# A Displacement Frequency Response Function-Based Approach for Locating Damage to Building Structures

Authors: Ching-Yun Kao; Xuan-Zhi Chen; Shih-Lin Hung
Journal: Advances in Civil Engineering
 (2020)

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## Abstract

Frequency response function (FRF) data can provide considerably more information on damage in the desired frequency range as compared to modal data extracted from a very limited range around resonances. Among structural health monitoring techniques, FRF-based methods have the potential to locate structural damage. Conventional structural damage detection technology collects structural response data using contact systems, such as displacement or acceleration transducers. However, installing these contact systems can be costly in terms of labor, cost, and time. Several noncontact measurement technologies, such as optical, laser, radar, and GPS, have been developed to overcome these obstacles. Given the rapid advances in optical imaging hardware technology, the use of digital photography in structural monitoring systems has attracted considerable attention. This study develops a displacement FRF-based approach to locate damage to building structures. The proposed damage location index,CurveFRFDI, improves the sensitivity of SubFRFDI, which is a substructure FRF-based damage location index proposed by Lin et al. (2012). Moreover, the feasibility of applying the proposed approach to locate damage to building structures using displacement measured by a digital camera combined with digital image correlation techniques is also investigated in this study. A numerical example and an experimental example are presented to demonstrate the feasibility of using the proposed approach to locate damage to building structures for single and multiple nonadjacent damage locations.


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## Body

## 1. Introduction

Vibration-based damage identification of structures refers to the in situ nondestructive sensing and analysis of the characteristics of a structure, including the structural response to external excitations, to detect changes that may indicate damage or degradation. The feasibility of applying various vibration characteristics, such as natural frequencies, mode shapes, mode shape curvatures, modal flexibility, and frequency response functions, to damage identification of structures has received considerable attention in the past few decades [1, 2].Lee and Shin [3] identified two main advantages of using the frequency response function (FRF) data. First, modal data can be contaminated by modal extraction errors in addition to measurement errors because they are derived datasets. Second, a complete set of modal data cannot be measured in all but the simplest structures. FRF data can provide considerably more information on damage in the desired frequency range as compared to modal data that are extracted from a very limited range around resonances. Some studies have shown that FRF-based methods are highly promising tools to detect damage to building structures [4–8]. Ni et al. [4] identified the seismic damage of a 38-story building using measured FRFs and neural networks. Their study used principal component analysis (PCA) to reduce dimension and eliminate the noise of measured FRFs. The PCA-compressed FRF data are then used as input to neural networks for damage identification. Furukawa et al. [5] developed a method to detect damage to building structures that uses uncertain FRFs based on a statistical bootstrap method. Kanwar et al. [6] detected damage of reinforced concrete buildings using FRFs. Hsu and Loh [7] detected damage to building structures subjected to earthquake ground excitation using FRFs of intact and damaged systems as well as system matrices of the intact system to derive the damage identification equations. Lin et al. [8] proposed a substructure-based acceleration FRF approach that uses the index, SubFRFDI, to locate damage to building structures.In addition to data analysis, data measurement is another problem that needs to be addressed in detecting damage to structures. Conventional structural damage detection technology collects structural response data by using contact systems, such as displacement or acceleration transducers. However, installing these contact systems can be costly in terms of labor, cost, and time [9]. Several noncontact measurement technologies, such as optical, laser, radar, and GPS, have been developed to overcome these obstacles [10]. The use of digital photography in structural monitoring systems has attracted considerable attention given the rapid advances in optical imaging hardware technology. Digital image correlation (DIC) is a measurement technique that extends the principles of photogrammetry to obtain full-field surface displacement measurements of an object using digital cameras. Several researchers have applied DIC to detect damage to building structures using dynamic responses [11–16]. Shih et al. [11, 12] developed a low-cost digital image correlation method to measure the dynamic response of shear buildings. The accuracy of the DIC method is sufficiently high for several applications. Combe and Richefeu [13] used the DIC technique coupled with geometrical rules to develop an approach to track the nonsmooth trajectory of particles. Sieffert et al. [14] presented a digital correlation technique to capture the full-field displacement by using a high-speed camera of a full-scale structure tested on a shaking table. Lu et al. [15] presented a digital image processing approach with a unique hive triangle pattern by integrating subpixel analysis for noncontact measurement of structural dynamic response data. Hung and Lu [16] integrated this approach and GPU computing technique to save on computation time.Many civil structures, especially high-rise buildings and long-span bridges, have low fundamental response frequencies. Measuring displacement directly, as opposed to integrating acceleration records (and potentially introducing significant error [17]), provides the opportunity to capture low-frequency response [18]. Detecting structural damage using displacement FRF should be more accurate than using acceleration FRF because the fundamental frequency of a structure becomes lower when a damage occurs. Based on these reasons, the objective of this paper is to enhance the work of Lin et al. [8]. In this paper, a damage location index (CurveFRFDI) based on SubFRFDI curvature and displacement FRF is used to locate damage to building structures to enhance the sensitivity of SubFRFDI to damage. The study also aims at investigating the feasibility of applying the proposed approach to locate damage to building structures using displacement measured by digital camera combined with the DIC technique. Moreover, a numerical example and an experimental example demonstrate the feasibility of applying the proposed approach for locating damage to building structures.


## 2. Damage Location Strategy

Lin et al. [8] presented a substructure-based FRF approach to locate damage to building structures. As presented in Figure 1, the ith substructure is a structure containing the ith–Nth floors (or degree of freedoms, DOFs) for an N-story building structure. The substructure-based FRFs of the jth DOF in the ith substructure can be simplified as follows:(1)H˜jiω=X˜¨jX˜¨i−1,j=itoN,where X˜¨j and X˜¨i−1 are the Fourier transforms of x˜¨jt and x˜¨i−1t, respectively; x˜¨jt and x˜¨i−1t are the absolute acceleration of the jth and (i − 1)th DOF, respectively. Theoretically, when the damage is assumed to have occurred in the columns between the ith and (i − 1)th DOFs, the substructure-based FRF is altered significantly in the ith DOF as described by H˜iiω. For efficiency, only one substructure-based FRF, H˜iiω, is determined for each substructure to reduce the computational time. Damage location is based on the FRFs, H˜11ω, H˜22ω, …, H˜NNω, of all substructures.Figure 1
(a) Original complete structure and (b) theith substructure (reproduced from Lin et al. [8], under the Creative Commons Attribution license/public domain).
(a)
(b)The substructure-based FRF damage location index (SubFRFDI) for the ith substructure is defined as follows:(2)SubFRFDIi=1−Exp−ρ2∑ω=abNDFiω2n,where ρ, a, b, and n are working parameters and the NDFiω is expressed as follows:(3)NDFiω=P¯iωmaxH˜i,diωmax，H˜i,uiωmax,and the absolute dissimilarity P¯iω is defined as follows:(4)P¯iω=H˜i,diω−H˜i,uiω,where H˜i,diω and H˜i,uiω are the magnitudes of H˜ii in the damaged and undamaged states, respectively. These N dissimilarities, P¯1ω∼P¯Nω, can be calculated correspondingly for a shear building with N floors. In equation (2), the coefficient ρ is a control value that scales the index between 0 and 1. The damage index is not sensitive to locate damage for a small ρ. However, the sensitivity of the damage index to locate damage will not increase for a large ρ. The value of the coefficient ρ is suggested to be 1∼10, with 5 used in this work. The range of selected frequencies for calculating SubFRFDI is set to be a∼b, where a is the starting frequency of zero and b is the end frequency, which is equal to the first modal frequency (undamaged state). The value n equals (b-a) divided by sampling time. If the properties of a structural system do not change, then SubFRFDIi is close to zero. However, if the damage to the ith floor in a shear building is severe, then the value of SubFRFDIi is high. Thus, SubFRFDIi can be regarded as the SubFRFDI corresponding to location i (the ith floor).This study investigates the feasibility of applying displacement measured by digital camera combined with DIC technique to locate damage to building structures, and thus, the substructure-based FRF of thejth DOF in the ith substructure in equation (2) is modified to calculate SubFRFDIi:(5)H˜jiω=X˜jX˜i−1,j=itoN,where X˜j and X˜i−1 are the Fourier transforms of x˜jt and x˜i−1t, respectively; x˜jt and x˜i−1t are the absolute displacement of the jth and (i − 1)th DOF, respectively.In fact, theSubFRFDI values corresponding to damaged and undamaged locations increase with the increasing of damage extent. Thus, SubFRFDI is insensitive to damage with larger damage extent. Moreover, SubFRFDI can locate single and multiple nonadjacent damages but is unable to locate multiple adjacent damages to building structures. Damage occurred at the location corresponding to a larger SubFRFDI value than adjacent locations for single and multiple nonadjacent damage cases. That is, if the ith floor is damaged, and the (i − 1)th and (i + 1)th floors are undamaged, SubFRFDIi is larger than SubFRFDIi−1 and SubFRFDIi+1, and the curve connecting SubFRFDIi−1, SubFRFDIi, and SubFRFDIi+1 is open downward. Thus, damage occurred at the location corresponding to a negative SubFRFDI curvature rather than a nonnegative SubFRFDI curvature for single and multiple nonadjacent damages. The curvature of SubFRFDIi, KSubFRFDIi, can be simply defined as follows:(6)KSubFRFDIi=SubFRFDIi+1+SubFRFDIi−1−2SubFRFDIi,fori=1,2,…,N,where SubFRFDI0 and SubFRFDIN+1 are both equal to zero. In this study, a damage location index based on SubFRFDI curvature, CurveFRFDI, is used to locate damage to building structures to enhance the sensitivity of SubFRFDI to single and multiple nonadjacent damages. The CurveFRFDI for an N-story building structure is expressed as follows:(7)CurveFRFDIi=−KSubFRFDIi,ifKSubFRFDIi<0,KSubFRFDIiA,ifKSubFRFDIi≥0,where A is a control value that scales CurveFRFDIi to a very small positive value for positive SubFRFDIi curvature. The value of coefficient A is large compared with SubFRFDI curvature. It is suggested that A≥100, and A is set to be 100 in this study. The range of CurveFRFDIi is from 0 to ∞. Like SubFRFDI, damage occurred at the location corresponding to a larger value of CurveFRFDI. Figure 2 shows the flowchart of the proposed approach for locating damage to building structures.Figure 2
The flowchart of the proposed damage location approach.


