To detect seismic damage of moment resisting frame (MRF) structures, a data-driven method using the fractal dimension (FD) of time-frequency feature (TFF) of structural seismic dynamic responses at measured stories is extended and refined. The TFF is defined as the real part of Gabor wavelet transform of translational interstory displacement, and FD is used to give a quantitative value to describe the calculated TFF. Static condensation method is first used to reduce the degrees-of-freedom (DOFs) of MRF and to express the rotational displacements using translational displacements. For linear MRF, the FDs of TFFs at all stories are the same using the definition of TFF and modal superposition principle. For damaged MRF with plastic hinges at the ends of beams and columns, the force analogy method is implemented to establish transformation matrix from plastic hinge rotations to translational interstory inelastic displacements. Due to the sparseness of the transformation matrix, plastic hinges only generate interstory inelastic displacements, which are low-frequency contents, in the vicinity of plastic hinges. Correspondingly, the FDs of TFFs of interstory displacements with inelastic component are different from the FDs of TFFs of the interstory displacements that do not contain inelastic component. A numerical simulation on a 16-story MRF was conducted. The simulation included 10 cases such as no damage or linear structure, plastic hinges in single-story beams, plastic hinges in single-story columns, plastic hinges in single-story beams and columns, and plastic hinges in multiple story beams and columns. The robustness to measurement noise was also investigated. The seismic damage detection results demonstrated that the proposed method was capable of locating the stories where the plastic hinges occurred.

After big earthquakes occur, the national and local governments and owners, in order to take actions of postearthquake emergency rescue and reconstruction, need to know the safety and applicability of building structures damaged by strong ground motions [

Damage detection is the core part of SHM, and the first important information needed for damage assessment and seismic retrofit for structures shaken by strong ground motions is to determine whether the structures is damaged and where the locations of damage are. Structural seismic damage is divided into three levels as component-level damage, story-level damage and structure-level damage. Due to the complexity and huge degrees-of-freedoms (DOFs) of the structure, sensors are only installed on very limited stories, which makes the current seismic damage detection methods for real structures mainly focus on the story-level. The structure excited by strong ground motion will be in the nonlinear stage because the structural material yields under high stress and large deformation. The plastic hinges will exist at the ends of beams and columns, the cracks at components like beams, columns, and walls will open and close, and the force-displacement curves will show obvious hysteresis characteristics. These kinds of nonlinear behaviour of monitored structure make the damage detection methods based on linear system theory not suitable for detecting structural seismic damage. Thus, nonlinear indicator function methods, which belong to data-driven approaches, are proposed to detect these kinds of damages by extracting the nonlinear features in the dynamic vibration measurements. Farrar et al. gave a comprehensive review on damage detection using nonlinear system identification and nonlinear indicator functions approaches [^{th} and 9^{th} order wavelet coefficients [

In this paper, a data-driven approach using the time-frequency feature (TFF) of the interstory displacement is further developed for detecting story-level damage of moment resisting frame (MRF) structure subjected to earthquake. This nonlinear indicator function method was proposed by the author to detect seismic damage of shear-type buildings. For the integrity and understandability, some important principles and equations are also given in the paper. In Section

Wavelet transform is a well-developed time-frequency analysis tool, transforming time domain signal into time-frequency domain distribution and showing the frequency contents evolution. Unlike Fourier transform using the sine and cosine basis without time resolution, wavelet transform uses wavelets with good time and frequency localization and is good for analysing nonstationary signals like strong ground motion and structural response. Gabor wavelet is especially developed to demonstrate local time and frequency distribution since it minimizes the product of time resolution and frequency resolution. Gabor wavelet, as one kind of analytical wavelet, has the property that the real and imaginary part of the corresponding wavelet coefficients is a Hilbert transform pair. For other wavelet transform whose mother wavelet is not analytic, the above relationships are not true. The Gabor wavelet transform is given by [

As deduced in Reference [

To keep the linear superposition of TFF and to suppress the interference between different modes, the real part, not the norm of the wavelet coefficients of the signal, is used in this study. Meanwhile, for most damage detection purpose, the scale is chosen within the fundamental frequency of the analysed structure. The TFF of the damaged structure will show much energy in the frequency band below the structural fundamental frequency. The TFF can be treated a matrix whose columns stand for time and rows for frequency, and it can be visualized as a three-dimensional (3D) surface.

