Two-stage damage identification method based on fractal theory and whale optimization algorithm

Abstract

Structural damage identification based on time domain method of vibration response has been widely developed in the recent decades, however, it still confronts some difficulties, such as measurement noise and model error. This paper proposes a novel two-stage damage identification method based on fractal dimension and whale optimization algorithm (WOA). In this study, based on vibration data, the difference in curvature of fractal dimension (DCFD) is used as the damage index to identify the location of suspicious damage elements in the first stage. A new objective function is proposed based on the curvature of fractal dimension (CFD) of acceleration signal, and the WOA is used to estimate the severity of the suspicious damaged element in the second stage. Firstly, the validity of the proposed method is verified by a numerical simply supported beam, and the results exhibit good damage identification ability. Then different noise levels (5% ~ 20%) are introduced into the dynamic responses to verify its robustness, the result shows that the method is of good anti-noise ability in the first stage. Although the second stage is slightly sensitive to noise, it can still effectively identify the severity of damage. Secondly, the vibration testing of a steel I-beam is designed to verify the rationality of the method in the application of actual structure. Finally, based on the simulated vibration test data of the I-40 Bridge, the applicability of the method to complex civil structure is verified, which shows that the method still has good ability to identify the location and severity of damage in complex structure and is of great significance in practical application.

Keywords

Damage identification time domain fractal dimension whale optimization algorithm two-stage damage identification method

Introduction

Civil engineering, bridges, buildings and dams etc., bear dead load, live load, natural disasters and man-made disasters during their service periods (Seo et al., 2016). These loads or actions will cause cumulative damage to the structures and reduce their safety. In order to prevent catastrophic failure and lengthen the service life of the structures, early damage detection technology is especially crucial (Wang et al., 2019). Structural health monitoring (SHM) technology has been used to monitor the behavior of structures. Generally, damage detection technology could be divided into two categories: model-based method and model-free method. Many model-based methods used modal parameters or their derivatives to define structural damage index, including modal shape (Roy and Ray-Chaudhuri, 2013), sensitivity matrix (Naseralavi et al., 2010), modal flexibility (Sung et al., 2014) and modal strain energy (Tan et al., 2017). However, these damage indexes are not very sensitive to damage due to the error of finite element model and will lead to the uncertainty of damage location and quantification in the model-based damage detection method (Law et al., 2010). On the contrary, if the damage features are directly extracted from the dynamic response, the location and severity of damage could be more accurately identified (An et al., 2016). The model-free method is to extract the structural damage index directly and detect the corresponding damage by processing the structural vibration response time history signal, which could save computation cost and reduce optimization time.

In recent years, various signal processing techniques have been widely used in damage identification, allowing people to directly extract features from vibration signals for damage detection (Bagheri et al., 2017; Morovati and Kazemi, 2016). Among them, techniques based on frequency domain are often used for damage identification, such as support vector machine (SVM) (Huang et al., 2021a) and multiple signal classification (MUSIC) techniques (Gkoktsi and Giaralis, 2020); Wang et al. (2018) proposed a damage identification method characterized by sparse wavelet reconstruction residual (SWRR) based on wavelet packet transform and sparse representation theory. Nair and Kiremidjian (2007) modeled the structural vibration signal as an autoregressive moving average (ARMA) process which damage was detected by observing a migration of the extracted autoregressive moving coefficients with damage. In addition, Obrien et al. (2017) used the empirical mode decomposition (EMD) method to decompose the signal into its main components and proposed a damage detection method using intrinsic mode functions (IMFs). Furthermore, other model-free based methods have also attracted wide attention from researchers, such as wavelet analysis methods (Kesavan and Kiremidjian, 2012; Pnevmatikos, 2010), autoregressive moving average with exogenous inputs (ARMAX) model (Xing and Mita, 2012), and fractal dimension-based method (An et al., 2014). There are many applications of the vibration-based method to other civil structures. Nguyen et al. (2015) proposed a vibration-based damage detection method for the wind turbine tower (WTT) and the result showed that the mode-shape-based damage detection (MBDD) method was better than the frequency-based damage detection (FBDD) method, but the damage identification near the free-end of the WTT was not accurate. Lee et al. (2012) used autoregressive (AR) model for time-series analysis and power spectral density (PSD) for frequency-domain analysis to identify the damage in the caisson structure by using acceleration responses measured from a single point. The proposed methods were successful in identified damage to foundation-structure interface of the caisson in different conditions.

