dynamic characteristics of a curved cable-stayed bridge from strong motion records

8/7/2019 Dynamic characteristics of a curved cable-stayed bridge from strong motion records

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Engineering Structures 29 (2007) 2001–2017

www.elsevier.com/locate/engstruct

Dynamic characteristics of a curved cable-stayed bridge identified fromstrong motion records

Dionysius M. Siringoringo∗, Yozo Fujino

Department of Civil Engineering, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan

Received 26 April 2006; received in revised form 5 September 2006; accepted 16 October 2006

Available online 6 December 2006

Abstract

An assessment of dynamic characteristics of the 455 m Katsushika–Harp curved cable-stayed bridge is presented. Dynamics characteristics

such as natural frequencies, mode shapes and modal damping ratios are obtained from seismic response of the bridge by employing a time-domain

multi-input multi-output (MIMO) system identification (SI) technique. The technique makes use of base motions and superstructure accelerations

as pairs of inputs–outputs to realize the coefficients of state-space system matrices. The SI results indicate the occurrence of many closely spaced

modal frequencies with spatially complicated mode shapes. Fourteen global modes in the ranges of 0.45–2.5 Hz were identified, in which the

girder motion dominated most of the modes. The tower modes were associated with girder modes and were characterized by the lowly-damped

motion. Using identification results from six earthquakes, the effects of earthquake amplitude on modal damping ratios were observed.c 2006 Elsevier Ltd. All rights reserved.

Keywords: Curved cable-stayed bridge; Instrumented bridge; Seismic response; MIMO system identification; Katsushika–Harp bridge

1. Introduction

The availability of multi-channel permanent sensors allows

regular full-scale dynamic tests that are essential for continuous

monitoring of a bridge. In a seismically active region,

such as Japan, this instrumentation provides an opportunity

to use system identification (SI) techniques to explain

bridge performance during earthquakes. By employing the SI

technique, it is also possible to monitor any changes in the

bridge behavior without the presence of visually observable

damage. In the context of bridge monitoring, this excellent

opportunity is beneficial to evaluate the adequacy of bridge

seismic design code [1].

The general approach of an earthquake-induced SI is to

use the input–output relation to recreate structural models

that are capable of reproducing the actual responses. In one

early study Beck [2] employed the output-error minimization

method for a linear, time-invariant structural system with

classical damping. McVerry [3] proposed a frequency domain

∗ Corresponding author. Tel.: +81 03 5841 6097; fax: +81 03 5841 7454. E-mail addresses: [email protected](D.M. Siringoringo),

[email protected](Y. Fujino).

approach using transfer function to minimize the objectivefunction of output error. Chaudhary et al. [4] improved it,

for a more general problem of non-classical damping that

includes the structural model in addition to the modal model.

This method, while powerful and significantly insightful,

requires prior information of structural properties that are

typically unavailable and difficult to obtain, especially for large

and complex structures such as cable-stayed bridges. Most

conventional SI techniques were developed in the frequency

domain due to the common practice of using frequency analyzer

for data acquisition. These approaches offer advantages in

incorporating soil–structure interaction into analysis, but often

suffer from damping estimation especially when closely spacedmodes are present.

Compared to cable-stayed bridges with straight girder, the

curved cable-stayed bridges are relatively few. Examples of

well-known curved cable-stayed bridges are the La Arena

Viaduct in Spain [5], the Safti Link Bridge in Singapore [6],

the Rhine Bridge near Schahausen, Switzerland [7] and the

twin curved cable-stayed bridge at the Malpensa airport in

Milan, Italy [8]. In the analysis of a cable-stayed bridge under

seismic action, the aspects of three dimensionality, multi-modal

contribution, multiple-support excitations and modal coupling

0141-0296/$ - see front matter c 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.engstruct.2006.10.009

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2002 D.M. Siringoringo, Y. Fujino / Engineering Structures 29 (2007) 2001–2017

Fig. 1. View of the Katsushika–Harp bridge.

are of great importance [9]. Accordingly, the SI techniquesshould be developed to provide accurate information that

accommodate observation of such aspects.Challenges in the SI technique applied to a curved cable-

stayed bridge come from its nature as a large and continu-

ous structure, which makes it difficult to be represented with a

lumped-mass model. Moreover, owing to its complexity, many

modes are closely spaced in frequency. And due to its long span,

spatial variability of ground motions may be significant. Forthis latter reason, the SI technique needs to employ a scheme

of multiple-input excitations. More recent works on the SI of

long-span bridges under seismic excitation have attempted to

overcome the problems by employing time-domain algorithmsthat include the effect of multi-input [10–12].

The focus of the work described in this paper is twofold: (1)

providing a systematic procedure for SI of the Katsushika–Harpcurved cable-stayed bridge using seismic records, and (2)

evaluating the dynamic characteristics of the bridge based on

the SI results. The SI method adopted here is based on the

input–output mapping technique of the System Realization

using Information Matrix (SRIM) [13,14]. Organization of thispaper is such that a brief account of the bridge is presented first,

followed by explanation of the SI methodology, instrumentation

and seismic records. Afterwards, the results of identification are

explained, including the comparisons with an analytical finiteelement model, previous tests using ambient vibration, and

forced vibration. Next the identification of tower modes and the

observation from six earthquakes are presented. Conclusions of

the study are summarized at the end.

2. Description of Katsushika–Harp bridge

The Katsushika–Harp Bridge (Fig. 1) is a curved cable-

stayed bridge located in the Katsushika area of Tokyo

Metropolitan city. The bridge crosses over an estuarial area

between the Arakawa and Nakagawa Rivers. Its girder is made

of steel with the width of 23.5 m and the total length of 455 m,

consisting of 220 m main span, and three side spans of 40.5, 134

and 60.5 m (Figs. 2 and 3). The bridge’s unique asymmetrical

S-shaped girder is composed by two curves with a radius of

334 m and 270 m each. The bridge has two rectangular towersmade of steel; the main tower (65 m in height and 3 m in width)

is located in the middle of the curve and the smaller one (13.8 m

in height and 2.5 m in width) is located at the end of highway

approach to the Hirai Bridge. The bridge stays spring from the

main tower (17 cables) and the right tower (7 cables) to support

the curved girder. Construction of the bridge was completed in

1987, and since then it became the part of the Shutoo Chuo

Loop line in the Tokyo Metropolitan expressway.

For the purpose of monitoring, 32 channels of sensors were

permanently installed on the bridge. The sensors consist of

29 accelerometers deployed at 12 locations, (Fig. 4) and 3

displacement sensors. For system identification, only responses

from accelerometers were utilized. These accelerometers havea range of frequency between 0.05 and 35 Hz with an accuracy

of 15 microampere per cm/s2. Among these accelerometers,

six are located on the substructure (on the pile foundation and

pile caps) and the rest were installed on the superstructure

(towers, pier caps and girder). On the girder, sensors were

installed at eight locations along the girder centerline. These

sensors measure accelerations in vertical, transverse and

longitudinal directions, at a data-sampling rate of 100 Hz. It

should be mentioned that all sensors measure the motion in

directions that coincide with the local coordinates of structure

members (i.e. the x -direction of measurement coincides with

the centerline of the bridge).

