin of Taiwan University is used to realize the classification and regression of SVM.


## 5.1. Classification Algorithm of SVM

The whole process of classification of data by SVM is illustrated in Figure1. Consider a training data set as {(x1,y1),…,(xl,yl)}, where each xi∈Rn denotes the input space of the sample and has a corresponding target value yi∈-1,1 for i=1⋯l, where l corresponds to the size of the training data.Figure 1
Classification of data by SVM.If the training is linearly separable, the boundary can be expressed as follows:(14)w·x+b=0,w∈Rn,b∈R,where the vector w defines the boundary, x is the input vector of dimension n and b is a scalar threshold, and (·) denotes the dot product. At the margins, where the support vectors are located, the equations for classes A and B, respectively, are as follows:(15)w·x+b=1,w·x+b=-1For no-separable data, soft margin of SVM is defined by(16)Margin=2w2and the optimization of the solution will be(17a)min12w2+C∑i=1lξi(17b)Subjecttoyiw·x+b≥1-ξi,ξi≥0,where ξi is slack variable and C is the error penalty. By using Lagrange multipliers algorithm to solve the dual optimization problem as shown in (17a) and (17b), the nonlinear decision function will yield(18)fx=sign∑i=1NαiyiKx,xi+b,αi>0,whereKx,xi=exp-γxi-xj2,γ>0,where the K(x,xi) is the radial basis function and it is defined as the kernel function. By using this kernel function, it can analyze higher dimensional data.


## 5.2. Regression Algorithm of SVM

The main thought behind the regression problem is to determine an approximation which can determine or predict the future target values accurately or at least close to the expected values. In general, a linear estimate of regression function takes the form(19)fx=w·x+b,where w∈Rn, b∈R. The goal is to find the value of w and b such that values of yi can be determined by minimizing the regression risk:(20a)min12w2+C∑i=1lξi+ξi∗(20b)Subjecttoyi-w·xi-b≤ε+ξiw·xi+b-yi≤ε+ξi∗ξi,ξi∗≥0,where ε is the precision, ξi, ξi∗ is slack variable, and C is the constant.By using Lagrange multipliers algorithm to solve the dual optimization problem as shown in (20a) and (20b), the nonlinear decision function will yield(21)fx=∑i=1lαi-αi∗x·xi+b=∑i=1lαi-αi∗Kx,xi+b,where αi,αi∗∈[0,C).In this study, software packageLIBsvm [41] developed by professor Lin of Taiwan University is used to realize the classification and regression of SVM.


## 6. Damage Identification Procedure

A novel structural damage identification method based on TF and SVM is proposed herein. Each of two damage features (WPEV, PC) is used as input for SVM to identify the damage location and damage severity separately. The flowchart is shown in Figure2, and the proposed strategy is based on the following five steps:Figure 2
Flowchart of damage identification method based on TF and SVM.(1) Acquiring structure acceleration responses at measuring points(2) Calculating the TF: the TFs between acceleration responses at the adjacent measuring points are calculated according to (1) and used as the research signals(3) Constructing the damage features (WPEV, PC): the normalized wavelet packet energy damage feature (WPEV) is calculated according to (6). Similarly, firstm-order principal components of the amplitude of TF, whose cumulative contribution rate is more than 85%, are selected to compose the one-dimensional damage feature (PC) according to (13)(4) Structure damage identification based on the damage features (WPEV, PC) and SVM: each of two damage features (WPEV, PC) is used as sample set, respectively, and damage location is used as corresponding sample label; the classification algorithm of SVM is employed to identify the damage location. Then, each of the two damage features (WPEV, PC) is used as sample set, respectively, and damage severity is used as corresponding sample label; the regression algorithm of SVM is employed to identify the damage severity.


## 7. Numerical Simulation

In this numerical study, the newly proposed damage detection method based on TF and SVM will be illustrated and verified on an offshore platform structure.


### 7.1. Offshore Platform Model

As shown in Figure3, the offshore platform structure consists of 68 steel tubular members that comprise 20 columns, 32 beams, and 16 diagonal braces in vertical planes. The columns are made of round steel tubes of Q235 with the diameter of 22 mm and the wall thickness of 2.2 mm; the beams and diagonal braces are also made of round steel tubes of Q235 with the diameter of 12 mm and the wall thickness of 1.6 mm. The steel plate with side lengths of 0.65 × 0.55 m and a thickness of 12 mm is set up in the upper structure of the platform. The heights of the five stories are all 0.5 m, and the side lengths of the bottom and top floors are 0.7 m × 0.5 m and 0.4 m × 0.3 m, respectively. The essential material properties of the steel tubular members are as follows: elastic modulus is 2.06 × 1011 N/m2, the density of steel is 7800 kg/m3, and the Poisson’s ratio is 0.3. The structure is fixed at the ground. A commercial finite element code (ANSYS10.0) is employed to establish the finite element model of the offshore platform. BEAM4 element is used to simulate the columns, beams, and braces, and SHELL63 element is used to simulate the top steel plate; MASS21 element is used to simulate the lumped mass outside the top plate.Figure 3
A numerical model of offshore platform.The excitation is the white noise with the sampling frequency of 500 Hz loading at the center of the bottom of the numerical model inYdirection. Signals to be analyzed are the acceleration responses in Y direction on joints 2, 6, 10, 14, and 18 in Figure 3, which are gained from the transient analysis module in ANSYS10.0. The sampling frequency of the acceleration response is 500 Hz and the duration is 20 s.