In the 19th century, the fractal theory was often applied to the study of concrete materials and geotechnical soils. However, structural damage identification based on fractal theory was rare. Until the 20th century, many excellent damage identification methods based on fractal theory emerged in the field of structural damage identification. Among them, fractal dimension (FD) has proven to be an effective method to identify the damage in simple beam-type structures and plate-type structures (Cao and Qiao, 2009; Lei et al., 2019). Moreno-Gomez et al. (2020) studied the effectiveness of Katz fractal dimension (KFD), Higuchi fractal dimension (HFD), box fractal dimension (BFD), Petrosian fractal dimension (PFD) and Sevcik fractal dimension (SFD), and the results showed that Higuchi and Petrosian fractal dimension algorithms exceed the others nonlinear measurements in efficiency to discriminate between a healthy structure and a damage one caused by corrosion. Huang et al. (2018a) used a multi-task sparse Bayesian learning method to predict damage location based on the FD of Katz, and the results showed that both single-point and multi-point damage identification were very accurate. Shi and Qiao, 2018 proposed a new surface FD called edge perimeter dimension (EPD), from which an EPD-based window dimension locus (EPD-WDL) algorithm for irregularity or damage identification of plate-type structures was established. Tao et al. (2018) proposed a data-driven method using the FD of time-frequency feature (TFF) of structural seismic dynamic responses at measured, and the damage detection demonstrated that the proposed method could locate the floor where the plastic hinge occurs. Lee and Eun (2015) extracted the waveforms FD from the measured frequency response function (FRF) data in the frequency domain and identified the damage location and severity of steel beams by the curvature changes of waveforms.

Generally, in the process of damage quantification, the quantification of structural damage needs to be realized by using algorithms. Among them, some researchers locate and quantitatively analyze directly through the algorithm (called the direct method), but others locate the damage first and then perform damage quantification (two-stage method). The direct method is a traditional method, while the two-stage method is a relatively new method. The advantage of two-stage method is to reduce the number of parameters to be identified, thereby alleviating the instability and divergence (Hou et al., 2021). Dinh-Cong et al. (2021) used Modal kinetic energy change ratio (MKECR) method to construct damage equations and solve them for positioning in the first stage, and in the second stage, the lightning attachment procedure optimization (LAPO) was introduced to complete the damage assessment. Nick et al. (2021) determined the location of the loss based on the loss index method of modal strain energy, and an artificial neural network was subsequently used to evaluate the severity of the damage. Ahmadi-Nedushan and Fathnejat, 2020 identified suspicious damaged elements based on the element modal strain energy and the damage index of the diagonal element of the structural matrix, and by using the modal response of the structure, the modified teaching-learning-based optimization (MTLBO) algorithm was used to estimate the damage severity of the suspicious damaged element. Wang et al. (2020) located potential damaged elements based on modal energy-based damage index (MEBI) in the first stage, and used beetle antenna search (BAS) to estimate the damage severity of these potential elements. In summary, the two-stage damage identification method based on model and intelligent algorithm has been well applied. However, structural damage detection inevitably has uncertainties, such as measurement noise and modeling errors, which will affect the performance of damage identification (Hou et al., 2021). In the method proposed in this paper, damage location is not based on finite element model in the first stage, which can avoid the errors caused by the model. At the same time, the parameters that need to be identified are reduced based on the results of the first stage, which improves the computational efficiency and accuracy in the second stage.

In this paper, a novel two-stage damage identification method is proposed combining both the model-based and model-free methods. The main idea of this method can be summarized as follows: (1) Damaged elements of structures are identified through the difference in curvature of fractal dimension (DCFD) of acceleration data. (2) The severity of damaged elements identified in the previous step is calculated through whale optimization algorithm (WOA). Among them, the acceleration signal is processed through the fractal theory to complete the damage location in the first stage. The damage location process is independent of structural model, which can avoid the errors caused by finite model. At the same time, it can reduce the parameters that need to be identified in the second stage, which effectively improves the accuracy and stability of damage identification. In the second stage, a new objective function is constructed based on the curvature of fractal dimension (CFD) of acceleration signal, and WOA is used to evaluate the severity of the damaged elements. In order to verify the effectiveness of the method and its applicability to different structural damage, several different damage cases are designed for the numerical simulation of a simply supported beam and the I-40 Bridge. Additionally, a vibration test of the steel I-beam is also conducted, the location and severity of structural damage are successfully identified and the robustness of the method is verified.

Methodology

Time domain responses

In this study, the damage identification is performed by applying external excitation to a structure to extract the time-domain information. The differential equation of motion of a linear structure is

M X^{″} (t) + C X^{'} (t) + K X (t) = F (t)

(1)

where M, C and K denote the mass, damping and stiffness matrices of the structure respectively; $X ″, X', X$ and F(t) represent the nodal acceleration, velocity, displacement and external force vector respectively, which are all time-dependent. The Newmark-β method is a better algorithm among several commonly used linear acceleration methods. In the paper, the time-stepping Newmark-β method is adopted to solve the differential equation to obtain the time domain response. The related equations can be summarized as follows

X'_{n + 1} = X'_{n} + [(1 - δ) X ″_{n} + δ X ″_{n + 1}]

(2)

X_{n + 1} = X_{n} + X'_{n} Δ t + [(- \frac{1}{2} - α) X ″_{n} + α X ″_{n + 1}]

(3)

where n is the node number, $Δ t$ is the time step, α and δ are the parameters adjusted according to the accuracy and stability requirements of integration, respectively. Newmark-β method is originally proposed as an unconditionally-stable integral method, which is the constant average acceleration method, that is, when δ≥ 0.5, α≥ 0.25 (0.5+δ)², the Newmark-β method is unconditionally-stable, in this paper, δ = 0.5, α = 0.25. According to equations (2) and (3), the expression of $X ″_{n + 1}$ can be obtained as follows