3. System identification methodology

In this study, the SRIM-based SI is employed to identify the

complex modal frequencies, damping ratios and mode shapes

of the bridge system. The SI consists of two steps, namely,

the realization of system matrices from correlation of input and

output data, and estimation of modal parameters from identified

system matrices. The main advantage of this technique is

its ability to provide a systematic way to identify modal

parameters from multiple-input and multiple-output earthquake

data without prior knowledge of structural properties or models.

Moreover, it allows the identification of a classically as well asa non-classically damped structural system, and thus eliminates

the need of trial and error in updating the modal parameters to

achieve reasonable agreement with the recorded data.

3.1. Basic formulation of earthquake-induced vibration

The identification starts from a basic equation of motion of

N degree-of-freedom (DOF) linear, time invariant, viscously

damped system subjected to earthquake excitation ug (t ), in the

spatial coordinate {u(t )} and continuous time (t ). The equation

is given as

Mu(t ) + Cu(t ) + Ku(t ) = −Wug(t ). (1)



D.M. Siringoringo, Y. Fujino / Engineering Structures 29 (2007) 2001–2017 2003

Fig. 2. Dimension and general configuration of the Katsushika–Harp bridge.

(a) Main tower. (b) Right tower.

Fig. 3. Dimensions of girder and tower.

Fig. 4. Sensor positions and measuring directions of the Katsushika Bridge (note: measuring direction is the local coordinate of the structural member).




The quantities M, C and K ∈ R N × N denote mass, damping

and stiffness matrices respectively, while the W ∈ R N ×q

denotes the continuous time input matrix, with q being the

dimension of input vector ug(t ). This equation can be expressed

as a finite dimensional, discrete-time, linear, time-invariant,

state variable dynamical system:

x (k + 1) = A x (k ) + B z(k ) (2a)

y(k ) = R x(k ) + D z(k ). (2b)

The vector x (k ) = [u(k ) u(k )]T in the latter expression

i s a 2n × 1 state vector consisting of displacement and

velocity. Vector z(k) consists of input acceleration recorded at

q channels, and y(k) denotes structural acceleration responses

recorded from m output channels.

The integer k = 0, 1, 2, . . . l denotes the time-step number,

i.e. x(k + 1) = x (k (t ) + t ), with t being the time interval.

The quantities of A ∈ R2 N ×2 N , B ∈ R

2 N ×q , R ∈ Rm×2 N and

D ∈ R

m×q

are the discrete representation of system matrices.Their corresponding continuous system matrices are given as

A = eAct , B =

t

0e[ Ac]τ dτ Bc

,

R = L

−M−1K −M−1C

, D = LM−1. (3)

Matrix [L] ∈ Rm× N is a transformation matrix that connects

the position of system degree-of-freedom with the measured

output responses. The quantities Ac ∈ R2 N ×2 N and Bc ∈

R2 N ×q are the system matrix in continuous time and the input

influence matrix of order 2 N × q, respectively, which are both

defined as:

Ac =

0 I

−M−1K −M−1C

Bc =

0

−I

. (4)

The system matrices A ∈ R2 N ×2 N , B ∈ R

2 N ×q , R ∈

Rm×2 N , and D ∈ R

m×q are all unknowns and to be

determined from given earthquake input–output data. With

several algebraic manipulations, Eq. (2) can be rewritten for

various p time spans in a matrix form as

yp(k ) = [Op] x(k ) + [Tp]zp(k ) (5)

where the y p(k ), z p(k ), [Op] and [Tp] are defined as follows

yp(k ) =

y(k )

y(k + 1)

y(k + 2)...

y(k + p − 1)

[Op] =

R

RA

RA2

...

RAp−1

zp(k ) =

z(k )

z(k + 1)

z(k + 2)...

z(k + p − 1)

[Tp] =

D

RB D

RAB RB D...

... · · ·. . .

RAp−2B RAp−3B RAp−4B · · · D

. (6)

Matrix [Op] ∈ R pm×2 N is commonly called theobservability matrix. The matrix [Tp] ∈ R

pm× pq is a

generalized Toeplitz matrix composed by the system Markov

parameters (i.e. RAk−1B). It should be mentioned that time

discretization error that might appear may be made negligible

by using a sufficiently small time step. Since most earthquake

excitation is sampled at a frequency of 100 Hz, which is

sufficiently small as compared to the most natural frequencies

of civil structures (i.e. far below 50 Hz), discretization error in

this case may be considered negligible.

3.2. System realization using information matrix

The kernel of the SRIM algorithm is identification of

observability matrix [Op]. Let [Op](m+1 : pm, :) be defined as

a matrix consisting of the last ( p−1)m rows (from the (m +1)th

row until the ( pm)th row) and all columns of [Op]. Similarly,

let the [Op](1 : ( p − 1)m, :) be defined as a matrix consisting

of the first ( p − 1)m rows and all columns of [Op]. The above

definitions are rewritten in matrix form as follows:

[Op](m + 1 : pm, :) =

RA

RA2

RA3

...

RAp−1

and

[Op](1 : ( p − 1)m, :) =

R

RA

RA2

...

RAp−2

.

(7)

Now, following the equality:

[Op](m + 1 : pm, :) =

RA

RA2

RA3

...

RAp−1

=

R

RA

RA2

...

RAp−2

A

= [Op](1 : ( p − 1)m, :)A (8)

the state matrix A can be identified as:

A = [O∗p](1 : ( p − 1)m, :)[Op](m + 1 : pm, :) (9)

where the asterisk (*) denotes the pseudo-inverse matrix.

Following derivation of Eqs. (8) and (9), the choice of integer

p becomes obvious. The integer p should be selected such that

matrix [Op](m + 1 : pm, :) of dimension ( p − 1)m × 2 N has

rank larger than or equal to 2 N , hence p ≥2 N m + 1.




The procedure to obtain the observability matrix starts by

expanding the vector equation in (5), into a matrix equation, i.e.,

[Yp](k ) = [Op]X(k ) + [Tp][Zp](k ) (10)

where

X(k ) = [ x (k ) x(k + 1) · · · x (k + s − 1)],

[Yp] = [ y p (k ) y p (k + 1) · · · y p (k + s − 1) ]

=

y(k ) y(k + 1) · · · y(k + s − 1) y(k + 1) y(k + 2) · · · y(k + s)

.

.

....

. . ....

y(k + p − 1) y(k + p) · · · y(k + p + s − 2)

[Zp] = [ z p (k ) z p (k + 1) · · · z p (k + s − 1) ]

=

z(k ) z(k + 1) · · · z(k + s − 1) z(k + 1) z(k + 2) · · · z(k + s)

.

.

....

. . ....

z(k + p − 1) z(k + p) · · · z(k + p + s − 2)

. (11)

To ensure that the rank of [Yp] and [Zp] is at least

equal to that of matrix [Op], a sufficiently large integer s

must be selected. Thus, rank evaluation must be performedbefore selecting the proper value of integer s. Now let us

define the following quantities: Ryy, Rzz, Rxx, Ryz, Ryx, and

Rxz as correlations between input (z), output (y), and state (x)

respectively, where the subscript indices denote the product of

correlation in such a way that the correlation of Ap and Bp is

defined as:

Rab =1

sAp(k )BT

p (k ). (12)

If the integer s = l − p is sufficiently large with l being

the total length of data and p the data shifts, the quantities in

Eq. (12) approximate the expected values in a statistical sense.