### 7.2. Numerical Simulation Scenarios

In this numerical study, only the damage of single brace (member ②, ③, ④, ⑤) in profile II ofY direction is considered. The damaged braces are simulated by decreasing their elastic modulus. The damage severities of 20%, 30%, 40%, 60%, 70%, and 80% are considered. Damage scenarios are shown in Table 1; there are 8 training scenarios (C1–C8) and 16 test scenarios (S1–S16) separately.Table 1
Damage scenarios of the numerical simulation.
Scenarios Damaged brace Damage severity Classification training label Regression training label Classification testing label Regression testing label (floor) (%) (floor) (%) (floor) (%) C1 ② 30 2 30 - - C2 ② 70 2 70 - - C3 ③ 30 3 30 - - C4 ③ 70 3 70 - - C5 ④ 30 4 30 - - C6 ④ 70 4 70 - - C7 ⑤ 30 5 30 - - C8 ⑤ 70 5 70 - - S1 ② 20 - - 2 20 S2 ② 40 - - 2 40 S3 ② 60 - - 2 60 S4 ② 80 - - 2 80 S5 ③ 20 - - 3 20 S6 ③ 40 - - 3 40 S7 ③ 60 - - 3 60 S8 ③ 80 - - 3 80 S9 ④ 20 - - 4 20 S10 ④ 40 - - 4 40 S11 ④ 60 - - 4 60 S12 ④ 80 - - 4 80 S13 ⑤ 20 - - 5 20 S14 ⑤ 40 - - 5 40 S15 ⑤ 60 - - 5 60 S16 ⑤ 80 - - 5 80After the acceleration responses on joints 2, 6, 10, 14, and 18 are acquired, TFs (T26, T610, T1014, T1418) are calculated according to (1). The damage features (WPEV, PC) are obtained by analyzing the amplitude of the TFs with WPD and PCA separately, and each of which is used to identify the damage location and damage severity.In order to maximize damage identification results, the basis function and its decomposition level for WPD are identified by a trial and error method. So, the TFs are decomposed by the Db4 wavelet with 3 levels in this paper; the normalized wavelet packet energy damage features (WPEV) are calculated according to (6).As an example,T26 with respect to damage scenario C7 is selected to illustrate the analysis procedure of PCA. T26 with the length of 10000 frequency points is divided into 8 groups, each with 1250 frequency points, forming a matrix Y1250×8; the analysis results are shown in Table 2, including the first 4-order eigenvalues, contribution rates, and accumulative contribution rates.Table 2
Analysis results of PCA forT26 corresponding to damage scenario C7.
Principal component order Scenario C7 Eigenvalue Contribution rate (%) Accumulative contribution rate (%) 1 2.616 69.293 69.293 2 1.473 23.500 92.793 3 0.870 5.803 98.596 4 0.665 0.810 99.406It can be seen from Table2 that the accumulative contribution rate of the first 2-order principal components is more than 85%, and the same results can be achieved for the other damage scenarios. Therefore, the first 2-order principal components are selected to compose the one-dimensional damage feature PC with 16 elements for damage identification.The damage features (WPEV, PC) extracted from damage scenarios C1–C8 are used as training samples for SVM, and the numbers of floor where the damaged braces locate are taken as the sample labels for SVM. The damage features (WPEV, PC) extracted from damage scenarios S1–S16 are used as testing samples for SVM.In order to consider the influence of noise, a certain degree of normal distribution of random white noise is added to the acceleration response to simulate the effect of noise, and the simulation formula of the response with noise is(22)xnoise=x+RMSx×Nlevel×Nunit,where xnoise and x are the acceleration responses with noise and without noise, respectively. RMSx is the root mean square of the acceleration response without noise, Nunit is the random data of normal distribution with the mean value of 0 and the variance is 1, Nlevel is the noise degree, taking the value of 0.05 and 0.1 in this numerical simulation.


### 7.3. Numerical Simulation Results of the Proposed Damage Identification Method

The classification algorithm C-SVC inLIBsvm is employed to identify the damage location, the penalty parameter C = 500, and the value of radial basis kernel function γ=1.The regression algorithmε-SVR inLIBsvm is employed to identify the damage severity, the threefold cross-validation rule is adopted, the penalty parameter C = 500, the value of radial basis kernel function γ=1, and the value of loss function ε-SVR p=0.01.


#### 7.3.1. Results of the Damage Identification Method Based on WPEV and SVM

The identified damage locations based on WPEV and SVM are shown in Table3, and the detection accuracy is shown in Table 4.Table 3
Identified damage locations based on WPEV and SVM.
Scenarios Damage location(floor) Identified damage location(floor) Without noise With 5% noise With 10% noise T 2 6 T 6 10 T 10 14 T 14 18 T 2 6 T 6 10 T 10 14 T 14 18 T 2 6 T 6 10 T 10 14 T 14 18 S1 2 2 2 2 ④ 2 2 2 ③ 2 2 ③ ③ S2 2 2 ③ 2 2 2 2 2 ③ 2 2 2 2 S3 2 2 2 2 2 2 2 2 2 2 2 ③ 2 S4 2 2 2 2 2 2 2 2 2 2 ③ 2 2 S5 3 3 3 3 3 3 3 3 ② 3 3 ② ③ S6 3 3 3 3 3 3 3 3 3 3 3 3 ② S7 3 3 3 3 3 3 3 3 3 3 3 3 3 S8 3 3 ④ 3 3 3 3 3 3 3 ② 3 3 S9 4 4 4 4 4 4 4 ② ③ 4 4 4 ③ S10 4 4 4 4 4 4 4 4 4 4 4 4 ③ S11 4 4 4 4 4 4 4 4 4 4 4 4 4 S12 4 4 4 4 4 4 4 4 4 ③ 4 4 4 S13 5 5 5 5 ④ 5 5 ④ ④ 5 5 ④ 5 S14 5 5 5 5 5 5 5 5 5 5 5 5 5 S15 5 5 5 5 5 5 5 5 5 5 5 5 ④ S16 5 5 5 5 5 5 5 5 5 5 5 5 ④ Note. (1) T26 denotes the TF between nodes 2 and 6 (the rest are the same); (2) circled numbers indicates that the identified damage location is wrong in Tables 3, 6, 10, and 12.Table 4
Detection accuracy of identified damage locations based on WPEV and SVM.
Damage location (floor) Detection accuracy Without noise With 5% noise With 10% noise 5 93.75% (15/16) 87.50% (14/16) 81.25% (13/16) 4 100% (16/16) 87.50% (14/16) 81.25% (13/16) 3 93.75% (15/16) 87.50% (14/16) 75.00% (12/16) 2 87.50% (14/16) 87.50% (14/16) 75.00% (14/16)The WPEVs extracted from TFs (T26, T610, T1014, T1418) are used as inputs for SVM to identify the damage severity, respectively, and the average of identified damage severities is shown in Table 5.Table 5
Average of identified damage severities based on WPEV and SVM.
Scenarios DS (%) Without noise With 5% noise With 10% noise AIDS (%) RE (%) AIDS (%) RE (%) AIDS (%) RE (%) S1 20 26.18 30.88 26.37 31.84 29.36 46.79 S2 40 44.44 11.81 47.4 18.50 43.93 9.83 S3 60 64.98 8.30 65.09 8.48 65.66 9.43 S4 80 84.08 5.10 84.26 5.33 84.89 6.11 S5 20 26.37 31.84 24.14 20.70 27.28 36.42 S6 40 47.4 18.50 42.5 6.24 43.17 7.92 S7 60 65.09 8.48 64.61 7.69 63.7 6.16 S8 80 84.26 5.33 83.66 4.58 84.68 5.85 S9 20 24.14 20.70 24.42 22.08 28.41 42.03 S10 40 43.24 8.11 44.06 10.20 45.05 13.76 S11 60 64.61 7.69 62.81 4.69 63.72 6.20 S12 80 83.66 4.58 82.45 3.06 83.61 4.51 S13 20 24.42 22.08 24.99 24.95 26.9 34.48 S14 40 44.08 10.20 44.58 11.45 44.5 11.26 S15 60 62.81 4.69 62.81 4.69 65.1 8.49 S16 80 82.45 3.06 82.6 3.25 83.57 4.46 Note. DS denotes the damage severity; AIDS denotes the average of the identified damage severity; RE denotes the average relative error.