To quantify the calculated TFF of a signal, FD is introduced. FD is an indicator for describing the complexity or irregularity of natural objects, textures, images, and signals [

MRF has translational and rotational vibration. As static condensation method points out, rotational vibration can be deduced by translational vibration if the inertial effect of beams is ignored. The motion equation for linear MRF is given by

Substituting Equation (

The relative translational displacements of the condensed linear MRF can be given as follows with the help of modal decomposition technique:

Using the property of FD, which is that the FD of a constant times a fractal is the same as the original fractal, the following equation can be deduced:

When MRF structure is excited by strong ground motion, plastic hinges will be formed at the ends of beams and columns. It is these plastic hinges that make the horizontal displacement of the structure contain inelastic displacements. The force analogy method establishes an explicit expression between the inelastic displacement and the plastic hinge rotation of the beams and columns. The force analogy method is a numerical method for solving dynamic responses of nonlinear structures. It was first proposed by Lin and developed and refined by Hart [

The nonequilibrium restoring force in the translational DOF generated by the plastic rotations at the ends of beams and columns is given by

Element with plastic hinges at both ends.

From Figure

Equation (

then the explicit expression between interstory inelastic displacement in the master DOF and the plastic hinge rotation is given by

When the MRF structure is in linear stage and there is no plastic hinge at the ends of beams and columns, then, based on Equation (

The 3D finite element model was established by an open source software called the Open System for Earthquake Engineering Simulation (OpenSees) [

16-story MRF structure model. (a) Geometrical size of the building (unit: mm). (b) Section of columns and beams (unit: mm).

The input ground motions were chosen from PEER’s NGA database. The record at El Centro site in October 15, 1979, Imperial Valley earthquake, was chosen as far-field strong ground motion. The record code was E12140. The PGA was 143 gal, the peak frequency was 1.76 Hz, and the original sampling interval was 0.05 s. This record was abbreviated as El Centro. The record at Ridge RT site, in January 18, 1994, Northridge earthquake, was chosen as near-field strong ground motion. The record code was ORR090. The PGA was 580 gal, the peak frequency was 1.22 Hz, and the original sampling interval was 0.02 s. This record was abbreviated as Northridge. In order to simulate no damage or different damage in specific stories, the chosen ground motions were scaled up or down, and the yielding stress of beams and columns were carefully set. The simulation included 10 cases such as no damage or linear structure, plastic hinges in single-story beams, plastic hinges in single-story columns, plastic hinges in single-story beams and columns, plastic hinges in multiple story beams, and columns as shown in Table

Location of nonlinearity and input ground motion.

Case | Yielding stress at weak component | Input ground motion |
---|---|---|

1 | Beams and columns 205 MPa | PGA = 28.6 gal El Centro |

2 | Beams at 1^{st} story 180 MPa, others 300 MPa |
PGA = 429 gal El Centro |

3 | Columns at 1^{st} story 180 MPa, others 300 MPa |
PGA = 429 gal El Centro |

4 | Beams and columns at 1^{st} story 180 MPa, others 300 MPa |
PGA = 429 gal El Centro |

5 | Beams and columns 205 MPa | PGA = 715 gal El Centro |

6 | Beams and columns 205 MPa | PGA = 11.6 gal Northridge |

7 | Beams at 1^{st} story 180 MPa, others 300 MPa |
PGA = 348 gal Northridge |

8 | Columns at 1^{st} story 180 MPa, others 300 MPa |
PGA = 348 gal Northridge |

9 | Beams and columns at 1^{st} story 180 MPa, others 300 MPa |
PGA = 348 gal Northridge |

10 | Beams and columns 205 MPa | PGA = 464 gal Northridge |

To detect damage and locate the weak stories, the translational displacements at each translational DOF were extracted from the OpenSees dynamic response analysis. The plastic rotation at the ends of beams and columns were also extracted to demonstrate where the weak stories were truly located. The Gabor wavelet transform program used was WAVELAB, a MATLAB wavelet toolkit provided by David Donoho’s group at Stanford University [

In case 1 when the MRF structure was excited by PGA = 28.6 gal El Centro ground motion, the structure was in linear range without yielded element. The first two modal frequencies were 2.13 Hz and 6.53 Hz, respectively. The contour map of time-frequency distribution, Fast Fourier Transform (FFT) spectra, and the interstory displacement is shown in Figure ^{st}, 7^{th}, and 15^{th} stories were shown here. It could be seen from Figure

Time histories, TFFs, and the FFT spectra of interstory displacements of the structure under PGA = 28.6 gal El Centro ground motion (Case 1). (a) Story 1, (b) Story 7, and (c) Story 15.

Log-log plot of the box counting of the TFF of the structure under PGA = 28.6 gal El Centro ground motion (Case 1). (a) Story 1, (b) Story 7, and (c) Story 15.

FDs of the TFFs of interstory displacements of the structure under PGA = 28.6 gal El Centro ground motion (Case 1).