To determine the observability matrix [Op], one can start byobtaining the data correlations of input–output matrices:

Rhh = [Op]Rxx[Op]T (13)

where the quantities Rhh and Rxx are defined as follows

Rhh = Ryy − RyzR−1zz RT

yz (14a)

Rxx = Rxx − RxzR−1zz RT

xz. (14b)

The quantity Rhh is determined from the output autocorrela-

tion matrix Ryy, the output–input cross-correlation matrix Ryz

and the inverse of the input autocorrelation matrix, which all

are available from input–output data. It should be mentioned

that the quantity of Rhh exists only if the input autocorrela-

tion matrix Rzz is a non-singular matrix. This requirement will

be satisfied if the input is persistent and rich enough for z(k ),

with k = 0, 1, . . . , l or in other words the input block matrix

Zp(k) is of full rank (i.e. qp). If we assume that the input and

the state vectors are two uncorrelated quantities, then the cross-

correlation matrix between them Rxz becomes a 2 N × q p zero

matrix, and Eq. (14b) thus reduces to Rxx = Rxx. This condi-

tion can be achieved if the input is a sufficiently long Gaussian

random signal. With this assumption, Eq. (14a) can be solved

to obtain matrix [Op], given the matrix Rhh computed from

input–output data.

To obtain the solution for matrix [Op], Eq. (14a) needs

to be factored into three matrices where the first matrix is

the transpose product of the third matrix. The singular value

decomposition is a logical choice for this matrix factorization.In this decomposition, only the first size ( pm × ( p − 1)m) row

and column of Rhh matrix is decomposed using singular value

decomposition. Hence, the new form of decomposition is:Rhh(:, 1 : ( p − 1)m) = HΣ

2VT =

H2 N H0

×

Σ

22 N 02 Nxno

0mo x2 N 0mo xn o

VT

2 N

VTo

= V2 N Σ

22 N V

T2 N (15)

where the dimension of Rhh(:, 1 : ( p−1)m) is ( pm ×( p−1)m).The integer no indicates the number of zero singular values

and also the number of columns in matrix Vo. The integer mo

denotes the numbers of columns in Ho that are orthogonal to

the column in H2 N .For noisy data, there are no zero singular values, that is,

no = 0. If no singular values are truncated, mo = m is

obtained. If some singular values are truncated, mo becomesthe sum of m. In both conditions of singular values truncation,

the partial decomposition method guaranties that there are atleast m columns of Ho that are orthogonal to the columns of

H2 N in Eq. (15). Further factorization of Eq. (15) produces,

Rhh[:, 1 : ( p − 1)m] = [Op]Rxx[Op]T

× [:, 1 : ( p − 1)m] = H2 N Σ 22 N V

T2 N . (16)

Finally following the identity in Eq. (16), the observability

matrix can be obtained:

[Op] = H2 N and

Rxx[Op]T[:, 1 : ( p − 1)m] = Σ 22 N VT2 N .

(17)

Given the observability matrix, the system matrix A can be

identified from Eq. (9). Subsequently, the modal parametersof the structural system can be estimated by solving the

eigenvalues problem of matrix A as:

A = . (18)

Matrices and represent the eigenvalues and eigenvec-

tors of matrix A, respectively. The eigenvalues and eigenvectors

can be real or complex, where in the latter case, they appear ascomplex conjugate pairs. The eigenvalues λi are actually ex-

pressed in z-domain, and therefore can be related to the modal

characteristics of dynamical system using the following trans-formation

λi = ln(λi )/t . (19)

After transformation, the natural frequency (ωi ) and modal

damping ratio (ξi ) can be estimated:

ωi =

Re(λi )2 + Im(λi )2, ξi = −Re(λi )/ωi . (20)

The mode shapes matrix in a coordinate system is obtainedby transforming the eigenvectors in z-domain into a coordinate-

domain using the output-transformation matrix R,

= R. (21)




Table 1

List of earthquakes used in analysis

Earthquake Magnitude Input acceleration Total length

(Triggered time) ( M j) Max- x (cm/s2) Max- y (cm/s2) Max- z (cm/s2) (s)

Feb 20, 1990 (15:53) 6.5 5.50 5.80 2.00 210.5

Feb 2, 1992 (04:04) 5.9 32.60 43.38 17.26 107.00

Oct 23, 2003 (17:57) 6.8 12.80 11.80 6.10 46.75Oct 23, 2003 (18:35) 6.8 11.40 9.55 4.36 39.75

Jul 23, 2005 (16:35) 6.0 52.61 29.21 6.93 45.00

Aug 16, 2005 (11:48) 7.2 14.86 15.99 4.80 70.25

In summary, the SI technique involves the following

procedures: (1) computation of the correlation matrices (Ryy,

Ryz and Rzz), using the p sampled data points of output block

matrix [Yp] and input block matrix [Zp]. (2) Calculation and

then factorization of the information matrix Rhh using singular-

value decomposition. The order of the system is determined by

examining the singular values of matrix Rhh. (3) Realization

of the observability matrix [Op], using the results of H2 N . (4)

Finally, realization of the state matrix A and R, followed by the

estimations of modal parameters by solving the eigenvalues and

eigenvectors of the realized state matrix.

The algorithm has been implemented in a numerical

simulation using a three-dimensional FEM of cable-stayed

bridge to examine its accuracy and efficiency. Results of

numerical simulation, which was presented elsewhere [15],

show that under a reasonable number of output sensors, modal

parameters of the bridge subjected to simulated filtered-short

ground motion can be estimated with great accuracy. The errors

of natural frequency estimates were found less than 2%, while

the error of damping estimates, which depends on the length of

data, signal-to-noise ratio and non-stationary, reach up to 15%.Implementation of the algorithm to seismic records from long-

span cable-stayed bridge and the evaluation of the results can

also be found in Ref. [10].

4. Seismic records

The present work utilizes ground motions and acceleration

responses obtained from six earthquakes (Table 1); among them

are the two sets of main shocks from the October 23, 2004

Chuetsu Niigata Earthquake. The strongest earthquake, in terms

of response amplitude, is the July 23, 2005 earthquake. This

earthquake had the epicenter 25 km southeast of Tokyo with

the focal depth of 65.6 km [16]. In analysis, the total 210 s of response from the February 20, 1990 earthquake was divided

into three frames 60 s each in order to evaluate the effect of

earthquake amplitude to modal parameters. All records were

sampled using the standard sampling frequency of 100 Hz and

baseline-corrected before processing.

5. Data processing and analysis

Data processing, analysis and the implementation of SI

shown in this section utilize the records from the strongest

earthquake in July 23, 2005. Typical input and output responses

of the bridge from this largest earthquake are plotted in Fig. 5.