#### 7.3.2. Results of the Damage Identification Method Based on PC and SVM

The identified damage locations based on PC and SVM are shown in Table6, and the detection accuracy is shown in Table 7.Table 6
Identified damage locations based on PC and SVM.
Scenarios Damage location(floor) Identified damage location (floor) Without noise With 5% noise With 10% noise T 2 6 T 6 10 T 10 14 T 14 18 T 2 6 T 6 10 T 10 14 T 14 18 T 2 6 T 6 10 T 10 14 T 14 18 S1 2 2 2 ③ 2 2 2 2 ③ 2 2 2 ④ S2 2 2 2 2 ③ 2 2 2 ③ 2 ④ 2 2 S3 2 2 2 2 2 2 2 2 2 2 2 2 2 S4 2 2 2 2 2 2 2 2 2 2 2 2 ④ S5 3 3 3 3 3 3 3 3 ② 3 3 ④ 3 S6 3 3 3 3 3 3 3 3 3 3 3 3 ④ S7 3 3 3 3 3 3 3 3 3 3 3 3 3 S8 3 3 3 3 3 3 3 3 3 3 ② 3 3 S9 4 4 4 4 4 4 4 4 ③ 4 4 4 4 S10 4 4 4 4 4 4 4 4 4 4 4 4 ③ S11 4 4 4 4 4 4 4 4 4 4 4 4 4 S12 4 4 4 4 4 4 4 4 4 4 4 4 4 S13 5 5 5 5 5 5 5 5 ④ 5 5 5 ④ S14 5 5 5 5 5 5 5 5 5 5 5 5 5 S15 5 5 5 5 5 5 5 5 5 5 5 5 5 S16 5 5 5 5 5 5 5 5 5 5 5 5 5Table 7
Detection accuracy of identified damage locations based on PC and SVM.
Damage location (floor) Accuracy Without noise With 5% noise With 10% noise 5 100% (16/16) 93.75% (15/16) 87.50% (15/16) 4 100% (16/16) 93.75% (15/16) 87.50% (15/16) 3 100% (16/16) 93.75% (15/16) 81.25% (13/16) 2 87.50% (14/16) 87.50% (14/16) 81.25% (13/16)The PCs extracted from TFs (T26, T610, T1014, T1418) are used as inputs for SVM to identify the damage severity, respectively, and the average of identified damage severities is shown in Table 8.Table 8
Average of identified damage severities based on PC and SVM.
Scenarios DS (%) Without noise With 5% noise With 10% noise AIDS (%) RE (%) AIDS (%) RE (%) AIDS (%) RE (%) S1 20 23.52 17.60 24.71 23.56 28.12 40.58 S2 40 42.38 5.96 42.16 5.41 44.93 12.32 S3 60 62.35 3.91 64.3 7.17 65.97 9.95 S4 80 81.58 1.97 81.09 1.36 85.93 7.41 S5 20 22.85 14.26 26.14 30.69 24.43 22.16 S6 40 41.07 2.68 42.54 6.34 42.66 6.64 S7 60 61.23 2.05 62.84 4.73 61.51 2.51 S8 80 80.54 0.67 82.34 2.93 82.3 2.87 S9 20 21.48 7.38 23.15 15.76 31.43 57.13 S10 40 41.53 3.82 43.34 8.42 43 7.49 S11 60 61.43 2.39 62.23 3.72 63.77 6.28 S12 80 80.78 0.98 81.36 1.70 84.67 5.84 S13 20 20.43 2.16 22.64 13.18 22.32 11.61 S14 40 40.26 0.64 40.9 2.26 40.59 1.47 S15 60 60.42 0.70 61.51 2.52 61.23 2.05 S16 80 80.02 0.02 80.72 0.90 82.53 3.16By analyzing the above damage identification results of the numerical simulation, the following can be found.(1) The detection accuracy of the damage location is improving with the enlargement of the damage severity. The results of damage location identification with the IFs (T26,T610) are superior to that with the IFs (T1014,T1418), which means that the acceleration responses close to the upper structure contain more information of damage and are more fit for the damage identification.(2) The relative error of the identified damage severity is relatively high when the brace is lightly damaged.(3) The detection accuracy of the damage location decreases with the increase of the noise level; however, things are different for the relative errors of the identified damage severities, which are fluctuated with the increase of the noise level.(4) The detection accuracy of the proposed method with damage feature constructed by PCA is superior to that constructed by WPD.