X ″_{n + 1} = \frac{1}{α Δ t^{2}} (X_{n + 1} - X_{n}) - \frac{1}{α Δ t} X'_{n} - (\frac{1}{2 α} - 1) X ″_{n}

(4)

According to the Newmark-β method, the displacement at step n+1 can be calculated according to equation (5)

X_{n + 1} = {[\tilde{K}]}^{- 1} \times [\tilde{F}]

(5)

where [K] and [F] are the equivalent stiffness matrix and effective load of the structure, respectively, which can be calculated according to equations (6) and (7) respectively

[\tilde{K}] = [K] + a_{0} [M] + a_{1} [C]

(6)

\begin{matrix} [\tilde{F}] = [F] + [M] (a_{0} X_{n} - a_{2} X'_{n} - a_{3} X ″_{n}) \\ + [C] (a_{1} X_{n} - a_{4} X'_{n} + a_{5} X ″_{n}) \end{matrix}

(7)

From the above equations, the acceleration and velocity can be expressed as follows

X ″_{n + 1} = a_{0} (X_{n + 1} - X_{n}) - a_{2} X'_{n} - a_{3} X ″_{n}

(8)

X'_{n + 1} = X'_{n} - a_{6} X ″_{n} - a_{7} X ″_{n + 1}

(9)

where the factors a_i (i = 1, … ,8) are given as

\begin{matrix} a_{0} = \frac{1}{α Δ t^{2}}, a_{1} = \frac{δ}{α Δ t}, a_{2} = \frac{1}{α Δ t}, a_{3} = \frac{1}{2 α} - 1 \\ a_{4} = \frac{δ}{α} - 1, a_{5} = \frac{Δ t}{2} (\frac{δ}{α} - 2), a_{6} = (1 - δ) Δ t, a_{7} = δ Δ t \end{matrix}

(10)

The time-dependent acceleration of a structure contains very comprehensive and useful information that can be used to identify the damage in the structure. Structural damage will lead to the change in dynamical responses such as the acceleration of the structure (Fallahian et al., 2018). equation (7) expresses that the acceleration response of the structure is obtained by applying dynamic loads. In this research, the acceleration response is analyzed using fractal theory, and damage identification is then performed. It is worth noting that the following FD represent the FD of acceleration in the paper.

Fundamentals of fractal dimension

The FD is the key of fractal theory, which can quantitate the irregularity of signal and describe the complexity of the fractal set (Hadjileontiadis et al., 2005). The FD in structural damage identification includes modal FD and time-series signal FD, which can effectively describe the state of a system. The time-domain response is one of the main characteristics of the structure. It can be obtained by direct measurement. The box dimension is a commonly used dimension of time-domain signals for analysis. The calculation process is expressed as following: First, supposing R is a bounded figure on the plane, and taking a small box with side length a to cover the figure R. Second, part of the box is empty, which covers a part of the curve. When a→ 0, the box dimension equation is obtained

D = \frac{\ln R (a)}{\ln a}

(11)

where D is the box dimension, R(a) is the number of non-empty boxes, and a is the side length of the box.

According to equation (11), the classification box dimension of the signal is calculated in the following way to introduce the box dimension into structural damage identification (An and Ou 2012)

{\begin{matrix} FD = - lim_{Δ T \to 0} [\frac{\ln (\sum | a_{i + 1} - a_{i} | / | a_{i + 1} - a_{i} | Δ T Δ T)}{\ln Δ T}] \\ FD ≅ - \frac{\ln (\sum_{i = 1}^{N - 1} | a_{i + 1} - a_{i} | / | a_{i + 1} - a_{i} | Δ T Δ T)}{\ln Δ T} \end{matrix}

(12)

where T is the total sampling time, ΔT is the sampling time interval, a₁, a₂, … , a_N are the collected structural vibration signal in time domain, and N is the number of sampling points.

To discuss the sensitivity of FD response as time-domain data to structural damage, a numerical case of a simply supported beam is introduced, which is shown as Figure 1, its length, width, and height are 5 m, 0.6 m, and 0.12 m respectively. The blue point is the time domain response measuring point, and the numbers in the circle are the number of the measuring points. The elastic modulus of the structure is 30 GPa, the cross-sectional area is 0.072 m², the mass density is 2360 kg/m³, and the section moment of inertia is 8.64×10⁻⁵ m⁴. A vertical harmonic excitation with an amplitude of 200 N is applied at the impact point (measuring point eight) and the acceleration of the simply supported beam is recorded at 15 measuring points as the time history response. In this example, damage is defined as a relative reduction of stiffness in each element as follows

K = \sum_{i = 1}^{nele} (1 - θ) k_{i}

(13)

Figure 1.

