Engineering Structures 29 (2007) 21432157

www.elsevier.com/locate/engstruct

A story damage index of seismically-excited buildings based on modal

frequency and mode shape
Jer-Fu Wang 1 , Chi-Chang Lin , Shih-Min Yen
Department of Civil Engineering, National Chung Hsing University, Taichung, 40227, Taiwan, ROC

Received 18 July 2006; received in revised form 23 October 2006; accepted 23 October 2006
Available online 20 December 2006

Abstract

In this paper, a story damage index (SDI) is developed and expressed as a simple formula based on modal frequency and mode shape obtained
from real earthquake records. It is useful because only one set of modal parameters is required for the calculation of the SDI to show the degree
of damage of the storey in question. According to numerical simulation results, it is shown that the proposed SDI has both high accuracy and
high reliability. The approximate SDI, named ASDI, is more convenient and can be applied without the floor mass information. It is verified
by some numerical simulations and the experimental data analysis for a benchmark (IASC-ASCE Phase II) model. This ASDI is also applied
to the damage assessment of a 7-storey reinforced concrete hotel building in Van Nuys, California, which experienced severe structural damage
during the 1994 Northridge earthquake. With both its fundamental frequency and mode shape identified by the SRIM (System Realization using
Information Matrix) identification technique, it is shown that the ASDI agrees fairly well with the results of the visual inspection, and is valuable
in practical application.
c 2006 Elsevier Ltd. All rights reserved.

Keywords: System identification; Damage assessment; Damage index; Earthquake record

1. Introduction With these measurements, the damage of the structure can then
be assessed by means of calculating damage indices. Damage
It is well-known that the earthquake resistant design of indices that have been proposed basically can be classified
buildings usually allows for a building to experience repairable into vibration response-based and structural parameter-based
damage during moderate and large earthquakes. Sometimes, the indices. The vibration response-based indices [1,2] used the
damage of a building is probably unrepairable and may even structural response measurements in a single excitation event
gradually lead to collapse. Therefore, it is important to measure and from that calculated the damage-related physical factors,
the damage immediately after the earthquake to ensure safety such as peak acceleration, peak velocity, energy, etc. The
of a building. To this end, nondestructive test (NDT) has a great structural parameter-based damage indices [37] estimated the
potential for damage assessment. change of structural parameters based on two independent
During the past decades, a great amount of research excitation events: before and after the damage occurrence. To
about the NDT of structures has been conducted because employ the latter indices, the structural parameters are expected
of the significant development of powerful systems for data to remain time-invariant during one of these two events. The
acquisition and signal processing. The procedure requires linear system identification technique can then be applied
that the dynamic responses of the structure and the external to identify the structural modal parameters, representing the
excitations be measured using sensors (usually accelerometers). structural properties during that duration. After obtaining the
identified results, a comparison of two sets of structural modal
parameters, before and after the damage, are then carried out
Corresponding author. Tel.: +886 4 22840438x225; fax: +886 4 22851992.
using various methodologies.
E-mail addresses: jerfu@ms16.hinet.net (J.-F. Wang),
cclin3@dragon.nchu.edu.tw (C.-C. Lin). Generally speaking, the vibration response-based damage
1 Tel.: +886 4 22862181. index indicates the damage of an entire structure or a structural

c 2006 Elsevier Ltd. All rights reserved.

0141-0296/$ - see front matter
doi:10.1016/j.engstruct.2006.10.018
2144 J.-F. Wang et al. / Engineering Structures 29 (2007) 21432157

component by a single value; the well-known index proposed index MFDI, it shows that the proposed SDI and ASDI are
by Park and Ang [2] is a good example. This index is useful for more accurate and more reliable. Besides verified through the
representing the damage degree of a single structural element. test data of the IASC (International Association for Structural
However, the monotonic forcedeformation curve of each Control)ASCE Phase II benchmark model, the index ASDI
element is required. On the other hand, the structural parameter- was also employed for the damage assessment of a 7-story
based damage index is developed specifically for the detection reinforced concrete hotel building in Van Nuys, California,
of damage location, but the procedure is more complicated which experienced severe structural damage during the 1994
than that for the former index. The structural parameter-based Northridge earthquake. Using the modal frequency and the
indices are further classified into three types, depending on the mode shape identified through the SRIM system identification
parameters used: modal frequency, mode shape, or both. technique [12,13], the damage index ASDI can easily be
Calculating the change of modal frequency to detect damage obtained. Compared with the results of the visual inspection,
is popular in structural health monitoring (SHM) systems its value in practical applications will clearly be revealed.
because damage is always accompanied by a reduction of
stiffness as well as modal frequency. This approach has been 2. Theoretical derivation
widely applied in damage alarming. Damage in different
locations and components actually leads to different frequency 2.1. Dynamic and characteristic equations
changes in various modes. Nevertheless, it remains difficult to
determine the damage location just by observing the changes Considering an N floor planar shear building frame with
of modal frequencies. Among the modal parameters of a mass m l at the lth floor and with stiffness kl and damping cl
structure system, the mode shape is obviously the only location- at the lth story, the equation of motion of the linear building
related parameter. Therefore, many researchers have attempted frame under ground acceleration u g (t) can be written as:
to establish mode-shape-based indices, such as modal curvature
Mx(t) + Cx(t) + Kx(t) = Mru g (t) (1)
index [5,7], index MAC (Modal Assurance Criterion) [3] and
index COMAC (Coordinate Modal Assurance Criterion) [4], where M, C and K are the N N mass, damping and stiffness
to identify damage and its locations. All above indices have matrices respectively:
simple expressions and have been applied in identifying the
location of damage. However, it has been shown that they have
m
1 0 0
low sensitivity to damage in some cases [8,9]. Considering both 0 m2
.. ..
modal frequencies and mode shapes to detect the occurrence
. .