In case 2 when the MRF structure was excited by PGA = 429 gal El Centro ground motion, the structure was damaged with plastic hinges in the beams at the 1^{st} story. The rotation of plastic hinges at the ends of columns and beams is shown in Figure ^{rd} story, 6 indicated the right end of a beam on the 3^{rd} story, and so on so forth. From Figure ^{st} story were damaged, and the columns at the 1^{st} story and beams and columns at other stories did not yield. The contour map of time-frequency distribution, FFT spectra, and the interstory displacement is shown in Figure ^{st} story was different from those at other stories, especially between 1.2 and 1.5 Hz in the frequency range and 10 and 12 s in the time range. The corresponding FDs in the scale interval ^{st} and 2^{nd} stories were a little different from those at other stories, and this is because plastic hinges introduced inelastic displacement at the 1^{st} story as inferred from Equation (

Rotation of plastic hinges at the ends of columns and beams under PGA = 429 gal El Centro ground motion (Case 2). (a) Rotation of plastic hinges at the ends of columns. (b) Rotation of plastic hinges at the ends of beams.

Time histories, TFFs, and the FFT spectra of interstory displacements of the structure under PGA = 429 gal El Centro ground motion (Case 2). (a) Story 1, (b) Story 7, and (c) Story 15.

Log-log plot of the box counting of the TFF of the structure under PGA = 429 gal El Centro ground motion (Case 2). (a) Story 1, (b) Story 7, and (c) Story 15.

FDs of the TFFs of interstory displacements of the structure under PGA = 429 gal El Centro ground motion (Case 2).

In case 3 when the MRF structure was excited by PGA = 429 gal El Centro ground motion, the structure was damaged with plastic hinges in the columns at the 1^{st} story. The rotation of plastic hinges at the ends of columns and beams is shown in Figure ^{st} story were damaged and the beams at the 1^{st} story and beams and columns at other stories did not yield. The contour map of time-frequency distribution, FFT spectra, and the interstory displacement is shown in Figure ^{st} story was quite different from those at other stories, especially between 1.2 and 1.7 Hz in the frequency range and 8 and 12 s in the time range. The corresponding FDs in the scale interval ^{st} story was quite different from those at other stories, and this could be interpreted by the transformation matrix

Rotation of plastic hinges at the ends of columns and beams under PGA = 429 gal El Centro ground motion (Case 3). (a) Rotation of plastic hinges at the ends of columns. (b) Rotation of plastic hinges at the ends of beams.

Time histories, TFFs, and the FFT spectra of interstory displacements of the structure under PGA = 429 gal El Centro ground motion (Case 3). (a) Story 1, (b) Story 7, and (c) Story 15.

Log-log plot of the box counting of the TFF of the structure under PGA = 429 gal El Centro ground motion (Case 3). (a) Story 1, (b) Story 7, and (c) Story 15.

FDs of the TFFs of interstory displacements of the structure under PGA = 429 gal El Centro ground motion (Case 3).

The coefficients at the first row were much higher than those at other rows, which meant the plastic hinges in the columns at the 1^{st} story mainly caused interstory inelastic displacement at the 1^{st} story, and thus, the FD of TFF at the 1^{st} story was different from others.

In case 4 when the MRF structure was excited by PGA = 429 gal El Centro ground motion, the structure was damaged with plastic hinges in the beams and columns at the 1^{st} story. The rotation of plastic hinges at the ends of columns and beams is shown in Figure ^{st} story were damaged and the beams and columns at other stories did not yield. The contour map of time-frequency distribution, FFT spectra, and the interstory displacement is shown in Figure ^{st} story was quite different from those at other stories, especially between 0 and 1.0 Hz in the frequency range and 5 and 20 s in the time range. The corresponding FDs in the scale interval ^{st} story were quite different from those at other stories, and it was because the plastic hinges in the beams and columns at the 1^{st} story mainly caused interstory inelastic displacement at the 1^{st} story.

Rotation of plastic hinges at the ends of columns and beams under PGA = 429 gal El Centro ground motion (Case 4). (a) Rotation of plastic hinges at the ends of columns. (b) Rotation of plastic hinges at the ends of beams.

Time histories, TFFs, and the FFT spectra of interstory displacements of the structure under PGA = 429 gal El Centro ground motion (Case 4). (a) Story 1, (b) Story 7, and (c) Story 15.

Log-log plot of the box counting of the TFF of the structure under PGA = 429 gal El Centro ground motion (Case 4). (a) Story 1, (b) Story 7, and (c) Story 15.

FDs of the TFFs of interstory displacements of the structure under PGA = 429 gal El Centro ground motion (Case 4).