## 7.1. Offshore Platform Model

As shown in Figure3, the offshore platform structure consists of 68 steel tubular members that comprise 20 columns, 32 beams, and 16 diagonal braces in vertical planes. The columns are made of round steel tubes of Q235 with the diameter of 22 mm and the wall thickness of 2.2 mm; the beams and diagonal braces are also made of round steel tubes of Q235 with the diameter of 12 mm and the wall thickness of 1.6 mm. The steel plate with side lengths of 0.65 × 0.55 m and a thickness of 12 mm is set up in the upper structure of the platform. The heights of the five stories are all 0.5 m, and the side lengths of the bottom and top floors are 0.7 m × 0.5 m and 0.4 m × 0.3 m, respectively. The essential material properties of the steel tubular members are as follows: elastic modulus is 2.06 × 1011 N/m2, the density of steel is 7800 kg/m3, and the Poisson’s ratio is 0.3. The structure is fixed at the ground. A commercial finite element code (ANSYS10.0) is employed to establish the finite element model of the offshore platform. BEAM4 element is used to simulate the columns, beams, and braces, and SHELL63 element is used to simulate the top steel plate; MASS21 element is used to simulate the lumped mass outside the top plate.Figure 3
A numerical model of offshore platform.The excitation is the white noise with the sampling frequency of 500 Hz loading at the center of the bottom of the numerical model inYdirection. Signals to be analyzed are the acceleration responses in Y direction on joints 2, 6, 10, 14, and 18 in Figure 3, which are gained from the transient analysis module in ANSYS10.0. The sampling frequency of the acceleration response is 500 Hz and the duration is 20 s.


## 7.2. Numerical Simulation Scenarios

In this numerical study, only the damage of single brace (member ②, ③, ④, ⑤) in profile II ofY direction is considered. The damaged braces are simulated by decreasing their elastic modulus. The damage severities of 20%, 30%, 40%, 60%, 70%, and 80% are considered. Damage scenarios are shown in Table 1; there are 8 training scenarios (C1–C8) and 16 test scenarios (S1–S16) separately.Table 1
Damage scenarios of the numerical simulation.
Scenarios Damaged brace Damage severity Classification training label Regression training label Classification testing label Regression testing label (floor) (%) (floor) (%) (floor) (%) C1 ② 30 2 30 - - C2 ② 70 2 70 - - C3 ③ 30 3 30 - - C4 ③ 70 3 70 - - C5 ④ 30 4 30 - - C6 ④ 70 4 70 - - C7 ⑤ 30 5 30 - - C8 ⑤ 70 5 70 - - S1 ② 20 - - 2 20 S2 ② 40 - - 2 40 S3 ② 60 - - 2 60 S4 ② 80 - - 2 80 S5 ③ 20 - - 3 20 S6 ③ 40 - - 3 40 S7 ③ 60 - - 3 60 S8 ③ 80 - - 3 80 S9 ④ 20 - - 4 20 S10 ④ 40 - - 4 40 S11 ④ 60 - - 4 60 S12 ④ 80 - - 4 80 S13 ⑤ 20 - - 5 20 S14 ⑤ 40 - - 5 40 S15 ⑤ 60 - - 5 60 S16 ⑤ 80 - - 5 80After the acceleration responses on joints 2, 6, 10, 14, and 18 are acquired, TFs (T26, T610, T1014, T1418) are calculated according to (1). The damage features (WPEV, PC) are obtained by analyzing the amplitude of the TFs with WPD and PCA separately, and each of which is used to identify the damage location and damage severity.In order to maximize damage identification results, the basis function and its decomposition level for WPD are identified by a trial and error method. So, the TFs are decomposed by the Db4 wavelet with 3 levels in this paper; the normalized wavelet packet energy damage features (WPEV) are calculated according to (6).As an example,T26 with respect to damage scenario C7 is selected to illustrate the analysis procedure of PCA. T26 with the length of 10000 frequency points is divided into 8 groups, each with 1250 frequency points, forming a matrix Y1250×8; the analysis results are shown in Table 2, including the first 4-order eigenvalues, contribution rates, and accumulative contribution rates.Table 2
Analysis results of PCA forT26 corresponding to damage scenario C7.
Principal component order Scenario C7 Eigenvalue Contribution rate (%) Accumulative contribution rate (%) 1 2.616 69.293 69.293 2 1.473 23.500 92.793 3 0.870 5.803 98.596 4 0.665 0.810 99.406It can be seen from Table2 that the accumulative contribution rate of the first 2-order principal components is more than 85%, and the same results can be achieved for the other damage scenarios. Therefore, the first 2-order principal components are selected to compose the one-dimensional damage feature PC with 16 elements for damage identification.The damage features (WPEV, PC) extracted from damage scenarios C1–C8 are used as training samples for SVM, and the numbers of floor where the damaged braces locate are taken as the sample labels for SVM. The damage features (WPEV, PC) extracted from damage scenarios S1–S16 are used as testing samples for SVM.In order to consider the influence of noise, a certain degree of normal distribution of random white noise is added to the acceleration response to simulate the effect of noise, and the simulation formula of the response with noise is(22)xnoise=x+RMSx×Nlevel×Nunit,where xnoise and x are the acceleration responses with noise and without noise, respectively. RMSx is the root mean square of the acceleration response without noise, Nunit is the random data of normal distribution with the mean value of 0 and the variance is 1, Nlevel is the noise degree, taking the value of 0.05 and 0.1 in this numerical simulation.


## 7.3. Numerical Simulation Results of the Proposed Damage Identification Method

The classification algorithm C-SVC inLIBsvm is employed to identify the damage location, the penalty parameter C = 500, and the value of radial basis kernel function γ=1.The regression algorithmε-SVR inLIBsvm is employed to identify the damage severity, the threefold cross-validation rule is adopted, the penalty parameter C = 500, the value of radial basis kernel function γ=1, and the value of loss function ε-SVR p=0.01.


