## EM

_{2002}15th ASCE Engineering Mechanics Conference

June 2-5, 2002, Columbia University, New York, NY

**APPLICATION OF THE DAMAGE INDEX METHOD TO PHASE II OF THE **

**ANALYTICAL SHM BENCHMARK PROBLEM**

Luciana R. Barroso1_{, Member ASCE }

and Ramses Rodriguez2

**ABSTRACT **

This paper presents research that utilizes the Damage Index Method to detect the location and severity of damage to the Second Phase of Analytical Studies of the Structural Health Monitoring Benchmark Problem developed by the ASCE Task Group on Structural Health Monitoring (TGoSHM). This method utilizes the ratio of curvature in the mode shape of the damaged structure vs. the undamaged structure. A shear-building model is utilized for damage detection of the unbraced frame system, both with full and partial sensor data. The location and severity of damage is carried out using the Damage Index Method for both existing damage patterns established by TGoSHM.

**Keywords: **damage index method, structural health monitoring, benchmark problem

**INTRODUCTION **

Earthquakes occurring in or near urban regions can cause severe damage to structural systems, as illustrated in recent seismic events such as those in Turkey and Taiwan. The maintenance and inspection of the affected structures can be both costly and time-consuming. Currently, engineers rely on visual inspection to detect deficiencies and estimate structural conditions. The quality of such inspections relies heavily on the knowledge and experience of the inspection crew. Another complicating factor is that damaged members may be obscured or hidden within the structure, below water, or below the ground surface.

Recent technological advances have resulted in a significant interest in development of new diagnostic techniques for monitoring the integrity of both existing and new structures in real time with minimum human involvement. To facilitate comparison between different algorithms developed to process vibration data to measure a structure’s health status, the IASC-ASCE Structural Health Monitoring Group developed a series of benchmark problems. The focus of the problem is a four-story model of an existing physical structural model at the University of British Columbia.

This research utilizes the Damage Index Method to detect the location and severity of damage to the Second Phase of Analytical Studies of the Structural Health Monitoring Benchmark Problem. This method utilizes the ratio of curvature in the mode shape of the damaged structure

1_{ Dept. of Civil Eng., Texas A&M University, College Station, TX 77843-3136. E-mail: lbarroso@tamu.edu }

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vs. the undamaged structure. In addition, the Damage Index method offers the following advantages: 1) no need to unit-mass normalization of the mode shapes, 2) good results for structures in which only few modes are available (3 or less), and 3) application can be extended up to determine the structural reliability level using some statistical approach to predict consequences in terms of reliability of life of the structure. This approach has been successfully applied to identify damage to Phase I of the Analytical Studies of the Structural Health Monitoring Benchmark Problem (Barroso and Rodriguez 2002). In this study, shear-building model is utilized for damage detection of the unbraced frame system of the Phase II studies, both with full and partial sensor data.

**DAMAGE DETECTION FRAMEWORK **

As only the dynamic responses (i.e. accelerations) are to be utilized, the modal information must be extracted from the data. The approach selected for this problem a frequency domain decomposition approach on the undamaged response data to extract the modal parameters (Brincker et al. 2000). After the dynamic information is extracted, the Damage Index Method is utilized for damage location and severity estimates. Both methods are described in the following sections.

**Extraction of Baseline from Undamaged Response Data **

Modal identification from the dynamic response of the undamaged structure was performed through frequency domain decomposition (FDD) approach proposed by Brinker et al. (2000). This approach can be utilized in the presence of unknown input excitation and has been shown to be accurate even in the presence of strong noise. The power spectral density function of the

output, *Gyy*(*j*ω), is calculated to be decomposed in the frequency domain through Singular Value

Decomposition (SVD) as follows:

( ) *H*

*yy* *i* *i i* *i*

*G* *jw* =*U S U* (1)

where the matrix *Ui* =

### [

*ui*

_{1},

*ui*

_{2}, "

*uim*

### ]

is a unitary matrix holding the singular vectors uij, Siis a diagonal matrix holding the scalar singular values, sij, and m is the rank of the matrix *Gyy*(*j*ω).

Natural frequencies are found at the corresponding peaks when plotting frequency again singular

values. Assuming a random excitation input, then the matrix *Gyy*(*j*ω) can be obtained at each

frequency step as follows:

( ) ( *T*)( )

*yy*

*G* *j*ω = *Y* *Y* (2)

where *Y* is the fast Fourier transform of the acceleration response matrix and _{Y}T_{ is the }

transpose of the complex conjugate of the matrix *Y*. Natural frequencies are found at the

corresponding peaks when plotting frequency again singular values.