The simply supported beam.

where K is the global stiffness matrix, k_i and θ_i are i-th element stiffness matrix in the global coordinate and its damage severity, respectively, nele is total number of elements. Simultaneously, actual structures are often affected by environmental factors, such as temperature variations and weather changes, which have an impact on the performance of damage identification. Studies shown that the temperature variations will cause the vibration features of structures, which make inaccurate damage identification (Huynh et al., 2015 and Huang et al., 2020a). Considering the temperature variation, the relationship between the ambient temperature and Young’s modulus is shown in Figure 2 (Huang et al., 2018b). In the finite element model of civil structures, structural damage can be simulated by stiffness changes. Introducing the temperature effect into the element stiffness matrix of structure, the problem of damage identification considering ambient temperature variations can be written as follows (Huang et al., 2019)

K (T) = \sum_{i = 1}^{nele} (1 - θ_{i}) k_{i} (T)

(14)

Figure 2.

The relationship between the material’s elastic modulus and temperature variation: (a) Concrete and (b) Steel.

where, K(T) and k_i(T) are global and i-th element stiffness matrix in the global coordinate when the ambient temperature is T. The influence of temperature variation on damage identification performance can be largely eliminated by equation (14). It is assumed that the 7-th element is damaged by 40%, the acceleration response signal of 7-th element and CFD of the intact 15-element are shown in Figure 3.

Figure 3.

Vibration data: (a) Acceleration response signal of 7-th element and (b) The FD curve of the intact 15-element.

Figure 3(a) shows that the acceleration of the damaged element changed significantly. The FD after damage is greater than the FD before damage and changes obviously at the damaged element points, which proves that the FD can reflect the FD before and after damage (Figure 3(b)). The deviation degree of the curve from a straight line can usually be expressed by curvature. The following equation is defined as the FD waveform concerning curvature

C_{i} = \frac{D_{i - 1} - 2 D_{i} + D_{i + 1}}{h^{2}}

(15)

where C_i is the curvature of FD waveform at node i; D_i is the FD at node i, and h is the element length in numerical model.

Similarly, this paper uses CFD as damage identification index, which is calculated as follows

CF D_{i} = \frac{F D_{i - 1} - 2 F D_{i} + F D_{i + 1}}{h^{2}}

(16)

The curvature difference value of CFD (DCFD) at node i before and after damage is calculated according to equation (17)

DCF D_{i} = CF D_{ai} - CF D_{bi}

(17)

where DCFD_i is the difference in curvature of fractal dimension, CFD_ai is the pre-damage curvature of fractal dimension, CFD_bi is the post-damage curvature of fractal dimension, and i is the measuring point number.

All negative values of curvature difference should be eliminated. The reason is that the FD of the acceleration signal at point i after damage is larger than the value before damage. According to equations (16) and (17), the CFD_i is smaller than that before damage. Therefore, the curvature difference at the damaged elements should be a positive value.

Objective function

The process of damage identification is usually realized by minimizing the objective function based on the damage identification indicators. Therefore, the choice of objective function is the key in damage identification, which plays a fundamental role in the convergence of optimization algorithms (Seyedpoor et al., 2019). Many scholars have studied different indicators to establish the objective function for quantitative analysis of structural damage. This paper defines an objective function based on the CFD of acceleration

f (x) = \frac{{‖ CF D^{E} - CF D^{A} (x) ‖}^{2}}{{‖ CF D^{E} ‖}^{2}}

(18)

where CFD represent the matrix calculated by equation (16), the superscripts E and A represent the experimental and the analytical, respectively, || || represents the Frobenius norm of the matrix, and x is the vector of design variables, which stands for the damage severity of structural elements.

Assuming the number of measuring points is n, the FD vectors (CFD and CFD(x)) would be compiled by n FD vectors that corresponding to n measuring point, which can be defined as follows

CF D_{1} = {}, \dots, CF D_{n} = {} \Rightarrow {\begin{matrix} CF D_{1} \\ CF D_{2} \\ : \\ CF D_{n} \end{matrix}}

(19)

The optimization algorithm

WOA is a population-based optimization algorithm designed by Mirjalili and Lewis (2016), which simulates the hunting behavior of humpback whales. In the predation process, three operators are used to simulate the predation, hunting, and foraging behaviors of whales. Therefore, according to the characteristics of the bubble-net predation method of humpback whales, WOA can be divided into three different stages: surrounding prey, bubble-net attack and searching for prey (Huang et al., 2021b). The WOA algorithm is implemented through the following steps:

1) Initialization. The original parameters, constants, and initial population are recognized. Like other evolutionary algorithms, WOA starts the search procedure from the initial population, which is arbitrarily generated in the search space as

X^{l} \leq X_{i} \leq X^{u}, 1, 2, \dots, dn

(20)

where X^l and X^u are the lower and upper limits of the variable, respectively, nd is the number of initial populations.