5.1. Spectra of the acceleration responses

Fig. 6 shows the Fourier spectra of input accelerations

recorded at the bottom of the main and the right tower. Inputs at

both towers were characterized by low frequency components

(0–1.6 Hz) in all three directions. The spectra of verticalinput at both towers reveal similar trend in all frequencies of

interest. On contrary, the spectra of both towers in longitudinal

and lateral directions differ from the frequency of 1.0 Hzonward. The differences here imply non-uniformities of input

excitations at the bottom of each tower. Therefore, these non-

uniformities should be considered as the case of multi-input

when implementing the system identification.The spectra of accelerations obtained from three channels

at different locations on the girder are shown in Fig. 7. Many

peaks were observed within a range of 0.5–2 Hz suggesting the

existence of closely-spaced natural frequencies of the bridge,

the case that is common in a large and flexible structure such

as a cable-stayed bridge. Most of the peaks that appear on the

longitudinal direction also appear on the lateral direction with

the almost equal amplitude, indicating the coupling of motions

between longitudinal and lateral direction. Spectrum of the

girder’s vertical responses reveals six distinct peaks constantly

appear on the three vertical channels. Some of the peaks in

vertical components have strong corresponding peaks at the

lateral and longitudinal, except for the first and the third peak at

the frequency of 0.46 Hz and 0.81 Hz respectively. Later, it wasconfirmed from SI results that these peaks are associated with

the pure vertical modes.After observing the spectra of girder responses in three

directions, it was decided that the range of frequency interest

for SI is between 0 and 2.5 Hz, since most dominant

peaks are contained within this range. Furthermore, owing

to the unique shape of the bridge girder, it was foundthat the identification of longitudinal and lateral modes is

difficult to be performed separately. Due to this reason, a

three-dimensional system arrangement is chosen, instead of

separating them into three in-plane motion systems like in

the application of modal identification to a straight-girder

cable-stayed bridge [10]. Using this system configuration the

influence of modal displacement and directions in a mode can

be visually observed.

5.2. Input–output data set

A system of 6 inputs and 23 outputs (6I–23O) is employed

for identification. The output channels are obtained from the




Fig. 5. Typical acceleration time histories recorded at (a) the bottom of main tower, (b) middle of girder recorded during the July 23, 2005 earthquake.

Fig. 6. Fourier amplitude of input acceleration measured on the bottom of the main tower and right tower. (a) Longitudinal direction, (b) lateral direction, (c) vertical

direction, recorded during the July 23, 2005 earthquake.

sensors at the girder, piers and on the top of the towers

(see Fig. 4). The girder sensors consist of 19 accelerometers

mounted at 8 locations along the girder horizontal centerline.

At the towers, four sensors are deployed to identify the

tower modes, consisting of two longitudinal and two lateral

accelerometers at each tower. Six accelerometers on the footing

of the main tower and the right tower are utilized as inputs.

These channels measure response in vertical, lateral and

longitudinal directions at the bottom of the towers and thus can

be viewed as direct inputs to the superstructures.




Fig. 7. Fourier amplitude of girder acceleration in (a) longitudinal direction, (b) lateral direction, (c) vertical direction, recorded during the July 23, 2005 earthquake.

As clearly shown in Eq. (18), the total number of modes

generated by the SI depends on the size of matrix A.

Furthermore, the size of matrix A is determined by the

number of singular values of matrix Rhh. The large singularvalues represent the strength of dominating modes, while

the remaining values consist of either noise or fictitious

computational modes. After excluding the entire zero (or small)

singular values that are associated with the noise or fictitiousmodes during decomposition of the Rhh matrix, the system

matrix A is realized. To facilitate the selection, a plot of singular

values is utilized, and the singular value cutoff is obtained from

a typical sudden drop of the values in this plot.

5.3. Realization of global modes

As explained in the previous section, every step in realiza-

tion uses an observation of singular value to decide the cutoff

values for the second realization of information matrix Rhh. In

the case of no-noise measurement, the cutoff value can easilybe determined by observing a significant drop in the singular

value plot. In practice, however no such case exists, since there

can hardly be a case of no-noise measurement. The inevitable

presence of noise and small non-linearity can distort the systemto some extent and results in the non-distinguishable singular

value plot. To overcome this limitation, a scheme of singular-

value truncation is implemented. All modes that are associated

with zero or smaller were eliminated (Fig. 8). The second re-alization was performed by using the retained singular values

and subsequently all corresponding modes were generated. This

will result in many modes comprising the real ones and the

computational modes. To distinguish real modes from fictitiousmodes, a modal supervision technique is utilized.

The technique selects certain modes by assigning threshold

values and modal supervisor, by using logic operators, which

Fig. 8. Singular values plot of the information matrix Rhh recorded during the

July 23, 2005 earthquake with p = 100.

are defined by the user. So at first, all modes are generated

then the resulted modes are put into a batch for comparison.

In this batch the resulting modes are passed through threshold

values. The threshold values are (i.e. all modes with the

criteria below were rejected): (1) negative damping ratio or/and

uncharacteristically high damping ratio, (2) Extended Modal

Assurance Criteria (EMAC) [17] less than 90%, (3) modes that

have uncharacteristic shapes or are incomparable with their

FEM counterparts, when compared to the FEM mode shapes

using the Modal Assurance Criterion (MAC) values.




Fig. 9. Consistency of frequency and damping estimates with respect to p-values.

To better illustrate the case, one can refer to the example

given in Fig. 8. This figure shows an example of the singular

value plot for the total bridge system. One can observe that

singular values beyond 100 are almost or less than zero.

Therefore, the value of 100 is selected as the singular value

cutoff. Using this number, the total number of modes obtainedfrom matrix A is 100/2 = 50, since they all appear in

conjugate frequencies. Upon observing the 50 modes from

the first generation and by adopting the modal supervision

technique explained above, only 14 modes were identified with

confidence. The other modes were perceived as fictitious modes

or noise-generated modes.

5.4. Consistency of modal parameters

The theory states that the minimum value of p ≥ 2 N /m + 1

will yield to an exact solution. However, due to measurement

noise that propagates into system, the actual minimum p-values might be found larger than the ideal minimum, and

thus need to be investigated in order to minimize bias that

might appear in calculation. In order to study the sensitivity

of modal parameters with respect to the length of input–output

in the block matrix ([Yp] and [Zp]), several records with

various lengths were evaluated. For this purpose, the number

of columns in both [Yp] and [Zp] block matrices were fixed at

3500 for all frames, while the number of rows is incremented

from 10 until all available numbers of rows are utilized. Modal

parameters were calculated separately for these different block

matrices and the modal supervision technique was applied

afterwards.

The results of modal parameters for various p-values

show higher estimates of both frequencies and dampings at

smaller p-values, before they decrease and finally stabilize

at certain values. The stabilized values are thus selected as

the representative estimates. It was found that most of the

modes were identified using p = 50. The p-values that givethe stabilized estimates may vary from one mode to another.

However, the observations show that, generally, the estimates

stabilize around p = 100. Fig. 9 shows typical examples

of the consistency of frequency and damping estimates with

respect to the p-values for the first vertical and the first lateral

mode of Katsushika–Harp Bridge. The figure clearly indicates

that steady results of natural frequency estimates are obtained

around the value of p = 100. To obtain a stabilized estimate

of damping ratio, however, a longer data set and thus larger

p-value is required. Sometimes, even after longer data was

included, a relatively small scatter of damping estimates are

still present. The reasons of the scatter estimates are related

to the facts that the small-valued damping is usually very

sensitive to the non-stationary nature of seismic excitation, and

the presence of measurement noise. Therefore to quantify a

representative damping estimate of one mode, an average value

and its standard deviation near the stabilized values are utilized

(see Table 3, for example).