and location of damage may be a more reliable way than relying . ..
..
M= ml . (2a)

on either one of them. The modal flexibility damage index
.. ..

(MFDI) [6] may be the most well-known one. The principle of
. .


this method is on the basis of the comparison of the flexibility
m N 1 0

matrices obtained from two sets of mode shapes. Moreover, 0 0 mN
this method involves the normalization of mode shape since the
mode shape values are not fixed. c1 + c2 c2 0 0

c2 c2 + c3 c3
Many researchers developed other damage indices for



.. .. .. .. .
various types of structures. Brasiliano et al. [9] evaluated the
. . . . .
0
.

residual error method in the movement equation to verify
C=
cl cl + cl+1 cl+1



its efficiency when applied to continuous beams and frame .

. .. .. .. ..


. . . .
structures. Kim and Chun [10] derived an index to apply .

0

c N 1 c N 1 + c N c N

to buildings. Kim et al. [11] employed frequency-based and

mode-shape-based damage detection methods for locating and 0 0 c N cN

quantifying damage in prestressed concrete beams. All of these (2b)

methodologies certainly have more accurate results than the
k1 + k2 k2 0 0

previous damage indices, but involve complicated expressions.

k2 k2 + k3 k3



In this study, a story damage index (SDI) with a simple
.. .. .. .. .
.

. . . .

0 .
expression for building damage assessment is developed. This



K= kl kl + kl+1 kl+1
index is expressed in terms of the floor mass, modal frequency


. .. .. .. ..

and mode shape of a particular mode. An approximate story .

. . . . . 0


damage index, named ASDI, is also presented. The index ASDI

k N 1 k N 1 + k N k N


is easy to calculate and as such more useful. In addition, its 0 0 k N kN
calculating error is limited. It is easy to apply, because only (2c)
one set of modal parameters are required for the calculation of
the ASDI value to show the degree of damage of the particular x(t) represents the N 1 vector of the floor displacements
story, or interval between measured floors in case of partial relative to the ground at a time t; and r indicates the N 1
measurements. Compared with the index COMAC and the influence vector. With other words, assuming that j and j
 J.-F. Wang et al. / Engineering Structures 29 (2007) 21432157 2145

are the jth eigenvalue and the N 1 nonzero eigenvector of the 2.3. Story stiffness with considering damping
building system, respectively, the characteristic equation can be
written as: It is shown that Eq. (3) is a general form of the relationaship
(2j M + j C + K) j = 0. (3) between the physical and modal parameters of a damped
structural system no matter its damping is proportional or
Eq. (3) can be rearranged into statespace form as nonproportional. Substituting Eqs. (2a)(2c) into Eq. (3)

M1 C M1 K j j
  
I 0 jj
  gives
= j . (4)
I 0 j 0 I j
a1,1 a1,2 0 0

The jth eigenvalue can be represented by a pair of conjugate a2,1 a2,2 a2,3
. .. .. ..

complex in terms of jth modal frequency, j , and damping .. . . .


ratio, j , as

0 al,l1 al,l al,l+1 0


q
.. .. ..
( j )1,2 = j Cm1 j ; Cm1 j = j i 1 2j . . .

(5)
.

..

with its squared value a N 1,N 2 a N 1,N 1 a N 1,N
0 0 a N ,N 1 a N ,N
(2j )1,2 = 2j Cm2 j ;
1 j


q
0
Cm2 j = (1 2 2j ) i2 j 1 2j 2 j


(6)








0

.. .


.




. .






where i = 1. Cm1 j and Cm2 j are functions of the jth modal




damping ratio j . When a structure has no damping, Cm1 j and l j = 0 (10)
.. ..

Cm2 j reduce to i and 1, respectively. 2j becomes 2j and Eq.

. .