### 7.3.1. Results of the Damage Identification Method Based on WPEV and SVM

The identified damage locations based on WPEV and SVM are shown in Table3, and the detection accuracy is shown in Table 4.Table 3
Identified damage locations based on WPEV and SVM.
Scenarios Damage location(floor) Identified damage location(floor) Without noise With 5% noise With 10% noise T 2 6 T 6 10 T 10 14 T 14 18 T 2 6 T 6 10 T 10 14 T 14 18 T 2 6 T 6 10 T 10 14 T 14 18 S1 2 2 2 2 ④ 2 2 2 ③ 2 2 ③ ③ S2 2 2 ③ 2 2 2 2 2 ③ 2 2 2 2 S3 2 2 2 2 2 2 2 2 2 2 2 ③ 2 S4 2 2 2 2 2 2 2 2 2 2 ③ 2 2 S5 3 3 3 3 3 3 3 3 ② 3 3 ② ③ S6 3 3 3 3 3 3 3 3 3 3 3 3 ② S7 3 3 3 3 3 3 3 3 3 3 3 3 3 S8 3 3 ④ 3 3 3 3 3 3 3 ② 3 3 S9 4 4 4 4 4 4 4 ② ③ 4 4 4 ③ S10 4 4 4 4 4 4 4 4 4 4 4 4 ③ S11 4 4 4 4 4 4 4 4 4 4 4 4 4 S12 4 4 4 4 4 4 4 4 4 ③ 4 4 4 S13 5 5 5 5 ④ 5 5 ④ ④ 5 5 ④ 5 S14 5 5 5 5 5 5 5 5 5 5 5 5 5 S15 5 5 5 5 5 5 5 5 5 5 5 5 ④ S16 5 5 5 5 5 5 5 5 5 5 5 5 ④ Note. (1) T26 denotes the TF between nodes 2 and 6 (the rest are the same); (2) circled numbers indicates that the identified damage location is wrong in Tables 3, 6, 10, and 12.Table 4
Detection accuracy of identified damage locations based on WPEV and SVM.
Damage location (floor) Detection accuracy Without noise With 5% noise With 10% noise 5 93.75% (15/16) 87.50% (14/16) 81.25% (13/16) 4 100% (16/16) 87.50% (14/16) 81.25% (13/16) 3 93.75% (15/16) 87.50% (14/16) 75.00% (12/16) 2 87.50% (14/16) 87.50% (14/16) 75.00% (14/16)The WPEVs extracted from TFs (T26, T610, T1014, T1418) are used as inputs for SVM to identify the damage severity, respectively, and the average of identified damage severities is shown in Table 5.Table 5
Average of identified damage severities based on WPEV and SVM.
Scenarios DS (%) Without noise With 5% noise With 10% noise AIDS (%) RE (%) AIDS (%) RE (%) AIDS (%) RE (%) S1 20 26.18 30.88 26.37 31.84 29.36 46.79 S2 40 44.44 11.81 47.4 18.50 43.93 9.83 S3 60 64.98 8.30 65.09 8.48 65.66 9.43 S4 80 84.08 5.10 84.26 5.33 84.89 6.11 S5 20 26.37 31.84 24.14 20.70 27.28 36.42 S6 40 47.4 18.50 42.5 6.24 43.17 7.92 S7 60 65.09 8.48 64.61 7.69 63.7 6.16 S8 80 84.26 5.33 83.66 4.58 84.68 5.85 S9 20 24.14 20.70 24.42 22.08 28.41 42.03 S10 40 43.24 8.11 44.06 10.20 45.05 13.76 S11 60 64.61 7.69 62.81 4.69 63.72 6.20 S12 80 83.66 4.58 82.45 3.06 83.61 4.51 S13 20 24.42 22.08 24.99 24.95 26.9 34.48 S14 40 44.08 10.20 44.58 11.45 44.5 11.26 S15 60 62.81 4.69 62.81 4.69 65.1 8.49 S16 80 82.45 3.06 82.6 3.25 83.57 4.46 Note. DS denotes the damage severity; AIDS denotes the average of the identified damage severity; RE denotes the average relative error.


### 7.3.2. Results of the Damage Identification Method Based on PC and SVM

The identified damage locations based on PC and SVM are shown in Table6, and the detection accuracy is shown in Table 7.Table 6
Identified damage locations based on PC and SVM.
Scenarios Damage location(floor) Identified damage location (floor) Without noise With 5% noise With 10% noise T 2 6 T 6 10 T 10 14 T 14 18 T 2 6 T 6 10 T 10 14 T 14 18 T 2 6 T 6 10 T 10 14 T 14 18 S1 2 2 2 ③ 2 2 2 2 ③ 2 2 2 ④ S2 2 2 2 2 ③ 2 2 2 ③ 2 ④ 2 2 S3 2 2 2 2 2 2 2 2 2 2 2 2 2 S4 2 2 2 2 2 2 2 2 2 2 2 2 ④ S5 3 3 3 3 3 3 3 3 ② 3 3 ④ 3 S6 3 3 3 3 3 3 3 3 3 3 3 3 ④ S7 3 3 3 3 3 3 3 3 3 3 3 3 3 S8 3 3 3 3 3 3 3 3 3 3 ② 3 3 S9 4 4 4 4 4 4 4 4 ③ 4 4 4 4 S10 4 4 4 4 4 4 4 4 4 4 4 4 ③ S11 4 4 4 4 4 4 4 4 4 4 4 4 4 S12 4 4 4 4 4 4 4 4 4 4 4 4 4 S13 5 5 5 5 5 5 5 5 ④ 5 5 5 ④ S14 5 5 5 5 5 5 5 5 5 5 5 5 5 S15 5 5 5 5 5 5 5 5 5 5 5 5 5 S16 5 5 5 5 5 5 5 5 5 5 5 5 5Table 7
Detection accuracy of identified damage locations based on PC and SVM.
Damage location (floor) Accuracy Without noise With 5% noise With 10% noise 5 100% (16/16) 93.75% (15/16) 87.50% (15/16) 4 100% (16/16) 93.75% (15/16) 87.50% (15/16) 3 100% (16/16) 93.75% (15/16) 81.25% (13/16) 2 87.50% (14/16) 87.50% (14/16) 81.25% (13/16)The PCs extracted from TFs (T26, T610, T1014, T1418) are used as inputs for SVM to identify the damage severity, respectively, and the average of identified damage severities is shown in Table 8.Table 8
Average of identified damage severities based on PC and SVM.
Scenarios DS (%) Without noise With 5% noise With 10% noise AIDS (%) RE (%) AIDS (%) RE (%) AIDS (%) RE (%) S1 20 23.52 17.60 24.71 23.56 28.12 40.58 S2 40 42.38 5.96 42.16 5.41 44.93 12.32 S3 60 62.35 3.91 64.3 7.17 65.97 9.95 S4 80 81.58 1.97 81.09 1.36 85.93 7.41 S5 20 22.85 14.26 26.14 30.69 24.43 22.16 S6 40 41.07 2.68 42.54 6.34 42.66 6.64 S7 60 61.23 2.05 62.84 4.73 61.51 2.51 S8 80 80.54 0.67 82.34 2.93 82.3 2.87 S9 20 21.48 7.38 23.15 15.76 31.43 57.13 S10 40 41.53 3.82 43.34 8.42 43 7.49 S11 60 61.43 2.39 62.23 3.72 63.77 6.28 S12 80 80.78 0.98 81.36 1.70 84.67 5.84 S13 20 20.43 2.16 22.64 13.18 22.32 11.61 S14 40 40.26 0.64 40.9 2.26 40.59 1.47 S15 60 60.42 0.70 61.51 2.52 61.23 2.05 S16 80 80.02 0.02 80.72 0.90 82.53 3.16By analyzing the above damage identification results of the numerical simulation, the following can be found.(1) The detection accuracy of the damage location is improving with the enlargement of the damage severity. The results of damage location identification with the IFs (T26,T610) are superior to that with the IFs (T1014,T1418), which means that the acceleration responses close to the upper structure contain more information of damage and are more fit for the damage identification.(2) The relative error of the identified damage severity is relatively high when the brace is lightly damaged.(3) The detection accuracy of the damage location decreases with the increase of the noise level; however, things are different for the relative errors of the identified damage severities, which are fluctuated with the increase of the noise level.(4) The detection accuracy of the proposed method with damage feature constructed by PCA is superior to that constructed by WPD.


