Advances in Civil Engineering

Volume 2018, Article ID 3183067, 20 pages

https://doi.org/10.1155/2018/3183067

## A Combined Modal Correlation Criterion for Structural Damage Identification with Noisy Modal Data

^{1}Institute of Structural Analysis and Antiseismic Research, School of Civil Engineering, National Technical University of Athens, Zografou Campus, Athens 15780, Greece^{2}Department of Civil Engineering and Energy Technology, OsloMet—Oslo Metropolitan University, Pilestredet 35, Oslo 0166, Norway

Correspondence should be addressed to Vagelis Plevris; on.temolso@ilegav

Received 22 August 2017; Revised 15 February 2018; Accepted 27 February 2018; Published 14 May 2018

Academic Editor: Dimitris Rizos

Copyright © 2018 Manolis Georgioudakis and Vagelis Plevris. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Structural damage identification is a scientific field that has attracted a lot of interest in the scientific community during the recent years. There have been many studies intending to find a reliable method to identify damage in structural elements both in location and extent. Most damage identification methods are based on the changes of dynamic characteristics and static responses, but the incompleteness of the test data is a great obstacle for both. In this paper, a structural damage identification method based on the finite element model updating is proposed, in order to provide the location and the extent of structural damage using incomplete modal data of a damaged structure. The structural damage identification problem is treated as an unconstrained optimization problem which is solved using the differential evolution search algorithm. The objective function used in the optimization process is based on a combination of two modal correlation criteria, providing a measure of consistency and correlation between estimations of mode shape vectors. The performance and robustness of the proposed approach are evaluated with two numerical examples: a simply supported concrete beam and a concrete frame under several damage scenarios. The obtained results exhibit high efficiency of the proposed approach for accurately identifying the location and extent of structural damage.

#### 1. Introduction

Structural damage identification has drawn increasing academic interest, as witnessed by the significant number of relevant journal and conference papers, during the recent years [1–5]. The necessity of detecting and repairing structural damage at its early stage has become imperative, and considerable effort has been devoted to developing nondestructive testing and evaluation (NDT&E) techniques [6]. Among the NDT&E techniques that have received significant attention in the computational mechanics field are those based on vibration signature analysis in order to obtain global information about the condition or state of health of the structural models using measured dynamic data. Such techniques use vibration characteristics of the structures including frequency response functions, natural frequencies, mode shapes, modal curvatures, and modal flexibilities to identify the occurrence of the structural damage.

Generally speaking, the existing methods of damage identification techniques based on modal testing can be classified into two major categories: direct and inverse methods. The direct methods utilize the change in modal measurement to instantly detect structural damage without the need of iterative computational procedures. In contrast, the second category of damage identification techniques poses the whole process as an inverse problem, in which the structural damage is identified by optimizing the correlation between the theoretical and the experimental modal parametric change. The level of correlation between the measured and the predicted modal characteristics is quantified using the modal correlation criteria that form useful mathematical tools identifying also the location and the extent of structural damage.

A class of damage identification methods belonging to the second category is based on the modification of structural model properties such as mass, stiffness, and damping to reproduce as closely as possible the measured static and dynamic responses corresponding to the experimental data [4]. These methods update the physical parameters of a finite element model of the structure by minimizing an objective function expressing the difference between finite element predicted and experimentally identified structural dynamic properties that are sensitive to damage such as natural frequencies and natural mode shapes. A thorough study of the finite element model updating approach was conducted by Friswell and Mottershead [7, 8]. Many successful applications of damage identification for detecting multiple structural damages based on the latter approach have been reported in the literature, implementing modern optimization techniques such as the evolutionary algorithms, among others [9–14].