(3) is reduced to:

(N 1) j


0





N j

0
(K 2j M) j = 0 (7)
which is the famous characteristic equation for an undamped where
system.
j m l + j cl + j cl+1 + kl + kl+1
2

2.2. Story stiffness without considering damping al,l = for l = 1, 2, . . . , N 1 (11a)

j m l + j cl + kl for l = N
2
Substituting Eqs. (2a) and (2c) into Eq. (7) results in the
equation given in Box I. al,l+1 = j cl+1 kl+1 (l = 1, 2, . . . , N 1) (11b)
Solving the equation in Box I from the last to the first rows, al,l1 = al1,l (l = 2, 3, . . . , N ). (11c)
a general expression for the lth story stiffness can be obtained
as: The last row of Eq. (10) can be written as:

N
X m n n j a N ,N 1 (N 1) j + a N ,N N j = 0. (12)
kl = 2j ; j = 1, 2, . . . , N (8)
n=l
1l j Substituting Eqs. (11a) and (11c) into Eq. (12) gives
where (c N j k N )(N 1) j + (m N 2j + c N j + k N ) N j = 0. (13)
l j (l1) j for l = 2, 3, . . . , N

1l j = (9) Solving Eq. (10) from the last to the first rows, a general
l j for l = 1.
equation for the stiffness of the lth story is:
To interpret Eq. (8) from a physical point of view, one
N
can imagine a building vibrating in its jth mode. In this X m n n j
condition, the vibration frequency of the building is just the kl = 2j cl j . (14)
1l j
jth modal frequency, j , and the displacement at the lth floor n=l

is the lth row of the jth mode shape, l j . Therefore, the lth Since j and 2j can be represented by Eqs. (5) and (6), the
floor acceleration equals to 2j l j . Also, the inertial force of above equation becomes:
the lth floor will be the mass multiplied by the acceleration,
m l 2j l j . Since the resultant force applying to the lth story is the N
X m n n j
PN kl = 2j Cm2 j cl j Cm1 j . (15)
summation of the inertial force above the story, n=l 2j m n n j , 1l j
n=l
and the lth story drift is equal to the relative displacement
between the lth and the (l 1)th floors, l j (l1) j , then it As previously discussed, Eq. (15) converges to Eq. (8) for an
is evident that the stiffness of the lth story can be obtained by undamped structure. Since the modal damping ratio of general
dividing the resultant force by the story drift and Eq. (8) can be building structures is quite small. It is adequate to apply Eq. (8)
solved easily. for the calculation of story stiffness.
2146 J.-F. Wang et al. / Engineering Structures 29 (2007) 21432157

(k + k ) 2 m k2 0 0
1 2 j 1
..
(k2 + k3 ) 2j m 2 .

k2 k3
.. .. ..

. . .

0 0
..

(kl + kl+1 ) 2j m l .

kl kl+1
.. .. ..

. . .

0
..


. k N 1 (k N 1 + k N ) 2j m N 1 k N
0 ... 0 k N k N 2j m N
0
1j
2 j 0
.. .
.
. .


l j = 0

.. .
.
. .


(N 1) j 0

N j 0

Box I.

2.4. Definition of story damage index Table 1

Floor mass ratio distributions for five-story shear buildings
Define that the damage of a building as the reduction in
Floor no. Floor mass ratio
percentage of story stiffness before and after damage. Based
Type 1 Type 2 Type 3 Type 4
on Eq. (8), the damage index of the lth story, termed SDIl , can
be expressed as: m5 2 1 1 1 1 2 1 1 1 2 1 1 1
m4 1 2 1 1 1 2 2 1 1 1 2 1 2
N m i ij m3 1 1 2 1 1 1 2 2 1 2 1 2 3
2
P
m2 1 1 1 2 1 1 1 2 2 1 2 1 4
kl j m l 1lj
i=l m1 1 1 1 1 2 1 1 1 2 1 1 2 5
SDIl = 1 =1 (16)
kl N
m i i j
j 2
P
m l 1l j
i=l stiffness reductions in the first and third stories are considered.
where the asterisk (*) denotes the damage state. The value of the Since the ASDIl is derived based on the assumption of uni-
SDIl is shown between 0 (no damage) and 1 (collapse), which form floor mass distribution, several floor mass distributions, as
is a convenient way to express the degree of story damage. For shown in Table 1, are used to validate the index ASDIl . Fig. 1(a)
most buildings, the floor mass distribution is generally uniform, illustrates the damage indices SDIl and ASDIl based on the
so the approximate value of the SDIl , represented as ASDIl , can first modal parameters for the first to fourth (l = 14) sto-
be written as: ries, respectively, versus the percentage reduction in the stiff-
ness at the first and third stories. Note that these two damaged
N ij stories are assumed to have the same stiffness reduction, and
2
P
j 1lj that the ordinate represents the values of SDIl (solid line) or
i=l
ASDIl = 1 . (17) ASDIl (different symbols for various floor mass distributions).
N
i j
2j
P
1l j It shows that SDI1 and SDI3 can accurately detect the degree of
i=l
story damage. Both SDI2 and SDI4 are equal to zero no matter
Obviously, the value of ASDIl can be easily obtained what degree of damage experienced in the first and third stories.
based on only one set of modal parameters ( j and j ). This Furthermore, ASDIl is still very close to the exact index value,
is beneficial for practical applications, because the modal SDIl , even though floor mass distribution is nonuniform. The
parameters are usually evaluated through system identification error increases as story damage becomes severe. However, at a
techniques, in which only the first few modal parameters can be degree of damage of 100%, the error remains smaller than 10%.
identified due to the noise contaminated in the measurements. Fig. 1(b) shows the story damage indices calculated using
the second modal parameters to investigate the applicability
3. Validations of SDI and ASDI
of SDI and ASDI formulae based on higher structural mode.
3.1. Simulation of a five-story building It is evident that the SDI index for each story still can detect
the actual damage degree exactly. However, the ASDI values
To verify the efficiency and accuracy of the proposed SDI for certain building and at certain story have significant errors,
and ASDI formulas, five-story shear buildings with varying in particular for the stories near the damage locations and
 J.-F. Wang et al. / Engineering Structures 29 (2007) 21432157 2147

