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The paper describes a new damage index for the seismic analysis of rc members. Experimental validation is made.
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This article was downloaded by: [UNAM]On: 4 August 2010Access details: Access Details: [subscription number 918399494]Publisher Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
Journal of Earthquake EngineeringPublication details, including instructions for authors and subscription information:http://www.informaworld.com/smpp/title~content=t741771161
A Damage Index for the Seismic Analysis of Reinforced Concrete MembersMario E. Rodrigueza; Daniel Padillaa
a National University of Mexico, Instituto de Ingenieria, Mexico City, Mexico
To cite this Article Rodriguez, Mario E. and Padilla, Daniel(2009) 'A Damage Index for the Seismic Analysis of ReinforcedConcrete Members', Journal of Earthquake Engineering, 13: 3, 364 — 383To link to this Article: DOI: 10.1080/13632460802597893URL: http://dx.doi.org/10.1080/13632460802597893
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The publisher does not give any warranty express or implied or make any representation that the contentswill be complete or accurate or up to date. The accuracy of any instructions, formulae and drug dosesshould be independently verified with primary sources. The publisher shall not be liable for any loss,actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directlyor indirectly in connection with or arising out of the use of this material.
Journal of Earthquake Engineering, 13:364–383, 2009
Copyright � A.S. Elnashai & N.N. Ambraseys
ISSN: 1363-2469 print / 1559-808X online
DOI: 10.1080/13632460802597893
A Damage Index for the Seismic Analysisof Reinforced Concrete Members
MARIO E. RODRIGUEZ and DANIEL PADILLA
Instituto de Ingenieria, National University of Mexico, Mexico City, Mexico
This article proposes a damage index for the seismic analysis of Reinforced Concrete membersusing the hysteretic energy dissipated by a structural member and a drift ratio related to failure inthe structure. The index was calibrated against observed damage in laboratory tests of 76 RCcolumn units under various protocols. Values obtained in this calibration had acceptable agreementwith the levels of damage observed in the test specimens. An analysis of the parameters involved inthe definition of the proposed damage index shows the importance of displacement history in thedrift ratio capacity of structures.
Keywords Seismic Damage; Damage Index; Seismic Analysis; Reinforced Concrete Elements
1. Introduction
A measure of structural damage for a postulated earthquake is relevant for the seismic
analysis of both new and existing building and bridge structures. Several damage indices
have been proposed in the literature to quantify these measures. With respect to the
existing structures, such damage indices provide important information that could be
implemented in the initial assessment and retrofit decision-making process. They can
also be used for performance-based engineering approaches. Such indices have been
reviewed in the literature [Cosenza et al., 1993; Williams and Sexsmith, 1995; Ghobarah
et al., 1999; Teran-Gilmore and Jirsa, 2005], evidently stressing the need for better
damage indices. Several damages indices have been recently proposed. Erduran and
Yakut [2004] proposed a damage measure expressed in terms of interstory drift ratio and
the effect of displacement history is not taken into account. Colombo and Negro [2005]
proposed a damage index defined as the ratio of the initial and the reduced resistance of a
structure, and requires the definition of several parameters related to ductility and energy
dissipation. Kim et al. [2005] proposed a damage measure based on results of finite
element analyses, in which material models were modified to consider fatigue damage
based on results of numerical tests. In the present article, only the damage index of Park
and Ang [1985] is discussed because quantities involved in that index are also involved
in the damage index later proposed.
Park and Ang [1985] proposed the damage index, IPA, that is widely used in the literature.
This index is perhaps one of the earliest and the most popular damage indeces defined as
IPA ¼um
uu
þ �R
d EH
m ry uu
(1)
Received 2 July 2007; accepted 2 July 2008.
Address correspondence to Mario E. Rodriguez, Instituto de Ingenieria, National University of Mexico,
Mexico City, Mexico; E-mail: [email protected]
364
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where parameter um is the maximum displacement of a SDOF system responding to an
earthquake and uu is the ultimate displacement under monotonic loading. Parameters EH
and ry are the hysteretic energy dissipated by the SDOF system and the yield resistance of
this system, respectively. Parameter b considers the effect of repeated loading. It is
assumed that this system has a mass m, and a maximum displacement ductility ratio
equal to mm. The natural circular frequency of this system, o, is equal to
! ¼ffiffiffiffik
m
r(2)
where k is the lateral stiffness of the elastic system.
Some disadvantages of using index IPA have been discussed in the literature [Cosenza
et al., 1993; Williams and Sexsmith, 1995; Ghobarah et al., 1999]. Among these disadvan-
tages, it is noted that the index IPA is not normalized as it is not equal to 0 when um is equal to
uy, where the latter parameter is the structural displacement at yielding, and the index IPA is
not equal to 1 when the structure fails under monotonic loading.