6. Identification results

6.1. Global modes and natural frequencies

Owing to the double curvature of the bridge girder,

the vibration modes are all three dimensional in principle.




Table 2

Value of the mdp index and the mode type recorded during the July 23, 2005 earthquake

Mode Identified mdp index of the girder Type of mode

no. freq (Hz) X -direction Y -direction Z -direction

1 0.46 0.05 0.04 1.00 1st Vertical (Sym)

2 0.61 0.32 1.00 0.57 1st Lateral + Vertical 1st Sym

3 0.66 0.371.00

0.30 2nd Lateral4 0.81 0.11 0.11 1.00 2nd Vertical (Asym)

5 0.93 0.24 1.00 0.28 3rd Lateral

6 0.99 0.24 1.00 0.21 4th Lateral

7 1.17 0.15 0.82 1.00 3rd Vertical (Sym) + Lateral

8 1.28 1.00 0.43 0.50 Longitudinal dominant mode

9 1.38 0.65 0.95 1.00 1st Torsion

10 1.64 0.27 1.00 0.56 5th Lateral

11 1.68 0.32 0.15 1.00 4th Vertical (Asym)

12 1.81 0.91 1.00 0.98 2nd Torsion

13 2.23 0.34 0.53 1.00 5th Vertical (Sym)

14 2.41 0.19 1.00 0.14 6th Lateral

To determine the type of a mode, a modal displacement

participation index (shortened hereafter as: mdp) is computedusing the following equation:

mdp( x , y, z)(φ j ) =|max{φ

j( x , y, z)}|

|max{φj

1: N ,( x , y, z)}|. (22)

Eq. (22) implies that for the j -th mode, the index calculates

ratio between the maximum absolute modal displacements of

structural component in one direction divided by the maximum

modal displacement of the structure. By employing the above

index, the participation or contribution of a motion in one

direction to the overall modal displacement can be quantified.

Therefore, the predominant direction and thus type of the mode

is determined. The dominating direction is the direction that

gives the mdp value equal or very close to unity.

Table 2 lists the results of identification with their mdp

index. Since the modes are all girder-dominant, the mdp index

calculates the participation of girder modal displacement in

three principal directions. Accordingly, the modes are classified

into four types of motion, namely: vertical, lateral, longitudinal

and torsion. The mode that is dominated by a motion in one

direction only, (i.e. vertical, lateral and longitudinal) has a

characteristic of mdp value equal to 1 in that corresponding

direction, and accompanied by weak participation (less than

0.5) of motions in other directions. A mode is classified as

torsional mode when mdp values for all directions exceed 0.5.One can notice from Table 2 that except for the first and the

second vertical modes, all other modes are characterized by

relatively strong participation of modal displacement in other

directions beside the predominant one. This indicates that due

to the effect of the curved girder, most of the modes were highly

coupled.

The SI generates a total of fourteen vibration modes within

the frequency range of 0.4–2.5 Hz. Using the mdp index the

fourteen modes are classified as: five vertical predominant

modes, six lateral modes, one longitudinally predominant

mode and two torsion modes. Although the spectral peaks

indicate closely spaced natural frequencies, all modes were well

separated in the identification and their shapes fit their FEM

counterparts mode shapes.The fundamental mode of the bridge is the first vertical

bending with a natural frequency of 0.45 Hz. This mode

exhibits a zero node symmetrical pattern of girder between the

main tower and the lower one (main span), while the side span

(Yotsugi approach) moving out-of-phase vertically. The mode

shows a pure vertical bending pattern, in which the participation

of motion in both longitudinal and lateral directions are very

weak as evidenced by the mdp < 0.05. The second vertical

mode was identified at the frequency of 0.81 Hz. The mode has

the characteristic of zero node and symmetrical main span, with

the side span moving in-phase. The participations of motion in

other directions were found stronger than they were in the first

vertical mode. The third mode was recovered at 1.17 Hz, with a

single node asymmetrical pattern of the main span. Unlike the

first two vertical modes, strong participation of lateral motion

appears in this mode (mdp = 0.8).

In the lateral direction, the first mode was found at the

frequency of 0.61 Hz. In this mode both towers move laterally

in-phase with each other as well as with the lateral motion of the

main span. The lateral motion of the main span was coupled

with vertical component (mdp = 0.57). Slightly higher, at

0.66 Hz, the second lateral mode was identified. The mode

has a characteristic of a zero node symmetric shape of the

main span in lateral direction, while the towers move out-of-

phase with each other. The first two lateral modes show thestrong participation of both towers but no participation of end-

piers in motion. More lateral modes were recovered at higher

frequency (0.93, 0.99, 1.64 and 2.41 Hz). They exhibit a single

node asymmetric pattern of the main span with the strong

participation of tower and weak modal displacement of both

end-piers.

As indicated in Table 3, the first torsional mode was

identified at 1.38 Hz. The fact that all sensors were installed

at the centerline of the bridge longitudinal axis makes it rather

difficult to visually observe the torsional nature of this mode.

However, its mdp values suggest the torsional characteristics.

In addition, when compared to the frequency range of the




Table 3

Comparison of the identified natural frequencies and damping ratios obtained from the seismic records with other tests and FEM

Mode Identified from earthquake Ambient Forced FEM Type of mode

no. 23-Jul-05 vibration vibration

test test

Frequency (Hz) Damping (%)a EMAC (%)

1 0.46 1.58 (0.78) 98.5 0.45 0.45 0.45 1st Vertical (Sym)2 0.61 1.64 (0.80) 97.0 0.62 0.60 0.60 1st Lateral + Vertical 1st Sym

3 0.66 3.34 (0.26) 99.0 0.70 0.67 0.67 2nd Lateral

4 0.81 1.46 (0.20) 96.8 0.83 0.82 0.82 2nd Vertical (Asym)

5 0.93 2.06 (0.32) 96.2 0.98 0.96 0.92 3rd Lateral

6 0.99 0.65 (0.03) 96.5 1.01 1.01 1.00 4th Lateral

7 1.17 3.64 (0.21) 93.6 1.17 1.18 1.17 3rd Vertical (Sym) + Lateral

8 1.28 1.79 (0.80) 91.8 – – N/A Longitudinal dominant mode

9 1.38 2.61 (0.74) 96.7 1.31 1.38 1.37 1st Torsion

10 1.64 1.33 (0.45) 94.5 1.71 1.63 1.64 5th Lateral

11 1.68 1.09 (0.30) 93.5 1.67 1.67 1.68 4th Vertical (Asym)

12 1.81 2.79 (0.63) 92.9 1.82 1.79 1.81 2nd Torsion

13 2.23 2.62 (0.10) 92.1 2.23 2.22 2.24 5th Vertical (Sym)

14 2.41 1.52 (0.43) 93.4 2.40 2.41 2.41 6th Lateral

Note: –: Not identified, N/A: not available.

a Values in the brackets denote the standard deviation.

first torsion mode identified in ambient vibration test, forced

vibration and also as predicted from FEM, this mode was

indeed identified within the same frequency range of the first

torsional mode.