In the present study, a structural damage identification method based on the finite element model updating is proposed, in order to provide the location and the extent of structural damage using incomplete modal data of a damaged structure. The problem of damage identification in structures is treated as an unconstrained single-objective optimization problem. The objective function of the problem to be minimized utilizes two modal correlation criteria, namely, the modal flexibility assurance criterion (MACFLEX) and the modified total modal assurance criterion (MTMAC). The combined criterion showed a superior performance in comparison to the performance of the individual criteria [15]. The performance and robustness of the proposed approach are evaluated with two numerical examples: a simply supported concrete beam and a concrete frame in several damage scenarios. The obtained results exhibit the high efficiency of the proposed approach for accurately identifying the location and extent of structural damage.

#### 2. Structural Damage Identification

The problem of damage identification can be classified into four levels [16]: (A) detection, (B) localization, (C) quantification, and (D) prediction of future damage (damage prognosis). At the level of damage detection (Level A), the existence of damage can be detected, while its location and severity remain unknown. Information about location of the damage can be provided by localization techniques (Level B). At the damage quantification level (Level C), both the location and severity of damage are estimated. Finally, at the prediction level (Level D), the remaining life of the structure is estimated based on the (identified) current damage state and future loads as well as predicted damage propagation. This study reaches the third level of damage identification, which means it investigates the ability to detect, localize, and estimate the severity of damage in structures.

##### 2.1. Damage Identification Model

It is proven that changes in the vibration characteristics of a structure are related to damage occurrence. Specifically, changes in the modal parameters, for example, natural frequencies and mode shapes, can provide an accurate indication of the presence of damage in a structure. Since modal parameters are dependent on the physical properties of the structure, that is, stiffness and mass, the finite element method (FEM) can be used as a tool for locating and quantifying damaged elements in a structure through an updating procedure of modal parameters, even for large-scale structures.

Mathematically speaking, structural damage affects the stiffness and physically the dynamic properties of the structure, such as natural frequencies and mode shapes [17]. It can be assumed that the global mass matrix remains the same in both the undamaged and the damaged state of the structure. This assumption can be considered quite accurate for the majority of practical applications. The eigenvalue problem of a structure with *n* active degrees of freedom (DOFs) can be written as follows:where is the global stiffness matrix of the structure ([*n *×* n*]), is the global mass matrix ([*n *×* n*]), is the *i*-th natural mode vector of the structure ([*n *×* *1]) corresponding to the natural frequency, and is the total number of natural modes to be obtained .

For a damaged structure, (1) takes the following form, respectively:where is the global stiffness matrix of the damaged structure ([*n *×* n*]) and is the *i*-th natural mode vector of the damaged structure ([*n *×* *1]) corresponding to the natural frequency. Equation (2) forms the basis of the damage identification method used in the present study.

##### 2.2. Modal Correlation Criteria

We consider two structures A and B with *n* active DOFs each, with eigenvalues and and natural frequencies and , where is the total number of natural modes obtained . The corresponding mode shape vectors are and ([*n *×* *1] each) for structures A and B, respectively. In order to compare two sets of values for the two structures, the use of modal correlation criteria is imperative, providing a measure of consistency and correlation between estimations of mode shape vectors.

###### 2.2.1. The Modal Assurance Criterion (MAC)

The modal assurance criterion (MAC) [18, 19] is one of the most popular tools for the quantitative comparison of mode shape vectors. The purpose of this criterion is to indicate the correlation between two sets of natural modes. Considering two mode shape vectors ([*n *×* *1]) and ([*n *×* *1]), for structures A and B, respectively, the element of the **MAC** matrix ([*m *×* m*]) is given by

takes values from zero, representing no consistent correspondence, to one, representing a consistent correspondence between the two mode shape vectors under consideration. In this manner, if the mode shape vectors and truly exhibit a consistent relationship, the modal assurance criterion element approaches unity. By calculating for all , we obtain the **MAC** matrix. In addition, we define ([1* *×* m*]) as the vector holding the diagonal terms of **MAC** matrix which can be easily calculated by setting in (3) [15].