## 3. Digital Image Correlation

Digital image correlation (DIC) is an easy optical method to measure continuous deformation by tracking the position of the same physical points shown in a reference image and a deformed image. A rectangular subset of pixels is identified on the speckle pattern around a point of interest (POI) on a reference image and their corresponding location determined on the deformed image to achieve tracking.This study employs the DIC approach presented in the work of Lu et al. [15] for displacement measurement in the experimental example. The approach used a unique hive triangle pattern by integrating subpixel analysis for noncontact measurement of structural dynamic response data. As shown in Figures 3(b) and 3(c), a fixed-size rectangular subset of pixels that make up hive triangle patterns is the region of interest (ROI), contained both within the reference (source) and within the deformed (target) images and marked with a red color box. Meanwhile, in Figure 3(a), they are designated as I and I′ in source and target images, respectively. The x0,y0 and x1,y1 are the coordinates of the points at the left-top of the ROI of the source image and the after deformed target images, respectively. Hence, u and v are the relative deformations of a particular point in image space. The coordinates of x0,y0 and x1,y1 are figured out in the following steps. Two images are first selected as S and T. The image S is an undeformed (source) image, and image T is a slightly deformed image (target) relative to image S. The difference between the two images, S and T, can be estimated through a simple difference method via pixel-wise computation, as shown in the following equation:(8)Di,j=Si,j−Ti,j,where i and j are the index of a pixel in the ith row and jth column of the source and target images based on the origin at the top-left corner. If the intensity of each pixel is from 0 to 255, the difference of each pixel between S and T is maximal when the equation value is 255, and S and T are exactly identical when the equation value is 0. If two images have the same background, the Di,j of the pixels in the background area is close to zero. Therefore, if an ROI changes the position due to deformation, the Di,j of these pixels covered in the ROI is relatively large.Figure 3
(a) Simplified diagram of source and target images, showing displacement variation of image block. (b) Source image. (c) Target image (reproduced from Lu et al. [15], under the Creative Commons Attribution license/public domain).
(a)
(b)
(c)Moreover, the coordinate,x0,y0, of the ROI of the source image can be computed using a mean-max method represented in equation (9). The method first calculates the means of the intensities of all columns and rows pixels in the source and one of the target images having slightly deformation. The index with the maximum mean values of the columns and rows indicates the x-axis and y-axis coordinates, respectively:(9)x0is the index ofmaxXi¯fori=1∼w,y0is the index ofmaxYj¯,forj=1∼h,where Xi¯=∑j=1hDi,j/h and Yj¯=∑i=1wDi,j/w in which w and h are the width and height of the source and target images in pixel. Particularly, the approach of Lu et al. employs an ideal hive triangle pattern as the ROI in calculating the correlation coefficient with the source image. The correlation coefficient used in the work is defined in the following equation:(10)CC=ΣΣf−f⋅g−gΣΣf−f2⋅ΣΣg−g21/2,where f and g are the pixel value of ROIs for the source and target images, respectively. The sign ⋅ denotes the mean operator and f and g are the means of ROIs in the source and target images, respectively. The coordinate, x1,y1, of the point at the left-top of the ROI of the after deformed target images can be figured out based on the following equation:(11)x1,y1=maxCCIx0,y0,Ix1,y1′,where I and I′ denote the aforementioned ROIs for source and target images, and CC refers to the correlation coefficient function to evaluate with the two ROIs. The relative displacement in the unit pixels (i.e., the pixel displacement), dx,dy, in the x-axis and y-axis coordinates can be estimated to be x1−x0,y1−y0. The precision of the pixel displacement, dx,dy, is an integer pixel value with original images. The actual length (L) of the pattern is a known quantity, and the pixel width (W) of the pattern has been evaluated by the measurement system. The actual displacement, u,v, can be calculated by using estimated pixel displacement, dx,dy, to multiply by a pixel ratio (Rp) from equation (12). At this point, the time-history displacements are estimated completely in the digital image measurement system:(12)Rp=LW,u,v=dx,dy×Rp.The subpixel analysis can improve the precision based on the subpixel estimation of a target image. The results of Lu et al. [15] indicated that the measurement system increases the precision to a pixel value of 0.1, or even 0.01, and the precision achieves 0.01 mm if Rp is less than 0.1 mm. Moreover, the maximum correlation coefficient is higher than 0.95 in the correct search and lower than 0.75 in the fail search. In the fail situation, another target image is select to evaluate the maximum correlation coefficient again.By employing a noncontacted approach for measurement of structural dynamic response data, the record can be divided into 9,000 image frames if the time history of displacements is recorded as a video that contains 90 seconds with 100 fps data. The first frame will be set as a source (reference) image, and other frames are the target images and are processed sequentially to obtain the coordinates (x1, y1) in each target image; consequently, the time history of displacements can be figured out. Figure 4 graphically presents a series of frames in a video and the corresponding computed time history of displacements.Figure 4
A series of frames in a video and the corresponding computed time history of displacements.


## 4. Examples

To confirm the feasibility of the proposed approach for locating damage to building structures, two examples, a numerical example and an experimental example, are studied.


### 4.1. Numerical Example

As shown in Figure5, a six-story shear plane frame structure is studied in this example to evaluate the feasibility of the proposed approach for locating damage. Each floor consists of one beam and two columns. The cross-sectional size of the beam is 240 mm × 30 mm. The length of the beam is 360 mm. The cross-sectional size of the column is 240 mm × 15 mm. The length of the column is 180 mm. All members (beams and columns) are assumed to be made from the same material, ASTM A992 steel, with a yield stress of fy=3500kgf/cm2. In this example, SAP2000 is used for structural analysis. The natural frequencies of the first six modes are 1.653 Hz, 4.975 Hz, 8.341 Hz, 11.472 Hz, 14.130 Hz, and 15.938 Hz. The damage in this example is simulated as reduced floor stiffness. Single and multiple damage locations are studied. Table 1 presents the simulated damage cases. The 1995 Kobe earthquake record is used as the external excitation, but normalized to 100 gal as the peak ground acceleration (PGA).Figure 5
The six-story shear frame model.Table 1
Damage scenarios of the numerical example.
Case
Reduced percentage of stiffness
k
1
k
2
k
3
k
4
k
5
k6
Dam_2F5%
0
5
0
0
0
0
Dam_2F10%
0
10
0
0
0
0
Dam_2F15%
0
15
0
0
0
0
Dam_2F20%
0
20
0
0
0
0
Dam_2F25%
0
25
0
0
0
0
Dam_3F5%
0
0
5
0
0
0
Dam_3F10%
0
0
10
0
0
0
Dam_3F15%
0
0
15
0
0
0
Dam_3F20%
0
0
20
0
0
0
Dam_3F25%
0
0
25
0
0
0
Dam_4F5%
0
0
0
5
0
0
Dam_4F10%
0
0
0
10
0
0
Dam_4F15%
0
0
0
15
0
0
Dam_4F20%
0
0
0
20
0
0
Dam_4F25%
0
0
0
25
0
0
Dam_2F&4F5%
0
5
0
5
0
0
Dam_2F&4F10%
0
10
0
10
0
0
Dam_2F&4F15%
0
15
0
15
0
0
Dam_2F&4F20%
0
20
0
20
0
0
Dam_2F&4F25%
0
25
0
25
0
0
Dam_2F10%&4F15%
0
10
0
15
0
0
Dam_2F10%&4F20%
0
10
0
20
0
0
Dam_2F15%&4F10%
0
15
0
10
0
0
Dam_2F20%&4F10%
0
20
0
10
0
0
Dam_3F&5F5%
0
0
5
0
5
0
Dam_3F&5F10%
0
0
10
0
10
0
Dam_3F&5F15%
0
0
15
0
15
0
Dam_3F&5F20%
0
0
20
0
20
0
Dam_3F&5F25%
0
0
25
0
25
0
Dam_3F10%&5F15%
0
0
10
0
15
0
Dam_3F10%&5F20%
0
0
10
0
20
0
Dam_3F15%&5F10%
0
0
15
0
10
0
Dam_3F20%&5F10%
0
0
20
0
10
0
Dam_2F&3F15%
0
15
15
0
0
0
Dam_2F&4 F&6F5%
0
5
0
5
0
5
Dam_2F&4 F&6F10%
0
10
0
10
0
10
Dam_2F&4 F&6F15%
0
15
0
15
0
15
Dam_2F&4 F&6F20%
0
20
0
20
0
20
Dam_2F&4 F&6F25%
0
25
0
25
0
25
Dam_2F5%&4F15%&6F10%
0
5
0
15
0
10
Dam_2F10%&4F15%&6F5%
0
10
0
15
0
5
Dam_2F15%&4F10%&6F5%
0
15
0
10
0
5
Dam_2F&3 F&4F15%
0
15
15
15
0
0
Note. ki (i = 1∼6) represents the stiffness of the ith floor.