Fig. 1(a). SDI and ASDI using the first modal parameters of a five-story shear building with various damage in the first and third stories and various floor mass
distributions.

floors with larger mass. Such errors result from the fact that ASDI, the modal frequency and real-value mode shape of
the vibration shape of higher mode is more tortuous than that a specific mode are obviously the required parameters for
of the first mode. The index ASDI becomes sensitive to the calculation. With the complex eigenvalue and eigenvector, the
variation of floor mass distribution if such higher modal data corresponding modal frequency can be obtained easily based
are employed. Therefore, it is concluded that the damage index on Eq. (5). However, the mode shape corresponding to the
ASDIl based on the fundamental mode is sufficiently accurate. displacement coordinate extracting from the complex-value
When floor mass distribution is known, SDI is applicable no eigenvector is also complex if the system is nonproportionally
matter the structural parameters of which mode are used. damped. Therefore, to apply SDI and ASDI formulae, the
absolute value of the eigenvector with the same sign as the real
3.2. Effect of nonproportional damping part of the eigenvector is used as the required real-value mode
shape.
For a given building, the modal parameters are usually Fig. 2 illustrates the indices SDI and ASDI based on the first
obtained by solving the complex eigenproblem of a statespace and the second modal parameters for the first to fourth stories
system matrix extracted through the system identification of the example building with first modal damping ratio ranging
technique. It is recognized that the complex mode-shape is from 0% to 10%. In this stidy, it is assumed that 50% damage
related to the structural damping, which is different from the at the first story and 30% damage at the third story occur
normal mode-shape used in the formula of SDI or ASDI. simultaneously. It shows that the index SDI, represented by a
Therefore, it is necessary to investigate the range of damping solid line, for each story remains almost the correct value as the
ratio where indices SDI and ASDI can be suitably applied. modal damping ratio increases. This result gives a confidence
Use a Type-4 five-story building, where the floor mass, story that the light structural damping (say, 2%5%) will not affect
damping, and story stiffness ratios along height are the same the applicability of the proposed SDI formula. As to the index
(i.e., 5:4:3:2:1, see Table 1), as a numerical example. The ASDI (dashdotdash line), it is shown that the first modal
system matrices M, C, and K are formed first, and the parameters can result in good accuracy, but the second modal
statespace eigenproblem, as described in Section 2.1, is then parameters will induce large error for the locations near the
solved. According to the formulae of the indices SDI and damage stories.
2148 J.-F. Wang et al. / Engineering Structures 29 (2007) 21432157

Fig. 1(b). SDI and ASDI using the second modal parameters of a five-story shear building with various damage in the first and third stories and various floor mass
distributions.

These numerical simulations demonstrate that the index SDI shapes are identical in coordinate l. This index was applied to
can successfully be used to assess the damage degree of any identify the damage location [9] since it was thought that the
story. With the fundamental modal parameters, the approximate damage changes the consistence of the mode shapes of intact
index ASDI still agrees with the real damage state in a building. and damaged structures at the location of the damage. Since
COMAC = 1 corresponds to a no damage state, which is
4. Comparison of various damage indices contrary to the definition of indices SDI and ASDI, 1COMAC
is therefore used to indicate the degree of damage in this paper.
4.1. Damage indices considered
(2) Modal Flexibility Damage Index (MFDI)
Besides the proposed damage indices SDI and ASDI, the The presence of cracks in a structure leads to an increase
following indices are also considered to compare their accuracy. in structural flexibility. Therefore, the changes observed in the
flexibility matrix can be interpreted as a damage indication
(1) Co-Ordinate Modal Assurance Criterion (COMAC)
in the structure, and as such allow for the evaluation and
The index COMAC [4], which is defined as
location of damage [6]. This method is based on a comparison
Nm
!2 of flexibility matrices obtained from two sets of experimental
l j l j
mode shapes. The method is applicable only if the mode shapes
P
j=1 are mass-normalized to unity ( Tj M j = 1), which implies that
COMACl = (18)
Nm Nm the estimation of structural mass is required. It has been derived
l2 l2j
P P
j that the diagonal terms of the modal flexibility matrix can be
j=1 j=1
expressed as;
is a mode shape-based index, and was originally used to
Nm
indicate the correlation among all mode shapes for two X
Fl = l2j /2j . (19)
structures at a common coordinate l. Note that the symbols in
j=1
Eq. (18) are the same as those defined previously, and that Nm
denotes the total number of modes considered. The maximum In Eq. (19), Fl represents the static displacement due to a unit
value of COMAC is 1, which indicates that two sets of mode static load applied at the lth degree-of-freedom (DOF), which
 J.-F. Wang et al. / Engineering Structures 29 (2007) 21432157 2149

Fig. 2. SDI and ASDI of a Type-4 five-story shear building with 50% damage in the first and 30% damage in the third stories versus various first modal damping
ratio.