In case 5 when the MRF structure was excited by PGA = 715 gal El Centro ground motion, the structure was damaged with plastic hinges in the beams and columns at multiple stories. The rotation of plastic hinges at the ends of columns and beams is shown in Figure

Rotation of plastic hinges at the ends of columns and beams under PGA = 715 gal El Centro ground motion (Case 5). (a) Rotation of plastic hinges at the ends of columns. (b) Rotation of plastic hinges at the ends of beams.

Time histories, TFFs and the FFT spectra of interstory displacements of the structure under PGA = 715 gal El Centro ground motion (Case 5). (a) Story 1, (b) Story 2, (c) Story 3, (d) Story 4, (e) Story 5, (f) Story 6, (g) Story 7, (h) Story 10, and (i) Story 15.

Log-log plot of the box counting of the TFF of the structure under PGA = 715 gal El Centro ground motion (Case 5). (a) Story 1, (b) Story 2, (c) Story 3, (d) Story 4, (e) Story 5, (f) Story 6, (g) Story 7, (h) Story 10, and (i) Story 15.

FDs of the TFFs of interstory displacements of the structure under PGA = 715 gal El Centro ground motion (Case 5).

To investigate the robustness of the approach to measurement noise, white Gaussian noise with 5%, 10%, 15%, and 20% root mean square (RMS) noise-to-signal were added to the relative displacements at each story. The results for Case 5 are shown here. Time histories, TFFs, and the FFT spectra of the interstory displacement at Story 1 under different noise levels are shown in Figure

Time histories, TFFs, and the FFT spectra of interstory displacements at Story 1 under different noise levels (Case 5). (a) Noise level of 5%, (b) noise level of 10%, (c) noise level of 15%, and (d) noise level of 20%.

FDs of the TFFs of interstory displacements under different noise levels (Case 5).

The results of the FDs of the TFFs of the MRF structure for Cases 6–10 under Northridge ground motion are shown in Figure

FDs of the TFFs of interstory displacements of the structure under Northridge ground motion for Cases 6–10. (a) Case 6, (b) Case 7, (c) Case 8, (d) Case 9, and (e) Case 10.

The damage detection results for the MRF excited by other strong ground motions were also investigated. The ground motions included TCU052W in September 20, 1999, Chichi earthquake abbreviated as Chichi, CUE90 in January 16, 1995, Kobe earthquake abbreviated as Kobe, C02065 in June 28, 1966, Parkfield earthquake abbreviated as Parkfield, and TAF111 in July 21, 1952, Kern County earthquake abbreviated as Taft, and the time histories and FFT spectra thereof are shown in Figure

Time histories, and the FFT spectra of strong ground motions. (a) Chichi, (b) Kobe, (c) Parkfield, and (d) Taft.

FDs of the TFFs of interstory displacements of the structure under strong ground motions when all the yielding stress of beams and columns as 205 MPa. (a) Chichi, (b) Kobe, (c) Parkfield, and (d) Taft.

The results when the structure was subjected to Chichi, Kobe, and Parkfield ground motions were similar as the results when subjected to El Centro or Northridge ground motions, and it could be seen that the lower stories were damaged in these cases. However, the FD of TFFs under Taft ground motion along the structure was a zigzag line, and it could be seen that the structure was damaged but the location of damage could not be determined. According to the definition of TFF given by Equation (

In this paper, an extension of a data-driven approach proposed by the authors for detecting damage of shear-type building structure is refined and further developed to detect seismic damage of MRF subjected to far-field or near-field strong ground motions. The method is one kind of vibration-based approach, which uses the measured displacements to detect damage and locate stories with plastic hinges at the ends of beams and columns. The principle is that the FD of TFF of interstory displacements with inelastic component are different from the FD of TFF of the interstory displacements that do not contain inelastic component. The numerical simulation on a 16-story MRF indicates that the method can locate the stories in which there are plastic hinges at the end of beams and columns. The damaged story location accuracy is better when the plastic hinges are at the end of columns than those at the end of beams. As the seismic damage of real MRF structure will introduce plastic hinges at the ends of beams or columns, the method may be utilized for detecting and localizing these kinds of damage. Further study will include experiment validation on a MRF in shaking table test, and testing on in-situ MRF under real ground motions will also be conducted.

The input ground motions were obtained from Pacific Earthquake Engineering Research (PEER) strong-motion database at

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported by the Science Foundation of the Institute of Engineering Mechanics, China Earthquake Administration (Grant nos. 2014B08 and 2016A03), National Natural Science Foundation of China (Grant no. 5150082083), and China Scholarship Council (Grant no. 201704190039). The first author is grateful to Professor Zhenming Wang at Kentucky Geological Survey, University of Kentucky, for his great support and guidance and help during visit.