#### 4.1.1. Damage Location UsingSubFRFDI

Figure6 compares the substructure-based FRFs for the 2nd and the 5th substructures (H˜22ω and H˜55ω) for case Dam_2F15%. Significant changes were observed only in the substructure-based FRF of substructure 2, H˜22ω, which demonstrates the ability of the substructure-based FRF to characterize and locate damage.Figure 6
The substructure-based FRFs in the second substructure for case Dam_2F15%. (a) The substructure-based FRFs of the 2nd DOF in the second substructure H˜22ω. (b) The substructure-based FRFs of the 5th DOF in the fifth substructure H˜55ω.
(a)
(b)Figures7–15 present the comparison of the SubFRFDI values for damage cases listed in Table 1. Results show that although the damage locations of all cases shown in Figures 7–9 are predicted correctly by SubFRFDI, false detection could occur if the number of damage location is unknown. For example, the 1st and 3rd floors for case Dam_2F&4F15% (see Figure 8(a)), the 2nd floor for case Dam_3F&5F15% (see Figure 8(b)), the 1st floor for case Dam_2F&3F15% (see Figure 8(c)), and the 1st floor for case Dam_2F&3F&4F15% (see Figure 9(b)) may be detected falsely because their corresponding SubFRFDI values are large. Figures 10–12 show that SubFRFDI values corresponding to damaged and undamaged locations increase with the increase in damage extent for cases of single and multiple damage locations. Thus, false detection could occur for large damage extent cases if the number of damage location is unknown.Figure 7
SubFRFDI values for cases Dam_2F15%, Dam_3F15%, and Dam_4F15%.Figure 8
SubFRFDI values for (a) case Dam_2F&4F15%, (b) case Dam_3F&5F15%, and (c) case Dam_2F&3F15%.
(a)
(b)
(c)Figure 9
SubFRFDI values for (a) case Dam_2F&4F&6F15% and (b) case Dam_2F&3F&4F15%.
(a)
(b)Figure 10
The comparison of theSubFRFDI for 5%, 10%, 15%, 20%, and 25% stiffness reduction in (a) the 2nd floor, (b) the 3rd floor, and (c) the 4th floor.
(a)
(b)
(c)Figure 11
The comparison of theSubFRFDI for 5%, 10%, 15%, 20%, and 25% stiffness reduction in (a) the 2nd and the 4th floors and (b) the 3rd and the 5th floors.
(a)
(b)Figure 12
The comparison of theSubFRFDI for 5%, 10%, 15%, 20%, and 25% stiffness reduction in the 2nd, the 4th, and the 6th floors.Figure 13
SubFRFDI values for (a) case Dam_2F10%&4F15%, (b) case Dam_2F10%&4F20%, (c) case Dam_2F15%&4F10%, and (d) case Dam_2F20%&4F10%.
(a)
(b)
(c)
(d)Figure 14
SubFRFDI values for (a) case Dam_3F10%&5F15%, (b) case Dam_3F10%&5F20%, (c) case Dam_3F15%&5F10%, and (d) case Dam_3F20%&5F10%.
(a)
(b)
(c)
(d)Figure 15
SubFRFDI values for (a) case Dam_2F5%&4F15%&6F10%, (b) case Dam_2F10%&4F15%&6F5%, and (c) case Dam_2F15%&4F10%&6F5%.
(a)
(b)
(c)For cases of multiple nonadjacent damage locations (cases Dam_2F&4F15% (Figure8(a)), Dam_3F&5F15% (Figure 8(b)), and Dam_2F&4F&6F15% (Figure 9(a))), the damage locations can be predicted by the location (the lowest floor of the substructure) corresponding to the local maximum value of SubFRFDI. However, this criterion is not suitable for cases of multiple adjacent damage locations such as cases Dam_2F&3F15% (Figure 8(c)) and Dam_2F&3F&4F15% (Figure 9(b)).The stiffness reduction of the 2nd floor (10%) is smaller than that of the 4th floor (15%), but SubFRFDI2 is larger than SubFRFDI4 for case Dam_2F10%&4F15% (see Figure 13(a)). However, the stiffness reduction of the 2nd floor (10%) is smaller than that of the 4th floor (20%) and SubFRFDI2 is smaller than SubFRFDI4 for case Dam_2F10%&4F20% (see Figure 13(b)). The results imply that the damage extent of the ith floor is not absolutely larger than that of the jth floor if SubFRFDIi is larger than SubFRFDIj. Moreover, false detection could occur for cases Dam_2F20%&4F10%, Dam_3F15%&5F10%, Dam_3F20%&5F10%, and Dam_2F15%&4F10%&6F5% because SubFRFDI1 (0.293) is a little larger than SubFRFDI4 (0.286) for case Dam_2F20%&4F10% (see Figure 13(d)), SubFRFDI2 (0.400) is just a little smaller than SubFRFDI5 (0.443) for case Dam_3F15%&5F10% (see Figure 14(c)), SubFRFDI2 (0.592) is larger than SubFRFDI5 (0.438) for case Dam_3F20%&5F10% (see Figure 14(d)), and SubFRFDI3 (0.199) is just a little smaller than SubFRFDI6 (0.214) for case Dam_2F15%&4F10%&6F5% (see Figure 15(c)). The results imply that SubFRFDI is unable to locate damage for cases of two (three) damage locations when the damage extent (stiffness reduction) is equal to and larger than 10% (5%).


#### 4.1.2. Damage Location UsingCurveFRFDI

Figures16–24 present the comparison of the CurveFRFDI values for damage cases listed in Table 1. Results show that CurveFRFDI has higher sensitivity to damage than SubFRFDI for cases of single and multiple damage locations as compared Figures 16–24 with Figures 7–15. Nevertheless, results imply that CurveFRFDI is also unable to locate damage for cases of multiple adjacent damage locations. For example, the 3rd floor for case Dam_2F&3F15% (see Figure 17(c)) and the 3rd floor for case Dam_2F&3F&4F15% (see Figure 18(b)) are falsely detected.Figure 16
CurveFRFDI values for cases Dam_2F15%, Dam_3F15%, and Dam_4F15%.Figure 17
CurveFRFDI values for (a) case Dam_2F&4F15%, (b) case Dam_3F&5F15%, and (c) case Dam_2F&3F15%.
(a)
(b)
(c)Figure 18
CurveFRFDI values for (a) case Dam_2F&4F&6F15% and (b) case Dam_2F&3F&4F15%.
(a)
(b)Figure 19
The comparison of theCurveFRFDI for 5%, 10%, 15%, 20%, and 25% stiffness reduction in (a) the 2nd floor, (b) the 3rd floor, and (c) the 4th floor.
(a)
(b)
(c)Figure 20
The comparison of theCurveFRFDI for 5%, 10%, 15%, 20%, and 25% stiffness reduction in (a) the 2nd and the 4th floors and (b) the 3rd and the 5th floors.
(a)
(b)Figure 21
The comparison of theCurveFRFDI for 5%, 10%, 15%, 20%, and 25% stiffness reduction in the 2nd, the 4th, and the 6th floors.Figure 22
CurveFRFDI values for (a) case Dam_2F10%&4F15%, (b) case Dam_2F10%&4F20%, (c) case Dam_2F15%&4F10%, and (d) case Dam_2F20%&4F10%.
(a)
(b)
(c)
(d)Figure 23
CurveFRFDI values for (a) case Dam_3F10%&5F15%, (b) case Dam_3F10%&5F20%, (c) case Dam_3F15%&5F10%, and (d) case Dam_3F20%&5F10%.
(a)
(b)
(c)
(d)Figure 24
CurveFRFDI values for (a) case Dam_2F5%&4F15%&6F10%, (b) case Dam_2F10%&4F15%&6F5%, and (c) case Dam_2F15%&4F10%&6F5%.
(a)
(b)
(c)For most cases in Figures19–21, the CurveFRFDI value corresponding to the damaged location increases with the increase in damage extent while that corresponding to the undamaged location is almost zero and varies slightly with the increase in damage extent (only in Figures 19(c) and 20(b), the CurveFRFDI value corresponding to the second floor wrongly increases with the damage extent). It implies that CurveFRFDI can locate damage regardless of intensity (extent) for most cases. Notably, detecting small extent damage is an important issue for early warning of structural health monitoring. The excellent ability of CurveFRFDI to locate small extent damage can be investigated from case Dam_2F5% in Figure 19(a), case Dam_3F5% in Figure 19(b), case Dam_4F5% in Figure 19(c), case Dam_2F&4F5% in Figure 20(a), case Dam_3F&5F5% in Figure 20(b), and case Dam_2F&4F&6F5% in Figure 21.