was used directly as a damage indicator [14]. To indicate the Fig. 3(a) plots damage indices SDIl and ASDIl for the first
degree of damage ranging from 0 (no damage) to 1 (collapse), story, the first-to-third (1F3F) floor interval and the third-
the damage index for the lth story using modal flexibility is to-fifth (3F5F) floor interval of the Type-4 building, with
defined as measurements at 1F, 3F and 5F under the considerations
Nm
of various stiffness reductions in the first and third stories.
l2j /2j For comparison, 1 COMAC and MFDI are also shown in
P
Fl j=1 Fig. 3(a). It is evident that SDI1 and ASDI1 are close to
MFDIl = 1 = 1 . (20)
Fl Nm being in agreement with the real values, even though only
l j / j
2 2
P
partial measurements are used. Between 1F and 3F, it is
j=1
shown that SDI1F3F and ASDI1F3F are slightly smaller than
It seems that the MFDIl has the similar but reciprocal form of the degree of damage occurring in the 3rd story. This is
ASDIl in the second term of Eq. (20). However, they are totally reasonable because the undamaged 2nd story dilutes the degree
different since the summation is performed with respect to the of damage. Moreover, from the near-zero value of SDI3F5F
mode number for MFDIl and to the DOF number for ASDIl . and ASDI3F5F , it shows that no damage occurs in the 4th and
5th stories, which agrees with the real situation. These results
4.2. Numerical simulation indicate that the indices SDI and ASDI can accurately display
the degree of damage between measured floors.
In practical application, it is generally impossible to acquire Comparing other damage indices, the square-labeled line,
full measurements because of large number of DOFs for a showing the result of 1 COMAC, obviously can not indicate
real building. Consequently, the sensors are usually selectively the actual damage situation. In particular, 1COMAC1 can not
installed at the lower, intermediate and upper floors, and only detect the occurrence of damage at the first story no matter what
partial measurements are available for most of the instrumented degree of damage occurs. As to the index MFDI, represented
buildings. Since response of the non-instrumented floors is by circle-labelled lines in Fig. 3(a), it appears that MFDI1 and
not available, the damage index tends to show, in an average MFDI1F3F , which perform better than those from COMAC,
sense, the degree of damage between two measured floors. generally can detect the occurrence and degree of damage
2150 J.-F. Wang et al. / Engineering Structures 29 (2007) 21432157

Fig. 3(a). Comparisons of various damage indices of Type-4 building with

partial measurements with damages in the 1st and 3rd stories. Fig. 3(b). Comparisons of various damage indices of Type-4 building with
partial measurements as damages at all stories.
of the first story and 1F3F interval, respectively. However,
MFI3F5F makes an erroneous judgment for the undamaged is the m 1 output vector corresponding to m measurements,
3F5F interval. Fig. 3(b) compares the three damage indices for and u(k) is the r 1 input vector corresponding to r inputs. A,
the same building with uniform damage. Because the damaged B, C, and D are system parameter matrices with dimensions
and undamaged systems have different modal frequencies but of n n, n r , m n, and m r , respectively. According to
the same mode shape, 1 COMAC of each story becomes the Refs. [12,13], the SRIM system identification technique is
zero. This shows that damage assessment based on both modal performed by establishing the matrices Y p (k) and U p (k) based
frequency and mode shape simultaneously is necessary. on the measured input u(k) and output y(k) vectors, where
5. Real building damage assessment
y(k) y(k + 1) y(k + Nr 1)

y(k + 1) y(k + 2) y(k + Nr )
5.1. Identification of building modal parameters Y p (k) = .. .. .. ..



. . . .

In this study, the SRIM system identification technique y(k + p 1) y(k + p) y(k + p + Nr 2)
is employed to estimate the dynamic properties (modal = [ y p (k) y p (k + 1) y p (k + Nr 1) ] (23)
frequencies and mode shapes) of the 7-story Holiday
u(k) u(k + 1) u(k + Nr 1)

Inn building in Van Nuys, California, based on response u(k + 1) u(k + 2) u(k + Nr )
U p (k) = .. .. ..

measurements from three earthquake events. The SRIM is ..

. . . .

applicable for a linearly time-invariant structural system u(k + p 1) u(k + p) u(k + p + Nr 2)
represented by the discrete-time statespace model as
= [ u p (k) u p (k + 1) u p (k + Nr 1) ]. (24)
x(k + 1) = Ax(k) + Bu(k) (21)
In the above matrices, y p (k) and u p (k) are mp 1 and
y(k) = C x(k) + Du(k) (22)
r p 1 vectors composed of p intervals of output and input
where x(k) is the n 1 state vector at time k1t and 1t is the measurements, respectively. By repeating the same procedure
constant sampling time interval corresponding to n DOFs, y(k) Nr times, except for shifting the starting point Nr times from
 J.-F. Wang et al. / Engineering Structures 29 (2007) 21432157 2151

Fig. 4. Illustration of the composition of y p (k) for the SRIM system

identification technique.