2) Encircling prey. Humpback whales can recognize the location of their preys and surround them. Since the location of the predation behavior in the search space is unknown, WOA assumes that the best candidate of the current solution is the target prey. After determining the location of the prey, other whales in the population will try to update their locations according to the location of the prey. This behavior is represented by the following equation

\vec{D} = | \vec{C} \cdot {\vec{X}}^{*} (t) - \vec{X} (t) |

(21)

\vec{X} (t + 1) = {\vec{X}}^{*} (t) - \vec{A} \cdot \vec{D}

(22)

where t indicates the current iteration, $\vec{A}$ and $\vec{C}$ are the coefficient vectors, $\vec{X^{*}}$ is the position vector of the best solution obtained so far, $\vec{X}$ is the position vector, | | is the absolute value, and is an element-by-element multiplication. It is worth mentioning here that $\vec{X^{*}}$ should be updated in each iteration if there is a better solution.

3) Bubble-net attacking method. Two procedures including shrinking encircling and spiraling updating are designed to mathematically model the bubble-net attacking behavior, where there is a probability of 50% to choose shrinking encircling and spiraling updating. The mathematical model is shown as follows

\vec{X} (t + 1) = {\begin{matrix} \vec{X^{*}} (t) - \vec{A} \cdot \vec{D} & p < 0.5 & (1) \\ \vec{D'} \cdot e^{bl} \cos (2 π l) + \vec{X^{*}} (t) & p < 0.5 & (2) \end{matrix}

(23)

where $\vec{D'} = | \vec{X^{*}} (t) - \vec{D} (t) |$ indicates the distance of the i-th whale to the prey (best solution obtained so far), b is a constant for defining the shape of the logarithmic spiral, l is a random number in [−1,1], p is a random number in [0,1].

4) Search for prey. In the prey exploration stage, unlike the development stage, when $\vec{A}$ satisfies $\vec{A} \geq 0$ the whale does not update its position solely based on the position of the prey, but will update its position based on the position of the whale selected randomly. The mathematical model is as follows

\vec{D} = | \vec{C} \cdot {\vec{X}}_{rand} - \vec{X} |

(24)

\vec{X} (t + 1) = {\vec{X}}_{rand} - \vec{A} \cdot \vec{D}

(25)

where ${\vec{X}}_{rand}$ is a random position vector (a random whale) chosen from the current population.

The proposed damage identification method

The main steps of the two-stage structural damage identification method based on FD and WOA are as follows, and its specific process is shown in Figure 4:

1) The DCFD is used to judge the damage location when a structure may be damaged, reduce the parameters to be identified, and improve the accuracy of damage location;

2) Only the suspicious damaged elements detected in the first stage are adopted to identify the severity of damage, and the WOA is used to find the optimal solution based on constructed objective function.

Figure 4.

The flow chart of damage identification.

Numerical example

In order to verify the effectiveness of the proposed two-stage damage identification method, damage identification of the simply supported beam mentioned above is carried out. Under the same condition, the tests are conducted by introducing the following three damage cases: single-point damage, two-point damage and three-point damage (Table 1). In practical applications, noise will interfere with the measurement of data, thereby affect the accuracy of identification results. To verify the noise robustness of this method, a noise with different levels of 5%, 10% and 20% is imposed to contaminate the acceleration (Fallahian et al., 2018; Seyedpoor et al., 2019). The effect of measurement noise on acceleration is calculated as

a_{d}^{noise} = a_{d} \times [1 + 2 (Random - 1) \times noise]

(26)

Table 1.

Damage cases of the simply supported beam.

Case	Element	Preset damage severity (%)
1	7	20
2	3,6	20,10
3	4,5,10	15,10,5

where, noise is the level of measurement noise, $a_{d}^{noise}$ and a_d are the noisy damaged acceleration and noise-free damaged acceleration, respectively, Random is the random vector with zero mean, and all elements of this vector are the random values between −1 and 1. Because noise is random when different levels of measurement noise are added to contaminate the acceleration signal, different acceleration signals will be generated for each generation. The identification results of damage location in the first stage are shown in Figure 5, and the damage quantitative results in the second stage are shown in Figure 6 and Table 2.

Figure 5.

Damage location in the first stage for considering noise levels of 5%, 10% and 20%: (a) Case 1, (b) Case 2 and (c) Case 3.

Figure 6.

Damage quantification results in the second stage considering noise levels of 5%, 10% and 20%: (a) Case 1, (b) Case 2 and (c) Case 3.

Table 2.

Identification results of the simply supported beam.

Case	Element	Preset damage severity (%)	Noise level (%)	Identified damage severity (%)
1	7	20	5	20.32
			10	20.83
			20	21.22
2	3,6	20, 10	5	20.32, 10.26
			10	21.08, 10.51
			20	22.31, 11.25
3	4,5,10	15, 10, 5%	5	15.34, 10.27, 5.19

It is shown in Figure 5 that the damage location results of three damage cases considering three different noise levels. In Case 1, it can be found that there is one perk in the DCFD curve (element 7, Figure 5(a)). In Case 2, there are two perks in the DCFD curve (elements 3 and 6, Figure 5(b)). In Case 3, there are three perks in the DCFD curve (elements 4, 5 and 10, Figure 5(c)). In the above three damage cases, it can be seen that different noise levels produced little effect on the damage location results. It is proved that the method could accurately locate the damage in the first stage and exhibited great robustness. The damage severities of the three cases of the simply supported beam polluted by different noise levels are shown in Figure 6(a)–(c). And the damage identification errors are summarized in Table 2.