### 4.2. Experimental Example

In this example, the dynamic responses of an eight-story downscale steel frame (Figure25), subjected to the Chi-Chi earthquake for different values of PGA (50 gal, 100 gal, 200 gal, 500 gal, and 1200 gal) shaking table tests, are processed to demonstrate the feasibility of the proposed approach for locating damage. Each floor of the steel frame has dimensions of 1500 mm in length, 1100 mm in width, and 1180 mm in height (Figure 26). Each floor consists of one slab and four columns. Each slab consists of two C 100 × 50 × 5 with length 1198 mm, two C 100 × 50 × 5 with length 798 mm, one Plate 1500 × 1100 × 20, and four Plate 150 × 150 × 50. Each column consists of four H 100 × 100 × 6 × 8 with length 1060 mm, four Plate 1060 × 150 × 25, and eight Plate 150 × 150 × 25. Table 2 lists the steel types used to construct the eight-story steel frame. These series of shaking table tests were undertaken by the National Center for Research on Earthquake Engineering in Taiwan. The displacements response histories of each floor are measured during the shaking table tests through linear variation differential transformation (LVDT) and a digital camera (Basler A504kc, sampling rate of 500 Hz) combined with the DIC approach presented in the work of Lu et al. [15], abbreviated as LVDT-measured data and DIC-measured data below. The damage in this example is simulated by reducing the cross section of certain columns as shown in Figure 27. Single and multiple nonadjacent damage locations are studied. Table 3 presents the simulated damage cases.Figure 25
The eight-story steel frame.Figure 26
A single floor of the eight-story steel frame: (a) front view and (b) side view.
(a)
(b)Table 2
Steel types of the six-story shear frame.
Member
Steel type
Type
Number
Type
Size (mm)
Number
Slab
8
C 100 × 50 × 5
Length: 1198
16
C 100 × 50 × 5
Length: 798
16
Plate 1500 × 1100 × 20
Length: 1500Width: 1100Thickness: 20
8
Plate 150 × 150 × 50
Length: 150Width: 150Thickness: 50
32
Column
32
H 100 × 100 × 6 × 8
Length: 1060
32
Plate 1060 × 150 × 25
Length: 1060Width: 150Thickness: 25
32
Plate 150 × 150 × 25
Length: 150Width: 150Thickness: 25
64Figure 27
The damaged columns.
(a)
(b)
(c)Table 3
Damage scenarios of the experimental example.
Case
Damage location
External excitation
Dam50_1F
The 1st floor
The Chi-Chi earthquake with PGA = 50 gal
Dam100_1F
The 1st floor
The Chi-Chi earthquake with PGA = 100 gal
Dam50_1F&3F
The 1st and the 3rd floors
The Chi-Chi earthquake with PGA = 50 gal
Dam100_1F&3F
The 1st and the 3rd floors
The Chi-Chi earthquake with PGA = 100 gal
Dam200_1F&3F
The 1st and the 3rd floors
The Chi-Chi earthquake with PGA = 200 gal
Dam500_1F&3F
The 1st and the 3rd floors
The Chi-Chi earthquake with PGA = 500 gal
Dam1200_1F&3F
The 1st and the 3rd floors
The Chi-Chi earthquake with PGA = 1200 gal


#### 4.2.1. Damage Location Using LVDT-Measured Data

Figure28 presents the comparison of the SubFRFDI values of LVDT-measured data for cases of single damage location, Dam50_1F and Dam100_1F. The damage locations of the two cases are predicted correctly by the SubFRFDI of LVDT-measured data. Figure 29 shows the comparison of the SubFRFDI values of LVDT-measured data for cases of two damage locations, Dam50_1F&3F, Dam100_1F&3F, Dam200_1F&3F, Dam500_1F&3F, and Dam1200_1F&3F. It shows that SubFRFDI1 and SubFRFDI3 are larger than others, and the damage locations (the 1st and the 3rd floors) of these cases are predicted correctly by the SubFRFDI if the number of damage location is known in advance. Nevertheless, SubFRFDI3 is much smaller than SubFRFDI1 and false detection of the 3rd floor may be occurred if the number of damage location is unknown.Figure 28
SubFRFDI values of LVDT-measured data for (a) case Dam50_1F and (b) case Dam100_1F.
(a)
(b)Figure 29
SubFRFDI values of LVDT-measured data for (a) case Dam50_1F&3F, (b) case Dam100_1F&3F, (c) case Dam200_1F&3F, (d) case Dam500_1F&3F, and (e) case Dam1200_1F&3F.
(a)
(b)
(c)
(d)
(e)Figure30 presents the comparison of the CurveFRFDI values of LVDT-measured data for cases of single damage location, Dam50_1F and Dam100_1F. The damage locations of the two cases are also predicted correctly by the CurveFRFDI of LVDT-measured data. Figure 31 shows the comparison of the CurveFRFDI values of LVDT-measured data for cases of two damage locations, Dam50_1F&3F, Dam100_1F&3F, Dam200_1F&3F, Dam500_1F&3F, and Dam1200_1F&3F. Although CurveFRFDI3 is much smaller than CurveFRFDI1 and false detection of the 3rd floor may be occurred if the number of damage location is unknown, CurveFRFDI has higher sensitivity to damage than SubFRFDI as compared Figures 30–31 with Figures 28–29. To solve this difficulty, a threshold CurveFRFDI value is suggested to be set to locate damage. The threshold CurveFRFDI value can be determined after numerous numerical simulations of damage scenarios for a certain building structure. For example, the threshold CurveFRFDI value can set to be 0.08 for this experimental example.Figure 30
CurveFRFDI values of LVDT-measured data for (a) case Dam50_1F and (b) case Dam100_1F.
(a)
(b)Figure 31
CurveFRFDI values of LVDT-measured data for (a) case Dam50_1F&3F, (b) case Dam100_1F&3F, (c) case Dam200_1F&3F, (d) case Dam500_1F&3F, and (e) case Dam1200_1F&3F.
(a)
(b)
(c)
(d)
(e)


#### 4.2.2. Damage Location Using DIC-Measured Data

Figure32 presents the comparison of the SubFRFDI values of DIC-measured data for case Dam100_1F. Figure 33 shows the comparison of the SubFRFDI values of DIC-measured data for cases of two damage locations, Dam200_1F&3F and Dam500_1F&3F. Figure 34 presents the comparison of the CurveFRFDI values of DIC-measured data for case Dam100_1F. Figure 35 shows the comparison of the CurveFRFDI values of DIC-measured data for cases of two damage locations, Dam200_1F&3F and Dam500_1F&3F. The results also show that the CurveFRFDI has higher sensitivity to damage than SubFRFDI as compared Figures 34–35 with Figures 32–33. The results further indicate that applying the proposed approach to locate single and multiple nonadjacent damages to building structures is feasible by using DIC-measured displacements.Figure 32
SubFRFDI values of DIC-measured data for case Dam100_1F.Figure 33
SubFRFDI values of DIC-measured data for (a) case Dam200_1F&3F and (b) case Dam500_1F&3F.
(a)
(b)Figure 34
CurveFRFDI values of DIC-measured data for case Dam100_1F.Figure 35
CurveFRFDI values of DIC-measured data for (a) case Dam200_1F&3F and (b) case Dam500_1F&3F.
(a)
(b)The experimental example proves the feasibility of the proposed approach for location damage to building structures using DIC-measured displacement response. These data are supposed to be noise free or low noise corrupted. Future work should apply the proposed approach to measurements in the field (actual cases) to investigate its capacity to deal with DIC-measured displacement response corrupted by high noise.