time k to time (k + Nr 1), the matrices Y p (k) (mp Nr ) and Fig. 5. Illustration of Phase II experimental benchmark model and distribution
U p (k)(r p Nr ) can be formed, as shown in Fig. 4. Through of sensors.
some mathematical processes [12,13], the system matrix A
can be carried out so that the modal frequencies, damping considered. Accelerometers were placed throughout the struc-
ratios and mode shapes can be obtained. It must be noted that ture to provide horizontal response measurements.
the selection of factors p and Nr is not unique. Therefore, In this paper, three other configurations of Phase II
a confirmation procedure, in which the curves between the benchmark are choosen as damaged buildings to domenstrate
the accuracy of index ASDI: (1) Config. 2: removal of all braces
calculated structural responses based on the measured inputs
on one of the y-face; (2) Config. 5: removal of the lefthand-
and the identified system matrices A, B, C, D and the actual
side brace in the first story on one of the y-face; (3) Config. 7:
structural measurements are compared, is performed to ensure
removal of all braces.
that the identified system is able to represent the real system.
Because employing the SRIM system identification tech-
nique requires the time-history responses of the inputs and out-
5.2. Damage assessment of phase II experimental benchmark puts of a system, the ambient vibration experimental data were
model chosen in this study. Meanwhile, it is assumed that each floor
of the benchmark model is rigid along the floor in-plane di-
In view of the situation that there is a difficulty of compar- rection. Two translational accelerations in x- and y-axes and
ing the merits of different SHM techniques, a series of bench- one torsional acceleration about the z-axis at the center of floor
mark studies were sponsored by the International Association and base masses are then calculated to represent the dynamic
for Structural Control (IASC)ASCE Task Group on Struc- behaviour of the building based on the three acceleration mea-
tural Health Monitoring, beginning with a relatively simple surements at each level (as shown in Fig. 5). Since each floor
benchmark problem and proceeding on to more realistic prob- was instrumented, this experiment becomes a full-measurement
lems, to provide a common basis for comparison of different case. To apply ASDI, the mass-centre accelerations along
techniques [15]. The benchmark studies consist of Phases I x- and y-directions are analyzed separately. This will be more
and II simulated and experimental benchmark problems. The suitable for Config. 1 and Config. 7 which have more symmet-
benchmark model is a four-story, two-bay by two-bay steel- rical configuration than Config. 2 and Config. 5.
frame scale structure built in the Earthquake Engineering Re- Beginning with SRIM building parameter identifications
search Laboratory at the University of British Columbia (UBC), for Configs. 1 (undamaged case) and 7 (damaged case)
Canada. The Phase II of the experimental benchmark stud- respectively, the identified first modal frequencies and mode
ies conducted on August 47, 2002, followed previous nu- shapes in x- and y-directions are summarized in Table 2, in
merical and experimental benchmark problems developed by which the validations of identified responses of the fourth
the ASCE Task Group. Various configurations were considered floor are shown in Fig. 6. It is shown that the fundamental
where damage was simulated by removing bracing or loosening frequency along the strong (x) axis is actually larger than
bolts in the benchmark model. Config. 1, as illustrated in Fig. 5, that along the weak (y) axis, and moreover, removal of all
is the reference (undamaged) case in which x-direction is the braces makes the fundamental frequency of the model structure
strong direction of the columns and each bay of outer frame reduce significantly. The ASDI indices of the first to fourth
is braced with a steel bar. Three types of excitation, electro- stories for both x- and y-directions are illustrated in Table 2.
dynamic shaker, impact hammer, and ambient vibration, were Another index, the reduction ratio of squared frequency which
2152 J.-F. Wang et al. / Engineering Structures 29 (2007) 21432157

Fig. 6. Comparison of identified responses and real measurements of the fourth floor of Phase II experimental benchmark model under ambient vibration.

represents the stiffness reduction ratio of entire building model, Table 2

is also calculated. Because the mode shape before and after Identified modal parameters and damage index ASDI of the Phase II
removing all braces remains identical, it can be seen that experimental benchmark model for Config. 1 and Config. 7
the influence of mode-shape value in Eq. (17) disappeared Parameters x-direction y-direction
and ASDI1 , ASDI2 , ASDI3 , and ASDI4 have the same value
with the reduction ratio of squared frequency, as discussed in Model Config. 1/Config. 7 Config. 1/Config. 7
configuration
Section 3.2. In addition, the damage degrees of 88% and 80%
in y- and x-directions respectively show that the each of four 1 /1 (Hz) 8.090/3.659 7.547/2.646
story braces provide 12% and 20% lateral stiffness. The greater
41 41

1.000
1.000 1.000
1.000
damage degree in y-direction reflects that the same number of



31

0.848
0.830 0.863
0.842
story braces has more stiffness contribution to the weaker axis 31
/ / /
21

of a structure. From these results, it is also concluded that the 21 0.635 0.635 0.637 0.615













11




brace acting like an in-filled wall provides most of the lateral




0.316

0.292

0.278

0.230

11
stiffness of a story even though its cross section area is quite 12
1 0.80 0.88
smaller than that of the column. 12
The results for second and third configuration pairs are ASDI4 0.82 0.89
illustrated in Tables 3 and 4. For the second pair (Configs. ASDI3 0.78 0.88
1 and 5), a 20% stiffness reduction occurs at the first story
ASDI2 0.81 0.89
in x-direction only. Obviously, ASDI1 (=0.21) can accurately
detected the damage degree in which 1 12 /12 (=0.103) ASDI1 0.78 0.86
does not successfully assess. However, ASDI4 in x-direction Config. 1: Full braced; Config. 7: No braced.
 J.-F. Wang et al. / Engineering Structures 29 (2007) 21432157 2153