###### 2.2.2. The Modified Total Modal Assurance Criterion (MTMAC)

One limitation of the MAC criterion is that it takes into account only the eigenvectors (mode shapes) and not the eigenvalues of the structures. This means that, in the case of uniform-like damage, the MAC criterion will not be able to detect almost any change; as in this case, the structure becomes more flexible (i.e., the eigenperiod increases), but there is no significant difference in the eigenvectors which remain almost unchanged. The natural frequencies provide global information of the structure, and they can be accurately identified through dynamic measurements.

The modified total modal assurance criterion (MTMAC) [20] is based on the MAC criterion but it also takes the corresponding eigenvalues into account. The MTMAC vector ([1* *×* m*]) is defined as follows:where is the *i*-th component of the vector. It should be noted that the MTMAC can be easily defined also as a matrix ([*m *×* m*]), whose diagonal is again the vector, as was the case with MAC. is a row vector with as many values as the number of natural modes considered . By multiplying the individual values of the vector, we obtain the scalar value as follows:

###### 2.2.3. The Modal Flexibility Assurance Criterion (MACFLEX)

It is well known that damage affects the stiffness matrix of the structure and more specifically it reduces the stiffness of the individual damaged elements. In other words, a reduction in the stiffness is equivalent to an increase in the structural flexibility. In structural health monitoring [21], it is advantageous to use changes in flexibility as an indicator of damage rather than using stiffness perturbations. This is due to the following reasons [22]:(i)The flexibility matrix is dominated by the lower modes, and so good approximations can be obtained even when only a few lower modes are employed.(ii)The flexibility matrices are directly attainable through the modes and mode shapes determined by the system identification process.(iii)Iterative algorithms usually converge the fastest to high eigenvalues.(iv)In flexibility-based methods, these eigenvalues correspond to the dominant low-frequency components in structural vibrations.

Therefore, the dynamically measured flexibility matrix which is calculated from the identified modal parameters can be used as a damage identification measure [14]. The flexibility matrix ([*n *×* n*]) for the structure is given bywhere is a matrix ([*n *×* m*]) containing all the mode shape vectors ([*n *×* *1] each) and is a diagonal matrix ([*m *×* m*]) which holds the eigenvalues on its diagonal. The elements of matrix can be also obtained separately using the following formula:

Each column of the flexibility matrix represents the displacement pattern of a structure associated with a unit force applied to the associated degree of freedom. As shown in (7), as the value of frequency decreases (i.e., the eigenperiod increases), the modal contribution to the flexibility matrix also increases. As a result, a good estimate of the flexibility matrix can be calculated even with a small number of the first low-frequency modes [15].

In order to compare the values of the flexibility matrix of the two structures and , the modal flexibility assurance criterion (MACFLEX) is applied. The *i*-th component of the vector ([1* *×* n*]) can be calculated as follows:where and are the *i*-th column vectors ([*n *×* *1]) of the flexibility matrices and , for structures and , respectively. is a vector with as many values as the number of columns in the flexibility matrices. Again, we could consider a full **MACFLEX** matrix by taking different vectors into account, instead of the *i*-th vector for both structures, but there is no point in that as again the diagonal terms of the **MACFLEX** matrix are the important ones. By multiplying the *n* individual values of the vector, we obtain the scalar value as follows:

#### 3. Damage Identification as an Optimization Problem

##### 3.1. Problem Formulation

The structural damage identification problem is treated as an unconstrained optimization problem. The objective function to be minimized utilizes the two modal correlation criteria , while the design variable describes the extent of damage of the *i*-th element of the structure. In this sense, the number of design variables is equal to the number of elements in the structure.

In this study, the standard beam finite elements are used to represent the structures of the numerical examples. It has been assumed that no alteration of the mass occurs before and after damage, which is acceptable for most engineering applications. Therefore, the parameterization of the damage is represented by a reduction factor or damage index of the element bending stiffness. This damage index, , for a damaged element represents the relative variation of the damaged element bending stiffness, to the initial (undamaged) bending stiffness , as follows:

This definition of the damage index for each element of the structure allows estimating not only the damage extent but also the damage location since the damage identification is carried out at the individual element level. The damage index can take values between 0 (no damage) and 1 (100% damage), although for numerical stability purposes, in the present study, the maximum damage has been limited to a value slightly below unity (i.e., 0.999), otherwise the structure will become a mechanism that cannot be analyzed and numerical instabilities will occur.