## 4.1. Numerical Example

As shown in Figure5, a six-story shear plane frame structure is studied in this example to evaluate the feasibility of the proposed approach for locating damage. Each floor consists of one beam and two columns. The cross-sectional size of the beam is 240 mm × 30 mm. The length of the beam is 360 mm. The cross-sectional size of the column is 240 mm × 15 mm. The length of the column is 180 mm. All members (beams and columns) are assumed to be made from the same material, ASTM A992 steel, with a yield stress of fy=3500kgf/cm2. In this example, SAP2000 is used for structural analysis. The natural frequencies of the first six modes are 1.653 Hz, 4.975 Hz, 8.341 Hz, 11.472 Hz, 14.130 Hz, and 15.938 Hz. The damage in this example is simulated as reduced floor stiffness. Single and multiple damage locations are studied. Table 1 presents the simulated damage cases. The 1995 Kobe earthquake record is used as the external excitation, but normalized to 100 gal as the peak ground acceleration (PGA).Figure 5
The six-story shear frame model.Table 1
Damage scenarios of the numerical example.
Case
Reduced percentage of stiffness
k
1
k
2
k
3
k
4
k
5
k6
Dam_2F5%
0
5
0
0
0
0
Dam_2F10%
0
10
0
0
0
0
Dam_2F15%
0
15
0
0
0
0
Dam_2F20%
0
20
0
0
0
0
Dam_2F25%
0
25
0
0
0
0
Dam_3F5%
0
0
5
0
0
0
Dam_3F10%
0
0
10
0
0
0
Dam_3F15%
0
0
15
0
0
0
Dam_3F20%
0
0
20
0
0
0
Dam_3F25%
0
0
25
0
0
0
Dam_4F5%
0
0
0
5
0
0
Dam_4F10%
0
0
0
10
0
0
Dam_4F15%
0
0
0
15
0
0
Dam_4F20%
0
0
0
20
0
0
Dam_4F25%
0
0
0
25
0
0
Dam_2F&4F5%
0
5
0
5
0
0
Dam_2F&4F10%
0
10
0
10
0
0
Dam_2F&4F15%
0
15
0
15
0
0
Dam_2F&4F20%
0
20
0
20
0
0
Dam_2F&4F25%
0
25
0
25
0
0
Dam_2F10%&4F15%
0
10
0
15
0
0
Dam_2F10%&4F20%
0
10
0
20
0
0
Dam_2F15%&4F10%
0
15
0
10
0
0
Dam_2F20%&4F10%
0
20
0
10
0
0
Dam_3F&5F5%
0
0
5
0
5
0
Dam_3F&5F10%
0
0
10
0
10
0
Dam_3F&5F15%
0
0
15
0
15
0
Dam_3F&5F20%
0
0
20
0
20
0
Dam_3F&5F25%
0
0
25
0
25
0
Dam_3F10%&5F15%
0
0
10
0
15
0
Dam_3F10%&5F20%
0
0
10
0
20
0
Dam_3F15%&5F10%
0
0
15
0
10
0
Dam_3F20%&5F10%
0
0
20
0
10
0
Dam_2F&3F15%
0
15
15
0
0
0
Dam_2F&4 F&6F5%
0
5
0
5
0
5
Dam_2F&4 F&6F10%
0
10
0
10
0
10
Dam_2F&4 F&6F15%
0
15
0
15
0
15
Dam_2F&4 F&6F20%
0
20
0
20
0
20
Dam_2F&4 F&6F25%
0
25
0
25
0
25
Dam_2F5%&4F15%&6F10%
0
5
0
15
0
10
Dam_2F10%&4F15%&6F5%
0
10
0
15
0
5
Dam_2F15%&4F10%&6F5%
0
15
0
10
0
5
Dam_2F&3 F&4F15%
0
15
15
15
0
0
Note. ki (i = 1∼6) represents the stiffness of the ith floor.


### 4.1.1. Damage Location UsingSubFRFDI

Figure6 compares the substructure-based FRFs for the 2nd and the 5th substructures (H˜22ω and H˜55ω) for case Dam_2F15%. Significant changes were observed only in the substructure-based FRF of substructure 2, H˜22ω, which demonstrates the ability of the substructure-based FRF to characterize and locate damage.Figure 6
The substructure-based FRFs in the second substructure for case Dam_2F15%. (a) The substructure-based FRFs of the 2nd DOF in the second substructure H˜22ω. (b) The substructure-based FRFs of the 5th DOF in the fifth substructure H˜55ω.
(a)
(b)Figures7–15 present the comparison of the SubFRFDI values for damage cases listed in Table 1. Results show that although the damage locations of all cases shown in Figures 7–9 are predicted correctly by SubFRFDI, false detection could occur if the number of damage location is unknown. For example, the 1st and 3rd floors for case Dam_2F&4F15% (see Figure 8(a)), the 2nd floor for case Dam_3F&5F15% (see Figure 8(b)), the 1st floor for case Dam_2F&3F15% (see Figure 8(c)), and the 1st floor for case Dam_2F&3F&4F15% (see Figure 9(b)) may be detected falsely because their corresponding SubFRFDI values are large. Figures 10–12 show that SubFRFDI values corresponding to damaged and undamaged locations increase with the increase in damage extent for cases of single and multiple damage locations. Thus, false detection could occur for large damage extent cases if the number of damage location is unknown.Figure 7
SubFRFDI values for cases Dam_2F15%, Dam_3F15%, and Dam_4F15%.Figure 8
SubFRFDI values for (a) case Dam_2F&4F15%, (b) case Dam_3F&5F15%, and (c) case Dam_2F&3F15%.
(a)
(b)
(c)Figure 9
SubFRFDI values for (a) case Dam_2F&4F&6F15% and (b) case Dam_2F&3F&4F15%.
(a)
(b)Figure 10
The comparison of theSubFRFDI for 5%, 10%, 15%, 20%, and 25% stiffness reduction in (a) the 2nd floor, (b) the 3rd floor, and (c) the 4th floor.
(a)
(b)
(c)Figure 11
The comparison of theSubFRFDI for 5%, 10%, 15%, 20%, and 25% stiffness reduction in (a) the 2nd and the 4th floors and (b) the 3rd and the 5th floors.
(a)
(b)Figure 12
The comparison of theSubFRFDI for 5%, 10%, 15%, 20%, and 25% stiffness reduction in the 2nd, the 4th, and the 6th floors.Figure 13
SubFRFDI values for (a) case Dam_2F10%&4F15%, (b) case Dam_2F10%&4F20%, (c) case Dam_2F15%&4F10%, and (d) case Dam_2F20%&4F10%.
(a)
(b)
(c)
(d)Figure 14
SubFRFDI values for (a) case Dam_3F10%&5F15%, (b) case Dam_3F10%&5F20%, (c) case Dam_3F15%&5F10%, and (d) case Dam_3F20%&5F10%.
(a)
(b)
(c)
(d)Figure 15
SubFRFDI values for (a) case Dam_2F5%&4F15%&6F10%, (b) case Dam_2F10%&4F15%&6F5%, and (c) case Dam_2F15%&4F10%&6F5%.
(a)
(b)
(c)For cases of multiple nonadjacent damage locations (cases Dam_2F&4F15% (Figure8(a)), Dam_3F&5F15% (Figure 8(b)), and Dam_2F&4F&6F15% (Figure 9(a))), the damage locations can be predicted by the location (the lowest floor of the substructure) corresponding to the local maximum value of SubFRFDI. However, this criterion is not suitable for cases of multiple adjacent damage locations such as cases Dam_2F&3F15% (Figure 8(c)) and Dam_2F&3F&4F15% (Figure 9(b)).The stiffness reduction of the 2nd floor (10%) is smaller than that of the 4th floor (15%), but SubFRFDI2 is larger than SubFRFDI4 for case Dam_2F10%&4F15% (see Figure 13(a)). However, the stiffness reduction of the 2nd floor (10%) is smaller than that of the 4th floor (20%) and SubFRFDI2 is smaller than SubFRFDI4 for case Dam_2F10%&4F20% (see Figure 13(b)). The results imply that the damage extent of the ith floor is not absolutely larger than that of the jth floor if SubFRFDIi is larger than SubFRFDIj. Moreover, false detection could occur for cases Dam_2F20%&4F10%, Dam_3F15%&5F10%, Dam_3F20%&5F10%, and Dam_2F15%&4F10%&6F5% because SubFRFDI1 (0.293) is a little larger than SubFRFDI4 (0.286) for case Dam_2F20%&4F10% (see Figure 13(d)), SubFRFDI2 (0.400) is just a little smaller than SubFRFDI5 (0.443) for case Dam_3F15%&5F10% (see Figure 14(c)), SubFRFDI2 (0.592) is larger than SubFRFDI5 (0.438) for case Dam_3F20%&5F10% (see Figure 14(d)), and SubFRFDI3 (0.199) is just a little smaller than SubFRFDI6 (0.214) for case Dam_2F15%&4F10%&6F5% (see Figure 15(c)). The results imply that SubFRFDI is unable to locate damage for cases of two (three) damage locations when the damage extent (stiffness reduction) is equal to and larger than 10% (5%).