Table 3
Identified modal parameters and damage index ASDI of the Phase II
experimental benchmark model for Config. 1 and Config. 5
Parameters x-direction y-direction
Model Config. 1/Config. 5 Config. 1/Config. 5
configuration
1 /1 (H z) 8.090/7.661 7.547/7.545

41 41

1.000
1.000 1.000 1.000



31 31
0.848

0.876 0.863 0.848
/ / /
21

21

0.635 0.674

0.637 0.636




11 11

0.316 0.374 0.278 0.306
12 Fig. 7. Distribution of sensors in the 7-story RC Holiday Inn building.
1 0.103 0.001
12
ASDI4 0.10 (0.00) 0.10 (0.00) theoretical model makes the ASDI index overestimate the
ASDI3 0.04 (0.00) 0.06 (0.00) damage degree of an asymmetrical building, it is shown that
ASDI2 0.02 (0.00) 0.08 (0.00) the ASDI still can detect the trend of damage without floor mass
ASDI1 0.21 (0.20) 0.09 (0.00) information and complicated mathematical calculation.
Config. 1: Full braced; Config. 5: Removal of the lefthand-side brace in the first
story on one of the y-face. The value in parentheses followed the ASDI value 5.3. Damage assessment of a 7-story reinforced concrete
represents the actual story damage degree. building

Table 4 The earthquake response measurements of the 7-story

Identified modal parameters and damage index ASDI of the Phase II
experimental benchmark model for Config. 1 and Config. 2
Holiday Inn building located at the city of Van Nuys of
the Los Angeles metropolitan area in California are used to
Parameters x-direction y-direction
investigate the applicability of the proposed damage index
Model Config. 1/Config. 2 Config. 1/Config. 2 ASDI. The building was designed as a reinforced concrete (RC)
configuration moment resisting frame, and was built in 1966. It is one of
1 /1 (Hz) 8.090/5.419 7.547/7.762 the 170 instrument-monitored buildings under the California



1.000

1.000

1.000 1.000
Strong Motion Instrumentation Program (CSMIP), in which
41
41
16 accelerographs distributed at the ground floor (GF), second



0.848 0.882 0.863
0.856
31 31
/ / / floor (2F), third floor (3F), sixth floor (6F) and roof (RF) were
21 21 0.635 0.621
0.637

0.608

installed, as shown in Fig. 7. It is seen that only one sensor was

11 11

0.316 0.336 0.278 0.290
12
installed at the base to measure the vertical motion.
1 0.55 0.06 The building was severly damaged by the 1994 Northridge
12
ASDI4 0.42 (0.40) 0.01 (0.00) earthquake (ML = 6.4) and was declared as unsafe and
ASDI3 0.63 (0.40) 0.04 (0.00) red-tagged by the Los Angeles Housing Authorities. The
ASDI2 0.49 (0.40) 0.18 (0.00) structural damage was extensive in the exterior north and south
ASDI1 0.57 (0.40) 0.01 (0.00) side frames (longitudinal (y) direction). No major damage
Config. 1: Full braced. Config. 2: Removal of all braces on one of the y-face. in the longitudinal interior frames was found. However, the
The value in parentheses followed the ASDI value represents the actual story nonstructural masonry brick walls were significantly damaged.
damage degree. A description of the damage to this building is given in
Table 5 and Fig. 8 [16]. At the base, peak values of 0.46g
and ASDI4 in y-direction appear about 10 per cent mistake. in the y direction, 0.40g in the transverse (x) direction and
This may result from the ignoring of the asymmetry of model in 0.28g in the vertical (z) were observed, whereas 0.59g along
Config. 5 and the inaccuracy of identified system parameters. In the y axis and 0.58g along the x axis were recorded at the
third pair (Configs. 1 and 2), all the stories have 40% stiffness roof. Since the building received damage during the whole
reduction in x-direction, which leads to large irregularity for duration of the earthquake excitation, the signatures after
model in Config. 2. Since it can induce torsionally coupled (TC) the strong motion part of the measurements can represent
effect that makes the fundamental frequency of model structure the behaviour of the damaged building. In addition, the
smaller than that without TC effect, it is shown that the ASDI measurements of this building recorded during the 1992
indices in x-direction overestimate the story damage degree Landers earthquake and its aftershock (Big Bear) earthquake
due to the overevaluation of squared-frequency reduction were also used. With small base accelerations (as shown in
ratio. Table 6), no damage of this building was reported during
Above investigations on Phase II benchmark model have these two earthquakes. Therefore, these measurements can
shown that the ASDI index is more applicable for damage represent the undamaged state before the 1994 Northridge
assessment of symmetrical buildings. Although the simple earthquake.
2154 J.-F. Wang et al. / Engineering Structures 29 (2007) 21432157