2. Proposed Seismic Damage Index
The proposed damage index has evolved from the seismic damage parameter introduced by
Rodriguez [1994]. This parameter considers a level of acceptable seismic performance. In
fact, such a parameter was calibrated using several earthquake records and a specific level of
seismic performance not severe enough to cause a collapse. Results obtained using that
parameter were found consistent with observed seismic damage in 11 earthquakes experi-
enced in the past in different countries [Rodriguez and Aristizabal, 1999].
It is known that the ratio drift to height is relevant in the seismic behavior of structures.
As shown later, the proposed damage index is calibrated against experimental results using
measured drift ratios in lateral load tests of RC column units. It follows that it is highly
useful to express a damage index in terms of drift ratio. This can be done by expressing the
displacement u, involved in the derivation of Eq. (1), in terms of drift ratio, y, that is
� ¼ u
h(3)
where h is the column height.
Although, in this article, the authors do not address how to extend their findings to the
case of regular multistory buildings, the following derivation of the proposed damage index
has also been used for assessing seismic damage in existing multistory buildings [Rodriguez
and Padilla, 2006]. It is useful to express the restoring force r(t) of the SDOF system as a
function of the restoring base overturning moment of the system, M(t), that is:
rðtÞ ¼ MðtÞh
: (4)
For the elastic response, the moment M(t) is given by
MðtÞ ¼ k� �: (5)
From Eqs. (4) and (5), and the definition of k we obtain:
k� ¼ k h2: (6)
From Eqs (2) and (6) we obtain an expression for o in terms of ky:
! ¼ffiffiffiffiffiffiffiffik�
mh2
r: (7)
A Damage Index for the Seismic Analysis of Reinforced Concrete Members 365
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The proposed damage index, Id, for the SDOF system previously described is defined as
Id ¼EH
E�; (8)
where parameter El (shown in Fig. 1) is defined as
E� ¼ k� �2c : (9)
In the above expression, parameter yc is the maximum drift ratio in an elastic SDOF (with
the same frequency of the analyzed system) such that the energy absorbed in this elastic
system, El, is equal to the hysteretic energy at collapse (EH = El). In the seismic demand-
to-capacity problem, parameters EH and El would correspond to the demand and capacity
terms, respectively. Therefore, if the plastic work demand is smaller than the capacity,
then according to Eq. (8), Id < 1.
From Eqs. (8) and (9) we obtain
Id ¼EH
k� �2c
: (10)
Another interpretation of parameter yc is given in the following. At collapse (EH = El)
that is when Id = 1, from Eq. (10) we obtain
�2c ¼
E�
k�: (11)
Since ky is the elastic energy absorbed by the SDOF system when it reaches the rotation
y = ±1 (see Fig. 1), �2c can be interpreted as the fraction of this energy that is dissipated
by the nonlinear system at collapse (EH = El) in the form of plastic work. By analyzing
results from a column database, it is shown later that parameter �2c appears to be constant
for RC members with similar structural characteristics, regardless of the displacement
history applied up to collapse. It follows that since �2c is the dimensionless plastic work
capacity in an RC member, see Eq. (11), for RC members with similar structural
characteristics, this capacity can be approximately considered an invariant property.
A second form of the proposed seismic damage index, Id, is proposed using a dimen-
sionless form of parameter EH, which is expressed with parameter g, and is computed from
[Rodriguez, 1994]
� ¼
ffiffiffiffiffiEH
m
q! �m h
; (12)
θ
M Kθ
θm1 θc θm2 1
θc θm1θm21
Eλ
FIGURE 1 Parameters involved in the definition of the proposed damage index.
366 M. E. Rodriguez and D. Padilla
Downloaded By: [UNAM] At: 23:11 4 August 2010
where parameter ym is defined as:
�m ¼um
h: (13)
It is a matter of interest that parameter g has also been used by Fajfar [1992] along with a
seismic design procedure which addresses the effect of cumulative seismic damage.
Parameter g has been evaluated for a set of ground motions and was found that it is
dependant on structural and ground motion parameters [Fajfar, 1992].
From Eqs. (7) and (12) we obtain:
�2 �2m ¼
EH
k�: (14)
The above expression indicates that �2 �2m is equal to the dimensionless plastic work
demand in the SDOF system. Since k� �2m is the energy absorbed in the elastic system
(with the same frequency of the analyzed system) when it reaches a maximum drift ratio
equal to ym, Eq. (14) indicates that when a nonlinear system reaches collapse (EH = El) at
a drift ratio ym, it will dissipate an hysteretic energy at collapse that is equal to g 2 times
that elastic energy. Furthermore, at collapse (EH = El), from Eqs. (11) and (14) we obtain:
�c ¼ � �m: (15)
To illustrate Eq. (15), let us analyze two cases of a SDOF system that reaches collapse (EH =
El) at either ym = ym1 or ym = ym2. For ym = ym1, see Fig. 1, �m � �c, and therefore � � 1,
which indicates that in the system the hysteretic energy at collapse (EH = El, see shaded area
in Fig. 1) is larger than the elastic energy absorbed in the elastic system when it reaches the
maximum drift ratio ym1 (see Fig. 1). For ym = ym2, see Fig. 1, �m � �c, and therefore � � 1,
which indicates that the hysteretic energy at collapse (EH = El) is smaller than the energy
absorbed by the elastic system when it reaches the maximum drift ratio ym2 (see Fig. 1).