## 7.3.1. Results of the Damage Identification Method Based on WPEV and SVM

The identified damage locations based on WPEV and SVM are shown in Table3, and the detection accuracy is shown in Table 4.Table 3
Identified damage locations based on WPEV and SVM.
Scenarios Damage location(floor) Identified damage location(floor) Without noise With 5% noise With 10% noise T 2 6 T 6 10 T 10 14 T 14 18 T 2 6 T 6 10 T 10 14 T 14 18 T 2 6 T 6 10 T 10 14 T 14 18 S1 2 2 2 2 ④ 2 2 2 ③ 2 2 ③ ③ S2 2 2 ③ 2 2 2 2 2 ③ 2 2 2 2 S3 2 2 2 2 2 2 2 2 2 2 2 ③ 2 S4 2 2 2 2 2 2 2 2 2 2 ③ 2 2 S5 3 3 3 3 3 3 3 3 ② 3 3 ② ③ S6 3 3 3 3 3 3 3 3 3 3 3 3 ② S7 3 3 3 3 3 3 3 3 3 3 3 3 3 S8 3 3 ④ 3 3 3 3 3 3 3 ② 3 3 S9 4 4 4 4 4 4 4 ② ③ 4 4 4 ③ S10 4 4 4 4 4 4 4 4 4 4 4 4 ③ S11 4 4 4 4 4 4 4 4 4 4 4 4 4 S12 4 4 4 4 4 4 4 4 4 ③ 4 4 4 S13 5 5 5 5 ④ 5 5 ④ ④ 5 5 ④ 5 S14 5 5 5 5 5 5 5 5 5 5 5 5 5 S15 5 5 5 5 5 5 5 5 5 5 5 5 ④ S16 5 5 5 5 5 5 5 5 5 5 5 5 ④ Note. (1) T26 denotes the TF between nodes 2 and 6 (the rest are the same); (2) circled numbers indicates that the identified damage location is wrong in Tables 3, 6, 10, and 12.Table 4
Detection accuracy of identified damage locations based on WPEV and SVM.
Damage location (floor) Detection accuracy Without noise With 5% noise With 10% noise 5 93.75% (15/16) 87.50% (14/16) 81.25% (13/16) 4 100% (16/16) 87.50% (14/16) 81.25% (13/16) 3 93.75% (15/16) 87.50% (14/16) 75.00% (12/16) 2 87.50% (14/16) 87.50% (14/16) 75.00% (14/16)The WPEVs extracted from TFs (T26, T610, T1014, T1418) are used as inputs for SVM to identify the damage severity, respectively, and the average of identified damage severities is shown in Table 5.Table 5
Average of identified damage severities based on WPEV and SVM.
Scenarios DS (%) Without noise With 5% noise With 10% noise AIDS (%) RE (%) AIDS (%) RE (%) AIDS (%) RE (%) S1 20 26.18 30.88 26.37 31.84 29.36 46.79 S2 40 44.44 11.81 47.4 18.50 43.93 9.83 S3 60 64.98 8.30 65.09 8.48 65.66 9.43 S4 80 84.08 5.10 84.26 5.33 84.89 6.11 S5 20 26.37 31.84 24.14 20.70 27.28 36.42 S6 40 47.4 18.50 42.5 6.24 43.17 7.92 S7 60 65.09 8.48 64.61 7.69 63.7 6.16 S8 80 84.26 5.33 83.66 4.58 84.68 5.85 S9 20 24.14 20.70 24.42 22.08 28.41 42.03 S10 40 43.24 8.11 44.06 10.20 45.05 13.76 S11 60 64.61 7.69 62.81 4.69 63.72 6.20 S12 80 83.66 4.58 82.45 3.06 83.61 4.51 S13 20 24.42 22.08 24.99 24.95 26.9 34.48 S14 40 44.08 10.20 44.58 11.45 44.5 11.26 S15 60 62.81 4.69 62.81 4.69 65.1 8.49 S16 80 82.45 3.06 82.6 3.25 83.57 4.46 Note. DS denotes the damage severity; AIDS denotes the average of the identified damage severity; RE denotes the average relative error.