The two torsion modes were found in relatively high-

frequency modes (i.e. at 1.38 and 1.81 Hz), which imply that the

torsional effect is not so dominant in this curved cable-stayed

bridge. The reason is related to the fact that while the girder

is essentially curved in the horizontal, it is supported at four

locations: both towers and the approach end-piers. Moreover,

the large radii of both girder curves and the position of the main

tower in the transitional point of the curves might have, to some

extent, reduced the effect of torsional motion.

It should be noted that there is one mode that was not

identified either during the forced vibration or ambient test,

but was identified from earthquake records. This mode is the

longitudinal predominant mode identified at the frequency of

1.28 Hz. The mode has a characteristic of longitudinal swing

motion, in which the girder, pier and tower move longitudinally

toward one direction along the centerline of the girder. The mdp

index reveals that the maximum girder modal displacement

occurred in the longitudinal direction. It appears that the

excitation level provided during ambient and forced vibration

tests were insufficient to produce reasonably strong vibrationto excite this mode. The similar fact was observed during

the study of SI applied to seismic response of a large cable-

stayed bridge with a straight girder [10,15]. In both references,

the longitudinal dominant mode was significantly insightful

to assess the performance of isolation devices between the

pier and girder. In this work, however, due to the lack of

sensors in pier-caps, such a performance assessment cannot be

performed.

Fig. 10 illustrates the mode shapes in two types of view:

the front and the plan view: From this figure the relationship

between components (i.e. vertical, lateral and longitudinal) in

the mode can easily be observed. Table 3 lists the frequency

estimates identified from the strongest earthquake (July 23,

2005). It is shown in this table that the natural frequencies

estimated from seismic response are in good agreement with

the results from ambient vibration and forced vibration tests

conducted after the completion of the construction, [18] as

well as the finite element model. Generally, natural frequencies

estimated from the ambient vibration test are slightly higher

than those estimated from the seismic records with the

maximum discrepancy between the two being 5.7%. This

suggests that, during the small level of ambient excitation the

structure is seemingly more rigid than it is during larger levels

of excitation.

6.2. Damping ratio estimates

In the conventional frequency domain SI, the modal

damping ratio can be estimated using the half-power bandwidth

of the Fourier spectrum peaks. In the case of this bridge,

however, estimating damping ratios from half-power bandwidth

are difficult since the spectrum peaks are closely spaced

or overlapped by adjacent peaks, especially in lateral and

longitudinal directions.One advantage of the SRIM identification algorithm is that

damping estimates are derived directly from the eigenvalues

of the state matrix A. Hence, the values are well separated

since each eigenvalue is associated with a unique mode. In

implementation of the identification algorithm, however, the

initial estimates of damping ratio were rather scattered and

require larger p-values to obtain stabilized results. This fact

was not found during the estimation of natural frequency.

The reasons of the scatter estimates are related to the facts

that the small-valued damping is usually very sensitive to the

non-stationary nature of seismic excitation, the presence of

measurement noise, the selection of p-values and the length

of data. Nevertheless, the representative values of damping for

each mode were obtained by observing the stabilized values

over several ranges of p-values as indicated in the example




Fig. 10. The first eight mode shapes of the Katsushika–Harp Bridge, identified from the July 23, 2005 earthquake records (note: dotted lines represent the stationaryposition, i.e. centerline of the girder, towers and piers).

(Fig. 9). The damping estimates were found within the range

of 0.5%–4% with average value and standard deviation of 2±

0.8%.

6.3. Tower modes

While the identification method can simultaneously generate

the global modes, the local modes of structural members can

also be identified by employing the sensors associated only

with those of the members. In this section, the tower modes

and their participation in the global modes are examined.

For this purpose, sensors located on the top of both towers

are utilized as outputs. These sensors measure accelerations

in longitudinal and lateral directions. Fig. 11 shows typical

longitudinal and lateral accelerations recorded on the top of

the main and the right tower. Considering the slender shape

of the tower, especially the main one, and the material they

are made of, a higher level of vibration is expected. In both

towers accelerations in lateral direction were found larger than

the accelerations in longitudinal direction. The identification




Table 4

Identified modal parameters of the main tower and their comparison with the global modes recorded during the July 23, 2005 earthquake

Mode no. Freq (Hz) Main tower Right tower Associated global mode

Damp (%)a Mode angle (◦) Damp (%)a Mode angle (◦)

1 0.45 1.61 (0.38) 58.7 1.32 (0.19) 148.38 1st Vertical (Sym)

2 0.59 0.94 (0.10) 264.3 0.56 (0.09) 265.27 1st Lateral + Vertical 1st Sym

3 0.65 2.55 (0.17) 264.8 1.67 (0.20) 85.70 2nd Lateral4 0.81 0.82 (0.09) 83.5 0.99 (0.18) 237.55 2nd Vertical (Asym)

5 0.92 2.23 (0.24) 266.5 2.95 (0.31) 87.13 3rd Lateral

6 1.09 0.85 (0.17) 81.6 2.47 (0.26) 67.5 4th Lateral

7 1.16 0.73 (0.15) 89.2 0.48 (0.17) 98.6 3rd Vertical (Sym) + Lateral

8 1.26 0.41 (0.35) 286.5 1.11 (0.33) 97.1 Longitudinal dominant mode

9 1.38 0.78 (0.18) 271.4 0.66 (0.14) 100.52 1st Torsion

a Values in the brackets denote the standard deviation.

Fig. 11. Typical acceleration responses of the tower recorded during the Feb 2,

1992 earthquake.

employs a system of six inputs and two outputs (6I–2O) for

each tower. The inputs were selected from the bottom of the

tower, while the outputs were the two channels on the top of

the main and the right tower. By observing the stabilized plot,

p = 100 was selected. Based on this selection the system

matrix A is realized and its singular values were examined

afterwards. Fig. 8 shows an example of the singular value

plot of the main tower system. One can observe that singularvalues beyond 60 are almost or less than zero. Therefore,

the value of 60 is selected as the singular value cutoff.

Using this number, the total number of modes obtained from

matrix A is 60/2 = 30, since they all appear in conjugate

frequencies. Afterwards, upon observing the 30 modes from

the first generation and by adopting the modal supervision

technique explained above, only 9 modes were identified with

confidence.

The results of SI of the main tower (Table 4) indicate that

all tower modes at the frequency range of 0–1.4 Hz were

associated with the girder motion. The interactions between

the girder motion and tower exist in such a way that the

Fig. 12. Fourier amplitude of the towers’ acceleration responses recorded

during the Feb 2, 1992 earthquake.

tower modes were resulted from their girder’s predominant

counterpart modes. For example the fundamental tower mode

at 0.45 Hz is the result of the girder–tower interaction mode

in vertical direction. This mode was found as the fundamental

mode in both towers. The alignments of modal displacement

(see mode angle in Table 4) indicate that both towers are leaning

towards the main span. The second and third modes were

dominated by lateral motion, with the difference in the phase.The second mode exhibits an in-phase motion of both towers,

while the third one shows the out-of-phase motion.