In summary, whether there is a single-point damage, double-point damage or multi-point damage, the method shows a very good identification performance in the first stage and almost unaffected by noise. In the second stage, although identification error increases with the increase of noise level, damage severity can still be identified. Though it is slightly sensitive to noise when minor damage (5% damage) is polluted by 10% noise, and the damage severity identification error is only about 7.8%, which is relatively acceptable. When 20% noise level is considered, the damage identification results begin to show slight divergence and inaccuracy. It can be seen that the damage can be located and quantified accurately and exhibited a good damage identification ability. Comparing the damage identification performance of the two-stage method proposed in this paper and the traditional direct method under different noises, the cases are shown in Table 3. Figure 7 shows the damage quantitative results of the two-stage method and the traditional direct method.

Table 3.

Damage cases of the simply supported beam.

Case	Element	Preset damage severity (%)	Noise level (%)
1	3,6	20, 10	5
2	3,6	20, 10	10
3	3,6	20, 10	20

Figure 7.

Damage identification results of the simply supported beam: (a) Case 1, (b) Case 2 and (c) Case 3.

Figure 5 and Figure 7 indicated that noise has an effect on damage identification. The maximum error of two-stage method when quantifying the severity of damage is 12.5% which is much lower than that of traditional direct method (21.7%). The result reveals that two-stage method has stronger convergence, which can more accurately locate the damage. In addition, the two-stage method and the traditional direct method take 16.32s and 263.87s to complete the damage identification process respectively. It proves that the two-stage method can greatly improve the computational efficiency.

The steel I-beam experimental example

Experiment description

The I-beam simply supported steel beam is shown in Figure 8(a). Its Young’s modulus, mass density, length and cross-sectional area are 2.0×10¹¹ Pa, 7900 kg/m³, 5 m, and 0.0014 m², respectively. In the experimental example, the excitation method is impact load. The I-beam is equally divided into eight elements, where the sensor is placed at the midpoint of each element. Two damage cases are introduced by cutting different width in flanges of the beam, where the percentage of stiffness reduction corresponding to each case can be calculated according to the following equation (Huang et al., 2020b)

β = \frac{l \times w}{l_{e} \times w_{e}}

(27)

Figure 8.

The experimental steel I-beam: (a) Overall view, (b) Typical cross section, (c) Damage introduced, (d) Data acquisition system, and (e) Accelerometer.

where l_e and l are the lengths of the intact and cut elements, respectively, w_e and w are the widths of the intact and cut elements, respectively. The influence of ambient temperature is achieved by equation (14). Two groups of damage conditions are conducted in this experiment: in the case of single-point damage (Case 1), the preset damage of element two is 5%; in the case of two-point damage (Case 2), the preset damage of element two and element 4 both are 5%.

Vibration experiments are carried out indoors and the dynamic characteristics of the I-beam is obtained, where the first four natural frequencies of the I-beam under undamaged conditions are shown in Table 4. The weight of the sensor is not negligible, so the MATLAB platform is employed to establish a finite element model, where the sensor is converted into a concentrated mass and applied to the finite element model. Table 4 summarizes the small errors between the experimental data and the finite element analysis data, which are mainly caused by finite element model errors and environmental temperature effects. In order to reduce the error, the parameters of finite element model are revised under the ambient temperature of 32°C during the vibration experiment (Huang et al., 2018a). The frequency of the revised model is basically consistent with the experimental results, which indicates that the model is sufficiently accurate and can be used as baseline finite element model of damage identification.

Table 4.

Experimental and analytical natural frequencies of the simple supported beam.

Mode	Experimental (Hz)	Before updating		After updating
Mode	Experimental (Hz)	Analytical (Hz)	Error (%)	Analytical (Hz)	Error (%)
1	12.207	12.342	1.11	12.295	0.72
2	53.711	54.826	2.08	53.152	1.04
3	114.746	116.511	1.54	115.761	0.88
4	207.520	211.621	1.77	209.325	0.87

Results and discussions

The DCFD curves of the I-beam is shown in Figure 9(a)–(b), where the damage is accurately located under the two damage conditions, which proves that the damage location method in the first stage is applicable in the application of actual laboratory structures.

Figure 9.

Damage location of (a) case 1 and (b) case 2 in the first stage, and damage quantification of (c) case 1 and (d) case 2 in the second stage.

The damage quantification results in the second stage are shown in Figure 9(c)–(d). In Case 1, the damage identification error of element two is 4.6%; In Case 2, the damage identification error of element two and four are 7.2% and 5%, respectively. Although there is small error in the damage identification results, which may arise from the fact that the beam is roughly supported by two movable supports, the identification performance is reasonably acceptable in experimental research. In summary, the two-stage damage identification method proposed in this paper is of certain application value in the application of actual engineering structures.