**Damage Localization and Severity: Damage Index Method **

Consider a linear, undamaged structure with *e* elements and *n* nodes. The *i*th_{ modal stiffness }

of this structure, *K _{i}*, is given by (Stubbs et al. 1992):

*T*

*i* *i* *i*

3

where Φ* _{i}* is the

*i*th

_{ modal vector and }

_{K}_{ is the system stiffness matrix. The contribution of the }

*th*

_{j}member to the *i*th_{ modal stiffness, }

*ij*

*k* , is given by:

*T*
*ij* *i* *j* *i*

*k* = Φ *k* Φ (4)

where *k _{j}* is the contribution of the

*j*th

_{ member to the system stiffness matrix. The sensitivity of }

the *j*th member to the *i*th mode is determined by the fraction of the modal energy of the *i*th mode

that is concentrated in the *j*th_{ member. This scalar value is given by: }

*ij*
*ij*
*i*
*k*
*F*
*K*

= (5)

The above modal parameters can be determined for both the undamaged and damaged structures, where an asterisk will indicate that the parameter refers to the damaged structure. For the damaged structure, the member sensitivity can also be expressed as follows:

1

1 *e*

*ij*

*ij* *ij* *ik* *k*

*k*
*i*

*k*

*F* *F* *A*

*K* α
∗
∗
∗
=
= ≅ _{} + _{}

### ∑

(6)where *A _{ik}*is a set of coefficients associated with the

*ith*

_{ element and location }

_{k}_{, and }

*k*

α is the

fraction of damage at location *k*. Dr. Stubbs et al. (1992) utilized the above information and

determined a damage indicator for the *j*th_{ element, }

*j*

β , obtained through the following:

* * * *
() ()
1 1
*
*
() ()
1 1
*NM* *e*
*T* *T*

*i* *j* *i* *i* *k* *i* *i*

*i* *k* *j*

*j* _{NM}_{e}

*j*

*T* *T*

*i* *j* *i* *i* *k* *i* *i*

*i* *k*

*K* *K* *K*

*k*
*B*

*k*

*K* *K* *K*

= =
= =
_{Φ} _{Φ +} _{Φ} _{Φ}
= =
_{Φ} _{Φ +} _{Φ} _{Φ}

### ∑

### ∑

### ∑

### ∑

j=1,2...e (7)

where *K _{j}*

_{( )}contains only the geometric information of the system’s stiffness matrix (all material

properties have been filtered out). These indicators are then normalized to provide a more robust

statistical criteria for damage localization. The normalized indicator for the *j*th member, *z _{j}*, is

given by:

### (

*j*

### )

*j*
*z*
β
β β
σ
−

= (8)

where β and σ_{β} represent the mean and the standard deviation of all the indicators,

respectively.

**BENCHMARK RESULTS **

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models are available: a fully braced model and a completely unbraced model. The damage patterns in the fully braced model involve the loss of stiffness in the bracing system. In contrast, the damage patterns in the unbraced model involve the loss of rotational stiffness at beam-column connections. Currently, the unbraced model has been evaluated both in the presence of both full and partial sensor information. In the partial sensor case, only the sensors

at the 2nd_{ and 4}th_{ stories are available. }

In Phase II of the benchmark problems, the complexity of the problem is incremented from Phase I in that connection flexibility is considered. The operating assumption is that joint flexibility is not known from an inspection of the drawings and is thus treated as unknown to the user. The rotational stiffness of the springs in the simulation model used to generate the problem data are equal to a nominal value multiplied by a ‘variability’, ranging from 0.75 to 1.25. While the exact amount of variability is unknown to the user, the factor is selected from a uniform distribution and is fixed by a seed value for all problem participants.

Another difference from Phase I studies lies in the mass definition. The mass on each floor differs from the nominal value by a factor, ranging from 0.9 to 1.1, which is unknown to the user. The position of the center of mass also differs somewhat from the nominal values. As the plan dimension is 2.5m by 2.5 m, the values for the eccentricities are selected from a uniform distribution with limits [-0.25m to 0.25m].

The two damage patterns defined for the structure are: (1) loss of rotational stiffness at 3 connections in level 1 and 2 connections in level 2 in each one of the perimeter frames of the strong (x-x) direction, and (2) loss of rotational stiffness at 2 connections in level 1 in each one of the perimeter frames of the strong direction. The damage identification is performed using a shear-building model, and only modal parameters from the first mode were used for the damage detection process. Each story at each face of the structure was modeled individually, resulting in a total of 4 individual shear frames as shown in Fig. 1(b).

Acceleration data records were obtained for all cases corresponding to the unbraced model with 10% RMS noise. Modal properties were extracted using the Frequency Domain Decomposition approach. The extracted natural frequencies, in Hertz, are given in Tables 1 and 2, with the results obtained from full sensor data being given in Table 1 while Table 2 gives the results obtained when only partial sensor data is available.

