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

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. 1is study develops a displacement FRF-based approach to locate damage to building structures. 1e 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.


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. e 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][5][6][7][8]. Ni et al. [4] identified the seismic damage of a 38-story building using measured FRFs and neural networks. eir study used principal component analysis (PCA) to reduce dimension and eliminate the noise of measured FRFs. e 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]. e 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][12][13][14][15][16]. Shih et al. [11,12] developed a low-cost digital image correlation method to measure the dynamic response of shear buildings. e 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. e 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.

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 i th substructure is a structure containing the i th -N th floors (or degree of freedoms, DOFs) for an N-story building structure. e substructure-based FRFs of the j th DOF in the i th substructure can be simplified as follows: where € X j and € X i− 1 are the Fourier transforms of € x j (t) and € x i− 1 (t), respectively; € x j (t) and € x i− 1 (t) are the absolute acceleration of the j th and (i − 1) th DOF, respectively. eoretically, when the damage is assumed to have occurred in the columns between the i th and (i − 1) th DOFs, the substructure-based FRF is altered significantly in the i th DOF as described by H e substructure-based FRF damage location index (SubFRFDI) for the i th substructure is defined as follows: where ρ, a, b, and n are working parameters and the NDF i (ω) is expressed as follows: and the absolute dissimilarity P i (ω) is defined as follows: where |H i in the damaged and undamaged states, respectively. ese N dissimilarities, P 1 (ω) ∼ P N (ω), 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. e 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. e value of the coefficient ρ is suggested to be 1∼10, with 5 used in this work. e 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). e value n equals (b-a) divided by sampling time. If the properties of a structural 2 Advances in Civil Engineering system do not change, then SubFRFDI i is close to zero. However, if the damage to the i th floor in a shear building is severe, then the value of SubFRFDI i is high. us, SubFRFDI i can be regarded as the SubFRFDI corresponding to location i (the i th floor).
is 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 the j th DOF in the i th substructure in equation (2) is modified to calculate SubFRFDI i : where X j and X i− 1 are the Fourier transforms of x j (t) and x i− 1 (t), respectively; x j (t) and x i− 1 (t) are the absolute displacement of the j th and (i − 1) th DOF, respectively. In fact, the SubFRFDI values corresponding to damaged and undamaged locations increase with the increasing of damage extent.
us, 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. at is, if the i th floor is damaged, and the (i − 1) th and (i + 1) th floors are undamaged, SubFRFDI i is larger than SubFRFDI i− 1 and SubFRFDI i+1 , and the curve connecting SubFRFDI i− 1 , SubFRFDI i , and SubFRFDI i+1 is open downward. us, damage occurred at the location corresponding to a negative SubFRFDI curvature rather than a nonnegative SubFRFDI curvature for single and multiple nonadjacent damages. e curvature of SubFRFDI i , K(SubFRFDI i ), can be simply defined as follows: where SubFRFDI 0 and SubFRFDI N+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. e CurveFRFDI for an N-story building structure is expressed as follows: where A is a control value that scales CurveFRFDI i to a very small positive value for positive SubFRFDI i curvature. e 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. e range of CurveFRFDI i 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.

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. is study employs the DIC approach presented in the work of Lu et al. [15] for displacement measurement in the experimental example. e 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. e (x 0 , y 0 ) and (x 1 , y 1 ) 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. e coordinates of (x 0 , y 0 ) and (x 1 , y 1 ) are figured out in the following steps. Two images are first selected as S and T. e image S is an undeformed (source) image, and image T is a slightly deformed image (target) relative to image S. e difference between the two images, S and T, can be estimated through a simple difference method via pixelwise computation, as shown in the following equation: 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 D(i, j) of the pixels in the background area is close to zero. erefore, if an ROI changes the position due to deformation, the D(i, j) of these pixels covered in the ROI is relatively large.
Moreover, the coordinate, (x 0 , y 0 ), of the ROI of the source image can be computed using a mean-max method represented in equation (9). e 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. e index with the maximum mean values of the columns and rows indicates the x-axis and y-axis coordinates, respectively: e correlation coefficient used in the work is defined in the following equation: ] where f and g are the pixel value of ROIs for the source and target images, respectively. e sign 〈·〉 denotes the mean operator and 〈f〉 and 〈g〉 are the means of ROIs in the source and target images, respectively. e coordinate, (x 1 , y 1 ), 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: Measuring displacement responses of the building structure in damaged and undamaged states

Start
Calculating The ith floor is undamaged The ith floor is damaged damaged and undamaged states, Calculating the damage location index CurveFRFDI i based on the curvature of  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. e relative displacement in the unit pixels (i.e., the pixel displacement), (d x , d y ), in the x-axis and y-axis coordinates can be estimated to be (x 1 − x 0 , y 1 − y 0 ). e precision of the pixel displacement, (d x , d y ), is an integer pixel value with original images. e 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. e actual displacement, (u, v), can be calculated by using estimated pixel displacement, (d x , d y ), to multiply by a pixel ratio (R p ) from equation (12). At this point, the time-history displacements are estimated completely in the digital image measurement system: e subpixel analysis can improve the precision based on the subpixel estimation of a target image. e 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 R p 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.
e 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 (x 1 , y 1 ) 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.