Finally, a relationship between Id and g can be obtained by combining Eqs. (10) and
(14), after which we obtain:
Id ¼� �m
�c
� �2
: (16)
For the case EH < El, using Eqs. (11) and (14) leads to � �m � �c, which in Eq. (16)
implies once again that Id � 1 .
It is of interest that according to the definition of Id, see Eq. (16), knowing the plastic
work capacity of a RC member (related to parameter yc) and the expected shape of the
hysteresis loops, we could predict whether or not a RC member would fail for a target
displacement history and deformation ym. This can be done by comparing the plastic
work capacity (k� �2c , see Eq. 11) and the plastic work demand (k� �
2 �2m, see Eq. 14) for
the target displacement history.
Usually, Force-Displacement relationships are used for the seismic analysis of SDOF
systems. However, the above derivation, based on Moment-Rotation relationships could
be also used in the Force-Displacement domain. These can be done considering that
parameter EH computed in either of the two mentioned domains are equal. This is due to
the fact that the incremental plastic work using the former type of relationship, DF Du, is
A Damage Index for the Seismic Analysis of Reinforced Concrete Members 367
Downloaded By: [UNAM] At: 23:11 4 August 2010
equal to the incremental plastic work using the latter relationship, DM Dq. For example, if
we want to express Eq. (11) in the Force-Displacement domain, using the given defini-
tions of parameters y and ky, this equation can be expressed as
u2c ¼
E�
k(17)
where uc is equal to yc h.
3. Description of the Column Database
The parameters involved in the definition of the proposed damage index were evaluated
using an experimental database consisting of a set of different RC columns tested in a
laboratory under axial and cyclic lateral loading. This column database is described in the
following. The experimental database used in this study for the calibration of Id consisted
of 76 RC columns tested in the last 30 years in different laboratories of USA, Japan, New
Zealand, and Canada. Figure 2 shows three different types of column test setups found in
the column database. A detailed description of the database can be found in Rodriguez
and Padilla [2006]. Only a brief description of this database is given in the following.
Table 1 lists several characteristics of the columns of the database such as compressive
strength of concrete f’c; tensile stress at yielding of longitudinal and transverse reinforce-
ment, fy and fyt, respectively; axial load ratio P/Ag f’c, where P is the axial load and Ag is
the column section area; ratio of longitudinal reinforcement, rl; ratio of volume of
transverse reinforcement to volume of column core, rt ; and ratio rt /rACI, where rACI
is the amount of transverse reinforcement prescribed by Ch. 21 of the ACI 318-05 [2005].
Table 1 also lists the column height, h (in columns with single or double curvature, see
Fig. 2), the shear span ratio M/VD, where M and V are the maximum flexural moment and
shear acting in a critical column section, respectively, and D is the depth of the column
cross section. In addition, Table 1 shows some test results such as the lateral stiffness of
the test unit, k; maximum drift ratio reached during testing at an ultimate level of damage
later described, ym; and hysteretic energy, EH, computed up to that drift ratio, with a
procedure which is also described later. The initial elastic lateral stiffness in column test
units, k, was obtained from an envelope of the measured lateral load-deformation hyster-
esis loops assuming a bilinear inelastic rule and a secant initial lateral stiffness at about
3/4 of the maximum measured lateral force.
Table 2 lists a summary of several characteristics of columns of the database such as
section type, axial load ratio, type of observed failure, type of lateral loading, and whether
h
P
P
h
P
h
h
(F, u)(F, u)
(F, u)(F, u
a) Single curvature b) Double curvature c) Single curvature with center stub
FIGURE 2 Types of test setups in the column database.
368 M. E. Rodriguez and D. Padilla
Downloaded By: [UNAM] At: 23:11 4 August 2010
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369
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1.6
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.06
370
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46
SO
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NA
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NA
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NA
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NA
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NA
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NA
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(Co
nti
nu
ed)
371
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TA
BL
E1
(Co
nti
nu
ed)
N�
Des
ign
atio
nR
efer
ence
Sec
tio
n
typ
e
fc
Mp
a
fy
Mp
a
fyt
Mp
a
P/A
g
f’c
rl
rt
rt/r
AC
I
h mm
M/V
D
K
kN/m
m
ym (ra
d)
EH
(m/s
ec)2
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10
)(1
1)
(12
)(1
3)
(14
)(1
5)
(16
)
73
ZA
HN
86
U7
1R
28
.32
01
.54
66
.00
.22
30
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51
0.0
15
62
.85
16
00
4.0
19
.60
.05
16
83
.03
74
ZH
O1
24
81
R1
9.8
10
0.5
55
9.0
0.8
01
0.0
24
50
.01
75
5.4
93
20
1.0
10
1.5
0.0
19
22
.63
75
ZH
O2
14
08
1R
21
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55
9.0
0.8
00
0.0
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50
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40
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04
1.4
0
76
ZH
O2
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09
1R
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2.1
2
Ref
eren
ce:
1)
Ref
eren
ce:
Tay
lor
eta
l.N
IST
IR5
28
5(1
99
3).