### 4.1.2. Damage Location UsingCurveFRFDI

Figures16–24 present the comparison of the CurveFRFDI values for damage cases listed in Table 1. Results show that CurveFRFDI has higher sensitivity to damage than SubFRFDI for cases of single and multiple damage locations as compared Figures 16–24 with Figures 7–15. Nevertheless, results imply that CurveFRFDI is also unable to locate damage for cases of multiple adjacent damage locations. For example, the 3rd floor for case Dam_2F&3F15% (see Figure 17(c)) and the 3rd floor for case Dam_2F&3F&4F15% (see Figure 18(b)) are falsely detected.Figure 16
CurveFRFDI values for cases Dam_2F15%, Dam_3F15%, and Dam_4F15%.Figure 17
CurveFRFDI values for (a) case Dam_2F&4F15%, (b) case Dam_3F&5F15%, and (c) case Dam_2F&3F15%.
(a)
(b)
(c)Figure 18
CurveFRFDI values for (a) case Dam_2F&4F&6F15% and (b) case Dam_2F&3F&4F15%.
(a)
(b)Figure 19
The comparison of theCurveFRFDI for 5%, 10%, 15%, 20%, and 25% stiffness reduction in (a) the 2nd floor, (b) the 3rd floor, and (c) the 4th floor.
(a)
(b)
(c)Figure 20
The comparison of theCurveFRFDI for 5%, 10%, 15%, 20%, and 25% stiffness reduction in (a) the 2nd and the 4th floors and (b) the 3rd and the 5th floors.
(a)
(b)Figure 21
The comparison of theCurveFRFDI for 5%, 10%, 15%, 20%, and 25% stiffness reduction in the 2nd, the 4th, and the 6th floors.Figure 22
CurveFRFDI values for (a) case Dam_2F10%&4F15%, (b) case Dam_2F10%&4F20%, (c) case Dam_2F15%&4F10%, and (d) case Dam_2F20%&4F10%.
(a)
(b)
(c)
(d)Figure 23
CurveFRFDI values for (a) case Dam_3F10%&5F15%, (b) case Dam_3F10%&5F20%, (c) case Dam_3F15%&5F10%, and (d) case Dam_3F20%&5F10%.
(a)
(b)
(c)
(d)Figure 24
CurveFRFDI values for (a) case Dam_2F5%&4F15%&6F10%, (b) case Dam_2F10%&4F15%&6F5%, and (c) case Dam_2F15%&4F10%&6F5%.
(a)
(b)
(c)For most cases in Figures19–21, the CurveFRFDI value corresponding to the damaged location increases with the increase in damage extent while that corresponding to the undamaged location is almost zero and varies slightly with the increase in damage extent (only in Figures 19(c) and 20(b), the CurveFRFDI value corresponding to the second floor wrongly increases with the damage extent). It implies that CurveFRFDI can locate damage regardless of intensity (extent) for most cases. Notably, detecting small extent damage is an important issue for early warning of structural health monitoring. The excellent ability of CurveFRFDI to locate small extent damage can be investigated from case Dam_2F5% in Figure 19(a), case Dam_3F5% in Figure 19(b), case Dam_4F5% in Figure 19(c), case Dam_2F&4F5% in Figure 20(a), case Dam_3F&5F5% in Figure 20(b), and case Dam_2F&4F&6F5% in Figure 21.


## 4.1.1. Damage Location UsingSubFRFDI

Figure6 compares the substructure-based FRFs for the 2nd and the 5th substructures (H˜22ω and H˜55ω) for case Dam_2F15%. Significant changes were observed only in the substructure-based FRF of substructure 2, H˜22ω, which demonstrates the ability of the substructure-based FRF to characterize and locate damage.Figure 6
The substructure-based FRFs in the second substructure for case Dam_2F15%. (a) The substructure-based FRFs of the 2nd DOF in the second substructure H˜22ω. (b) The substructure-based FRFs of the 5th DOF in the fifth substructure H˜55ω.
(a)
(b)Figures7–15 present the comparison of the SubFRFDI values for damage cases listed in Table 1. Results show that although the damage locations of all cases shown in Figures 7–9 are predicted correctly by SubFRFDI, false detection could occur if the number of damage location is unknown. For example, the 1st and 3rd floors for case Dam_2F&4F15% (see Figure 8(a)), the 2nd floor for case Dam_3F&5F15% (see Figure 8(b)), the 1st floor for case Dam_2F&3F15% (see Figure 8(c)), and the 1st floor for case Dam_2F&3F&4F15% (see Figure 9(b)) may be detected falsely because their corresponding SubFRFDI values are large. Figures 10–12 show that SubFRFDI values corresponding to damaged and undamaged locations increase with the increase in damage extent for cases of single and multiple damage locations. Thus, false detection could occur for large damage extent cases if the number of damage location is unknown.Figure 7
SubFRFDI values for cases Dam_2F15%, Dam_3F15%, and Dam_4F15%.Figure 8
SubFRFDI values for (a) case Dam_2F&4F15%, (b) case Dam_3F&5F15%, and (c) case Dam_2F&3F15%.
(a)
(b)
(c)Figure 9
SubFRFDI values for (a) case Dam_2F&4F&6F15% and (b) case Dam_2F&3F&4F15%.
(a)
(b)Figure 10
The comparison of theSubFRFDI for 5%, 10%, 15%, 20%, and 25% stiffness reduction in (a) the 2nd floor, (b) the 3rd floor, and (c) the 4th floor.
(a)
(b)
(c)Figure 11
The comparison of theSubFRFDI for 5%, 10%, 15%, 20%, and 25% stiffness reduction in (a) the 2nd and the 4th floors and (b) the 3rd and the 5th floors.
(a)
(b)Figure 12
The comparison of theSubFRFDI for 5%, 10%, 15%, 20%, and 25% stiffness reduction in the 2nd, the 4th, and the 6th floors.Figure 13
SubFRFDI values for (a) case Dam_2F10%&4F15%, (b) case Dam_2F10%&4F20%, (c) case Dam_2F15%&4F10%, and (d) case Dam_2F20%&4F10%.
(a)
(b)
(c)
(d)Figure 14
SubFRFDI values for (a) case Dam_3F10%&5F15%, (b) case Dam_3F10%&5F20%, (c) case Dam_3F15%&5F10%, and (d) case Dam_3F20%&5F10%.
(a)
(b)
(c)
(d)Figure 15
SubFRFDI values for (a) case Dam_2F5%&4F15%&6F10%, (b) case Dam_2F10%&4F15%&6F5%, and (c) case Dam_2F15%&4F10%&6F5%.
(a)
(b)
(c)For cases of multiple nonadjacent damage locations (cases Dam_2F&4F15% (Figure8(a)), Dam_3F&5F15% (Figure 8(b)), and Dam_2F&4F&6F15% (Figure 9(a))), the damage locations can be predicted by the location (the lowest floor of the substructure) corresponding to the local maximum value of SubFRFDI. However, this criterion is not suitable for cases of multiple adjacent damage locations such as cases Dam_2F&3F15% (Figure 8(c)) and Dam_2F&3F&4F15% (Figure 9(b)).The stiffness reduction of the 2nd floor (10%) is smaller than that of the 4th floor (15%), but SubFRFDI2 is larger than SubFRFDI4 for case Dam_2F10%&4F15% (see Figure 13(a)). However, the stiffness reduction of the 2nd floor (10%) is smaller than that of the 4th floor (20%) and SubFRFDI2 is smaller than SubFRFDI4 for case Dam_2F10%&4F20% (see Figure 13(b)). The results imply that the damage extent of the ith floor is not absolutely larger than that of the jth floor if SubFRFDIi is larger than SubFRFDIj. Moreover, false detection could occur for cases Dam_2F20%&4F10%, Dam_3F15%&5F10%, Dam_3F20%&5F10%, and Dam_2F15%&4F10%&6F5% because SubFRFDI1 (0.293) is a little larger than SubFRFDI4 (0.286) for case Dam_2F20%&4F10% (see Figure 13(d)), SubFRFDI2 (0.400) is just a little smaller than SubFRFDI5 (0.443) for case Dam_3F15%&5F10% (see Figure 14(c)), SubFRFDI2 (0.592) is larger than SubFRFDI5 (0.438) for case Dam_3F20%&5F10% (see Figure 14(d)), and SubFRFDI3 (0.199) is just a little smaller than SubFRFDI6 (0.214) for case Dam_2F15%&4F10%&6F5% (see Figure 15(c)). The results imply that SubFRFDI is unable to locate damage for cases of two (three) damage locations when the damage extent (stiffness reduction) is equal to and larger than 10% (5%).


