DAMAGE DETECTION OF REINFORCED
CONCRETE STRUCTURE BASED ON LIMITEDSTRONG MOTION RECORDS
Gong Maosheng∗
MEE07151

Supervisor:

Toshihide Kashima∗∗

ABSTRACTThe author presents one simplified technique and methodology for structural parameter identificationand global damage detection from limited number of strong motion records. The time-invariant andtime-varying parameters of structure are identified by using the off-line system identification method
ARX and the on-line system identification method RARX respectively. The presence of damage is
detected primarily by checking the changes of parameters during the earthquakes. Then, one procedure
is introduced to convert MDOF structural system to equivalent SDOF system. The integral damagelevel of the equivalent SDOF system is detected by using the ductility factor and damage index. The
damage index of the equivalent SDOF system can be considered as the integral damage level indicatorof the MDOF system. The method is applied to Hachinohe City Hall building which was lightly
damage in the Sanriku-oki Earthquake on Dec. 28, 1994. The parameters of the building are identified
and the damage level is detected from the strong motion records. The results show that the method haswith high efficiency, validity, applicability and practicability. The method can be implemented toassess the structural damage level and to judge if the damage is repairable or not after a big earthquake.
Furthermore, the method can be used to predict the failure of a whole structure and to evaluate theperformance in earthquake. The presented technique and method can be considered as a significant aid
for making structural retrofitting decision after devastating earthquake.

Keywords: Strong motion record, Damage detection, System identification.
INTRODUCTION
The structural damage detection aims to detect, localize and classify damage of structures and also to
predict and assess the safety and remaining service life of structures. Many kinds of techniques andmethods have been developed and applied for the purpose in the past decades (Doebling, 1996; Sohn,
2003; Farrar, 2007). In fact, the damage is not meaningful without comparison between undamaged
state and damaged state. That means the structural parameters before and after damage are criticalinformation for damage detection. Determining the dynamic properties of a structural system from theresponse data is well known as system identification, and many kinds of methods have also beendeveloped. Generally speaking, the damage detection is related to or based on the structural parameter
identification at a certain extent. To monitor the structural response and performance during
earthquakes, strong motion seismographs are installed in many buildings. The response data recordedby the strong motion observation system during earthquakes can be used to determine the structuralparameters and damage state (Loh, et al, 1996, 2000). It is very important to judge damage level after
earthquake especially for the buildings slightly or moderately damaged, but sometimes it is difficult tocheck the damage by visual inspection since the structural members are covered with finishingmaterials. In such case, it has remarkable meaning to detect the damage and determine the damagelevel by using the strong motion records. Such kind of post-earthquake damage assessment and∗Institute of Engineering Mechanics, China Earthquake Administration, Harbin, China.
International Institute of Seismology and Earthquake Engineering, Building Research Institute, Japan.

∗∗

damage detection has significant life-safety implication, because it helps to determine whether thebuilding is safe enough for reoccupation. Furthermore, it has great meaning to judge if the building
can be repaired and reoccupied from the economy issues. According to the analysis stated above, theauthor presents simplified methods to identify the parameters and diagnose the damage level after
huge earthquake from the limited strong motion records.
METHODOLOGY
System Identification
Two system identification methods are used to obtain the structural parameters. One is the off-line
ARX (Auto-Regression with eXogenous) method for time-invariant parameters, the other is RARX
(Recursive ARX) method for the time-varying parameters. The ARX model can be denoted as:y (t ) = ϕ T (t )θ + e(t )(1)where θ is the vector including unknown system parameters; ϕ (t ) is the vector including input andoutput samples. The solution of Eq. (1) can be estimated by the following Eq. (2) (Ljung, 1999):

θˆN =

N

ϕ (t )ϕ (t )

t =1

T

−1 N

ϕ (t ) y (t )

(2)

t =1

The modal parameters can be calculated once the system parameters are obtained as shown below:1− ln( Z r Z r∗ )(3)
ωr =lnZ r ln Z r* , ξ r =∆t
2 ln Z r ln Z r*
where ( Z r , Z r∗ ) is the r-th discrete complex conjugate eigenvalue pair and ∆t is the sampling period.
ω r and ξ r are the r-order modal frequency and damping ratio respectively.
To track changes of structural parameters during the vibration, RARX method is used to identify thetime-varying parameters. RARX is a system algorithm to estimate recursively parameters of ARX
model in Eq. (1) by using RLS method. The solution can be estimated as following (Ljung, 1999):(4)θˆ(t ) = θˆ(t − 1) + P (t )ϕ (t )ε (t )T
(5)ε (t ) = y (t ) − ϕ (t )θˆ(t − 1)P(t − 1)ϕ (t )ϕ T (t ) P(t − 1)λλ + ϕ T (t ) P(t − 1)ϕ (t )Then the equivalent stiffness Keq can be calculated by using the following Eq. (7):4π 2 MK eq =
T2
where, M is the total mass of the structure or system; T is the fundamental period of the structure.
P(t ) =1