## 7.3.2. Results of the Damage Identification Method Based on PC and SVM

The identified damage locations based on PC and SVM are shown in Table6, and the detection accuracy is shown in Table 7.Table 6
Identified damage locations based on PC and SVM.
Scenarios Damage location(floor) Identified damage location (floor) Without noise With 5% noise With 10% noise T 2 6 T 6 10 T 10 14 T 14 18 T 2 6 T 6 10 T 10 14 T 14 18 T 2 6 T 6 10 T 10 14 T 14 18 S1 2 2 2 ③ 2 2 2 2 ③ 2 2 2 ④ S2 2 2 2 2 ③ 2 2 2 ③ 2 ④ 2 2 S3 2 2 2 2 2 2 2 2 2 2 2 2 2 S4 2 2 2 2 2 2 2 2 2 2 2 2 ④ S5 3 3 3 3 3 3 3 3 ② 3 3 ④ 3 S6 3 3 3 3 3 3 3 3 3 3 3 3 ④ S7 3 3 3 3 3 3 3 3 3 3 3 3 3 S8 3 3 3 3 3 3 3 3 3 3 ② 3 3 S9 4 4 4 4 4 4 4 4 ③ 4 4 4 4 S10 4 4 4 4 4 4 4 4 4 4 4 4 ③ S11 4 4 4 4 4 4 4 4 4 4 4 4 4 S12 4 4 4 4 4 4 4 4 4 4 4 4 4 S13 5 5 5 5 5 5 5 5 ④ 5 5 5 ④ S14 5 5 5 5 5 5 5 5 5 5 5 5 5 S15 5 5 5 5 5 5 5 5 5 5 5 5 5 S16 5 5 5 5 5 5 5 5 5 5 5 5 5Table 7
Detection accuracy of identified damage locations based on PC and SVM.
Damage location (floor) Accuracy Without noise With 5% noise With 10% noise 5 100% (16/16) 93.75% (15/16) 87.50% (15/16) 4 100% (16/16) 93.75% (15/16) 87.50% (15/16) 3 100% (16/16) 93.75% (15/16) 81.25% (13/16) 2 87.50% (14/16) 87.50% (14/16) 81.25% (13/16)The PCs extracted from TFs (T26, T610, T1014, T1418) are used as inputs for SVM to identify the damage severity, respectively, and the average of identified damage severities is shown in Table 8.Table 8
Average of identified damage severities based on PC and SVM.
Scenarios DS (%) Without noise With 5% noise With 10% noise AIDS (%) RE (%) AIDS (%) RE (%) AIDS (%) RE (%) S1 20 23.52 17.60 24.71 23.56 28.12 40.58 S2 40 42.38 5.96 42.16 5.41 44.93 12.32 S3 60 62.35 3.91 64.3 7.17 65.97 9.95 S4 80 81.58 1.97 81.09 1.36 85.93 7.41 S5 20 22.85 14.26 26.14 30.69 24.43 22.16 S6 40 41.07 2.68 42.54 6.34 42.66 6.64 S7 60 61.23 2.05 62.84 4.73 61.51 2.51 S8 80 80.54 0.67 82.34 2.93 82.3 2.87 S9 20 21.48 7.38 23.15 15.76 31.43 57.13 S10 40 41.53 3.82 43.34 8.42 43 7.49 S11 60 61.43 2.39 62.23 3.72 63.77 6.28 S12 80 80.78 0.98 81.36 1.70 84.67 5.84 S13 20 20.43 2.16 22.64 13.18 22.32 11.61 S14 40 40.26 0.64 40.9 2.26 40.59 1.47 S15 60 60.42 0.70 61.51 2.52 61.23 2.05 S16 80 80.02 0.02 80.72 0.90 82.53 3.16By analyzing the above damage identification results of the numerical simulation, the following can be found.(1) The detection accuracy of the damage location is improving with the enlargement of the damage severity. The results of damage location identification with the IFs (T26,T610) are superior to that with the IFs (T1014,T1418), which means that the acceleration responses close to the upper structure contain more information of damage and are more fit for the damage identification.(2) The relative error of the identified damage severity is relatively high when the brace is lightly damaged.(3) The detection accuracy of the damage location decreases with the increase of the noise level; however, things are different for the relative errors of the identified damage severities, which are fluctuated with the increase of the noise level.(4) The detection accuracy of the proposed method with damage feature constructed by PCA is superior to that constructed by WPD.


## 8. Shaking Table Test

In order to further verify the feasibility of the damage identification method proposed in this study, a shaking table model test, as shown in Figure4, is carried out. The test model and parameters are consistent with the numerical simulation model.Figure 4
The offshore platform shaking table model test.The instruments consist of 8 capacitive uniaxial accelerometers (Model 2220-005, Silicon Designs Inc, USA) for response measurement, an electric shaking table (DY-1000-8, Suzhou Test Instrument General Plant, China), and a measurement system (PL16-DCB8, Integrated Measurement and Control Cooperation, Germany) for data acquisition. The accelerometers are mounted on the joints between beams and columns in Y direction.There are several spliced connections on members that will simulate the damage in the structure. As shown in Figure4, if the replacement part is installed, the member is damaged partly; if the four bolts and the shim in a spliced connection are removed, the member is damaged completely; if the bolts and shim are reinstalled, the damaged member recovers.Similar to the numerical study, only the damage of single brace in the side where the braces of M1 and M2 locate is considered. The damage scenarios are shown in Table9; there are 4 training scenarios (C1–C4) and 2 test scenarios (S1, S2).Table 9
Damage scenarios of the shaking table model test.
Scenarios Damaged brace (floor) Damage condition (%) Classification training label Regression training label Classification testing label Regression testing label (%) (floor) (%) (floor) C1 M 1 ( 4 ) Φ 8 × 1.2 (78) 4 78 - - C2 M 1 ( 4 ) Removed (100) 4 100 - - C3 M 2 ( 3 ) Φ 8 × 1.2 (78) 3 78 - - C4 M 2 ( 3 ) Removed (100) 3 100 - - S1 M 1 ( 4 ) Φ 7 × 1.0 (88) - - 4 88 S2 M 2 ( 3 ) Φ 7 × 1.0 (88) - - 3 88 Note. Φ8×1.2 (78) denotes that the diameter and thickness of the replacement part is 8 mm and 1.2 mm; figures in brackets refer to damage severity.The white noise with the sampling frequency of 250 Hz is used as the input ground excitation inY direction, and the horizontal acceleration responses of each floor are collected for each scenario. The sampling frequency is 500 Hz and the duration is 35 s. For each scenario, the data from each accelerometer is processed by subtracting its mean.The horizontal acceleration responses on joints 2, 6, 10, 14, and 18 from 2 s to 22 s are selected to calculate the TFs (T26,T610,T1014,T1418), and the rest of the analysis procedures are similar to the numerical simulation.The classification algorithm C-SVC inLIBsvm is employed to identify the damage location, the penalty parameter C = 650, and the value of radial basis kernel function γ=1.The regression algorithmε-SVR inLIBsvm is employed to identify the damage severity, the threefold cross-validation rule is adopted, the penalty parameter C = 650, the value of radial basis kernel function γ=1, and the value of loss function ε-SVR p=0.01.