2)
Ref
eren
ce:
Tay
lor
eta
l.N
IST
IR5
98
4(1
99
7).
3)
Ref
eren
ce:
Kaw
ash
ima
Ear
thq
uak
eE
ng
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ab.(
htt
p:/
/ww
w.c
e.w
ash
ing
ton
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/).
372
Downloaded By: [UNAM] At: 23:11 4 August 2010
the columns had low or high confinement. In addition, Table 2 lists the range of values
for typical design parameters in these columns, such as compressive strength of concrete
f’c, tensile stress at yielding of longitudinal and transverse reinforcement, fy and fyt,
respectively; axial load ratio P/Ag f’c; ratio of volume of transverse reinforcement to
volume of column core rt; confinement index (rt fyt/f’c); maximum drift ratio ym; and
shear span ratio M/VD. The range of values for this parameter and the observation of
damage at end of testing for the column units of the database suggest that their failure
mode were either flexure or flexure-shear dominated.
Figure 3 shows typical examples of lateral displacement histories applied in the 76
RC columns of the database. As seen here, these cyclic displacement histories cover a
wide range of displacement histories. Most column units of the database were tested
TABLE 2 Summary of characteristics of columns of the database
Characteristic Quantity Characteristic Range
Selected test specimens 76 f’c (Mpa) 19.8–115.8
Circular test specimens 24 fy (MPa) 240–511
Square test specimens 52 fyt (MPa) 255–792
Test specimens with detailed description
of damage
23 P/Ag f’c 0–0.9
Test specimens w/o detailed description
of damage
53 rt 0.001–0.039
Test specimens with rt fyt / f’c > 0.1 36 rt / rACI 0.17–.49
Test specimens with rt fyt / f’c < 0.1 40 rt fyt / f’c 0.02–0.5
Test specimens with P/Ag f’c < 0.1 29 ym (%) 0.5–10.8
Test specimens with 0.1< P/Ag f’c < 0.3 26 M/VD 1.0–10
Test specimens with P/Ag f’c > 0.3 21
Test specimens with monotonic loading 2
Test specimens with reversed cyclic lateral
loading
63
Test specimens with earthquake type loading 11
–400
4080
120
0–80
–40
0
0
0
4080
Δ (
mm
)
–90–45
04590
0
-120-60
060
120
Δ (
mm
)
–400
4080
120
0–40
04080
120
0
–40
–20
0
20
40
Δ (
mm
)
–20–10
01020
0
Δ (
mm
)Δ
(m
m)
Δ (
mm
)
–150–75
075150
0
Δ (
mm
)Δ
(m
m)
Δ (
mm
)
–12
–6
0
6
12
0
Δ (
mm
)
-150-75
0
75
150
0
Δ (
mm
)
–20
–10
0
10
20
0
Δ (
mm
)
FIGURE 3 Lateral displacement histories in the test specimens of the database.
A Damage Index for the Seismic Analysis of Reinforced Concrete Members 373
Downloaded By: [UNAM] At: 23:11 4 August 2010
statically and in other cases dynamically using earthquake excitations such as column
units No 21–26 (see Table 1).
4. Definition of Parameters Involved in Proposed Damage Index
4.1. Definition of Parameter qm
A significant factor pertaining to the evaluation of the database needs mentioning here. The
definition of failure considered in this study corresponded to the response of a structural
element at an ultimate drift ratio equal to ym, which was defined when the lateral strength of
the test unit had a decay of 20%. Beyond this level of strength decay, total repair is
generally needed [Park and Ang, 1985]. This definition of ym is illustrated in Fig. 4. It is a
matter of great interest that in most of the database selected in this study, testing was
terminated at levels of strength decay of about 20%. In fact, only 10 test specimens of the
database reached levels of strength decay of 50% or more, which would more likely
represent a collapse or an imminent collapse. With the given definition of ym, a set of
values for this parameter was generated using the experimental database listed on Table 1.
Quite interestingly, different authors have proposed different definitions for classifi-
cation of damage. For example, values found using the well known Park and Ang index
fell in a wide range of damage classification. According to this index a threshold value
between repairable and nonrepairable damage is 0.4, leading to a wide range of classi-
fication of severe damage before collapse from 0.4–1.0 [Park et al., 1985]. In fact, it is
also suggested that the value of 1.0 representing collapse in the Park and Ang index
should change to a value of 0.8 to represent collapse [Williams and Sexsmith, 1995].