## 4.1.2. Damage Location UsingCurveFRFDI

Figures16–24 present the comparison of the CurveFRFDI values for damage cases listed in Table 1. Results show that CurveFRFDI has higher sensitivity to damage than SubFRFDI for cases of single and multiple damage locations as compared Figures 16–24 with Figures 7–15. Nevertheless, results imply that CurveFRFDI is also unable to locate damage for cases of multiple adjacent damage locations. For example, the 3rd floor for case Dam_2F&3F15% (see Figure 17(c)) and the 3rd floor for case Dam_2F&3F&4F15% (see Figure 18(b)) are falsely detected.Figure 16
CurveFRFDI values for cases Dam_2F15%, Dam_3F15%, and Dam_4F15%.Figure 17
CurveFRFDI values for (a) case Dam_2F&4F15%, (b) case Dam_3F&5F15%, and (c) case Dam_2F&3F15%.
(a)
(b)
(c)Figure 18
CurveFRFDI values for (a) case Dam_2F&4F&6F15% and (b) case Dam_2F&3F&4F15%.
(a)
(b)Figure 19
The comparison of theCurveFRFDI for 5%, 10%, 15%, 20%, and 25% stiffness reduction in (a) the 2nd floor, (b) the 3rd floor, and (c) the 4th floor.
(a)
(b)
(c)Figure 20
The comparison of theCurveFRFDI for 5%, 10%, 15%, 20%, and 25% stiffness reduction in (a) the 2nd and the 4th floors and (b) the 3rd and the 5th floors.
(a)
(b)Figure 21
The comparison of theCurveFRFDI for 5%, 10%, 15%, 20%, and 25% stiffness reduction in the 2nd, the 4th, and the 6th floors.Figure 22
CurveFRFDI values for (a) case Dam_2F10%&4F15%, (b) case Dam_2F10%&4F20%, (c) case Dam_2F15%&4F10%, and (d) case Dam_2F20%&4F10%.
(a)
(b)
(c)
(d)Figure 23
CurveFRFDI values for (a) case Dam_3F10%&5F15%, (b) case Dam_3F10%&5F20%, (c) case Dam_3F15%&5F10%, and (d) case Dam_3F20%&5F10%.
(a)
(b)
(c)
(d)Figure 24
CurveFRFDI values for (a) case Dam_2F5%&4F15%&6F10%, (b) case Dam_2F10%&4F15%&6F5%, and (c) case Dam_2F15%&4F10%&6F5%.
(a)
(b)
(c)For most cases in Figures19–21, the CurveFRFDI value corresponding to the damaged location increases with the increase in damage extent while that corresponding to the undamaged location is almost zero and varies slightly with the increase in damage extent (only in Figures 19(c) and 20(b), the CurveFRFDI value corresponding to the second floor wrongly increases with the damage extent). It implies that CurveFRFDI can locate damage regardless of intensity (extent) for most cases. Notably, detecting small extent damage is an important issue for early warning of structural health monitoring. The excellent ability of CurveFRFDI to locate small extent damage can be investigated from case Dam_2F5% in Figure 19(a), case Dam_3F5% in Figure 19(b), case Dam_4F5% in Figure 19(c), case Dam_2F&4F5% in Figure 20(a), case Dam_3F&5F5% in Figure 20(b), and case Dam_2F&4F&6F5% in Figure 21.


## 4.2. Experimental Example

In this example, the dynamic responses of an eight-story downscale steel frame (Figure25), subjected to the Chi-Chi earthquake for different values of PGA (50 gal, 100 gal, 200 gal, 500 gal, and 1200 gal) shaking table tests, are processed to demonstrate the feasibility of the proposed approach for locating damage. Each floor of the steel frame has dimensions of 1500 mm in length, 1100 mm in width, and 1180 mm in height (Figure 26). Each floor consists of one slab and four columns. Each slab consists of two C 100 × 50 × 5 with length 1198 mm, two C 100 × 50 × 5 with length 798 mm, one Plate 1500 × 1100 × 20, and four Plate 150 × 150 × 50. Each column consists of four H 100 × 100 × 6 × 8 with length 1060 mm, four Plate 1060 × 150 × 25, and eight Plate 150 × 150 × 25. Table 2 lists the steel types used to construct the eight-story steel frame. These series of shaking table tests were undertaken by the National Center for Research on Earthquake Engineering in Taiwan. The displacements response histories of each floor are measured during the shaking table tests through linear variation differential transformation (LVDT) and a digital camera (Basler A504kc, sampling rate of 500 Hz) combined with the DIC approach presented in the work of Lu et al. [15], abbreviated as LVDT-measured data and DIC-measured data below. The damage in this example is simulated by reducing the cross section of certain columns as shown in Figure 27. Single and multiple nonadjacent damage locations are studied. Table 3 presents the simulated damage cases.Figure 25
The eight-story steel frame.Figure 26
A single floor of the eight-story steel frame: (a) front view and (b) side view.
(a)
(b)Table 2
Steel types of the six-story shear frame.
Member
Steel type
Type
Number
Type
Size (mm)
Number
Slab
8
C 100 × 50 × 5
Length: 1198
16
C 100 × 50 × 5
Length: 798
16
Plate 1500 × 1100 × 20
Length: 1500Width: 1100Thickness: 20
8
Plate 150 × 150 × 50
Length: 150Width: 150Thickness: 50
32
Column
32
H 100 × 100 × 6 × 8
Length: 1060
32
Plate 1060 × 150 × 25
Length: 1060Width: 150Thickness: 25
32
Plate 150 × 150 × 25
Length: 150Width: 150Thickness: 25
64Figure 27
The damaged columns.
(a)
(b)
(c)Table 3
Damage scenarios of the experimental example.
Case
Damage location
External excitation
Dam50_1F
The 1st floor
The Chi-Chi earthquake with PGA = 50 gal
Dam100_1F
The 1st floor
The Chi-Chi earthquake with PGA = 100 gal
Dam50_1F&3F
The 1st and the 3rd floors
The Chi-Chi earthquake with PGA = 50 gal
Dam100_1F&3F
The 1st and the 3rd floors
The Chi-Chi earthquake with PGA = 100 gal
Dam200_1F&3F
The 1st and the 3rd floors
The Chi-Chi earthquake with PGA = 200 gal
Dam500_1F&3F
The 1st and the 3rd floors
The Chi-Chi earthquake with PGA = 500 gal
Dam1200_1F&3F
The 1st and the 3rd floors
The Chi-Chi earthquake with PGA = 1200 gal


### 4.2.1. Damage Location Using LVDT-Measured Data

Figure28 presents the comparison of the SubFRFDI values of LVDT-measured data for cases of single damage location, Dam50_1F and Dam100_1F. The damage locations of the two cases are predicted correctly by the SubFRFDI of LVDT-measured data. Figure 29 shows the comparison of the SubFRFDI values of LVDT-measured data for cases of two damage locations, Dam50_1F&3F, Dam100_1F&3F, Dam200_1F&3F, Dam500_1F&3F, and Dam1200_1F&3F. It shows that SubFRFDI1 and SubFRFDI3 are larger than others, and the damage locations (the 1st and the 3rd floors) of these cases are predicted correctly by the SubFRFDI if the number of damage location is known in advance. Nevertheless, SubFRFDI3 is much smaller than SubFRFDI1 and false detection of the 3rd floor may be occurred if the number of damage location is unknown.Figure 28
SubFRFDI values of LVDT-measured data for (a) case Dam50_1F and (b) case Dam100_1F.
(a)
(b)Figure 29
SubFRFDI values of LVDT-measured data for (a) case Dam50_1F&3F, (b) case Dam100_1F&3F, (c) case Dam200_1F&3F, (d) case Dam500_1F&3F, and (e) case Dam1200_1F&3F.
(a)
(b)
(c)
(d)
(e)Figure30 presents the comparison of the CurveFRFDI values of LVDT-measured data for cases of single damage location, Dam50_1F and Dam100_1F. The damage locations of the two cases are also predicted correctly by the CurveFRFDI of LVDT-measured data. Figure 31 shows the comparison of the CurveFRFDI values of LVDT-measured data for cases of two damage locations, Dam50_1F&3F, Dam100_1F&3F, Dam200_1F&3F, Dam500_1F&3F, and Dam1200_1F&3F. Although CurveFRFDI3 is much smaller than CurveFRFDI1 and false detection of the 3rd floor may be occurred if the number of damage location is unknown, CurveFRFDI has higher sensitivity to damage than SubFRFDI as compared Figures 30–31 with Figures 28–29. To solve this difficulty, a threshold CurveFRFDI value is suggested to be set to locate damage. The threshold CurveFRFDI value can be determined after numerous numerical simulations of damage scenarios for a certain building structure. For example, the threshold CurveFRFDI value can set to be 0.08 for this experimental example.Figure 30
CurveFRFDI values of LVDT-measured data for (a) case Dam50_1F and (b) case Dam100_1F.
(a)
(b)Figure 31
CurveFRFDI values of LVDT-measured data for (a) case Dam50_1F&3F, (b) case Dam100_1F&3F, (c) case Dam200_1F&3F, (d) case Dam500_1F&3F, and (e) case Dam1200_1F&3F.
(a)
(b)
(c)
(d)
(e)