Fig. 12 shows the Fourier spectra of both towers in

longitudinal and lateral directions. Due to larger lateral motion,

peaks in lateral channels dominate the spectra. One can

notice from these spectra that the frequency peaks are indeed

associated with the girder dominant modes. Fig. 13 plots the

alignment of modes identified from the main tower. From the

angle of mode alignment, it can be seen that except for the

first mode (0.45 Hz), all main tower modes were dominated

by the lateral motion. It is interesting to see that even in the

modes that were associated with the girder vertical modes, the




Fig. 13. Mode shape vectors of the main tower. (Note: solid line indicates the principal alignment of the mode, the dotted line indicates the principal axis of the

tower.)

towers’ modal displacement are still lateral-predominant. This

characteristic was not found in the single-plane cable-stayed

bridge with the straight girder, such as the case of the Tsurumi-

Fairway Bridge [19].

In the identification process, damping estimates of tower

modes were stabilized easily and the scatter is confined within

a narrow range of estimates as shown by the small value of standard deviations (see Table 4). An essential feature observed

here is the small value of damping estimates of both towers.

In the main tower, for instance, except for the modes that are

associated with the 1st vertical, 2nd and 3rd lateral modes,

damping estimates were all identified less than 1%. The mean

value for all modes is 1.21 ± 0.74%. The damping estimates

of the right towers are also small, with the mean value of

1.35 ± 0.86. These results show that damping ratios of both

towers are lower than the damping estimates of the global

modes (mean value: 2.00 ± 0.87%).

The small damping of both towers can be understood as the

effect of the stay cables. Since the flexural rigidity of the cableis very small, the restraining effect especially in lateral direction

becomes considerably smaller. Therefore, with the slender

shape of both towers, they can be considered as freestanding

cantilever columns. It should be mentioned, however, that due

to the curved shape of the girder, the cables are not necessarily

on one plane. Consequently, there is still some portion of axial

rigidity from the cables acting on the tower in lateral direction.

This axial rigidity may contribute to the tower flexibility in

lateral motion. In a similar case of a single-plane cable-stayed

bridge and similar shape of steel box section tower, but with a

straight girder, an even lower damping ratio of the tower mode

(0%–1%) was observed [20].

7. Comparisons of modal parameters from six earthquakes

The seismic records available for this study cover a range of

accelerations with the maximum amplitude from 2 to 60 cm/s2.

And as previously mentioned, the identification procedure can

be performed over a relatively narrow time window. Therefore,

by employing several time windows from different levels of acceleration amplitude, variations of modal parameters can be

observed. In a long record such as the February 20, 1990

earthquake, the responses were divided into three frames of

60 s each, using the time-shifted windows. With this procedure,

the identified modal parameters can be compiled to detect the

possible structural changes during excitation process.

For this purpose, six earthquakes records were divided into

8 frames and the resulting modal parameters were compared.

The maximum input accelerations recorded at the base of

the main tower were found at the range of 2–52 cm/s2,

and their corresponding root-mean-squares were between

0.6 and 5.4 cm/s2. Table 5 shows the results of natural

frequency estimates identified from the eight frames. In thistable, the frequency estimates are listed with the standard

deviation values obtained from p-values around the stabilized

frequencies. In general, constant results of frequency estimates

were obtained in most of the frames. This clearly indicates

that the natural frequency of the structure remains constant

regardless of the level of input amplitude. It should also be

mentioned that in all modes variations of frequency estimates

are very small as noted by the maximum standard deviation that

equals 0.03 Hz.

The results of damping ratios shown in Table 6, on the other

hand, indicate larger variations of estimates. Mean values of

damping ratio of most modes were found within the range of




Table 5

Natural frequencies of the bridge identified from six earthquakes

Type of mode Identified natural frequency (Hz) and its standard deviation (in bracket)

Feb 20, 1990 Feb 2, 1992 Oct 23, 2004 Jul 23, 2005 Aug 16, 2005

Main Main

Frame Frame Frame shock shock

1 2 3 17:57 18:35

1st Vertical (Sym) 0.45 0.45 0.45 0.45 0.45 0.45 0.46 0.45

(0.00) (0.00) (0.00) (0.01) (0.00) (0.00) (0.00) (0.00)

1st Lateral + Vertical 1st 0.60 0.60 0.60 0.59 0.62 0.59 0.61 0.60

Sym (0.01) (0.00) (0.00) (0.00) (0.00) (0.00) (0.01) (0.01)

2nd Lateral 0.66 0.67 0.67 0.67 0.65 0.66 0.66 0.66

(0.01) (0.00) (0.00) (0.00) (0.01) (0.01) (0.00) (0.00)

2nd Vertical (Asym) 0.81 0.82 0.82 0.84 0.80 0.80 0.81 0.81

(0.00) (0.00) (0.00) (0.01) (0.01) (0.00) (0.01) (0.00)

3rd Lateral 0.96 – – 0.95 0.95 0.93 0.93 –

(0.00) – – (0.01) (0.00) (0.01) (0.01) –

4th Lateral 1.00 1.01 1.00 1.00 1.01 1.00 0.99 1.05

(0.00) (0.00) (0.01) (0.01) (0.02) (0.01) (0.01) (0.01)

3rd Vertical (Sym) + Lateral 1.16 1.18 1.17 1.19 1.16 1.14 1.17 1.16

(0.00) (0.01) (0.01) (0.01) (0.00) (0.01) (0.01) (0.01)

Longitudinal dominant mode 1.28 1.28 – 1.28 1.29 1.27 1.28 1.29

(0.02) (0.01) – (0.01) (0.01) (0.01) (0.01) (0.01)

1st Torsion 1.38 1.38 1.37 1.35 1.36 1.35 1.38 1.37

(0.01) (0.00) (0.01) (0.01) (0.01) (0.01) (0.02) (0.02)

5th Lateral 1.62 1.61 1.64 1.64 – 1.62 1.64 1.64

(0.01) (0.02) (0.01) (0.02) – (0.02) (0.01) (0.01)

4th Vertical (Asym) 1.68 1.67 1.68 1.68 1.68 1.68 1.68 1.68

(0.01) (0.02) (0.01) (0.01) (0.02) (0.02) (0.01) (0.02)

2nd Torsion 1.82 1.83 1.81 1.84 1.86 1.83 1.81 1.86

(0.01) (0.02) (0.03) (0.01) (0.01) (0.02) (0.01) (0.02)

5th Vertical (Sym) 2.22 2.22 – 2.20 2.22 2.22 2.23 2.22

(0.01) (0.01) – (0.02) (0.02) (0.02) (0.02) (0.02)

6th Lateral 2.40 2.41 2.41 2.40 2.41 2.37 2.41 2.44

(0.02) (0.02) (0.01) (0.02) (0.01) (0.03) (0.01) (0.02)

0%–5%. The standard deviation of the estimates, however, was

quite large compared to the mean values, indicating the scatterestimates.