4.2. Definition of Parameter EH
The hysteretic energy, EH, for the 76 test specimens of the database was computed by
numerical integration of the lateral force-drift hysteretic response cycles recorded in these
test specimens [Rodriguez and Padilla, 2006]. For each test specimen, parameter EH was
computed up to the cycle with the drift ratio equal to ym, with the additional consideration
that if a second cycle at that drift ratio dropped its strength to more than 20%, then that
cycle was not computed in the evaluation of EH.
4.3. Definition of Parameter qc
Parameter yc for the database was computed using Eq. (11), making El = EH and computing
EH as described in Sec. 4.2, and ky was computed using Eq. (6) with the definition of k given
in Sec. 3.
F
θm
0.8Fmax
Fmax
θ
FIGURE 4 Definition of maximum drift ratio ym.
374 M. E. Rodriguez and D. Padilla
Downloaded By: [UNAM] At: 23:11 4 August 2010
5. Evaluation of Parameters qc, qm, and g Using the Experimental Database
From the database, seven groups of test specimens were used for a detailed evaluation of
parameter yc. As shown in Table 3, each of these groups had similar structural properties,
defined with the axial load ratio P/Ag f’c, ratio rt/rACI, and shear span ratio M/VD.
However, specimens in each group were subjected to different displacement histories.
In some cases, within each group, parameters k and h, which define ky (see Eq. 6), had
different values (see Table 1). For these seven groups of test specimens, parameter g was
evaluated at the maximum drift ratio ym reached using Eq. (12) and results are shown in
TABLE 3 Analysis of the effect of displacement history on drift ratio capacity of similar
RC column
Group Test Unit g ym yc
(a) LEH1015 1.08 0.099 0.107 P/Agf c < 0.1
LEH815 1.09 0.091 0.100 82% <t /rACI < 94%
KOWAU1 1.67 0.062 0.103 5.3 < M/VD < 10
KOWAU2 0.96 0.108 0.104
(b) KUN7 1.14 0.059 0.067 P/Agf c < 0.1
KUN8 1.31 0.057 0.075 100% < rt /rACI < 126%
KUN9 1.18 0.066 0.078 M/VD = 4.5
KUN10 1.21 0.066 0.079
KUN11 1.32 0.055 0.073
KUN12 1.80 0.039 0.07
(c) KANSTC1 1.15 0.0461 0.0532 P/Agf c < 0.1
OHNO84L3 1.24 0.0456 0.0564 46% < rt /rACI < 76%
SOES86U1 0.81 0.0612 0.0498 3.1 < M/VD < 5
TP001 1.5 0.034 0.051
TP002 1.07 0.044 0.047
TP005 0.71 0.0724 0.051
(d) ANG81U4 1.05 0.0365 0.0383 0.2 < P/Agf c < 0.23
TANA90U1 0.96 0.0401 0.0386 276% < rt /rACI < 369%
TANA90U4 0.72 0.0487 0.0349 M/VD = 4
TANA90U2 0.80 0.0404 0.0322
ZAHN86U7 0.79 0.0516 0.0407
(e) ZHO22309 0.58 0.0383 0.0224 P/Agf c > 0.33
ARA82102 0.85 0.0340 0.0290 199% < rt /rACI < 549%
WAT89U9 1.14 0.0218 0.0248 3 < M/VD < 4
ANG81U3 0.91 0.0318 0.0290
ZHO1248 0.87 0.0366 0.0318
(f) SOES86U2 0.69 0.0314 0.0218 P/Agf c = 0.3
SOES86U3 0.71 0.0282 0.0200 40%<rt /rACI < 111%
SOES86U4 0.94 0.022 0.0207 M/VD = 4
(g) WAT89U6 0.98 0.0157 0.0154 P/Agf c > 0.5
WAT89U7 1.27 0.0084 0.0106 34% < rt /rACI < 103%
WAT89U8 1.15 0.0111 0.0127 M/VD = 4
A Damage Index for the Seismic Analysis of Reinforced Concrete Members 375
Downloaded By: [UNAM] At: 23:11 4 August 2010
Table 3. Calculated values of parameter yc using Eq. (11) are also shown in Table 3 and
plotted in Fig. 5. As seen there, for each group of test specimens, parameter yc is nearly
constant, which leads to two important observations. The first observation is that speci-
mens with similar structural properties (as listed above) subjected to different loading
protocols have different drift capacities (measured as ym) and an approximately constant
equivalent elastic drift yc. The second observation is that according to the definition of
parameter yc, in columns with similar structural properties subjected to different loading
protocols, the plastic work capacity appears to be constant (see Fig. 5), and this capacity
can be evaluated as k� �2c , see Eq. (11).
The line at 45o with the x axis in Fig. 5 indicates the case yc = ym . Therefore, the zone
above the line corresponds to the case yc > ym, and the zone below the line corresponds to
the case yc < ym. Accordingly, the former and later cases correspond to the cases g > 1 and
g < 1, respectively, as seen in Eq. (15). These results also give an insight into the effect of
displacement history on the ultimate deformation capacity of an RC element, which is
accounted for by parameter g. For example, it can be said that the condition g = 1.0 would
be related to a medium effect of displacement history, and g > 1.0 and g < 1.0 would be
related to a high and low effect of displacement history, respectively.