The I-40 bridge

The I-40 Bridge was a steel-concrete composite bridge over Rio Grande in New Mexico, and Farrar et al. (1994) tested this bridge and it is one of the most classical tests in the structural health monitoring field of bridges. This numerical example takes the I-40 Bridge is adopted as a numerical example to establishe the finite element model and verify the applicability of the damage identification method in complex structures.

Discription of the I-40 bridge

Figure 10 illustrates an elevation view of the portion of the bridge that was tested, which consisted of three spans. The end spans are of the equal length of 39.9 m, and the center span is 49.7 m.

Figure 10.

Elevation view of the portion of the I-40 bridge.

Finite element model of the I-40 bridge

The finite element model of the I-40 Bridge is established based on the MATLAB platform, which is shown in Figure 11. The model uses a spatial Cartesian rectangular coordinate system, where the X-direction is the longitudinal direction, the Y-direction is the vertical direction and the Z-direction is the transverse direction. The numerical model uses a 20-degree-of-freedom shell element to simulate the main girder, slab beam and concrete bridge deck, and a 6-degree-of-freedom beam element to simulate piers, cross beams, longitudinal beams and steel bars. The bridge support is simulated by a three-degree-of-freedom spring element. The single spring constants k_x = 3.17×10⁶ N/m, k_y=1.26×10⁶ N/m, k_z=4.29×10⁷ N/m. The structural material parameters are shown in Table 5. The finite element model established under these parameters has high accuracy and can be used as a standard model for damage identification of the I-40 Bridge (Huang et al., 2019). The first six analytical frequencies and modal assurance criterion (MAC) are extracted and shown in Table 6 (Huang et al., 2021b).

Figure 11.

The finite element model of the I-40 bridge.

Table 5.

Structural material properties of the finite-element model.

Material	Parameters	Value
Concrete	Mass density	2323 kg/m³
	Modulus of elasticity	2.482×10¹⁰ Pa
	Poisson ratio	0.2
	Shear modulus	8.946×10⁹ Pa
Steel	Mass density	7850 kg/m³
	Modulus of elasticity	2.1×10¹¹ Pa
	Poisson ratio	0.3
	Shear modulus	8.043×10¹⁰ Pa

Table 6.

Analytical and experimental frequencies and modal assurance criterion.

Mode	Experimental frequencies (Hz)	Analytical frequencies (Hz)	Error (%)	MAC
1	2.483	2.482	0.03	0.995
2	2.959	3.002	1.41	0.984
3	3.499	3.418	2.38	0.991
4	4.079	4.037	1.06	0.971
5	4.167	4.037	3.22	0.972
6	4.631	4.656	0.54	0.971

The bridge deck is divided into 144 shell elements and the north and south webs are divided into 48 shell elements. At the same time, the upper and lower flanges of the webs on both sides are divided into 192 beam elements, and the longitudinal beams are divided into 72 beam elements. In addition, the braces are divided into 50 beam elements, and the slab beams are divided into 100 beam elements. In order to match with the actual bridge and reduce the number of measuring points, two elements along the bridge direction (X-direction) are combined into one element in this numerical example and the vertical acceleration of the center of the element (Y-direction) is extracted. The acceleration signal points are distributed in four rows, and 12 signals points ae arranged on the upper and lower edges of the north and south webs, which indicates that there are 48 signal points as shown in Figure 12.

Figure 12.

Damage cases of the I-40 bridge.

Damage identification

The four damage cases are shown in Table 7. A simple harmonic excitation with an amplitude of 2000 N is applied at the point of impact (Figure 11), and the acceleration of the north and south beams of the I-40 Bridge is captured based on the 48 measuring points. The damage location results of I-40 Bridge in the first stage are shown in Figure 13 and Figure 14, and the damage quantitative results in the second stage are shown in Figure 15 and Table 8.

Table 7.

Damage case of the I-40 bridge.

Case	Element	Preset damage severity (%)
1	8	20
2	3,20	20, 10
3	5,6,23	15, 10, 5
4	4,7,15,22	20, 15, 10, 5

Figure 13.

Damage location results of the I-40 bridge in the first stage: (a) Case 1, (b) Case 2, (c) Case 3 and (d) Case 4. (The abscissas of the first and second lines are element 1~12, and the abscissas of the third and fourth lines are element 13–24.)

Figure 14.

Damage location results of the I-40 bridge in the first stage after improving the difference in curvature of fractal dimension: (a) Case 1, (b) Case 2, (c) Case 3 and (d) Case 4. (The abscissas of the first and second lines are element 1~12, and the abscissas of the third and fourth lines are element 13–24.)

Figure 15.

Damage quantification results in the second stage considering noise levels of 3%, 5% and 10%: (a) Case 1, (b) Case 2 and (c) Case 3.

Table 8.

Identification result of damage severity of I-40 bridge.