Taking into account all of the above, the mathematical formulation of the structural damage identification problem takes the following form:

##### 3.2. The Objective Function

Setting up an objective function, selecting the updating parameters, and using robust optimization algorithms are three crucial steps in structural identification. They require deep physical insight and usually also trial-and-error procedures. The objective function has to reflect the deviation between the numerical prediction and the real behavior of the structure. For this reason, an objective function may be formulated in terms of the discrepancy between the finite element model and experimental quantities.

In this study, a combined objective function has been implemented, which takes into account the values of two different modal correlation criteria that express the relationship between the real (experimentally measured) damage and the damage which is numerically predicted by the finite element model. In particular, the objective function of the optimization problem is the following:where and are the values of the corresponding individual objective functions

The minimum value (target value) for the objective function is zero, which is also the case for the individual objective functions and .

##### 3.3. Noise Simulation for Experimental Data

In a real-life scenario, the vibration characteristics (eigenvalues and eigenmodes) of the real damaged structure would have to be determined (measured) by experiment. In our case, for practical reasons and due to lack of appropriate experimental data, these properties are also calculated numerically using a finite element “real damage” model. This approach is acceptable for the purposes of the present study and does not cause any limitations to the proposed methodology. Thus, in the present study, the term “experimental data” has nothing to do with a real experiment but rather corresponds to simulated numerical data of the “damaged” structure. At this point, it needs to be clarified that although the real damage information is used in order to generate the “damaged” model and obtain the “experimental” data, this information is not made available to the optimization algorithm. The optimization algorithm only considers the “experimental” modal data and tries to find the damage based only on them.

Real experimental data acquired from experiments can never be perfectly accurate. During modal testing, it is usual to assume that the frequencies of vibration are accurately determined and that is in the determination of the amplitudes of the mode shapes that the experimental errors will occur. This assumption is usually valid since the frequency of shakers, even at resonance, can be quite accurately controlled. In order to simulate this phenomenon and be more compatible with real experimental values of modal parameters, the “experimental data” in the present study have been “enriched” by adding an artificial noise into them using the following equation [23]:where is the *i*-th component of the *j*-th experimental mode shape vector, is the *i*-th component of the *j*-th calculated mode shape vector, (noise ratio) is the percentage of noise added to the calculated data (usually 0–10%), and is a uniformly distributed random number in the range of (−1, 1).

##### 3.4. The Differential Evolution Algorithm

Choosing the proper search algorithm for solving an optimization problem is not a straightforward procedure. In the past, a number of studies have been published where structural optimization problems have been solved using metaheuristic search algorithms and especially those based on adopting Darwinian principles of evolutionary process. These algorithms achieve efficient performance for a wide range of combinatorial optimization problems. Among the plethora of such algorithms, the differential evolution (DE) algorithm is adopted in this study to solve the structural damage identification problem of Section 3.1. DE is considered a fast and efficient metaheuristic having gained increased popularity in the engineering optimization community, with numerous applications in various research fields [24].

Differential evolution (DE) is a stochastic population-based evolutionary algorithm for global optimization, introduced by Storn and Price [25]. It follows the standard evolutionary algorithm flow with some significant differences in the mutation and selection processes. The simplicity of DE algorithm is based on only three tunable parameters: the mutation factor , the crossover probability , and the total number of particles (population size) NP. The fundamental idea behind DE is the use of vector differences by choosing randomly selected vectors and then taking their difference as a means to perturb the parent vector with a special kind operator and probe the search space. Several variants of DE have been proposed so far [26], but the present study is focused on the nominal approach (DE/rand/1/bin). According to this, each of the members of the population undergoes mutation and crossover. Once crossover occurs, the offspring is compared to the parent and the one with the best fitness between the two moves to the next generation (selection process).