Table 5
Damage description of the 7-story Holiday Inn building due to 1994 Northridge earthquake

Location Damage description

1st story (a) 5 bays with cracks between bricks (north side frame)
(b) 3 column cracks due to short column effect (north side frame)
2nd floor (2F) (a) Diagonal cracks along a side column (north side frame)
(b) Diagonal cracks in beams near two beamcolumn joints (north side frame)
3rd floor (3F) (a) X shear cracks in one beamcolumn joint (north side frame)
(b) Crack through a beam near joint (north side frame)
(c) Cracks along a side column (north side frame)
3rd story (a) Cracks along a side column (north side frame)
4th floor (4F) (a) X shear cracks in 3 beamcolumn joints (north side frame)
(b) Crack through a beam near joint (north side frame)
5th floor (5F) (a) X shear cracks in 2 beamcolumn joints (north side frame)
(b) X shear cracks in 5 beamcolumn joints (south side frame)
Above 5th floor (5F) No visible damage

Fig. 8. Representation of damage in north and south views due to 1994 Northridge earthquake and sensor locations for CH01CH08 of Van Nuys Holiday Inn
Building [16].
 J.-F. Wang et al. / Engineering Structures 29 (2007) 21432157 2155

Table 6
Identified modal parameters and damage index ASDI of the Holiday Inn building based on the 1992 Lander and Big Bear earthquakes

Parameters x-direction y-direction

Earthquake event 1992 Landers 1992 Big bear 1992 Landers 1992 Big bear
PBAa (g) 0.042 0.020 0.041 0.020
Modal frequency 0.786 0.781 0.880 0.855
(Hz)
1st mode shape RF 1.000 1.000 1.000 1.000
6F 0.938 0.954 0.884 0.874
3F 0.432 0.506 0.354 0.365
2F 0.221 0.262 0.165 0.172
Damage index 6FRF 0.33 0.13
ASDI 3F6F 0.12 0.02
2nd story 0.11 0.08
1st story 0.13 0.09
a Peak Base Acceleration.

Fig. 9(a). Comparison of identified responses and real measurements of Holiday Inn building under three earthquakes at sensor CH03 (transverse axis).

To evaluate the damage of the Holiday Inn building, CH08, are used as outputs for the x axis, where the sensor
the building motions in x and y directions are assumed to number is shown in Fig. 7. Employing the SRIM identification
be independent of each other. CH16 is used as input, and technique, the first modal frequency and mode shape of the
CH09, CH10, CH11, CH12 are used as outputs for the y building in the x- and y-directions after the 1992 Landers
axis, while CH14 is used as input and CH03, CH04, CH06, earthquake and the 1992 Big Bear earthquake are identified
2156 J.-F. Wang et al. / Engineering Structures 29 (2007) 21432157

Fig. 9(b). Comparison of identified responses and real measurements of Holiday Inn building under three earthquakes at sensor CH09 (longitudinal axis).

Table 7
Identified modal parameters and damage index ASDI of the Holiday Inn building before and after the 1994 Northridge earthquake

Parameters x-direction y-direction

Earthquake event 1992 Landers 1994 Northridge 1992 Landers 1994 Northridge
PBAa (g) 0.042 0.400 0.041 0.460
Modal frequency (Hz) 0.786 0.590 0.880 0.512
1st mode shape RF 1.000 1.000 1.000 1.000
6F 0.932 0.961 0.884 0.947
3F 0.428 0.414 0.354 0.384
2F 0.220 0.171 0.165 0.163
Damage index 6FRF 0.01 0.25
ASDI 3F6F 0.47 0.67
2nd story 0.51 0.70
1st story 0.28 0.64
a Peak Base Acceleration.

and shown in Table 6, respectively. Figs. 9(a) and 9(b) since it is impossible that the story stiffness may increase in
illustrate the comparisons between the identified responses real condition. It can be seen that ASDI1 , ASDI2 , ASDI3F6F
and the measurements at CH03 and CH09 to validate the and ASDI6FRF are all small, which agree with the fact that
identified modal frequency and mode shape. Based on the no visible damage was reported after the 1992 Landers and
modal parameters after two earthquake events, the ASDI Big Bear earthquakes. Meanwhile, some index values around
damage indices of intervals between measured floors are also 0.1 are obtained. This may result from the neglect of the
calculated and presented in Table 6. It is noticed that ASDI = TC effect, the shear-building assumption, and soil compliance,
0 will be assigned if the calculated ASDI is less than zero etc.
 J.-F. Wang et al. / Engineering Structures 29 (2007) 21432157 2157

The Holiday Inn building experienced severe damage during Acknowledgments

the 1994 Northridge earthquake. In order to estimate the
dynamic properties of the building under the damaged state, This work was supported by the National Science Council
the signatures after the strong motion part of the whole of the Republic of China under Grants NSC 93-2625-Z-005-
measurements are extracted to represent the behavior of the 009 and NSC 93-2811-Z-005-002. These supports are greatly
building after the main shock. The identified first modal appreciated.
frequency and mode shape of the building are given in Table 7,
with the validation of the identified results shown in Figs. 9(a) References
and 9(b). The damage index of each interval between measured
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