Figure 6 shows a plot of calculated values of parameter �2c obtained using Eq. (11) for
the 76 test specimens of the database. These values are plotted on the y axis as a function
of ym, which is plotted on the x axis. Results shown in Fig. 6, and Eqs. (11) and (15),
indicate that the parabolic curve y = x2 defines the condition yc = ym or g = 1.0. With the
same reasoning, it can be shown that results above the parabolic curve would correspond
to the condition yc > ym and g > 1.0. Results below the parabolic curve would correspond
to the condition yc < ym and g < 1.0.
The above discussion gives ground to explain what has been commonly observed in
cyclic lateral load tests of similar specimens. A hysteretic energy at the ultimate damage
level dissipated with a large number of lateral load cycles would lead to less deformation
capacity than in the case when a hysteretic energy is dissipated in a similar specimen with
a smaller number of cycles at the same damage level. To elaborate on this finding, results
obtained in six identical RC column units, tested under various protocols [Takemura and
Kawashima, 1997] are discussed in the following.
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
θc
θm
4
10 5.35.3 8
4 .5 4 .54 .5
4 .54 .5
4 .5
3 .1
4
4 53 .1
3 .1 4
4
4 32 4
44
44
4
44
(a)
(b)
(c)
(d)(e)
(f)
(g)
Note: Numbers indicatevalues of parameter M/VD
θc= θm
FIGURE 5 Evaluation of yc as a function of ym for groups of test specimens with similar
structural characteristics.
376 M. E. Rodriguez and D. Padilla
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Figure 7 shows lateral force-drift ratio hysteresis loops obtained in the mentioned six
columns whose characteristics are also shown in Table 1. The insets in Fig. 7 show the
measured values of ultimate drift ratios, ym, the calculated values of yc using Eq. (11),
and the calculated values of parameter g at collapse (EH = El) obtained using Eq. (15).
The arrows in Fig. 7 correspond to the last cycle considered in the computation of EH for
the cyclic case. As can be seen in Fig. 7, different values of parameter ym were obtained
in these identical six test specimens. However, computed values for the equivalent elastic
drift yc appears to be constant for these six specimens, that is, about the same plastic work
capacity. It is important to note that that the test unit TP006 reached an ultimate drift
capacity of about three times that reached by test unit TP001.
Since they were not tested under a cyclic loading, test units TP004 and TP006 need
further explanation on how parameter yc was calculated (see Fig. 7). To be consistent
with the definition of failure given here for the cyclic case, failure in a monotonic type of
loading would correspond to a lateral strength decay of 20% at the first loading . As seen
in Fig. 7, before unloading, column units TP004 and TP006 did not undergo any strength
decay, therefore according to the given definition of failure, they did not fail, and their
results need to be evaluated properly. In fact, the testing of these column units is a
combination of both monotonic and cyclic cases.
To evaluate parameter yc for column units TP004 and TP006, it is assumed that a
cyclic loading would have at least one full incursion in the four quadrants of a plot lateral
force-displacement up to a strength decay of at least 20% at the maximum drift reached in
testing. For column unit TP004 these assumptions would lead to consider its plastic work
at the end of testing, which was equal to 51.9 kN-m. The plastic work computed for this
unit at the first quadrant was 14.4 kN-m (see Table 1), that is, the total plastic work of this
unit was about 3.6 times the plastic work of the ‘‘monotonic’’ response. Accordingly, for
the evaluation of parameter yc in column units TP004 and TP006, parameter EH was
computed considering the plastic work at the end of testing of column unit TP004. Based
on these results, it is suggested that a simplified procedure for the evaluation of EH is used
when computing Id for the monotonic case, in which EH is assumed equal to 4 times the
plastic work computed at the first quadrant. The monotonic case is further analyzed later.
0 0.02 0.04 0.06 0.08 0.1 0.12θm
θc2
0
0.003
0.006
0.009
0.012
0.015
γθc θm>
2 2θc θm=
( >1)
γθc θm<( <1)
FIGURE 6 Relationships between a dimensionless hysteretic energy at failure and ym.
A Damage Index for the Seismic Analysis of Reinforced Concrete Members 377
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Figure 8 plots values of parameter yc obtained as the ratio El / ky (see Eq. 11) for the
six test specimens as a function of observed parameter ym. Results shown in Fig. 8
indicate that the six test specimens had about the same value of equivalent elastic drift yc.
Considering Eq. (11) and that parameters k and h had similar values in this group of
specimens, these results indicate that these test units dissipated about the same hysteretic
energy at different levels of parameter ym.
According to Eq (10) for an evaluation of parameter Id, it is necessary to determine
the value of parameter yc. Results in Fig. 5 show a wide range of expected yc values.