Case	Element	Preset damage severity (%)	Noise level (%)	Identified damage severity (%)
1	8	20	3	20.63
			5	20.78
			10	21.22
2	3,20	20, 10	3	20.77, 10.21
			5	20.64, 10.52
			10	21.10, 10.76
3	5,6,23	15, 10, 5	3	15.55, 10.25, 5.3
			5	15.78, 10.61, 5.38
			10	15.95, 10.95, 5.46
4	4,7,15,22	20, 15, 10, 5	3	20.72, 15.51
			3	10.30, 5.28
			5	20.91, 15.63
			5	10.65, 5.40
			10	21.22, 16.1
			10	10.86, 5.45

The damage location results of four damage cases under external excitation are shown in Figure 13. It can be found that there are one, two, three and four perks in the DCFD curve in Case 1, Case 2, Case 3 and Case 4 respectively. Although the damaged element can be identified in the above four cases, there are some small peaks in the elements adjacent to the damaged elements because the web of the I-40 Bridge is the model built with shell elements, that is, there is a “proximity effect”. In order to eliminate the “proximity effect”, a new improvement method is proposed for the plate-type structure to improve the DCFD index and calculated according to the following equation

\begin{matrix} DCF D_{i} = DCF D_{i} \times α + min (DCF D_{i - 1}, DCF D_{i + 1}) \\ \times (1 - α) \end{matrix}

(28)

where i is the sampling point of the acceleration signal, α is the weight coefficient, which is determined according to the dichotomy: α=0.6∼0.7. The damage location results based on the improved method are shown in Figure 14.

The Figure 14 shows that the misjudgment of some adjacent elements can be eliminated based on the improved DCFD, but there is still a multitude of “proximity effect”. The reason may be that there are not enough elements on the north and south sides in X and Z direction. Therefore, the number of signal points adjacent to each signal point is small, and the signal points near the damaged element are affected by the damaged element. According to equation (28), when the DCFD value is large, there is no suitable small value in the DCFD value to reduce, which leads to the misjudgment for the adjacent elements. Hence, the improved DCFD index proposed in this paper can be used to reduce the “proximity effect” of plate-type structure. However, it will cause a small amount of misjudgment when locating the damage of the plate-type structure when the element number of a structure is small.

Identification results of damage severity of the I-40 Bridge are summarized in Figure 15 and Table 8. It is clearly shown that although the overall identification error of the method proposed in this paper increases with the increase of noise level, there is still of good performance for the identification of damage severity, and it is also applicable to more complex structures. Even in the case of noise level of 10%, the damage identification error is only 9% when the severity of damage is about 5%, which is relatively small for small structural damage. The results indicate that the proposed damage identification method is of strong damage identification ability and is potential in the application of practical engineering structures.

Conclusions

This paper proposed a two-stage damage identification method based on fractal theory and WOA. Firstly, the feasibility of the method is verified through a numerical simply supported beam example. Subsequently, an I-beam experiment is designed and performed to verify the rationality of the method in the application of actual engineering structure. Finally, the applicability is verified by establishing a finite element model for simulation based on an actual bridge, the I-40 Bridge. Furthermore, the noise levels (5% ∼ 20%) are applied to simply supported beam numerical examples to verify the robustness of the method combined with the actual project. The results indicates that the method shows good identification ability and robustness which makes it potential in practical engineering applications.

The conclusions are as follows

1) The damage identification results of a numerical simply supported beam and an experimental steel I-beam show that this method is very effective. When the simply supported beam condition is two-point damage (20% noise level), the two-stage method and the direct method take 16.32s and 263.87s to complete the damage identification, and their damage severity identification errors are 12.5% and 21.7%, respectively. Therefore, two-stage method firstly carried out to determine the suspicious damaged element, which can reduce the number of suspicious damaged elements for the next stage of the determination of damage severity and therefore improve the accuracy and computational efficiency of damage identification.

2) During the damage identification of the I-40 Bridge, it is found that there is a “proximity effect” because the web of the I-40 Bridge is the model built with shell elements, so the DCFD index is improved and the results shows that the method is not only suitable for simple structures, but also exhibits good damage identification ability for complex engineering structures.

3) To verify the robustness of noise, the acceleration data are polluted by different levels of noise for the numerical simply supported beam example and the I-40 Bridge. The results show that during the first stage of damage location, the proposed method exhibits great robustness. In the second stage of damage quantification, it shows obvious sensitive to noise when minor damage (5% damage) is polluted by 10% noise, but the damage severity identification error is acceptable. When the noise level reaches 20%, the damage identification error begins to increase.

Despite the feasibility of the proposed two-stage damage identification method based on fractal theory and WOA, there are several issues that need to be resolved. Firstly, the effect of excitation-point change on the damage identification results. In addition, when vertical harmonic excitation is loaded at the mid-span, the damage identification performance is better than that of other positions. Therefore, the other methods that can effectively eliminate the excitation-point should be studied extensively. Secondly, in equation (16), element length h is a constant, which means that elements require to be equally discretized. The sensors will need to be arranged at equal intervals in actual engineering applications. Finally, the current damage identification method is suitable for damage identification of structures with constant cross section. Future work will apply this method to damage identification of structures with variable cross section.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (52178300) and the Graduate Innovative Fund of Wuhan Institute of Technology (No: CX2020107).

ORCID iDs

Minshui Huang

Zian Xu

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