In more detail, we consider an optimization problem with *D* dimensions. First, all individuals of the population are initialized at random positions in the search space. After initialization, each member of the population undergoes mutation and a donor vector is generated such aswhere , , and are three individuals from the population at random, which must be distinct from each other and from individual .

In the next step, the crossover operator is applied by generating the trial vector . The *i*-th component of is defined either from the *i*-th component of or the *i*-th component of , with probability CR as follows:where is a random number with uniform distribution, , and *R* is a random integer in [1, 2, …, *D*], which ensures that, in any case, after the crossover operation at least one component is changed and thus . The last step of the generation procedure is the implementation of the selection operator where the target vector is compared to the trial vector . If the trial vector has a better fitness value than , then the individual is replaced in the population with the trial vector as follows:where is the objective function to be minimized and is the new design vector for the next generation. If any component of the vector obtained from (15) is out of the given bounds, then the specific component is reset by assigning a random value to it within the given bounds. This ensures that the final vector from (17) will always have all values within the given range.

The optimization procedure is finished when a desirable convergence criterion has been satisfied. In the present study, the termination criterion is met when the maximum number of generations has been reached and no other convergence criterion has been used, for reasons of uniformity.

#### 4. Numerical Examples

Two different structures have been investigated to show the effectiveness of the proposed methodology for identifying the location and extent of damage in structures. The first structure is a simply supported concrete beam, while the second is a concrete frame. The finite element model for both examples is based on an Euler–Bernoulli assumption of the planar elements with three degrees of freedom per node (two translational and one rotational). The mutation factor and the crossover probability CR of the DE algorithm are common in both problems with values = 0.6 and CR = 0.9, respectively. The maximum number of generations is also common for both problems, with value 3000. Due to the different numbers of design variables between the two problems, the population size is different, taken as NP = 40 for the beam problem and NP = 100 for the computationally more intensive frame problem.

To check the robustness of the proposed methodology, one to four vibration modes are considered, and due to the limited number of sensors, the mode shape vector and the vertical displacements are supposed to be only read at a limited number of locations corresponding to the vertical degrees of freedom of the structures (monitored DOFs). This means that the eigenmode information (displacements) that corresponds to the experimental data is obtained for these DOFs only. The discretization of the model is an important issue that can affect the results, to some extent, but if a dense FE mesh is used, then the damage location can be approximated with enough accuracy.

A network of sensors is needed to record the eigenvectors and eigenvalues of the damaged structure. In the present study, the DOF locations correspond to the locations of the sensors, but this does not have to be the case in general. The recorded locations do not have to be the same as the DOFs of the finite element model and can be completely different, although when compared, the two systems have to be consistent with each other (comparison at the same locations). This means that, for example, the FEM mesh can be denser than the network of sensors but the comparison has to be made at the same predefined locations.

Furthermore, in order to assess the performance of the optimization procedure, we introduce a metric which indicates how well the optimization algorithm manages to find the real damage, by means of a single scalar value. The prediction error between real damage vector and the predicted damage vector of the best individual of the current generation step is defined using the Euclidean norm as follows:

The smaller the value of the , the better the prediction. In case of a perfect prediction , the takes the value of zero.

##### 4.1. Simply Supported Beam

A simply supported beam [15, 27] is analyzed in this section to illustrate the effectiveness of the proposed methodology and the performance of the different objective functions . The geometry, boundary conditions, and the finite element mesh of the beam are shown in Figure 1. The beam has a total length of 6 m, and it is discretized by 10 equal length beam elements of rectangular cross section *b* (width) × *h* (height) = 0.25 m × 0.20 m. The beam is considered to have Young’s modulus *E* equal to 30 GPa and a density *ρ* equal to 2500 kg/m^{3}. The number of monitored DOFs is 9, and the first four eigenmodes of the undamaged structure are shown in Figure 2.