These values depend on three main structural properties: the axial load ratio P/Ag f’c, the
–200
–150
–100
–50
0
50
100
150
200
–0.1 –0.08 –0.06 –0.04 –0.02 0 0.02 0.04 0.06 0.08 0.1
θ
θ θ
F (kN)
F (kN) F (kN)
F (kN)
θ
–(a) Test Unit TP001 (θm = 0.034, θc = 0.051, γ = 1.5) (b) Test Unit TP002 (θ m = 0.044, θc = 0.047, γ = 1.07)
(c) Test Unit TP003 (θm = 0.058, θc = 0.043, γ = 0.74) (d) Test Unit TP004 (θm = 0.082, θc = 0.043, γ = 0.52)
(e) Test Unit TP005 (θm = 0.072, θc = 0.051, γ = 0.71) (f) Test Unit TP006 (θm = 0.087, θc = 0.047, γ = 0.54)
–200
–150
–100
–50
0
50
100
150
200
–0.1 –0.08 –0.06 –0.04 –0.02 0 0.02 0.04 0.06 0.08 0.1
–150
–100
–50
0
50
100
150
200
–0.1 –0.08 –0.06 –0.04 –0.02 0 0.02 0.04 0.06 0.08 0.1
–200
θ
F (kN)
–150
–100
–50
0
50
100
150
200
–0.1 –0.08 –0.06 –0.04 –0.02 0 0.02 0.04 0.06 0.08 0.1
–200
θ
F (kN)
–150
–100
–50
0
50
100
150
200
–0.1 –0.08 –0.06 –0.04 –0.02 0 0.02 0.04 0.06 0.08 0.1
–200
–150
–100
–50
0
50
100
150
200
–0.1 –0.08 –0.06 –0.04 –0.02 0 0.02 0.04 0.06 0.08 0.1
–200
FIGURE 7 Lateral force-drift ratio hysteresis cycles in identical columns tested by
Takemura and Kawashima [1997].
378 M. E. Rodriguez and D. Padilla
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confining reinforcement and the shear-span ratio M/VD. An accurate procedure for
evaluating parameter yc as a function of the mentioned structural properties is beyond
the scope of this paper. However, the authors suggest a simple procedure to obtain an
approximation of the expected value of parameter yc for a particular case. The
procedure is based on assuming that the case under study belongs to one of the
seven groups shown in Fig. 5, after which its corresponding value of yc can be
obtained. Given the complexity of the seismic damage analysis problem addressed in
this study, in which we need to consider not only energy dissipation capacity but also
energy demand, the use of this simple procedure for evaluating yc is useful for such
damage analysis.
The above findings suggest that both static and dynamic experimental responses are
not directly comparable, since displacement history is relevant for defining the capacity
for the drift ratio of RC members. As discussed before, in such a comparison the plastic
work demands (k� �2 �2
m, see Eq. 14) need to be computed, and compared to the plastic
work capacity (k� �2c , see Eq. 11).
6. Damage Analysis of RC Members that Fail Under Monotonic Loading
Figure 9 shows an idealized force-deformation relationship for a SDOF system that fails
under monotonic loading considering an elasto-plastic curve and an incursion in the
positive quadrant with a maximum drift ratio equal to ym. The drift ratio at yielding of
this system is yy and its elastic stiffness is equal to ky. Figure 9 also shows the parameter
yc and the energy absorbed in the elastic SDOF system at collapse (with the same
frequency of the analyzed system and EH = El) when reaches the deformation yc.
Parameter EH for the SDOF system in the monotonic case is computed with the previous
recommendation that EH is equal to four times the energy dissipated by a monotonic
loading, which leads to:
EH ¼ 4 k� �y ð�m � �yÞ: (18)
Considering the collapse level (EH = El), parameter yc is calculated using Eqs. (11) and
(18), which leads to:
0
0.02
0.04
0.06
0 0.02 0.04 0.06 0.08 0.1θm
θc
TP001
TP002
TP003
TP005
TP004
TP006
FIGURE 8 Relationship between an equivalent elastic drift and ym for identical columns
tested by Takemura and Kawashima [1997].
A Damage Index for the Seismic Analysis of Reinforced Concrete Members 379
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�2c ¼ 4 �y ð�m � �yÞ: (19)
From Eq. (19), the following relationship between yc and ym is obtained:
�2c ¼
4 �2m
�m
ð1� 1
�m
Þ: (20)
Equation (20) indicates that by using the parameters ym and mm obtained in testing an RC
member that fails under monotonic loading, we can get an approximation of the energy
dissipation capacity of a similar member when it fails under cyclic loading. Some
examples are given in the following which support this hypothesis.
Let us analyze the cases of test units TP004 and TP006, which failed under monotonic
loading and were discussed in Sec. 5 of this article along with their companion test units
with similar structural characteristics but tested using different displacement histories.
Measured values of parameters ym and mm at collapse found after monotonic testing of
column units TP004 and TP006 are shown in Table 4. Results of the computation of yc with
Eq. (20) using these values are also shown in this table. As seen in Table 4, these results are
close to the previously computed values of parameter yc shown in insets of Fig. 7.