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. Figure 5, 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. e cross-sectional size of the beam is 240 mm × 30 mm. e length of the beam is 360 mm. e cross-sectional size of the column is 240 mm × 15 mm. e 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 f y � 3500 kgf/cm 2 . In this example, SAP2000 is used for structural analysis. e natural frequencies of the first six modes are 1  Advances in Civil Engineering 5 simulated as reduced floor stiffness. Single and multiple damage locations are studied. Table 1 presents the simulated damage cases. e 1995 Kobe earthquake record is used as the external excitation, but normalized to 100 gal as the peak ground acceleration (PGA). Figure 6 compares the substructure-based FRFs for the 2 nd and the 5 th substructures (H

Damage Location Using SubFRFDI.
2 (ω) and H 5 (ω)) for case Dam_2F15%. Significant changes were observed only in the substructurebased FRF of substructure 2, H  Figure 8(a)), the 2 nd floor for case Dam_3F&5F15% (see Figure 8(b)), the 1 st floor for case Dam_2F&3F15% (see Figure 8(c)), and the 1 st 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. us, false detection could occur for large damage extent cases if the number of damage location is unknown.

4.1.2.
Damage Location Using CurveFRFDI. Figures 16-24 Figure 17(c)) and the 3 rd floor for case Dam_2F&3F&4F15% (see Figure 18(b)) are falsely detected. For most cases in Figures 19-21, 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. e 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.            Table 2 lists the steel types used to construct the eight-story steel frame. ese series of shaking table tests were undertaken by the National Center for Research on Earthquake Engineering in Taiwan. e 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. e damage in this example is simulated by reducing the cross section of certain columns as shown in Figure 27. Single and multiple nonadjacent 14 Advances in Civil Engineering damage locations are studied. Table 3 presents the simulated damage cases. Figure 28 presents the comparison of the SubFRFDI values of LVDT-measured data for cases of single damage location, Dam50_1F and Dam100_1F. e damage locations of the two cases are predicted correctly by the SubFRFDI of LVDTmeasured 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 SubFRFDI 1 and SubFRFDI 3 are larger than others, and the damage locations (the 1 st and the 3 rd floors) of these cases are predicted correctly by the SubFRFDI if the number of damage location is known in advance. Nevertheless, SubFRFDI 3 is much smaller than SubFRFDI 1 and false detection of the 3 rd floor may be occurred if the number of damage location is unknown. Figure 30 presents the comparison of the CurveFRFDI values of LVDT-measured data for cases of single damage location, Dam50_1F and Dam100_1F. e damage locations  of the two cases are also predicted correctly by the Cur-veFRFDI 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 CurveFRFDI 3 is much smaller than CurveFRFDI 1 and false detection of the 3 rd floor may be occurred if the number of damage location is unknown, CurveFRFDI has higher sensitivity to damage than Sub-FRFDI 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. e threshold Cur-veFRFDI 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 32 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      Figure 27: e damaged columns. 18 Advances in Civil Engineering   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. e results also show that the CurveFRFDI has higher sensitivity to damage than SubFRFDI as compared Figures 34-35 with Figures 32-33. e results further indicate that applying the proposed approach to locate single and multiple nonadjacent damages to building structures is feasible by using DICmeasured displacements. e experimental example proves the feasibility of the proposed approach for location damage to building structures using DIC-measured displacement response. ese 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.

Conclusions
e current work develops an approach for locating damage to building structures. e proposed damage location index, CurveFRFDI, improves the sensitivity of SubFRFDI, a substructure FRF-based damage location index proposed by Lin et al. [8]. e 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. e following important conclusions are drawn from the results: (1) Using SubFRFDI 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. us, false detection may be occurred for large damage extent cases if the number of damage locations is unknown. (2) In most cases, the CurveFRFDI 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, Cur-veFRFDI has higher sensitivity to damage than SubFRFDI. us, using CurveFRFDI can predict damage location more accurately than using Sub-FRFDI for cases of multiple damage locations with large damage extent.
(3) For cases of multiple damage locations, some Cur-veFRFDI 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. e 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) Both SubFRFDI 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.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.