Fig. 14 attempts to illustrate the trend of damping estimateswith respect to the maximum of input accelerations measured

at the base of the main tower. It is shown in this figure that,despite the scatter estimates, one can observe a trend of input

amplitude dependence of the mean damping estimates. In thefirst four modes, most damping estimates were found to be

within 0%–4% with the mean values increasing as the inputamplitude increased. The dependence of the damping estimates

on the input amplitude is quite apparent for the third and fourthmode, where the estimates do not show large scatter. For the

first and second mode, however, the large scatter of estimatesmakes it more difficult to arrive at conclusive results of

damping-input amplitude relationship. Generally, larger inputexcitation will produce larger structural response. And due to

the larger response, greater energy dissipation would take placeon the bridge component such as friction in bearings. Owing to

this mechanism and also by considering the effect of curvedgirder and the out-of-plane arrangement of the stays, larger

damping during larger input excitation is anticipated. However,since damping is small in value and complex in mechanism,

more comprehensive and detailed analysis are required before

arriving at a quantitative relationship between the mechanisms

and their contribution to the overall damping amplification,

especially for such a complex structure during large earthquake

excitation. This is beyond the scope of the current paper and

will be incorporated in future work.

8. Conclusions

The assessment of dynamic characteristics of the

Katsushika–Harp Bridge, a curved cable-stayed bridge using

seismic responses has been presented in this study. Owing tothe dense sensor deployment, excellent quality of measured re-

sponse, and the recent development of algorithms to deal with

the data, the dynamic characteristics of the bridge can be effi-

ciently and accurately estimated.

Fourteen modes were identified within the range of

0–2.5 Hz. Due to the curved shape of the girder; most of them

were closely spaced in frequency and having complicated 3D

motion-coupled mode shapes. Using the modal displacement

ratio, the participation of motion in one direction can be

determined and thus the type of mode is obtained according

to the dominant direction of motion. Based on observation of

these characteristics, the following conclusions are drawn:




Fig. 14. Variation of damping estimates with respect to the maximum input amplitude, recorded from 8 frames of six earthquakes.

1. The characteristics of the modes are somewhat similar to

those of cable-stayed bridge with the straight girder, in

which the girder motion dominated the global mode shapes.

High-coupled motions in three directions, however, were

observed that make it difficult to obtain the pure mode inone direction only. In all but the first vertical mode, the

effect of lateral motion was found significant. Torsional

effect was found to be relatively small as evidenced by the

high-frequency of torsional dominant modes. This might be

due to the fact that while the girder is essentially curved

in horizontal, it is well supported at four locations: both

towers and the approach end-piers. In addition, the large

radii of both girder curves and the position of the main tower

might have reduced the effect of the torsional effect in the

motion.

2. Tower modes identified within the same frequency range

of the global modes were all associated with girdermodes. Except for the first mode, all tower modes were

characterized by dominant lateral motions. Damping ratios

of tower modes were found lower than the global modes. The

average damping ratios identified at both towers are 1.21%

and 1.3%.

3. Using the response from six earthquakes with the level

of input amplitude between 2 and 52 cm/s2, the trend of

modal parameters was evaluated. It was found that while the

frequencies remain constant, damping ratios exhibit slight

dependencies on earthquake input amplitude. An increasing

trend of damping ratio was observed with the increase of the

earthquake input amplitude.

The scheme of system realization using information matrix

adopted for SI in this work has been an indispensable

instrument in dealing with voluminous seismic responses

recorded from multi-channels of sensors on the bridge. The

algorithm is straightforward and requires neither a priorstructural model, nor an optimization procedure to obtain

accurate estimates of dynamic characteristics of a complex

structure such as the curved cable-stayed bridge. In the view

of structural health monitoring, an efficient and accurate

identification technique are essential, especially for quick

structural assessment after the occurrence of earthquake

events. In the future, instrumentation of the cables should

be considered. This would be helpful to monitor the degree

of coupling between tower and girder caused by the radial

arrangement of the cables and also to monitor the tension of

cables during extreme shaking such as an earthquake.

Acknowledgement and disclaimer

The authors would like to express their gratitude to

the Tokyo Metropolitan Expressway Public Corporation for

providing the strong motion records and the drawings of

the bridge. Opinions, findings, and conclusions expressed in

this material are those of the authors and do not necessarily

reflect those of the Tokyo Metropolitan Expressway Public

Corporation. The first author gratefully acknowledges the

Japanese Ministry of Education, Culture, Sports, Science and

Technology, for financial support provided during this research

at the University of Tokyo.




Table 6

Damping ratio of the bridge identified from six earthquakes

Type of mode Identified damping ratio (%) and its standard deviation (%)

Feb 20, 1990 Feb 2, 1992 Oct 23, 2004 Jul 23, 2005 Aug 16, 2005

Main Main

Frame Frame Frame shock shock

1 2 3 17:57 18:35

1st Vertical (Sym) 1.53 0.52 1.19 2.26 2.22 1.33 1.58 1.26

(0.99) (0.46) (0.30) (1.10) (1.54) (0.34) (0.78) (0.36)

1st Lateral + Vertical 1st 2.16 1.18 – 1.97 2.49 0.74 1.64 2.72

Sym (0.73) (0.13) 0.47(0.31) (0.70) (0.39) (0.26) (0.80) (0.21)

2nd Lateral 1.36 1.65 2.19 1.53 1.98 3.68 3.34 2.23

(0.09) (0.08) (0.28) (0.29) (0.44) (0.24) (0.26) (0.22)

2nd Vertical (Asym) 0.61 0.83 0.66 0.44 0.52 1.04 1.46 1.12

(0.25) (0.10) (0.08) (0.14) (0.23) (0.38) (0.20) (0.09)

3rd Lateral 3.42 – – 2.39 2.89 4.34 2.06 –

(1.79) – – (0.45) (0.09) (0.36) (0.32) –

4th Lateral 2.86 4.36 5.44 2.71 2.15 1.56 0.65 2.67

(1.39) (1.67) (0.72) (1.18) (0.29) (0.34) (0.03) (0.36)

3rd Vertical (Sym) + 1.78 1.95 0.71 0.55 1.21 2.27 3.64 0.79

Lateral (1.06) (0.65) (0.56) (0.41) (0.21) (0.50) (0.21) (0.15)

Longitudinal dominant 2.47 2.33 – 2.58 3.70 2.28 1.79 2.33mode (0.21) (0.22) – (0.37) (0.43) (0.78) (0.80) (1.30)

1st Torsion 2.91 – 3.41 3.61 3.72 1.23 2.61 0.23

(2.17) 2.56 (-) (1.28) (0.65) (1.03) (0.65) (0.74) (0.17)

5th Lateral 4.79 5.51 1.74 3.30 – 1.72 1.33 2.11

(1.17) (1.07) (1.00) (0.55) – (0.45) (0.45) (0.18)

4th Vertical (Asym) 2.25 – 3.07 0.90 3.06 2.29 1.09 2.38

(0.55) 3.46 (-) (0.30) (0.52) (0.21) (0.85) (0.30) (0.59)

2nd Torsion 8.68 4.01 1.62 1.61 1.36 1.69 2.79 3.57

(0.79) (0.75) (0.61) (1.43) (0.11) (0.72) (0.63) (0.63)

5th Vertical (Sym) 3.63 0.90 2.58 1.62 3.23 1.45 2.62 3.06

(1.00) (0.96) (0.41) (0.91) (0.13) (0.96) (0.10) (0.44)

6th Lateral 1.38 1.76 1.06 1.73 2.47 1.35 1.52 2.88

(0.31) (0.71) (0.65) (0.80) (0.25) (0.35) (0.43) (0.88)

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dynamic characteristics of a curved cable-stayed bridge from strong motion records

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