### 4.2.2. Damage Location Using DIC-Measured Data

Figure32 presents the comparison of the SubFRFDI values of DIC-measured data for case Dam100_1F. Figure 33 shows the comparison of the SubFRFDI values of DIC-measured data for cases of two damage locations, Dam200_1F&3F and Dam500_1F&3F. Figure 34 presents the comparison of the CurveFRFDI values of DIC-measured data for case Dam100_1F. Figure 35 shows the comparison of the CurveFRFDI values of DIC-measured data for cases of two damage locations, Dam200_1F&3F and Dam500_1F&3F. The results also show that the CurveFRFDI has higher sensitivity to damage than SubFRFDI as compared Figures 34–35 with Figures 32–33. The results further indicate that applying the proposed approach to locate single and multiple nonadjacent damages to building structures is feasible by using DIC-measured displacements.Figure 32
SubFRFDI values of DIC-measured data for case Dam100_1F.Figure 33
SubFRFDI values of DIC-measured data for (a) case Dam200_1F&3F and (b) case Dam500_1F&3F.
(a)
(b)Figure 34
CurveFRFDI values of DIC-measured data for case Dam100_1F.Figure 35
CurveFRFDI values of DIC-measured data for (a) case Dam200_1F&3F and (b) case Dam500_1F&3F.
(a)
(b)The experimental example proves the feasibility of the proposed approach for location damage to building structures using DIC-measured displacement response. These data are supposed to be noise free or low noise corrupted. Future work should apply the proposed approach to measurements in the field (actual cases) to investigate its capacity to deal with DIC-measured displacement response corrupted by high noise.


## 4.2.1. Damage Location Using LVDT-Measured Data

Figure28 presents the comparison of the SubFRFDI values of LVDT-measured data for cases of single damage location, Dam50_1F and Dam100_1F. The damage locations of the two cases are predicted correctly by the SubFRFDI of LVDT-measured data. Figure 29 shows the comparison of the SubFRFDI values of LVDT-measured data for cases of two damage locations, Dam50_1F&3F, Dam100_1F&3F, Dam200_1F&3F, Dam500_1F&3F, and Dam1200_1F&3F. It shows that SubFRFDI1 and SubFRFDI3 are larger than others, and the damage locations (the 1st and the 3rd floors) of these cases are predicted correctly by the SubFRFDI if the number of damage location is known in advance. Nevertheless, SubFRFDI3 is much smaller than SubFRFDI1 and false detection of the 3rd floor may be occurred if the number of damage location is unknown.Figure 28
SubFRFDI values of LVDT-measured data for (a) case Dam50_1F and (b) case Dam100_1F.
(a)
(b)Figure 29
SubFRFDI values of LVDT-measured data for (a) case Dam50_1F&3F, (b) case Dam100_1F&3F, (c) case Dam200_1F&3F, (d) case Dam500_1F&3F, and (e) case Dam1200_1F&3F.
(a)
(b)
(c)
(d)
(e)Figure30 presents the comparison of the CurveFRFDI values of LVDT-measured data for cases of single damage location, Dam50_1F and Dam100_1F. The damage locations of the two cases are also predicted correctly by the CurveFRFDI of LVDT-measured data. Figure 31 shows the comparison of the CurveFRFDI values of LVDT-measured data for cases of two damage locations, Dam50_1F&3F, Dam100_1F&3F, Dam200_1F&3F, Dam500_1F&3F, and Dam1200_1F&3F. Although CurveFRFDI3 is much smaller than CurveFRFDI1 and false detection of the 3rd floor may be occurred if the number of damage location is unknown, CurveFRFDI has higher sensitivity to damage than SubFRFDI as compared Figures 30–31 with Figures 28–29. To solve this difficulty, a threshold CurveFRFDI value is suggested to be set to locate damage. The threshold CurveFRFDI value can be determined after numerous numerical simulations of damage scenarios for a certain building structure. For example, the threshold CurveFRFDI value can set to be 0.08 for this experimental example.Figure 30
CurveFRFDI values of LVDT-measured data for (a) case Dam50_1F and (b) case Dam100_1F.
(a)
(b)Figure 31
CurveFRFDI values of LVDT-measured data for (a) case Dam50_1F&3F, (b) case Dam100_1F&3F, (c) case Dam200_1F&3F, (d) case Dam500_1F&3F, and (e) case Dam1200_1F&3F.
(a)
(b)
(c)
(d)
(e)


## 4.2.2. Damage Location Using DIC-Measured Data

Figure32 presents the comparison of the SubFRFDI values of DIC-measured data for case Dam100_1F. Figure 33 shows the comparison of the SubFRFDI values of DIC-measured data for cases of two damage locations, Dam200_1F&3F and Dam500_1F&3F. Figure 34 presents the comparison of the CurveFRFDI values of DIC-measured data for case Dam100_1F. Figure 35 shows the comparison of the CurveFRFDI values of DIC-measured data for cases of two damage locations, Dam200_1F&3F and Dam500_1F&3F. The results also show that the CurveFRFDI has higher sensitivity to damage than SubFRFDI as compared Figures 34–35 with Figures 32–33. The results further indicate that applying the proposed approach to locate single and multiple nonadjacent damages to building structures is feasible by using DIC-measured displacements.Figure 32
SubFRFDI values of DIC-measured data for case Dam100_1F.Figure 33
SubFRFDI values of DIC-measured data for (a) case Dam200_1F&3F and (b) case Dam500_1F&3F.
(a)
(b)Figure 34
CurveFRFDI values of DIC-measured data for case Dam100_1F.Figure 35
CurveFRFDI values of DIC-measured data for (a) case Dam200_1F&3F and (b) case Dam500_1F&3F.
(a)
(b)The experimental example proves the feasibility of the proposed approach for location damage to building structures using DIC-measured displacement response. These data are supposed to be noise free or low noise corrupted. Future work should apply the proposed approach to measurements in the field (actual cases) to investigate its capacity to deal with DIC-measured displacement response corrupted by high noise.


## 5. Conclusions

The current work develops an approach for locating damage to building structures. The proposed damage location index,CurveFRFDI, improves the sensitivity of SubFRFDI, a substructure FRF-based damage location index proposed by Lin et al. [8]. The feasibility of applying the proposed approach to locate damage to building structures using DIC-measured displacement is investigated in this study. A numerical example and an experimental example are presented to demonstrate the feasibility of using the proposed approach to locate damage to building structures. The following important conclusions are drawn from the results:(1)
UsingSubFRFDI can predict damage location accurately for cases of single damage location. However, SubFRFDI cannot locate damage for cases of multiple damage locations with large damage extent. SubFRFDI values corresponding to damaged and undamaged locations increase with the increase in damage extent for cases of single and multiple damage locations. Thus, false detection may be occurred for large damage extent cases if the number of damage locations is unknown.(2)
In most cases, theCurveFRFDI value corresponding to damaged location increases with the increase in damage extent while that corresponding to undamaged location is almost zero and varies slightly with the increase in damage extent, which implies that CurveFRFDI can locate damage regardless of intensity (extent) for most cases. Moreover, CurveFRFDI has higher sensitivity to damage than SubFRFDI. Thus, using CurveFRFDI can predict damage location more accurately than using SubFRFDI for cases of multiple damage locations with large damage extent.(3)
For cases of multiple damage locations, someCurveFRFDI values corresponding to damaged location may be much smaller than others, making it difficult to detect those damage locations. To solve this difficulty, a threshold CurveFRFDI value is suggested to be set to locate damage. The threshold CurveFRFDI value can be determined after numerous numerical simulations of damage scenarios for a certain building structure.(4)
Applying the proposed approach and DIC-measured displacements to locate single and multiple nonadjacent damages to building structures is feasible.(5)
BothSubFRFDI and CurveFRFDI cannot locate damage for cases of multiple adjacent damage locations. How is the proposed approach applicable to the prediction of multiple adjacent damage locations should be investigated based on this research.



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*Source: 4509576-2020-03-17.xml*