P(t − 1) −

(6)

(7)

MDOF System to Equivalent SDOF System
To estimate global damage index of structures under the excitation of earthquake based on the limitedmstrong motion response records, the MDOF system is converted
F
to equivalent SDOF system. The total mass M is assumed as theequivalent mass Me of the SDOF system as shown in Eq. (8):
N

N

mj

Fj

Me =

Me
Fe

hj
he
m1

F1

VbM
MbM

VbS
MbS

Figure 1. Equivalent SDOF system of MDOF system

N
i =1

mi

(8)

where mi is the mass of the i-th story of MDOF system. Thenthe distribution of the earthquake shear force of the MDOF
system is assumed as the following Eq. (9) (AIJ code):

Ai = 1 +

1

αi

− αi

2T
1 + 3T(9)where the α i is the normalized weight of the i-th story, which is calculated as the weight above i-thstory divided by the weight above ground. T is the natural period. If the response data on roof isrecorded, the earthquake force on the i-th story can be estimated by the following Eq. (10):FN = mN a N max , Fi =

Ai
FN
AN

i = 1,2N(10)where m N is the mass of top story and a N max is the maximum acceleration on the top. FromVbS = VbM , the aSmax of the SDOF system under the same excitation can be estimated by Eq. (11):aSmax = Fe / M e =N
i =1

Fi / M e(11)

Ductility Factor and Damage Index Calculation
The structural damage can be described by the ductility factor and hysteretic energy dispersed by thebuilding in earthquake. Herein, the ductility factor is estimated from the nonlinear response spectra fordamage detection purpose. The concept is if we have the properties of the SDOF system, such as
period, damping and ductility factor, we can find out theSamaximum response from the response spectra. On theSaµ1 (T2)
contrary, the ductility factor can be determined from theconstant ductility nonlinear response spectra if the
maximum response of the SDOF is known.
Saµ1 (T)The
relationship between the equal ductility response
Saµ1
spectra and maximum response of the SDOF system can be
Saµ1 (T1)
Saµ2 (T1)expatiated by Figure 2 from which the ductility factor can
Saµ (T)aSmax
be obtained if the response spectra Sa 1 and Sa 2 of theSaµ2 (T)Saµ2 (T1)Saµ2strong motion record at base are known. From Figure 2, we
can have the following the calculations:

T − T1( S a 1 (T2 ) − S a 1 (T1 )) (12)T2 − T1
T − T1
S a 2 (T ) = S a 2 (T1 ) +( S a 2 (T2 ) − S a 2 (T1 )) (13)T2 − T1

S a 1 (T ) = S a 1 (T1 ) +
T1

T

T2

Figure 2. Interpolation relationshipThe ductility factor of the SDOF system can be approximately estimated by Eq. (14) from aS max .

µ = µ1 +S a 1 (T ) − S a (T )( µ 2 − µ1 ) = µ1 +S a 1 (T ) − aS max(14)( µ 2 − µ1 )S a 1 (T ) − S a 2 (T )S a 1 (T ) − S a 2 (T )A hybrid damage index of combination of ductility and hysteretic energy is adopted to predict integral
damage level and the damage index is calculated by using the Bispec software (Hachem, 2002, 2008).
PARAMETER IDENTIFICATION AND DAMAGE DETECTION
Hachinohe City Hall Building
The Hachinohe City Hall building (Figure 3) is used as
the example to determine the parameters and damagelevel from the strong motion records. The building is a
5-story with one story basement, and RC type structure
is adopted for bearing system. The building wasdamaged by Sanriku-oki Earthquake on Dec. 28, 1994
(denoted by EQ2). The response data were obtained in
the earthquake and its aftershock (denote by EQ3).
Before EQ2, the response data was recorded in one
small earthquake (denoted by EQ1) on Oct. 9, 1994.