**Table 1. Extracted Bending Natural Frequencies: Full Sensors **

**Undamaged ** **Damage 1 ** **Damage 2 **

**Bending **

**Mode ** _{y-y x-x y-y x-x y-y x-x }

1 3.10 4.01 3.10 3.40 3.10 3.86

2 9.75 13.23 9.75 12.78 9.75 13.23

3 16.71 25.02 16.71 24.87 16.71 25.02

4 23.51 39.54 23.51 38.78 23.51 39.09

These values correlate well with those found through the actual analytical model and given in the benchmark definition. Note that the changes in frequency in the damaged cases correspond to a decrease in stiffness in the x-x (strong) direction, where the damage occurred. Also, note that use of partial sensor data still results in very accurate estimates of the frequency. The

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A small discrepancy is observed at higher modes, but as only 1st_{ mode information will be }

utilized in the damage identification process this error should have no impact on the results.

**Table 2. Extracted Bending Natural Frequencies: Partial Sensors **

**Undamaged ** **Damage 1 ** **Damage 2 **

**Bending **

**Mode ** _{y-y x-x y-y x-x y-y x-x }

1 3.01 4.01 3.01 3.40 3.10 3.86 2 9.75 13.23 9.75 12.78 9.75 13.23 3 16.71 25.02 16.71 24.72 16.71 25.02

4 23.51 39.54 23.51 38.93 23.51 38.78

Once the modal information was extracted, the Damage Index Method was utilized to determine the location and severity of the damage. A shear-building model was utilized as the

damage identification model during this process. The normalized damage indicator, *zj*, was

determined and the results from these analyses are shown in Figures 1 and 2 for the full sensor cases. Damage index values greater than 1 correspond to possible damage locations. In both cases, the method accurately identifies the damage location in story level 1. In damage pattern 1, a small probability is given that damage is also found in story 2; however, that damage in incorrectly placed in the weak direction. Damage pattern 2 corresponds to a smaller degree of damage to the structure; however, the method has no difficulties identifying the damage location. The analysis based on the data from the partial sensor data gives similar results.

−2 −1 0 1 2 3 4

1 2 3 4

**Story Level**

−2 −1 0 1 2 3 4

1 2 3 4

−2 −1 0 1 2 3 4

1 2 3 4

**Story Level**

−2 −1 0 1 2 3 4

1 2 3 4

**Weak Direction, Frame 1** **Weak Direction, Frame 2**

**Strong Direction, Frame 1** **Strong Direction, Frame 2**

**z** **z**

**z** **z**

**Fig. 1: Damage index for pattern 1 **

−2 −1 0 1 2 3 4

1 2 3 4

**Story Level**

**Weak Direction, Frame 1**

−2 −1 0 1 2 3 4

1 2 3 4

−2 −1 0 1 2 3 4

1 2 3 4

**Story Level**

−2 −1 0 1 2 3 4

1 2 3 4

**Weak Direction, Frame 2**

**Strong Direction, Frame 1** **Strong Direction, Frame 2**

**z ** **z **

**z ** **z **

**Fig. 2: Damage index for pattern 2 **

**CONCLUSIONS **

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connected with the use of a shear-building damage identification model, which is not as sensitive to rotational information. Using a more sophisticated damage identification model that does account for rotational information should result in more accurate damage locations. Current research efforts are underway to explore the benefits of a modified damage identification model. The authors are also investigating the efficacy of this damage identification approach with a method that extracts undamaged modal information directly from the vibration data of the damaged structure.

**ACKNOWLEDGMENTS **

The authors would like to acknowledge the contributions from Dr. Stubbs in implementing the Damage Index Method to this problem.

**REFERENCES **

Barroso, L. R., and Rodriguez, R. (2002). "Stiffness-Mass Ratios Method for Baseline Determination and Damage Assessment of a Benchmark Structure." 2002 American Controls Conference, Anchorage, Alaska.

Bernal, D. (2002). "Phase II of the ASCE Benchmark Study."

http://www.ce.ust.hk/asce.shm/wp_home.asp.

Brincker, R., Zhang, L., and Andersen, P. (2000). "Modal Identification from Ambient Responses using Frequency Domain Decomposition." 18th International Modal Analysis

Conference, San Antonio, Texas, 625-630.

Stubbs, N., Kim, J.-T., and Tapole, K. (1992). "An Efficient and Robust Algorithm for Damage Localization in Offshore Platforms." ASCE 10th Structures Congress, San Antonio, Texas.

**APPENDIX I. NOTATION **

The following symbols are utilized in this paper:

*aij* = coefficients that are function of the mode shape and mass terms

**A ** = matrix containing unknown coefficients *aij*

**b** = Vector containing eigenvalues

*Gyy(j*ω) = power spectral density function

*i*

*k* = Lateral stiffness of *i*th_{ story }

**K** = Elastic system stiffness matrix

*i*

*m* = Mass of *i*th_{ floor }

**M** = System mass matrix

Si = Diagonal matrix holding the scalar singular values

*i*

*U* = Unitary matrix holding the singular vectors

**x** = Vector of the unknown stiffness to mass ratios

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*T*

*Y* = Transpose of the complex conjugate of the matrix *Y*

α*j* = Damage severity indicator for the *j*th element

*j*

β = Damage indicator for the *j*th_{ element }

λ = Eigenvalue

Φ = Μode shape

*j*