# Comparison of damage identification algorithms on experimental modal data from a bridge Page: 4 of 8

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Middle Span of North Girder East * West

N-1, N-2. N-3. N-4 . N-5. N-i. N-7. N-8 . N-9 * N-10. N-11

160 ft (48.8 m)

V Element 1 Element 160

(Nodes 1.2) (Nodes 160.161)

160 ft (48.8 m)

Fig. 4. Refined accelerometer locations.

North Grider East * West

1 . N"2 . N-3 . N-4 . N-5 . N"8 . N"7 . N-8 . N-9 . N-10. N-11. N-12. N"13.130 ft (39.6 m)

160 ft (48.8 m)

130 ft (39.6 m)

Damage Location

r Element 1 Element 210

(Nodes 1-2) (Nodes 210-211)

420 ft (128 m)

[-Fig. 5. Coarse accelerometer locations.

4. DESCRIPTION OF DAMAGE IDENTIFICATION

METHODS

A very brief description of the damage identification methods

that were used is given below. The reader is referred to the

appropriate references for a more detailed discussion of

these methods. A summary of their implementation for the

study reported herein can be found in [61.

4.1 DAMAGE INDEX METHOD

The Damage Index Method was developed by Stubbs and

Kim [7] to localize damage in structures given their

characteristic mode shapes before and after damage. For a

structure that can be represented as a beam, a damage

index, 3, based on changes in curvature of the ith mode at

location j is defined as

[yi"(x)]2 dx + [ri"(x)]2 dx) f' (" (x)]2 dx

[,"(x)]2 dx + J[W;"(x)]2dxJ [4"(x)]2dx

where yi("(x) and yi "(x) are the second derivatives of the

ith mode shape corresponding to the undamaged and

damaged structures, respectively. L is the length of the

beam. a and b are the limits of a segment of the beam where

damage is being evaluated. When more than one mode is

used the damage index is the sum of damage indices from

each mode. For mode shapes obtained from ambient data,

the modes are normalized such that"} [mln= 1,

(2)

where [m] is assumed to be the identity matrix [8].

To determine mode shape amplitudes at location between

sensors, the mode shapes are fit with a cubic polynomial.

As shown in Fig. 4 for the refined set of accelerometers, the

middle span of the north girder is divided into 160 1-ft (0.305-

m) segments. Modal amplitudes are interpolated for each of

the 161 nodes forming these segments. Similarly, for the

coarse set of accelerometers the entire length of the north

girder (all three spans) is divided into 210 2-ft (0.610 m)

segments with mode shape interpolation yielding amplitudes

at 211 node locations as shown in Fig. 5.. Statistical

methods are then used to examine changes in the damage

index and associate these changes with possible damage

locations. A normal distribution is fit to the damage indices,

and values falling two or more standard deviations from the

mean are assumed to be the most likely location of damage.

4.2 MODE SHAPE CURVATURE METHOD

Pandey, Biswas, and Samman [9] assume that structural

damage only affects the stiffness matrix and not the mass

matrix. Hence, for the undamaged and damaged condition

the eigenvalue problems are given as([K] - X[M]){yr;} = {0}, and

([K] - X;[M]){y;} = {0}, respectively,(3)

(4)where [K] = the stiffness matrix, X1 =the ith eigenvalue, [M]

= the mass matrix, yr; = the rth displacement eigenvector ofrsli ir

n/i ir

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Jauregui, D.V. & Farrar, C.R. Comparison of damage identification algorithms on experimental modal data from a bridge, article, December 31, 1995; New Mexico. (https://digital.library.unt.edu/ark:/67531/metadc668451/m1/4/: accessed April 19, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.