Figure 3. Hachinohe City Hall (Photograph by T. Kashima)

Time-invariant Parameters
Just for example, the time-invariant parameters in 164° direction identified from the data in EQ1 areshown in Figure 4. All the identification results are shown in Table 1 in which the parameters in both
directions are also compared. From Table 1 we can see, the structural natural frequency and equivalent
stiffness decreased but the damping ratios increased after the EQ2. The reduction degree of theparameters is shown in Figure 5. It can be concluded that the building was damaged by the EQ2.
200

f =3.2575Hz(164-B1F)-(164-06F)

1

12
10
Amp

10
a/cm⋅s-2

14

Measured data
Identified data
Error=0.18125

8
6
4

-10

2

-20
0

10

20

30
t/s

40

50

(a) Comparison of measured and identified data

0
0

10

20

f/Hz

30

40

50

(b) Transfer function between base and roof

Figure 4. ARX Identification results from EQ1 data
Time-varying Parameters
Table 1. Identified parameters from data in the 3 earthquakes
The presence of damage can bef (Hz)
Keq(109N/m)ξ (%)
Earthquake
detected by comparison of the
254°
164°
254°
164°
254°
164°
parameters in 2 small events(a) EQ1
3.25
3.26
2.72
4.87
4.11
4.14
before and after the big earth(b) EQ2
2.64
2.78
3.83
6.26
2.71
3.01
quake. However, we don’tknow when the damage occur(c) EQ3
2.66
2.78
2.96
4.88
2.75
3.01
ed and how is the damage level.
(b-a)/a(%)-18.77 -14.72 +40.81 +28.54 -34.06 -27.29
Even we don’t know damage(c-a)/a(%)-18.15 -14.72 +8.82+0.21 -33.09 -27.29
occurred during the big
110
earthquake if we don’t have small earthquake response-10
f 254 direction
data before the big earthquake for comparison. For
f 164 direction
100
0
K 254 direction
damage detection by only using strong motion records
K 164 direction
from one earthquake, the on-line RARX method is used to90
10detect the accurate damage time and the decline trend
during the earthquake. The time-varying parameters of
80
20
254° direction in EQ2 are shown in Figure 6. From which
we can see the natural frequency decreased gradually until
70
30
arrived at the lowest level, then, it was a little bit
recovered at the tail part of the strong motion records but
40
60
EQ3
EQ2
EQ1
not reach the original value. That means the building was
Figure 5. Variation of f and Keq
definitely damaged by the earthquake. The decrease of thefrequency indicates the damage process during the big earthquake.
Fundamental Frequency
Furthermore, it can be
800
5Measured data
Identified result
concluded that the building
Identified
data
600
Segment result
Error=7.5433e-05
started to be damaged at
4400around 20s and stopped at
200
3around 38s from the0variation of the parameters.
-200
2
From the above analysis,-400
we can know that by using
1
0
50
100
150
-600
on-line RARX method, the
0
50
100
150
t/s
t/s
(b) Time-varying frequency(a) Comparison
damage can be detected by
Figure6.
RARX
Identification
results
from
EQ2 data (254°)
only using strong motion
eq

f/Hz

a/cm⋅s-2

Reduction (%)

Percentage (%)

eqrecords from one earthquake. The RARX method has good qualities to detect the changes of theparameters in big earthquakes, and the parameters of the original state (undamaged) of the structure
can also be determined for comparison purpose. It can be concluded the presence of damage can also
be detected primarily by only using strong motion data from one earthquake.
Ductility Factor Estimation and Damage Detection
The strong motion data in 164° direction is used as the example to show the ductility factor estimationand damage detection. The building is converted to equivalent SDOF system as shown in Table 2 fromwhich we can see the natural period is 0.3067s which is from the identification results and the peakvalue of the acceleration of the SDOF system is 586.44cm/s2 which is used to calculate the ductility
factor from the nonlinear equal ductility response spectra, and the equivalent height is 13.5m.
Table 2. Parameters of the equivalent SDOF system for Hachinohe City Hall (164°)
N

story

Mi(t)

j =i

M j (t)

αi

T(s)

Ai

PA(cm/s2)

Fi(kN)

hi(m)

1

2141

9862

1.00

0.30671.000/

8.78E+03

4.50

2

2488

7721

0.78

0.3067

1.111/

9.76E+03

3.95

3

2070

5233

0.53

0.3067

1.269/

1.11E+04

3.95

4

1572

3163

0.32

0.3067

1.462/

1.28E+04

3.95

5

1591

1591

0.16

0.30671.744

962.601.53E+04

3.95

2Sa(cm/s )SDOF 9862//
0.3067/586.44
5.78E+04
13.5
The nonlinear response spectra of the record at the base of the building with different ductility factors
and with 5% damping ratio are shown in the Figure 7. The damping factor is adopted from theidentification results from EQ1 in Table 1. According to the T=0.3067s and the maximum acceleration
PA=586.44cm/s2 of the SDOF system, the ductility factor can be calculated from the equal ductility
spectra by using the interpolation method as shown in Table 3 from which we can know the ductility
factor of the equivalent SDOF system of Hachinohe City1200
µ =1.0Hall is 1.85. The ductility factor can be considered as theµ =1.51000
average ductility level of the building in Sanriku-okiµ =2.0µ =2.5
Earthquake on Dec. 28, 1994.
800µ =3.0
600
400
200
0
0.0