Furthermore, these computed values of parameter yc for the monotonic cases using Eq.
(20) are also close to those previously computed for the column units tested under cyclic
loading, namely, TP001, TP002, TP003, and TP005; see insets in Fig. 7.
7. Calibration of Proposed Damage Index Against Observed Evolution ofDamage in Test Specimens of the Database
A set of 21 test specimens was selected from the database chosen for this study for a
calibration of the proposed index and observed evolution of damage during testing
kθM
θθmθcθy
ryh
Eλ
FIGURE 9 Parameters involved in the evaluation of the proposed damage index for a
RC member that fails under monotonic loading.
TABLE 4 Evalution of parameter yc in monotonic tests
Column Unit ym mm yc (Eq 20) yc (Fig 7)
TP004 0.082 12 0.046 0.043
TP006 0.087 10 0.052 0.047
380 M. E. Rodriguez and D. Padilla
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[Rodriguez and Padilla, 2006]. This number of test specimens was arrived at based on the
detailed description of observed evolution of damage presently available. The following
classification of damage levels was considered in this calibration:
1. Localized minor cracking or first yielding of tensile reinforcement;
2. Light cracking throughout or first yielding of compressive reinforcement;
3. Severe localized cracking or onset of localized spalling;
4. Buckling of longitudinal reinforcement.
Figure 10 shows a plot of results found for Id using Eq. (10), where EH was computed at
the drift ratio yi, corresponding to the levels of damage A–D observed during the testing
of the 21 test units. These levels of damage are identified with the symbols shown in an
inset of Fig. 10. The results shown in Fig. 10 indicate some trends on the relationships
between observed damage and parameters Id and yi/ym. These trends are summarized in
an inset of Fig. 10, which indicates that minor damage would correspond to Id < 0.1;
moderate damage would correspond to 0.1 < Id < 0.6; and severe damage would
correspond to Id > 0.6. The inset of Fig. 10 also shows the values of yi/ym that would
correspond to each of these three levels of damage.
A calibration of the Park and Ang index, IPA, against the observed evolution of
damage in the abovementioned set of 21 test units is shown in Fig. 11. As seen there, the
trend of results using the Park and Ang index is comparable to that shown in Fig. 10 for
the proposed damage index. Results in Fig.11 indicate that minor damage would corre-
spond to IPA < 0.2; moderate damage would correspond to 0.2 < IPA < 0.6; and severe
damage would correspond to IPA > 0.6.
It must be pointed out that the proposed damage index does not have the previously
mentioned disadvantages of using the Park and Ang damage index, since when um is
equal to uy the index Id is equal to zero, and Id is equal to 1 when the structure fails under
monotonic loading
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0θi/θm
Id
(A)(B)(C)(D)Id = 0.1Id = 0.6
Id Damage
Classification i / mθθ
0 to 0.1 A,B 0 to 0.3
0.1 to 0.6 C 0.3 to 0.7
0.6 to 1.0 D 0.7 to 1.0
FIGURE 10 Calibration of proposed damage index and observed damage in 21 test
specimens of the database.
A Damage Index for the Seismic Analysis of Reinforced Concrete Members 381
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8. Conclusions
This article proposes a new damage measure for the seismic damage analysis of structural
RC members. The following conclusions were obtained in this study.
1. An evaluation of results observed in laboratory tests of a set of RC column units
under different loading protocols showed the importance of displacement history
for defining the capacity for the drift ratio of RC members in a seismic event. It
follows that there is no unique capacity for the drift ratio of an RC member (or
similar ones).
2. The plastic work capacity of an RC member is nearly constant for RC members
with similar structural properties. Since at failure this work is equal to the plastic
work demand (earthquake or laboratory test demands), this leads to a definition of
a rational seismic damage measure.
3. The proposed damage measure was calibrated against experimental results
from a column database selected for this study at failure and at levels of
damage preceding failure. Results of this calibration indicate a reasonable
agreement between predicted and observed damage in the analyzed column
database.
4. There is a relationship between displacement history and the capacity for the
drift ratio of RC members, which is captured by the parameter g (or yc/ym).
This relationship can be used for defining lateral loading protocols in laboratory
tests.
5. The proposed damage index has the convenient feature that is equal to zero when
the maximum displacement is equal to the displacement at yielding, and it is equal
to 1 when the RC member fails under monotonic loading.
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
IPA
(A)(B)(C)(D)IPA = 0.2IPA = 0.6
θi/θm
FIGURE 11 Calibration of Park and Ang damage index, IPA, and observed damage in 21
test specimens of the database.
382 M. E. Rodriguez and D. Padilla
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Acknowledgments
Professor S. Pujol from Purdue University, and four reviewers made useful suggestions
that helped to improve the manuscript. Their reviews and time are greatly appreciated.
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A Damage Index for the Seismic Analysis of Reinforced Concrete Members 383
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