### 8.1. Results of the Damage Identification Method Based on WPEV and SVM

The identified damage locations based on WPEV and SVM are shown in Table10, and the detection accuracy is 62.5% (5/8).Table 10
Identified damage locations based on WPEV and SVM.
Scenario Damage location Identified damage location (floor) (floor) T 2 6 T 6 10 T 10 14 T 14 18 S1 4 4 4 4 ③ S2 3 3 ④ ④ 3The WPEVs extracted from TFs (T26, T610, T1014, T1418) are used as inputs for SVM to identify damage severity, respectively, and the average of identified damage severities is shown in Table 11.Table 11
The average of identified damage severities based on WPEV and SVM.
Scenario DS (%) AIDS (%) RE (%) S1 88 93.36 6.09 S2 88 95.66 8.71


### 8.2. Results of the Damage Identification Method Based on PC and SVM

The identified damage locations based on PC and SVM are shown in Table12, and the detection accuracy is 75% (6/8).Table 12
Identified damage locations based on PC and SVM.
Scenario Damage location Identified damage location (floor) (floor) T 2 6 T 6 10 T 10 14 T 14 18 E2 (88%) 4 4 4 4 ③ E5 (88%) 3 3 3 ④ 3The PCs extracted from TFs (T26, T610, T1014, T1418) are used as inputs for SVM to identify the damage severity, respectively, and the average of identified damage severities is shown in Table 13.Table 13
Average of identified damage severities based on PC and SVM.
Scenario DS (%) AIDS (%) RE (%) E2 88 92.69 5.33 E5 88 94.89 7.83By analyzing the above damage identification results of the shaking table model test, the following can be found.(1) The results of identified damage location withT26 are superior to that with T1014 and T1418, which means that the acceleration responses measured from the upper structure contain more information of damage and are more fit for the damage identification.(2) Similar to the numerical simulation, the detection accuracy of the proposed method with damage feature constructed by PCA is superior to that constructed by WPD.(3) The result of the shaking table model test is inferior to that of the numerical simulation; the reason may be due to the influence of noise and operational errors.


## 8.1. Results of the Damage Identification Method Based on WPEV and SVM

The identified damage locations based on WPEV and SVM are shown in Table10, and the detection accuracy is 62.5% (5/8).Table 10
Identified damage locations based on WPEV and SVM.
Scenario Damage location Identified damage location (floor) (floor) T 2 6 T 6 10 T 10 14 T 14 18 S1 4 4 4 4 ③ S2 3 3 ④ ④ 3The WPEVs extracted from TFs (T26, T610, T1014, T1418) are used as inputs for SVM to identify damage severity, respectively, and the average of identified damage severities is shown in Table 11.Table 11
The average of identified damage severities based on WPEV and SVM.
Scenario DS (%) AIDS (%) RE (%) S1 88 93.36 6.09 S2 88 95.66 8.71


## 8.2. Results of the Damage Identification Method Based on PC and SVM

The identified damage locations based on PC and SVM are shown in Table12, and the detection accuracy is 75% (6/8).Table 12
Identified damage locations based on PC and SVM.
Scenario Damage location Identified damage location (floor) (floor) T 2 6 T 6 10 T 10 14 T 14 18 E2 (88%) 4 4 4 4 ③ E5 (88%) 3 3 3 ④ 3The PCs extracted from TFs (T26, T610, T1014, T1418) are used as inputs for SVM to identify the damage severity, respectively, and the average of identified damage severities is shown in Table 13.Table 13
Average of identified damage severities based on PC and SVM.
Scenario DS (%) AIDS (%) RE (%) E2 88 92.69 5.33 E5 88 94.89 7.83By analyzing the above damage identification results of the shaking table model test, the following can be found.(1) The results of identified damage location withT26 are superior to that with T1014 and T1418, which means that the acceleration responses measured from the upper structure contain more information of damage and are more fit for the damage identification.(2) Similar to the numerical simulation, the detection accuracy of the proposed method with damage feature constructed by PCA is superior to that constructed by WPD.(3) The result of the shaking table model test is inferior to that of the numerical simulation; the reason may be due to the influence of noise and operational errors.


## 9. Conclusions

In this study, a novel damage identification method based on TF and SVM is proposed. The damage features, extracted from the amplitude of TF by WPD and PCA, are used as inputs for SVM to identify the damage location and damage severity, respectively. The proposed method is verified with the numerical simulation and shaking table model test of an offshore platform under white noise excitation, and some conclusions are drawn as follows.(1) The acceleration responses measured from the upper structure contain more information of damage and are more suitable for the damage identification.(2) The detection accuracy of the proposed method with damage feature constructed by PCA is superior to that constructed by WPD.(3) The proposed method does not need the information of the excitation and the data from undamaged structure, needs only small size samples, and has certain antinoise ability.(4) The selection of the measuring joints has a great influence on the proposed damage identification method, so, how to select the locations of measuring joints needs further study in the future.



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*Source: 4892428-2018-06-11.xml*