0.5

1.0
T(s)

1.5

2.0

Figure 7. Equal ductility response spectra ofbase record 164° ( ξ = 5% )0.8

Damage Index

0.6

0.4

µ =1.0µ =1.5µ =2.0µ =2.5µ =3.0

Table 3. Ductility factor estimation in 164° direction
µ
T1
T2
T
T
0.30
0.40
0.3067/617.57896.76636.411.50
Saµ1550.17759.24564.28
2.00
Saµ2
/
/
586.441.85
SaµTable 4. Damage index in 164° direction
µ
T1
T2
T
T
0.30
0.40
0.3067/
0.09620.0555
0.093
1.50
DIµ1
0.1588
0.111
0.156
2.00
DIµ2
/
/
0.137
1.85
DIµ

The damage index spectra calculated from the baserecord(164°) are shown in Figure 8 from which the
0.2
damage index of the system can be estimated according
to the fundamental period T, damping ratio ξ and the0.0
ductility factor µ obtained in Table 3. The calculation
0.0
0.5
1.0
1.5
2.0
result is shown in Table 4 from which we can see theT(s)damage index of equivalent SDOF system is 0.137. By
Figure 8. Damage index of base record 164° ( ξ = 5% )using the same procedure, the damaged index in 254° direction is 0.083. Considering the realcircumstance, for the same building, the maximum damage index in both directions should be adopted
to describe the damage level. From this concept, the damage index of the building is 0.137. According
to study of Valles, et al (1996), the damage index 0.137 means the damage of the building is slight andthe damage is repairable for reoccupation after the earthquake.
By using same procedure and methodology, the damage of one 9-story RC building, Department of
the Architecture and Building Science of Tohoku University which was damaged in Miyagi-ken
Earthquake (Sep. 15, 1998), is detected. It is found that the damage index is 0.282 in the earthquake,
which means the building is moderately damaged by the earthquake and the damage is repairable.
From the example, it can be concluded that the structural damage caused by the earthquake can bedetected and assessed only by using the strong motion records on the roof and at the bottom of thestructure through implementing the procedure and methodology presented in the study.
CONCLUSIONS
The author presented one method to identify parameters and detect damage level of RC buildings byusing strong motion records at the top and the bottom. The parameters of Hachinohe City Hallbuilding are identified and damage level is detected. Some conclusions are summarized as follows:
1) The time-invariant and time-varying structural parameters including modal frequency,
damping ratio and equivalent stiffness can be determined from the strong motion records by using theoff-line ARX and on-line RARX system identification models. The changes of the parameters, such as
decreasing of the frequency and reduction of the equivalent stiffness, can be used to find out thepresence of damage with the structure.
2) The MDOF structural system can be converted to equivalent SDOF system for damage
detection purpose. The ductility factor and damage index can be obtained by using the interpolation
method from the constant ductility nonlinear response spectra and damage index spectra.
3) The ductility factor of the equivalent SDOF system can be considered as the average
ductility level of structure, and damage index of the equivalent SDOF system can be considered as
integral damage level indicator of the building.
4) The damage index of structure in the study is defined as the global damage indicator andcan be used to predict damage state or the failure of a whole structure after earthquake.
5) The method presented in the study can be implemented for structural performance
evaluation and for decision making of repair and continuous use of the instrumented buildings afterdevastating earthquakes.
ACKNOWLEDGEMENT
I would like to express my sincere and heartfelt gratitude to Dr. Taiki Saito for his continuous support,
valuable suggestion and guidance during my study.
REFERENCES
Doebling, S. W., et al, 1996, Los Alamos National Laboratory report, LA-13070-MS.
Farrar, C. R., and Lieven, N. A. J., 2007, Phil. Trans. R. Soc. A 365, 623-632.
Hachem, M. M., 2002, Doctoral Thesis, University of California, Berkeley, USA.
Hachem, M. M., 2008, http://www. ce.berkeley.edu/~hachem /bispec/index.html.
Ljung, L., 1999, PTR Prentice Hall, Upper Saddle River, N.J.
Loh, C. H., and Lin, H. M., 1996, Earthquake Engineering and Structural Dynamics, 25, 269-290.
Loh, C. H., Lin, C. Y., and Huang, C. C., 2000, Journal of Engineering Mechanics, 126(7), 693-703.
Sohn, H., et al., 2003, Los Alamos National Laboratory Report, LA-13976-MS.
Valles, R. E., Reinhorn, A. M., Kunnath, S. K., et al., 1996, NCEER-96-0010.